Yann Charles

Tuan Hung Nguyen

Kevin Ardon

Monique Gaspérini

Hung Tuan Nguyen

Monique Gasperini

Scale Transition in Finite Element Simulations of Hydrogen-Plasticity Interactions

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Introduction

The development of hydrogen-based energetic supply chain leads to increased interest in hydrogen-material interactions phenomenon, especially for metallic materials (see [START_REF] Djukic | Hydrogen embrittlement of industrial components: prediction, prevention, and models[END_REF][START_REF] Traidia | Review of hydrogen-assisted cracking models for application to service lifetime prediction and challenges in the oil and gas industry[END_REF] for a review of hydrogenassisted cracking models, [START_REF] Ghosh | Three-dimensional imaging of hydrogen blister in iron with neutron tomography[END_REF] for a global description of these interactions for pipeline and pressure vessel, [START_REF] Djukic | The synergistic action and interplay of hydrogen embrittlement mechanisms in steels and iron: localized plasticity and decohesion[END_REF] for a comprehensive review of the embrittlement mechanisms interactions in steels and iron, [START_REF] Martin | Enumeration of the hydrogen-enhanced localized plasticity mechanism for hydrogen embrittlement in structural materials[END_REF] for a review on hydrogen-enhanced localized plasticity experimental results, and the recent warning by [START_REF] Lynch | Discussion of some recent literature on hydrogen-embrittlement mechanisms: addressing common misunderstandings[END_REF]) on these mechanisms). Aside from the experimental aspect, driven by the progress of observation and characterization devices (such as nano-scales experiments (Alvaro et al., 2015;[START_REF] Barnoush | Examination of hydrogen embrittlement in FeAl by means of in situ electrochemical micropillar compression and nanoindentation techniques[END_REF][START_REF] Deng | Hydrogen embrittlement revealed via novel in situ fracture experiments using notched micro-cantilever specimens[END_REF][START_REF] Müller | The impact of hydrogen on mechanical properties; a new in situ nanoindentation testing method[END_REF], 3D hydrogen localization by atom probe [START_REF] Cheng | Direct observation of hydrogen-trapping sites in newly developed high-strength mooring chain steel by atom probe tomography[END_REF][START_REF] Koyama | Recent progress in microstructural hydrogen mapping in steels: quantification, kinetic analysis, and multi-scale characterisation[END_REF] or neutron tomography [START_REF] Ghosh | Three-dimensional imaging of hydrogen blister in iron with neutron tomography[END_REF][START_REF] Pfretzschner | Hydrogen in metals visualized by neutron imaging[END_REF], or 2D mapping by Kelvin Probe Force Microscopy [START_REF] Evers | Hydrogen detection in metals: a review and introduction of a Kelvin probe approach[END_REF][START_REF] Melitz | Kelvin probe force microscopy and its application[END_REF]), modeling tools have made notable progress. This progress was helped on the one hand by the increase of computational capacities, and on the other hand, by the development or the adaptation of specific approaches, at several scales (see [START_REF] Barrera | Understanding and mitigating hydrogen embrittlement of steels: a review of experimental, modelling and design progress from atomistic to continuum[END_REF] for a review). Extensive investigations, on several materials or systems, have been made from very local up to mesoscale using one the following numerical tools (see (Aubert et al., 2019) for a global picture of the multi-scale modeling approach):

-Density Functional Theory -DFT-: [START_REF] Lu | Energetics of hydrogen impurities in aluminum and their effect on mechanical properties[END_REF] investigates the effect of H atoms on pure aluminum mechanicals properties, while (Jiang and Carter, 2004a) focused on hydrogen diffusion in Iron; [START_REF] Metsue | Trapping/detrapping kinetic rates of hydrogen around a vacancy in nickel and some consequences on the hydrogenvacancy clusters thermodynamic equilibrium[END_REF] investigates the interactions between vacancies and hydrogen in Nickel. Traction Separation laws have been derived from DFT results (Van der [START_REF] Van Der Ven | Impurity-induced van der Waals transition during decohesion[END_REF] for various material, as aluminum [START_REF] Ehlers | Ab initio determination of the traction-separation curve for a metal grain boundary: a critical assessment of strategies[END_REF]2017), iron (Jiang and Carter, 2004b) or nickel (Alvaro et al., 2015)); -Molecular Dynamic -MD-: for instance, investigations on the hydrogen assisted failure have been focused on pre-cracked Nickel single [START_REF] Wen | Modeling of hydrogen embrittlement in single crystal Ni[END_REF] or bicrystals [START_REF] Song | A nanoscale mechanism of hydrogen embrittlement in metals[END_REF], or aluminum single crystal coated by alumina [START_REF] Verners | Comparative molecular dynamics study of fcc-Al hydrogen embrittlement[END_REF]. Hydrogen-material interactions have also been studied, as in pure iron nano-pillars [START_REF] Xu | Molecular dynamics study of effect of hydrogen atoms on mechanical properties of α-Fe nanowires[END_REF], as well as hydrogen-vacancy interplay in tungsten material [START_REF] Fu | The trapping and dissociation process of hydrogen in tungsten vacancy: a molecular dynamics study[END_REF]. A complete picture is given in [START_REF] Tehranchi | The role of atomistic simulations in probing hydrogen effects on plasticity and embrittlement in metals[END_REF])); -Kinetic Monte Carlo -KMC-: diffusion-related material parameters might be extracted from computations, as in zircon [START_REF] Zhang | Anisotropic hydrogen diffusion in α-Zr and Zircaloy predicted by accelerated kinetic Monte Carlo simulations[END_REF]. The influence of defects on diffusion might be accounted for too, as well as in [START_REF] Ramasubramaniam | Effect of atomic scale plasticity on hydrogen diffusion in iron: quantum mechanically informed and on-the-fly kinetic Monte Carlo simulations[END_REF] for dislocation in iron, or [START_REF] Oda | Kinetic Monte Carlo simulation on influence of vacancy on hydrogen diffusivity in tungsten[END_REF] for vacancies in tungsten.

At the component scale, Finite Element (FE) provides the most popular framework for performing numerical simulation on structure in presence of hydrogen (to reproduce hydrogen-sensitivity characterization test, as in [START_REF] Olden | Influence of hydrogen from cathodic protection on the fracture susceptibility of 25%Cr duplex stainless steel -constant load SENT testing and FE-modelling using hydrogen influenced cohesive zone elements[END_REF] for SENT, [START_REF] Charles | Numerical modeling of the Disk Pressure Test up to failure under gaseous hydrogen[END_REF] for Disk Pressure Test, (Charles et al., 2017a) for U-Bend, or (Ayadi et al., 2017) for shear and tensile test coupled with blistering). In Abaqus software, the hydrogen diffusion and trapping are usually sequentially solved, as in [START_REF] Moriconi | Cohesive zone modeling of fatigue crack propagation assisted by gaseous hydrogen in metals[END_REF] for instance, while classical elastoplasticity is considered (but in few recent works, in which gradient plasticity models have also been proposed [START_REF] Martínez-Pañeda | Strain gradient plasticity-based modeling of hydrogen environment assisted cracking[END_REF]). Fully coupled resolution scheme, as the one used in [START_REF] Charles | Finite element simulation of hydrogen transport during plastic bulging of iron submitted to gaseous hydrogen pressure[END_REF], are not that numerous in Abaqus.

Recently, investigations have been made at an intermediate scale: the single crystal one. This scale is strategically pertinent, because it fills the gap between low scales and high ones, allowing one to get a better picture of hydrogen effects on materials at the component scale, improving their reliability. Two main tools have been used in such an approach: dislocation dynamics (e.g., to get the influence of Hydrogen on dislocation motion in steel or iron in 2D [START_REF] Taketomi | Dislocation dynamics analysis of hydrogen embrittlement in alpha iron based on atomistic investigations[END_REF] or 3D [START_REF] Gu | Quantifying the effect of hydrogen on dislocation dynamics: a three-dimensional discrete dislocation dynamics framework[END_REF][START_REF] Yu | Discrete dislocation plasticity HELPs understand hydrogen effects in bcc materials[END_REF] configurations or to investigates the hydrogen dragging by mobile dislocations [START_REF] Sills | Solute drag on perfect and extended dislocations[END_REF])), and crystal plasticity finite element method (CPFEM). To be more specific, among the finite elements studies at the polycrystal scale, few use crystal plasticity while most of them only deals with hydrogen transport through heterogeneous mediums.

This chapter is mainly dedicated to the scale transition between hydrogen transport and plasticity-induced trapping modeling from the polycrystal scale (using CPFEM), up to the component one. At the crystal scale, mechanical properties and hardening are anisotropic ones, the medium being heterogeneous, while at the macro-scale, every property is isotropic and the model is homogeneous.

Hydrogen transport and trapping is classically described by the diffusion formulation proposed by [START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF] and later improved by [START_REF] Krom | Hydrogen transport near a blunting crack tip[END_REF][START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF]) (see below section 4.2.2), applied on a Small Scale Yielding configurations. This approach has been implemented in commercial or home-made finite element software, coupled (or not) with mechanical fields, and used in numerous studies to investigate specific features of hydrogen-material interactions in homogeneous structures, including embrittlement or interactions with thermal fields (not being exhaustive, e.g., to study the plastic strain localization during a tensile test on steel [START_REF] Miresmaeili | A coupled elastoplastic-transient hydrogen diffusion analysis to simulate the onset of necking in tension by using the finite element method[END_REF], to analyze the permeation test on steel samples [START_REF] Legrand | Computational analysis of geometrical factors affecting experimental data extracted from hydrogen permeation tests: II -consequences of trapping and an oxide layer[END_REF], to model or analyze crack propagation [START_REF] Takayama | Hydrogen diffusion analyses of a cracked steel pipe under internal pressure[END_REF] or hydrogen repartition after welding in steel pipes [START_REF] Yan | 3D modeling of the hydrogen distribution in X80 pipeline steel welded joints[END_REF], after welding in steel pipes, or to analyze the effect of a thermomechanical field in a tungsten plasma-facing component (Benannoune et al., 2019a)).

This transport and trapping equation is also used in all of the fewer studies conducted on polycrystalline configurations; among them, three main objectives might be found -the determination of polycrystalline aggregate effective diffusion properties; -an investigation of the hydrogen-assisted grain boundary failure; -the determination of the plastic activities in specific grains in conjunction with experimental results. These studies account nor for hydrogen diffusion neither for hydrogen-plasticity interactions.

This chapter is dedicated to the average diffusion processes in polycrystals, accounting for both mechanical fields (stress, and plastic strain), and non-Fickian effects. After quickly describing the simulations that can found in the literature, and related to hydrogen-assisted failure or plastic strain repartition in a polycrystal, a survey of the finite element works dedicated to the determination of effective diffusion properties through polycrystals will be presented.

CPFEM computations have been used to study the influence of stress and hydrogen concentration heterogeneities on failure. [START_REF] Rimoli | A three-dimensional multiscale model of intergranular hydrogen-assisted cracking[END_REF] has used a 3D pre-cracked regular polycrystal to model an AISI 4340 steel, using trap-free grain boundary diffusion and hydrogen-sensitive cohesive elements to model a transgranular crack propagation. [START_REF] Benedetti | Grain-boundary modelling of hydrogen assisted intergranular stress corrosion cracking[END_REF] performed similar computations on steel, without any cracks, and for a given load, as [START_REF] Benabou | Coupled stress-diffusion modelling of grain boundary segregation and dynamic embrittlement in a copper alloy[END_REF] for copper alloys. [START_REF] Yu | Cohesive zone simulation of grain size and misorientation effects on hydrogen embrittlement in nickel[END_REF] focused on the effect of grain boundary disorientation on its hydrogen-assisted failure, considering a bi-crystal made of steel with elastic anisotropy and cohesive zones for the grain boundary. Finally, [START_REF] Wu | Prediction of diffusion assisted hydrogen embrittlement failure in high strength martensitic steels[END_REF] conducted 2D computations considering a bulk hydrogen trapping and transport process, and intergranular embrittlement modeled by an overlapping element method.

Few diffusion-free CPFEM models have been used to correlate experimental data and numerical plastic localization (e.g., in steel (Aubert et al., 2016)) or failure (e.g., in aluminum alloys [START_REF] Pouillier | A study of intergranular fracture in an aluminium alloy due to hydrogen embrittlement[END_REF]).

While claiming to work at the polycrystal scale, numerous investigations are dedicated to the determination of the effective diffusion process through stress-free heterogeneous structures; as a consequence, only a classical Fick law is used with heterogeneous diffusion coefficients. In these studies, mainly in 2D, the grain morphology might be regular or based on a Voronoi tessellation, and the diffusion properties depending on the studied materials. For Nickel material, grains boundaries act as diffusion short-cut, following the pioneering work of [START_REF] Swiler | Heterogeneous diffusion effects in polycrystalline microstructures[END_REF] and latter [START_REF] Zhu | Microstructure dependence of diffusional transport[END_REF]; the grain boundary type [START_REF] Legrand | Numerical analysis of the influence of scale effects and microstructure on hydrogen diffusion in polycrystalline aggregates[END_REF], their density (Jothi et al., 2015a), and their connectivity [START_REF] Osman Hoch | Effects of grain-boundary networks on the macroscopic diffusivity of hydrogen in polycrystalline materials[END_REF] are the main features for extracting an effective diffusion coefficient. Trapping at grain boundaries have been introduced in pure Nickel [START_REF] Ilin | Effect of grain boundary trapping kinetics on diffusion in polycrystalline materials: hydrogen transport in Ni[END_REF] (Jothi et al., 2015c) or Nickel alloys [START_REF] Turk | Grain boundary carbides as hydrogen diffusion barrier in a Fe-Ni alloy: a thermal desorption and modelling study[END_REF], without considering a coupling with mechanical fields. For two-phase steel polycrystals, diffusion short-cut mechanism has also been introduced, not coupled with mechanical fields [START_REF] Yazdipour | 2D modelling of the effect of grain size on hydrogen diffusion in X70 steel[END_REF]; such work focuses mainly on the phase morphology influence on hydrogen transport, as in [START_REF] Sezgin | Anisotropy of cold-worked Type-304 austenitic stainless steel: focus on the hydrogen diffusivity[END_REF]. Similar studies have also been performed on 3D aluminum polycrystals [START_REF] Lacaille | Finite element analysis of the grain size effect on diffusion in polycrystalline materials[END_REF].

The impacts of mechanical fields (pressure, and trapping by dislocation) on the average diffusivity are not well studied, and most of the works do not include crystal plasticity. For instance, [START_REF] Olden | FE simulation of hydrogen diffusion in duplex stainless steel[END_REF] consider J 2 plasticity in a two-phased steel, and focused on the influence of grain morphology on diffusion and trapping kinetic, while [START_REF] Shibamoto | Modeling of hydrogen diffusion behavior considering the microstructure of duplex stainless steel weld metal[END_REF] used a pre-computed pressure field and a regular grain shape, for the same aim. Last, [START_REF] Jothi | Influence of grain boundary misorientation on hydrogen embrittlement in bi-crystal nickel[END_REF] only considers elastic anisotropy in Nickel polycrystals.

CPFEM is used in a 3D regular iron polycrystal in [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF], and in a 2D Voronoi steel one in [START_REF] Hassan | Micromechanical modelling of coupled crystal plasticity and hydrogen diffusion[END_REF]: in these two works, the influence of a predeformation on hydrogen diffusion is investigated, accounting for both pressure stress and dislocation induced traps. Last, [START_REF] Ilin | Simulation of hydrogen diffusion affected by stress-strain heterogeneity in polycrystalline stainless steel[END_REF] focused on hydrogen redistribution in a 2D steel polycrystal due to an applied load, and at the crystal scale, FE investigations have also been performed to investigate the experimentally measured diffusion anisotropy in Nickel single crystal (J.-X. [START_REF] Li | Anisotropy of hydrogen diffusion in nickel single crystals: the effects of self-stress and hydrogen concentration on diffusion[END_REF], considering elastic anisotropy and vacancy-induced strain. Very few works have investigated the reformulation of crystal plasticity laws to account for the hydrogen effect on slips [START_REF] Bal | On the utility of crystal plasticity modeling to uncover the individual roles of microdeformation mechanisms on the work hardening response of Fe-23Mn-0.5C TWIP steel in the presence of hydrogen[END_REF][START_REF] Birnbaum | Hydrogen-enhanced localized plasticitya mechanism for hydrogen-related fracture[END_REF][START_REF] Cailletaud | Une introduction à la plasticité cristalline interactions avec l'environnement[END_REF][START_REF] Kumar | Modelling of fatigue crack initiation in hydrogen charged polycrystalline nickel[END_REF][START_REF] Vasios | Crystal Plasticity[END_REF], or the correspondence between the diffusion process through a polycrystal and the equivalent homogeneous medium (Charles et al., 2018;Jothi et al., 2015b).

From the previous picture, it appears that CPFEM is not commonly used in computations, and especially for the effective diffusion coefficient evaluation.

The different modeling assumptions used in this chapter are first presented, as well as the tools developed required to perform multi-scale finite element computations (section 4.2). Afterward, two specific points are focused on to illustrate the consequences of scale transition in finite element modeling, while dealing with hydrogen transport and trapping:

-on section 4.3, a reformulation of the dislocation-related trap density function is proposed for iron material, to get the same transport kinetic through a polycrystal and the corresponding homogenous sample.

-section 4.4. is dedicated to the presentation of the model proposed by [START_REF] Dadfarnia | Modeling hydrogen transport by dislocations[END_REF], aiming at including hydrogen dragging by dislocation in a homogeneous medium. An adaptation of this approach at the crystal scale is proposed. The consequences at the polycrystal scale of such formalism, accounting for textures, have been investigated.

Modeling assumptions

The modeling assumptions at the polycrystal scale are directly adapted from the one used at the macroscopic scale, considering few adaptations linked to the specific way plasticity occurs in crystals.

Crystal plasticity mechanical behavior

The anisotropic elasticity is defined through C ij elastic constants.

The crystal plasticity is described by a classical viscous formulation (Asaro, 1983) for numerical purpose only. The slip rate ! γ α on the α th slip system, defined by its normal vector n α and its slip direction m α , is related to the resolved shear stress τ α by a power-law relationship

! γ α = ! a 0 τ α τ c α τ α τ c α n-1 [4.1]
τ c α is the critical resolved shear stress, ! a 0 is a reference strain rate, and n the strain-rate sensitivity, chosen high enough to avoid viscous effects. From slip rates on each system, the global plastic strain rate tensor is

! ε p = m α ⊗ n α ( ) α ∑ ! γ α [4.2]
τ α is obtained by

τ α = 1 2 m α ⊗ n α + n α ⊗ m α ( ) : σ [4.3]
where σ is the local stress field; ⊗ denotes the tensorial product, and ':' the tensorial contraction. The hardening law is described by [START_REF] Peirce | An analysis of nonuniform and localized deformation in ductile single crystals[END_REF])

! τ α = h αβ ! γ β α ∑ [4.4]
where h αβ represent the self-hardening and h αβ,α≠β = qh αα the latent one, with

h αα = h 0 sech 2 h 0 γ τ s -τ 0 [4.5] γ = ! γ α dt 0 t ∫ α ∑
! is the cumulated shear strain.! h0,! τ s ! and! τ 0 are material parameters. !

No influence of hydrogen on mechanical behavior is taken into account in the present work.

Hydrogen transport equation

The hydrogen transport equation is based on the local balance between hydrogen concentration in normal interstitial lattice sites (NILS) and hydrogen concentration in trapping sites [START_REF] Krom | Hydrogen transport near a blunting crack tip[END_REF][START_REF] Oriani | The diffusion and trapping of hydrogen in steel[END_REF]

)! K T θ L = θ T 1-θ T [4.6]
where K T represents the equilibrium constant with K T =exp(-W B /RT) (W B being the trap binding energy). NILS hydrogen concentration C L relates to NILS density N L by C L =N L θ L whereas trapped hydrogen density C T relates to trap density N T by C T =N T θ T ; θ L and θ T are respectively NILS and trapping sites occupancy, assuming that θ L <<1. The hydrogen diffusion equation, as proposed by Sofronis et al. [START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF]) is thus

∂ C T + C L ( ) ∂t + ∇ϕ L = 0 [4.7]
ϕ L is the Fickian flux, modified to account for the effect of hydrostatic pressure [START_REF] Bogkris | The effect of stress on the chemical potential of hydrogen in iron and steel[END_REF][START_REF] Dederichs | Anisotropic diffusion in stress fields[END_REF][START_REF] Li | The thermodynamics of stressed solids[END_REF][START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF]:

ϕ L = -D L ∇C L - V H RT C L ∇P H [4.8]
where D L is the hydrogen diffusion coefficient, R the perfect gas constant, T the absolute temperature, V H the partial molar volume of hydrogen in solid solution and P H the hydrostatic pressure, equal to -1/3 tr σ.

The global transport and trapping equation might be derived from Equations [4.7] and [4.8], as

C T 1-θ T ( ) + C L C L ∂C L ∂t + ∇ -D L ∇C L - V H RT C L ∇P H $ % & & ' ( ) ) +θ T dN T dε p ! ε p = 0 [4.9]
N T is here assumed to be only function of the equivalent plastic strain ε p [START_REF] Kumnick | Deep trapping states for hydrogen in deformed iron[END_REF], corresponding to a phenomenological description of the trapping by dislocations (see Equation [4.12] bellow). It is worth noting that Equation [4.6] might be replaced by the following kinetic reaction [START_REF] Mcnabb | A new analysis of the diffusion of hydrogen in iron and ferritic steels[END_REF]

∂θ T ∂t = kθ L 1-θ T ( ) -pθ T [4.10]
where k and p represents the probability of respectively trapping and detrapping processes, so that K T =k/p. In this work, however, Oriani's assumption has been considered (equation [4.6]).

Implementation

The previous constitutive laws have been implemented in Abaqus FE software, based on previous works [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF]2017a); the main features are recalled below. Several user subroutines (Simulia, 2011) have been developed:

-an UMAT one, based on the Huang one [START_REF] Huang | A user-material subroutine incorporating single crystal plasticity in the ABAQUS finite element program[END_REF][START_REF] Kysar | Addendum to "A user-material subroutine incorporating single crystal plasticity in the ABAQUS finite element program[END_REF], allowing to include anisotropic crystal plasticity in Abaqus (Equations Kinetic trapping (Equation [4.10]) [START_REF] Benannoune | Numerical simulation of the transient hydrogen trapping process using an analytical approximation of the McNabb and Foster equation[END_REF][START_REF] Charles | Adaptation of hydrogen transport models at the polycrystal scale and application to the U-bend test[END_REF], as well as transient thermal fields [START_REF] Benannoune | Numerical simulation by finite element modelling of diffusion and transient hydrogen trapping processes in plasma facing components[END_REF][START_REF] Vasikaran | Implantation of a reactiondiffusion Process in Abaqus software[END_REF] might also been included in computations.

Mechanical parameters

For all of the applications proposed in the present article, α-iron was considered. The anisotropic elasticity is described by the C 11 , C 21 , and C 44 constants for cubic symmetry (see [START_REF] Lord | Elastic stiffness coefficients of iron from 77° to 673[END_REF].

The values of h 0 , τ 0 and τ s were chosen to be representative of tensile test performed on pure iron single crystals [START_REF] Franciosi | Investigation of slip system activity in iron at room temperature by SEM and AFM in-situ tensile and compression tests of iron single crystals[END_REF][START_REF] Le | Etude par essais in situ MEB et AFM sur monocristaux des mécanismes de glissement à température ambiante de la ferrite de pureté commerciale[END_REF] (see Table 4.2). Only the 12 slip systems {110}〈111〉 for bcc structures were used for the sake of simplicity.

q h 0 (MPa) τ 0 (MPa) τ s (MPa)
1.1 175 30 100 ! a 0 and n values were chosen equal to respectively 0.001 s -1 and 110. Last, diffusion parameters at 300 K for α-iron are reported on Table 4.3 [START_REF] Hirth | Effects of hydrogen on the properties of iron and steel[END_REF][START_REF] Krom | Hydrogen transport near a blunting crack tip[END_REF][START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF], assuming their coincidence at both component and crystal scale.

D L (m 2 /s) V H (m 3 /mol) N L (atom/m 3 ) W B (kJ/mol) 1.27×10 -8 2×10 -6 8.46×10 28 -60
Table 4.3. Hydrogen-related parameters for α-iron.

The mechanical behavior of the polycrystal Representative Elementary Volume (REV) is defined based on a Voce law, the yield stress σ Y being related to the equivalent plastic strain ε p by

σ Y = σ 0 + R sat 1-e -Cε p " # $ % & ' [4.11]
σ 0 , R sat and C are material parameters, identified from the mean polycrystalline tensile curves [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF]. The obtained parameters are reported on At both scale, as a first approximation, the same trap density was used corresponding to the one identified for α-iron [START_REF] Johnson | Hydrogen in iron[END_REF][START_REF] Kumnick | Deep trapping states for hydrogen in deformed iron[END_REF][START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF] log N T = 23.26 -2.33e -5.5ε p [4.12]

Identification of a trap density function at the crystal scale

In a previous work [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF], we have shown that, if no trapping by dislocations is considered, the mechanical heterogeneities induced by the polycrystalline features do not influence the overall hydrogen diffusion in samples, whatever the grain numbers. If trapping is included, however, important differences were observed: if the diffusion parameters identified at the macro scale might be used at the crystal one, the trap density must be adapted and the formulation presented on equation [4.12] is no longer relevant.

We aim in the following to propose a reformulation of N T at the crystal scale so that the transport and plasticity-induced trapping process at both scales becomes equivalent: for a given polycrystal and its homogeneous equivalent model, several average plastic strains were considered, and the average diffusion processes were compared. Last, N T was identified at the crystal scale, based on the time-lag method [START_REF] Fallahmohammadi | Measurement of lattice and apparent diffusion coefficient of hydrogen in X65 and F22 pipeline steels[END_REF][START_REF] Kumnick | Deep trapping states for hydrogen in deformed iron[END_REF].

Geometry, mesh, and boundary conditions applied on the polycrystals

The considered polycrystalline aggregate is set as a parallelepipedic bar, with a square base surface equal to 0.001 2/3 mm 2 and a height h=0.43 mm, made of truncated octahedrons (see Figure 4.2). Each grain is meshed with 60 full integration 8 nodes tri-linear elements. [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF].

The polycrystal geometry has been generated by the Neper program [START_REF] Quey | Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing[END_REF] and imported in Abaqus CAE using python scripts (Simulia, 2011).

Grains orientation have been defined by an ORIENT User Subroutine (Simulia, 2011), using random sets of Euler angles (ϕ 1 ,Φ,ϕ 2 ), considering an overall isotropic texture.

Computations are made of two-steps: first a purely mechanical simulation, secondly a diffusion one under a constant mechanical loading.

Mechanical loading

In the first step, a displacement is imposed on the upper surface, for to a given uniaxial strain (0, 10, 20, 30, 40 50 and 60%). Symmetric boundary conditions have been set on the lower one (Figure 4.3a), and uniform mixed-orthogonal (or block) on the lateral ones, to model periodicity [START_REF] Charles | Modelling the competition between interface debonding and particle fracture using a plastic strain dependent cohesive zone[END_REF][START_REF] Salahouelhadj | Estimation of the size of the RVE for isotropic copper polycrystals by using elastic-plastic finite element homogenisation[END_REF]. strain. For h=0.43 mm, the bar is made of 167 grains [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF].

Diffusion and trapping

In the second step, hydrogen transport occurs under constant strain. A zero normal flux is imposed on each polycrystal faces but on the lower one, where C L =C 0 =2.084×10 21 atom/m 3 [START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF] is instantaneously imposed at the beginning of the diffusion step (Figure 4.3b). This value is computed from the Sivert law established in [START_REF] Johnson | Hydrogen gas embrittlement[END_REF], and corresponds to the quantity of the steady state hydrogen concentration in a sample surrounded by an dihydrogen gas, the pressure of which being 1 bar.

Hydrogen transport and trapping then occur, based on equation [4.9]; the computation is stopped as soon as the steady state is reached. (here, 465 nodes) [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF].

For all computations, the diffusive hydrogen concentration values C L have been saved for all of the upper face nodes, and at each computed time, from which its statistical features have been derived. To compute the average 〈C L 〉, the value of each node have been weighted by the surface of the corresponding cell in the Voronoi tessellation of the upper bar face (see Figure 4.4).

Results

The 〈C L /C 0 〉 evolution with time on the upper bar face has been computed for all of the considered applied strains, and for the two models (homogeneous and polycrystalline). To discuss the influence of trapping on hydrogen transport, the time lag method has been used [START_REF] Barrer | Diffusion in and through Solids[END_REF], the trap-free time lag t L and with trapping t T being extracted from computations results. These two times lag are supposed to be linked by the following linear relationship [START_REF] Fallahmohammadi | Measurement of lattice and apparent diffusion coefficient of hydrogen in X65 and F22 pipeline steels[END_REF][START_REF] Kumnick | Deep trapping states for hydrogen in deformed iron[END_REF] t

T t L -1 = α N T C 0 [4.13]
where α is a constant, wich depends on the considered test.

Identification of N T (ε p ) at the crystal scale

Plotting t L /t T versus N T for the homogeneous bar (Figure 4.5) shows a linear relationship, as expected, with a slope α=5.69; N T values have been computed using equation [4.12], knowing, for each computation, the value of the equivalent plastic stain ε p . α differs from the value obtained in the previous study [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF] for the same configuration (α=4), because in this latter, t L values has been computed using an explicit relationship [START_REF] Barrer | Diffusion in and through Solids[END_REF] and not extracted from computations. This α value might then be used to extract, from polycrystalline 〈C L /C 0 〉 curves, the trap density as a function of the average equivalent plastic strain in the polycrystal 〈ε p 〉 (Figure 4.6b). 〈ε p 〉 is computed as following

ε p = V i ε p i Gauss Point ∑ [4.14]
where V i and ε p i are respectively the volume and equivalent plastic strain associated to the integration point n°i.

As already observed [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF], the transport and trapping process in the polycrystalline structure is slower than in the equivalent homogeneous medium, inducing a higher identified trap density value. Plotting ε p versus 〈ε p 〉 leads to a linear relationship (Figure 4.6a): as a consequence, to have a coincidence between the transport and trapping process through a polycrystal and its equivalent homogeneous medium, it is only needed to impose at the crystal scale a trap density function of the equivalent plastic strain shcu that log N T = 23.26 -2.33e -5.5×0.76×ε p [4.15]

Validation

The comparison of the transport and trapping process in the homogenous and in the polycrystalline samples predeformed at 20% is shown on Figure 4.7. As it can be seen, the average transport processes in the two samples becomes equivalent while using the modified N T formulation for the polycrystal (Equation [4.15]). The steady state value reached by 〈C L /C 0 〉 on the polycrystal upper face is greater than 1, due to differences in term of mean P H value between the lower and the upper faces [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF]: as soon as the ∇P H contribution in Equation [4.9] is no longer accounted for (green triangles in Figure 4.7), differences between the average transport process through polycrystalline and homogeneous media vanished. This adaptation of N T is a first step toward a proper description of the trapping process at the crystal scale using diffusion-based equations. While especially dealing with trapping by dislocations, the scale transition computations point out that the equivalent plastic strain might not be the most relevant measure to describe the trap evolution; a much convincing one might be the dislocation density (see e.g. [START_REF] Oudriss | The diffusion and trapping of hydrogen along the grain boundaries in polycrystalline nickel[END_REF] for Nickel), but such an analysis is beyond the scope of the present numerical study. This trapped hydrogen, furthermore, can be transported by the dislocation movement. It is indeed showed by experimental works (see, among other, (Albrecht et al., 1982;[START_REF] Berkowitz | Dislocation transport of hydrogen in steel[END_REF][START_REF] Donovan | Accelerated evolution of hydrogen from metals during plastic deformation[END_REF][START_REF] Hirth | On the transport of hydrogen by dislocations[END_REF][START_REF] Hwang | Dislocation transport of hydrogen in iron single crystals[END_REF]1983)) that hydrogen trapped near dislocations can diffuse during dislocations movement, creating an additional diffusion flux, which contributes to faster hydrogen transport due to the higher dislocation velocity than hydrogen diffusion rate. This hydrogen transport mechanism is then expected to significantly modify the occurrence and kinetics of material-hydrogen interactions and hydrogen embrittlement, and has to be accounted for in the hydrogen transport simulations.

Adaptation of the Dadfarnia's model at the crystal scale

Recently, [START_REF] Dadfarnia | Modeling hydrogen transport by dislocations[END_REF] extended the Sofronis and McMeeking model [START_REF] Krom | Hydrogen transport near a blunting crack tip[END_REF][START_REF] Krom | Hydrogen trapping models in steel[END_REF][START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF] to include hydrogen transport by dislocation by adding an extra flux term. Applied to the simulation of hydrogen concentration ahead a crack tip in small scale yielding with isotropic elastoplastic material (as in previous works [START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF])), they showed through a parametric study the effect of the hydrogen transport by dislocation both for bcc and fcc materials, which permits to improve the simulation of hydrogen transport.

We do not aim here to review their assumptions and modeling, but just to propose a direct adaptation of this approach at the crystal scale, to illustrate the influence of discrete predetermined transport directions on this formalism on the one hand, and its consequence on the transport process through a polycrystalline media on the other hand.

Formulation at the polycrystal scale

Following [START_REF] Dadfarnia | Modeling hydrogen transport by dislocations[END_REF], equation [4.7] must be rewritten as

∂ C L + C T ( ) ∂t + ∇. ϕ L + ϕ d ( ) = 0 [4.16]
where ϕ d is the hydrogen flux associated with mobile dislocation, defined as θ T N T m v d . v d is the dislocation velocity, and N T m the part of N T associated with mobile dislocation. This expression might be generalized to various slip systems, so that

ϕ d = ϕ d α α ∑ [4.17] with ϕ d α = θ T N T m,α v d α [4.18]
ϕ d α is the hydrogen flux carried by the movement of mobile dislocations along the slip direction of the α th slip system. v d α represents the dislocation velocity in the slip plan α, in which the trap density associated with mobile dislocation is N T m,α .

Following [START_REF] Dadfarnia | Modeling hydrogen transport by dislocations[END_REF], this function can be calculated from mobile dislocation density ρ m α as follow:

N T m,α = λρ m α a [4.19]
where a is lattice parameter and λ=√2 or √3 for bcc or fcc lattice. Orowan's law links the slip rate in slip system α

! γ α = bρ m α v d α [4.20]
v d α is the velocity vector's magnitude and b the Burger's one. Hence,

v d α is determined by v d α = v d α m α
, where m α represents the slip direction of dislocations movement in the α th slip system. It is important to underline that we do not account for potential transport towards the -m α direction. From equation [4.19], one obtains

ρ m α = ! γ α b v d α [4.21] Substituting [4.20] into [4.21] leads to N T m,α v d α = λ ! γ α ba [4.22]
Therefore, the flux of hydrogen transported by mobile dislocations along the α th slip system is

ϕ d α = θ T ! γ α b v d α m α [4.23]
Then, using equation [4.23], the extension to various slip systems of the hydrogen transport by mobile dislocations in the hydrogen transport used by [START_REF] Dadfarnia | Modeling hydrogen transport by dislocations[END_REF] leads to

C T 1-θ T ( ) + C L C L ∂C L ∂t + ∇ -D L ∇C L - V H RT C L ∇P H $ % & & ' ( ) ) +θ T dN T dε p ! ε p + ∇ λθ T ba ! γ α m α α ∑ " # $ $ % & ' ' = 0 [4.24]
N T being here assumed to be only a function of the equivalent plastic strain ε p [START_REF] Kumnick | Deep trapping states for hydrogen in deformed iron[END_REF] and, assuming equilibrium between trapped and diffusive hydrogen [START_REF] Oriani | The diffusion and trapping of hydrogen in steel[END_REF]. The lattice parameter a and the burger vector b were set to 0.287 nm and 0.248 nm respectively [START_REF] Dadfarnia | Modeling hydrogen transport by dislocations[END_REF].

Application to single crystals

A α-Fe bcc single crystal is here considered; only the 12 slip systems {110}〈111〉 for bcc structures are considered in the present study (the numbering of these slip systems is given in Appendix 4.7). The considered single crystal is a cube of side 1 mm, regularly meshed with 1000 tri-linear full integration hexahedral elements (Figure 4.9). First, the applied (mechanical and hydrogen related) boundary conditions are presented; selected crystal orientations are then described, as are the used material parameters.

Boundary and initial conditions

Mechanical loading is applied on a single crystal pre-charged with hydrogen.

Mechanical boundary conditions

Mechanical symmetry boundary conditions are applied on the bottom face of the cube, while a 0.05 mm displacement is applied to the top one (corresponding to an average strain of 5% in the Y direction), to allow to capture the effect of plasticity triggering on hydrogen repartition. The applied displacement rate is set to 0.05 mm/s (corresponding to an applied strain rate of 5.10 -2 s -1 ); this high value is expected to decrease the purely Fickian contribution ϕ L in the global hydrogen flux, enhancing the ϕ d one.

Initial hydrogen concentration

The sample is precharged in hydrogen so that, before any mechanical loading, C L =C 0 everywhere in the bulk, with C 0 =2.084×10 21 atoms/m 3 [START_REF] Johnson | Hydrogen gas embrittlement[END_REF][START_REF] Sofronis | Numerical analysis of hydrogen transport near a blunting crack tip[END_REF][START_REF] Taha | A micromechanics approach to the study of hydrogen transport and embrittlement[END_REF]. Hydrogen-related symmetry conditions are applied on all external faces so as the normal hydrogen flux is zero (Figure 4.10). ∀ M in the single crystal, C L (t=0)=C 0 .

Plastic strain on slip systems induced by the mechanical loading is expected to redistribute the hydrogen in the crystal, due to the concomitant effects of trapping and extra term ϕ d , and the subsequent C L heterogeneous distribution. In the whole section 4.5, the contribution of the hydrostatic pressure to the global hydrogen flux has been neglected (i.e., V H has been set to 0).

Crystal orientations

To investigate the influence of slip system configurations on the hydrogen transport, two cases have been chosen for the macroscopic tensile direction Y -see Figure 4.10):

• 123 ! "
# $ as Y direction, for which a single glide is expected (n°9, see Appendix 4.7, leading to ϕ d = ϕ d 9 ). To get ϕ d in the XY plane, the Z vector is set to 143 : for such a coordinate system, m 9 //-0.78X+0.6Y.

• 111 ! " # $ as Y direction, where multiple slip systems are expected (n°1 to 6, see Appendix 4.7). Z is set parallel to 110 .

Results

The results of the hydrogen map evolutions due to plastic strain on the two above described configurations are presented, with a particular focus on the 123 direction.

Traction along the 123 direction

On Figure 4.11 is reported the spatial repartition the total cumulated shear strain γ and the one the slip system (9), γ 9 , showing it is the only activated system, as expected from the computations of the Schmid factors computation in a pure tensile configuration (Appendix 4.7). It can be seen that γ is six times greater than γ 1 ; in this symmetric orientation, six equivalent slip systems operate in this configuration (n°1 to 6), with the same submitted to a displacement in the 123 and 111 directions, for various applied strain rates. It has been checked that, if ϕd=0, CL=Cref everywhere in the sample, for the two configurations.

Influence of the applied strain rate

Computations have been performed for various applied strain rates (5.10 -3 , 5.10 -4 , 5.10 -5 and 5.10 -6 s -1 ), to exhibit the competition of between ϕ L and ϕ d in the two single crystals. The C L /C ref maps at the end of the mechanical loading are reported on Figure 4.15, in which C ref denotes the homogeneous concentration at the end of the mechanical loading in the reference case, defined by ϕ d =0.!

Consequences on hydrogen transport through a polycrystalline bar

The above computations suggest that the ϕ d contribution might have a significant impact on the global hydrogen transport process, depending on the single crystal orientation, the loading direction, and the applied strain rate. In [111] orientation, ϕ d was shown to contribute to increase hydrogen transport in the tensile direction, whereas it is not the case for the [ 123 ] orientation. Then at the polycrystal scale, these effects are expected to be influenced by the crystallographic texture of the polycrystal. To investigate this coupling, several computations have been performed on a parallelepipedic polycrystal (1×1×10 mm 3 ), on which both displacement rate and hydrogen concentrations conditions have been imposed (Figure 4.16a).

The bar is meshed with 10000 tri-linear linear cubic elements, so that one finite element represents one grain, for the sake of simplicity and computation cost. Such an aggregate contains enough grains to be considered as a representative elementary volume for the transport and trapping process [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF].

The applied boundary conditions have been set as following (Figure 4.16b):

1. an hydrogen concentration C L = C 0 is instantaneously imposed on the polycrystal lower face, as well as a constant displacement rate ! U on the upper face, up to an average strain in the traction direction of 5%; 2. on the lateral faces, uniform mixed-orthogonal boundary conditions are imposed [START_REF] Charles | Modelling the competition between interface debonding and particle fracture using a plastic strain dependent cohesive zone[END_REF][START_REF] Salahouelhadj | Estimation of the size of the RVE for isotropic copper polycrystals by using elastic-plastic finite element homogenisation[END_REF], as well as a zero normal hydrogen flux, to mimic mechanical and diffusion periodicity.

After all computations, the average 〈C L 〉 value of the diffusive hydrogen repartition on the bar upper face is extracted, following the same methodology than the one presented in [START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF], and its evolution with the normalized time t/t max is plotted, t max being the time needed to reach 5% strain.

Two parameters are focused on: the applied displacement rate (assuming an overall isotropic texture), and the texture (for a given applied displacement rate). To enhance the hydrogen transport by dislocations contribution to the total hydrogen flux ϕ, computed ϕ d have been multiplied by 100 in the computations presented in the next section, following Dadfarnia [START_REF] Dadfarnia | Modeling hydrogen transport by dislocations[END_REF].

Influence of the strain rate for an isotropic texture

The considered applied strain rates are the same than in the section 4.4.5. pole figures. The decreases of 〈CL/C0〉 for an applied strain rate of 10 -6 s -1 is due to the increase of trap density due to plastic strain development (see [START_REF] Benannoune | Numerical simulation of the transient hydrogen trapping process using an analytical approximation of the McNabb and Foster equation[END_REF]).

The main information extracted from these computations is that the ϕ d contribution has, in average, negligible effect on the global hydrogen transport process, recalling that this flux has been multiplied by 100 in the Figure 4.17a results. Such a result might be qualitatively explained by a statistical effect of the grain orientations over the polycrystal: diffusion acceleration due to a well-oriented grain (e.g, a 111 one) is counter-balanced by a bad oriented one (e.g. a 123 one). However this is not the case for non-isotropic textures. The results are similar for the sharpest fiber textures (a-c), but significantly differ from the isotropic case (h). Between the two, it can be noticed that the effect of the texture remains pronounced, even for textures with apparent moderate fiber reinforcements (see the differences between (g) and (h) for example) It can also be underlined that if no extra hydrogen transport is considered (black line (h) and blue dots on Figure 4.19), no differences appears in the 〈C L 〉 evolution with time, showing that there is, in average, no influence of the polycrystal texture on the hydrogen transport and trapping process, as for the isotropic texture. For the chosen configurations, the equivalent plastic strain state repartition appears to be statistically equivalent in the samples (see Figure 4.20), and therefore the effect of crystal orientations on trapping is limited effect since it is controlled by the local equivalent plastic strain. More precise accounting for the trapping process at the slip system scale would be needed to improve the modeling, which was out of scope of the present study.

Conclusion

In this chapter, we have focused on the hydrogen transport and trapping relationship, while considered at the crystal and component scales. We have linked, especially, the average time-to-diffusion thought deformed samples and the one along with an equivalent homogeneous sample. As we have shown, the hydrogen trapping in single crystal and its subsequent transport have non-trivial consequences at the component scale due to the polycrystal features (mechanical anisotropy, slip system, texture,…). The link we are developing under Abaqus software, using a fully mechanical-hydrogen transport and trapping resolution scheme, might help to redefine the global material parameters (as the effective diffusion coefficient) or evolution laws (as the diffusion equation), at the macro-scale.

The polycrystal scale in conjunction with crystal plasticity provides a convenient frame for studying the material-hydrogen interactions, and especially, the interplay between defects (as vacancies or dislocations) and hydrogen transport and trapping.

In the case of hydrogen dragging by dislocations, or any relevant phenomenon at that scale, as proposed in the literature, it is possible to include in computations the effect of a given crystal structure and material texture, opening the path to texture engineering for decreasing (or increasing) the transport process. Of course, the used formalism has to be challenged, and, especially, modified to account for the specificities of that scale (e.g., using the dislocation density as a parameter for the trap density). Further developments are thus needed, provided by both models and experiments at lower scales, to improve the description of the hydrogen and material interactions and to test assumptions extracted from experiments. Abaqus,3,10,13,35 CPFEM,4,5,6,See Polycrystalline,See Crystal Plasticity Crystal Plasticity,3,6,7,10,36,See CPFEM Diffusion,3,4,5,6,9,10,[START_REF] Song | A nanoscale mechanism of hydrogen embrittlement in metals[END_REF]12,13,14,19,26,32,33,34,35,See Transport Dislocation,3,6,7,19,20,21,36 FE,3,See Finite Element Finite Element,3 Hydrogen,2,3,4,5,6,7,8,9,12,14,15,18,19,20,21,22,23,24,25,26,27,28,30,31,32,33,34,35,36 Metal: Aluminum,2,5,6;Copper,5;Iron,2,3,6,7,[START_REF] Song | A nanoscale mechanism of hydrogen embrittlement in metals[END_REF]12;Nickel,2,5,6,19;Polycrystal,4;Single Crystal,2,3,6,22,23,26,27,30,34,35;Steel,3,4,5,6;Tungsten,3,2,6 Polycrystalline,4,12,13,15,16,17,18,19,20,30,31,See CPFEM Scale Transition,4,6,19 Transport,3,4,5,6,7,8,9,10,12,14,15,16,17,18,19,20,21,22,24,26,28,30,31,32,33,34,35,36,See Diffusion Trapping,3,4,5,6,7,8,9,10,[START_REF] Song | A nanoscale mechanism of hydrogen embrittlement in metals[END_REF]12,14,15,16,17,18,19,24,31,34,35,36 User Subroutine, 10, 13

Conclusion

In this chapter, we have focused on the hydrogen transport and trapping relationship, while considering the crystal and component scales. We have linked, especially the average time-to-diffusion thought deformed samples and the one along with an equivalent homogeneous sample. As detailed earlier, the hydrogen trapping in a single crystal and its subsequent transport have non-trivial consequences at the component scale due to the polycrystal features (mechanical anisotropy, slip system, texture, etc.). The link we are developing under the Abaqus software, using a fully mechanical-hydrogen transport and trapping resolution scheme, might help to redefine the global material parameters (as the effective diffusion coefficient) or evolution laws (as the diffusion equation), at the macro-scale.

The polycrystal scale in conjunction with crystal plasticity provides a convenient frame for studying the material-hydrogen interactions, especially the interplay between defects (as vacancies or dislocations) and hydrogen transport and trapping.

In the case of hydrogen dragging by dislocations, or any relevant phenomenon at that scale, as proposed in the literature, it is possible to include, in computations, the effect of a given crystal structure and material texture, opening the path to texture engineering for decreasing (or increasing) the transport process. Of course, the used formalism has to be challenged and, especially, modified to account for the specificities of that scale (e.g. using the dislocation density as a parameter for the trap density). Further developments are thus needed, provided by both models and experiments at lower scales, to improve the description of the hydrogen and material interactions and to test assumptions extracted from experiments.

  [4.1] to [4.5]); -an UMATHT one, to define the coupled transport (Equation [4.9]); -an ORIENT subroutine to impose a local crystal orientation.The flowchart of the user subroutine integration in Abaqus is presented on Figure4.1; the developments have been made in order to simultaneously solve the mechanical and diffusion problem. The procedure used is 'coupled temp-disp', based on the analogy between the thermal and Fick diffusion equations[START_REF] Oh | Coupled analysis of hydrogen transport using ABAQUS[END_REF].

Figure 4 . 1 .

 41 Figure 4.1. Coupled transport-trapping problem implementation in Abaqus. In red is underlined the data computed in the subroutines and transferred to other subroutines or Abaqus program.

Figure 4

 4 Figure 4.2. A truncated octahedron[START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF].

  Figure 4.3. Boundary conditions for (a) the mechanical computation up to a given macroscopic strain and (b) the diffusion computations considering a constant appliedstrain. For h=0.43 mm, the bar is made of 167 grains[START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF].

Figure 4 . 4 .

 44 Figure 4.4. Voronoi tessellation of the polycrystal upper face designed to get the upper face mean CL value evolution with time; each red seed corresponds to a node(here, 465 nodes)[START_REF] Charles | Comparison of hydrogen transport through pre-deformed synthetic polycrystals and homogeneous samples by finite element analysis[END_REF].

Figure 4 . 5 .

 45 Figure 4.5. Evolution of tT/tL with NT for several applied strains.

Figure 4

 4 Figure 4.6. (a) Relationship between 〈εp〉 and εp, respectively the average equivalent plastic strain in the polycrystalline model and the equivalent plastic strain in the homogenous one. (b) variation of NT with both 〈εp〉 and εp, computed using the timelag method and α=5.69. It is worth noting that this value has been identified so that the NT variation with εp extracted from the homogeneous model computations fits with the theoretical curve NT(εp).

Figure 4 . 7 .

 47 Figure 4.7. Comparison of the 〈CL/C0〉 evolution with time on the sample upper face for the polycrystalline (green line) and for the homogeneous samples (red line). For the polycrystalline model, and for the sake of comparison is also plotted the 〈CL/C0〉 evolution considering the initial NT formulation (dashed green line).

Figure 4 . 8 .

 48 Figure 4.8. Maps of CL/C0 at =2000 s for (a) the homogeneous and (b) the polycrystalline models. (c) 〈CL/C0〉 evolution with time for the two models (homogeneous in red and polycrystalline in green) on the samples upper face.

Figure 4 . 9 .

 49 Figure 4.9. Single crystal configuration. (XYZ) denotes the global orthonormal coordinate systems.

Figure 4 . 10 .

 410 Figure 4.10. Single crystal configuration. (XYZ) denotes the global orthonormal coordinate systems. Displacement is applied along the Y direction.

Figure 4 . 11 .

 411 Figure 4.11. Comparison at the end of the mechanical loading of the (a) γ and (b) γ 9repartition maps.

Figure 4 . 12 .Figure 4

 4124 Figure 4.12. Visualization of the hydrogen flux ϕ (∼ϕd). It can be observed that, as expected, this flux is collinear to m 9 , in the (XY) plane. The yellow dot represents the central node.

α α=1 6 ∑

 6 cumulated slip. In such a configuration, ϕ d / / m α α=1 6 ∑ = 3Y , which is consistent with Figure 4.14b: C L accumulates on the upper face of the sample, pointed out by the m vector. Such redistribution indicates a potential acceleration of the hydrogen transport process towards the traction direction.

Figure 4 . 15 .

 415 Figure 4.15. Dimensionless diffusive hydrogen repartition for the single crystals

Figure 4

 4 Figure 4.16. (a) Boundary conditions applied on the polycrystalline bar. (b) Temporal variation of U(t).

Figure 4

 4 Figure 4.17. (a) Evolution of 〈CL/C0〉 on the upper face for all of the used strain rates considering the ϕd contribution (dashed lines) or not (full lines). (b) Polycrystal 〈111〉pole figures. The decreases of 〈CL/C0〉 for an applied strain rate of 10 -6 s -1 is due to the increase of trap density due to plastic strain development (see[START_REF] Benannoune | Numerical simulation of the transient hydrogen trapping process using an analytical approximation of the McNabb and Foster equation[END_REF]).

Figure 4

 4 Figure 4.18. (a-g) considered 〈111〉 fiber textures with decreasing sharpness. (h) corresponds to the isotropic case of Figure 4.17b. For case (a) computations were performed with and without any transport by dislocations.

Figure 4 . 20 .

 420 Figure 4.20. Equivalent plastic strain repartition in the polycrystal at the end of the mechanical loading with the textures of (a) Figure 4.18a and of (b) Figure 4.18h. (c) Statistical distribution of the equivalent plastic strain in the two samples.

Table 4 .1).

 4

	C 11 (MPa)	C 12 (MPa)	C 44 (MPa)
	236900	140600	116000

Table 4 .

 4

1. Elastic parameters for the crystal elastoplastic mechanical behavior

Table 4 .

 4

2.

Hardening parameters for the crystal elastoplastic mechanical behavior.

Table 4 .

 4 4, assuming an overall isotropic elasticity is described with the Young modulus E and the Poisson ratio ν

	E (GPa)	ν	σ 0 (MPa)	R sat (MPa)	C
	215	0.3	70.3	220.4	7.4

Table 4

 4

.4 Mechanical parameters corresponding to the average polycrystal behavior.

Table 4 .

 4 Please clarify the usage of the term "thought" in th term "average time-to-diffusion thou deformed samples." 5. Slip plane numbering[START_REF] Huang | A user-material subroutine incorporating single crystal plasticity in the ABAQUS finite element program[END_REF] and Schmid factors in specific configurations4.6. ReferencesAlbrecht, J.,Bernstein, I. M., and Thompson, A. W. (1982). Evidence for dislocation transport of hydrogen in aluminum. Metallurgical Transactions A, 13, 811-820. doi: 10.1007/BF02642394. Alvaro, A., Thue Jensen, I., Kheradmand, N., Løvvik, O. M., and Olden, V. (2015). Hydrogen embrittlement in nickel, visited by first principles modeling, cohesive zone simulation and nanomechanical testing. International Journal of Hydrogen Energy, 40, 16892-16900. doi: 10.1016/j.ijhydene.2015.06.069. Asaro, R. J. (1983). Micromechanics of crystals and polycrystals. Advances in Applied Mechanics, 23, 1-115. doi: 10.1016/S0065-2156(08)70242-4.Aubert, I., Bouhattate, J., Charles, Y., Couvant, T., Dhondt, M., Duhamel, C., Fregonese, M., Gaspérini, M., Hénaff, G.,Metsue, A., and Saintier, N. (2019). Local-scale modeling of plasticity-environment interactions. InMechanics - Microstructure -Corrosion Coupling. Elsevier, pp. 413-438. doi: 10.1016/B978-1- 78548-309-7.50018-1. Aubert, I., Saintier, N., Olive, J.-M., and Plessier, F. (2016). A methodology to obtain data at the slip-band scale from atomic force microscopy observations and crystal plasticity simulations. Application to hydrogen-induced slip localization on AISI 316L stainless steel. ActaMaterialia, 104, 9-17. doi: 10.1016/j.actamat. 2015.11.042. Ayadi, S., Charles, Y., Gaspérini, M., Caron Lemaire, I., and Da Silva Botelho, T. (2017). Effect of loading mode on blistering in iron submitted to plastic prestrain before hydrogen cathodic charging. International Journal of HydrogenEnergy, 42, 10555-10567. doi: 10.1016Energy, 42, 10555-10567. doi: 10. /j.ijhydene.2017.02.048. .02.048.

			1 1 1 ]	0.00	0.27
	2	( 0 1 1 )	[ 1 1 1 ]	0.29	0.27
	3	( 1 0 1 )	[ 1 1 1 ]	0.35	0.27


Philippe Batifoulier

MAKING THE PATIENT PAY: A LUDICROUS POLICY

B) Health at the Cost of Vulgar Economics

Making the Patient Pay: A Ludicrous Policy

Philippe Batifoulier

Making the patient pay is a fashionable strategy, and coinsurance, medical deductibles, and fixed-fee medical billing are systems in widespread use. In France, as elsewhere, the financial contribution of patients towards their own health expenditure has increased considerably, with different situations arising in different countries based on the role played by private health insurance (see, for example, Quesnel-Vallée et al. (2012)).

This strategy is intentional. It is a matter of organizing cuts to healthcare for financial reasons so as to combat waste. The provision of a large national health service is seen to encourage a consumption of care that is of little benefit to people's health. In this context, a national health service is a perverse incentive and it would be better to cut health cover for the benefit of all. Policies in which medical costs are shared with patients (coinsurance, fixed fees, and medical deductibles-see boxed text) do not merely have as their object the occasional saving of a few million euros. Their aim is to discourage healthcare consumption-and this works! This strategy is indeed effective, as going without care for financial reasons is widespread, as is demonstrated by the Commonwealth Fund survey of 2010, in which 20% of respondents in the United States declared that, during the course of the last twelve months, they had had considerable difficulty in paying their health expenses or had been unable to do so. In the study, France was the European country with the highest rate, with 9%, and the United Kingdom had the lowest, with 2%. This strategy has grave political consequences. The belief that health insurance is a problem provides those in opposition to its extension with a valid line of defense. In the United States, this view of health insurance is frequently used in public debate in order to prevent any attempt at extending cover to include the millions of Americans who do without it [START_REF] Gladwell | The Moral-Hazard Myth. The Bad Idea behind our Failed Health-Care System[END_REF]. In Europe, since health insurance is essentially the responsibility of the state and mandatory, it is its withdrawal that is called for in the name of making the patient responsible. The argument in favor of a free health service therefore comes up against the financial interests of private insurance companies, which benefit from the withdrawal of public health insurance, in order to expand their market. Finally, emphasizing the involvement of the patient's insurance in the abuse of healthcare expenditure, oddly enough, exonerates doctors from any responsibility in this expenditure; it is, in fact, as though the patient is able to benefit from this care without going through a doctor to write out a prescription! Making the patient pay is a strategy illustrative of the situation as regards the power struggles at play in the health-related domain. The political power of doctors (or at least some of them) is unequal to that of their patients.

If the stakes are considerable, the consequences are tragic. Making patients pay is exceedingly detrimental to their state of health, fuelling inequality at the same time as causing new expenses, especially through schemes connected with the referral of healthcare to hospitals. Regardless of the harm, which I will clarify, this remains a strategy that is obstinately pursued by governments. This erroneous tenacity can largely be attributed to the strong theoretical foundations underlying the strategy, which are, however, to a large extent questionable.

The Slogan: Health Must Come at a Price

Standard economic theory is based on an obscure term used to criticize generous health cover, that of moral risk (or moral hazard in the original version), denoting healthcare consumption as a result of medical insurance. The term is all the more puzzling because there is no morality attached to this moral risk, but only rational behavior on the part of the patient. It is to the American economist, Mark V. [START_REF] Pauly | The Economics of Moral Hazard. Comment[END_REF], that we are indebted for laying the foundations of the theory of moral hazard in healthcare by endowing it with a negative slant. The idea of moral risk is in fact neutral. It is largely reliant on evidence (we consume more care when we are insured) and no mention is made of whether the consumption of care is justified or otherwise. We might, in fact, highlight the fact of consuming more care thanks to insurance, thus hoping for the existence of a moral risk.

However, Pauly's work would consist in persuading people that this moral risk is to be feared as health insurance lowers the price of the care, making healthcare more attractive to a rational consumer. In a market where commodities and services are bought without insurance, the choice between different commodities is dictated by a consumer's purse and the relationship between the usefulness of the commodity in question and its price. Insurance distorts a behavior perceived to be optimal, since a reduction in price means that more can be bought with the same degree of usefulness. Thus, a budget that has been drained by healthcare might have been used for something else. This incomplete and biased interpretation of moral risk as the classic price effect in consumer standard economic theory results in the loss of collective well-being. Patients benefit from an allocative inefficiency: insurance means that care is free (or affordable) and they, thus, consume even more.

Cost sharing, Excess, and Moral Risk: Some Definitions

The schemes for cost sharing between the insurer and the insured (the patient) are of four types:

-Deterrent fees (or coinsurance), corresponding to a percentage of the care, are widely used in France, Belgium, and Switzerland, both in general medical and specialist practices, as well as for medicines. In Germany, coinsurance is only used for medicines. In the United States, it is frequently used in private insurance and in numerous Medicare services (for patients over sixty-five).

-Copayment, which is a flat-rate contribution independent of cost, corresponds to a fixed-price entry-level cost. This scheme is used almost invariably for hospital care. It corresponds to an inclusive stay or the payment of a flat rate entitling the patient to certain types of healthcare.

IV

In the United States, copayment for visits to the doctor, paid through health insurance schemes, doubled between 2001 and 2006.

-A deductible is an annual default threshold for admission that is particularly widely used in Switzerland and the Netherlands, and in Sweden, for reimbursable medicines. It has recently made its appearance in France, being used as part of a copayment scheme for medicines and medical transportation. Reimbursement in its entirety is activated beyond an annual threshold of €50.00. Deductibles are in widespread use in the United States in private medical insurance as well as the state-operated Medicare program.

-Reference pricing involves capping the price for medical care paid by the state. The patient pays the difference in price between the cost borne by the state and the actual cost. This scheme is widely used in Europe for medicines.

These schemes increase the excess payment made by patients and it is they who are responsible for the healthcare costs not provided for by the national health service (the Sécurité Sociale in France). Increases in the patient's "excess" can be refinanced through private health insurance (in France this is a supplementary medical insurance, through a mutual insurance company for instance) and/or directly taken on by consumers ("out-of-pocket payments"). In all these cases, in the future, patients will have to consider devoting a more significant part of their income to financing their healthcare. This strategy is recommended by the economic theory of moral hazard, which must be distinguished from the notion of moral risk pure and simple. The notion of the patient's moral risk is allied to a healthcare consumption that is caused by a national health service. This notion is neutral and does not refer to the justifiable or unjustifiable nature of this consumption. The standard economic theory of moral hazard puts a negative slant on it: being well insured causes waste.

It is not so much the fact of its being free as its spirit of freedom that is considered to be an aberration. The economic theory of moral hazard is in open conflict with the sentiment so widely shared that health has no price. Such a sentiment imparts something special and noble to healthcare commodities and services. It stands in opposition to a commercialization that aims to put a price on everything and a value on nothing. The fact of its being free upsets the sovereignty of the consumer, making the dreamed-of world in which there is no insurance seem ugly (in theory, the situation in question is effectively one in which there is no health insurance). The "hatred of what is free" (Caillé and Chanial 2010) means that health is undervalued and is relegated to the rank of any old commodity, like cars, where insurance techniques have proved their mettle.

This economic theory of moral hazard goes beyond the simple notion of moral risk. By considering that health insurance is a problem because it leads to unnecessary consumption owing to the fact that it is by and large free, its existence is not under discussion, only its harmfulness. The consequences of this economic policy are immediate: we must reduce a person's health cover and resort to healthcare that is more expensive.

Building a Convenient Fiction

Do patients really behave as though they are freeloaders when it comes to a national health service? This idea is the direct result of the portrayal of people in the standard economic theory, namely, that people are natural opportunists who are capable of anything, including using illness to their own advantage in order to profit from their health cover. The patient here is the individual portrayed in the rational choice theory, according to which people seek to maximize their personal interests in each and every circumstance. This is why the theory of moral hazard discounts any morality, needing, for the sake of its internal cohesion, to discard any values that might influence the way people behave or render the theory null and void. It aims to prevent opportunism on the part of the insured party, which is considered to be the expected reaction of an individual if he or she is rational and intends to use health insurance to his or her own advantage. All rational individuals act in the same way and there is no need to appeal to an absence of morality for individuals to act as spongers on the national health service. The overconsumption of healthcare is, thus, a characteristic of insurance rather than of the individuals themselves; they are merely reacting to this incentive. If it is insurance that is at fault, the problem must be solved by cutting it back. The recognition of the existence of a "morality" among the stakeholders is potential suicide as far as the theory is concerned. If individuals obey moral obligations or social conventions preventing them from exploiting health insurance, then the fear of overconsumption is no longer justified and it is pointless to make the patient pay.

There is no reason to doubt that individuals are indeed capable of opportunism, but neither should we assume that this is automatically the case when they have an interest in the community in which they live. Social welfare systems highlight a particular type of relationship with others based on giving, bringing the individual's link with society to life by relegating an interest in what is profitable to the background [START_REF] Caillé | Dé-penser l'économique[END_REF]. Individuals are not devoid of social responsibility. This is the position adopted by [START_REF] Arrow | Uncertainty and the Welfare Economics of Medical Care[END_REF] when he considers that the fear of a moral risk (an overconsumption of care) can potentially be removed by "social obligations" or "institutions of trust" that are more effective than commercial incentives. It is by rejecting this agreed theory of trust that Pauly would later go on to construct the economic theory of moral hazard with regard to health,1 endowing it with the negative connotation expressed in policies aimed at making patients pay.

Making the patient pay is a strategy that is founded on a theory in which the patient has no depth. He or she does not make judgments; he or she calculates. This completely self-interested individual does not fit in with a network of social relationships capable of directing his or her behavior, putting the brakes on his or her opportunism. The economic theory of moral hazard is a political and social creation without society. Consequently, it is the vision of social welfare-a national health service in particular-that is misrepresented. In this economics-orientated point of view, national insurance must be concerned with the calculation of individual risk, discarding the aim of a shared world. Self-interest prevents mutual givingall for each and each for all-represented by national insurance (Chanial 2000). This marginalization of the fundamental characteristics of a national health service enables the creation of a convenient fiction in which the patient is summed up as a cheat par excellence, someone willing to exploit insurance for his own personal gain.

Sick People with No Sickness?

And yet, the context in which health insurance operates is one of sickness, and it is this which largely negates the opportunism expected of the patient. The cold calculation of the rational patient evaporates in the presence of emotions caused by sickness. A sick person must face up to dread, anguish, and fear, pushing possible overconsumption due to the existence of insurance into the background. These emotions are, to a very large extent, shared ones and uphold a common world that is in conflict with the crazed individualism of a scheming patient.

The economic theory of moral hazard is only applicable to sick people with no sickness. If this is not the case, then the conclusions drawn from it should be largely reassessed. Thus, the theory of moral hazard ex ante (before the illness) takes into consideration the fact that individuals will make no effort to prevent illness when they have effective health cover (insurance encourages risk taking). This idea can be largely scrapped because it is not only financial costs that are incurred by illness. Suffering, disability, and so forth are also involved. These other ill effects are not covered by insurance, however, with only the cost of healthcare being covered. This is why I fail to see where the interests of a patient in neglecting to make any efforts at prevention could possibly lie, efforts that would affect his or her own well-being and quality of life. Does a well-insured patient indulge in risk taking because he or she knows that, in the future, his cirrhosis or lung cancer will be treated? Someone with full medical coverage will rationally make preventative efforts; otherwise he or she would be gambling with his or her future health and his or her inclusion in the labor market. I fail to see, therefore, how full coverage can adversely affect preventative behaviors. In fact, the very opposite seems to be the case, with those with the best insurance taking the greatest care to prevent sickness. The theory of moral hazard ex ante is dangerous, moreover, if reduced coverage is capable of discouraging prevention. In fact, the voluntary act of prevention involves greater sensitivity to the cost of healthcare, especially owing to the fact that a more far-sighted view of the benefits of preventative care is taken in relation to curative care. It is, thus, the flat fees, the coinsurance, and the deductibles that are detrimental to prevention rather than the opposite, as is claimed by the theory (Batifoulier 2002).

If we focus on the theory of moral hazard ex post (after the illness), according to which insurance leads to an overconsumption of healthcare, we might also have serious doubts concerning its veracity. It is only the sick who claim on their insurance. The healthy do not need healthcare even if there is nothing to pay. Indeed, everyone is not clamoring for a triple bypass operation just because it is free and we might well be skeptical concerning the pleasure taken from waiting six hours in the ER purely because it is free. Health-related economics revealed very late (with Nyman ( 1999)) what patients had long known, namely that insurance enables access to care that is impossible without it. It provides the financial resources necessary to get better. It cannot be reduced to a problem of a cost that is so low that it means that those who are healthy use it, as is claimed by the standard economic theory. It is billed even more as a supplementary revenue offered to the sick in order to satisfy a need to which only healthcare is able to respond. It is the prohibitive excess fees that dissuade individuals from seeking treatment. By reducing them, insurance ensures that healthcare consumption, restrained until then, is effective.2

Paying Is Seriously Detrimental to People's Health

One of the strong points of the economic theory of moral hazard was that it demonstrated empirically that cost-sharing policies had no impact on the state of health. This was the chief contribution of the study carried out by the Rand Corporation (Health Insurance Experiment), sponsored by the US federal government in 1974, in which the effect of an organized reduction of the amount of health insurance on the health of numerous patients was measured by randomly distributing insurance policies with varying coinsurance rates (from 0% to 95%). Paying evidently meant taking less care, with economists referring to a price elasticity of -0.2: for a rise of 10% in the cost paid by the patient, the demand for care (recourse to a doctor) was reduced by 2%. The crux of the matter lies elsewhere, however. The study showed above all that a reduction in insurance in the order of 25% posed no threat to an individual's health [START_REF] Manning | Health Insurance and the Demand for Medical Care: Evidence from Randomized Experi ment[END_REF][START_REF]and the Insurance Experiment Group[END_REF]). 3 It, therefore, secured a policy in which costs were shared with the patient and legitimized the existence of deterrent fees, copayments, and deductibles as a means of cutting health insurance. This study is the only large-scale enquiry ever realized and it is not likely to be repeated owing to its very high cost (to the order of fifty million dollars). It, therefore, remains the point of reference on which all those who believe that it is appropriate to ask for contributions from patients rely. With Pauly's theory, this empirical study represents the intellectual framework behind policies that aim to make the patient pay. The empirical proof is as flawed as the theoretical approach is deficient. Firstly, the empirical evidence seems unreliable, leading to results today that are in direct opposition to those of the Rand Corporation. A first wave of criticisms relates to its methodological bias. The individuals, who agreed to participate in the experiment, were those who did not anticipate any specific health issues. Anticipating 3. The insurance policies were allocated randomly using a sample of six thousand people (under sixty-five) followed for between three and five years. The results showed that many people did not need healthcare for a year (this was the case for ambulatory care and even more so for hospital treatment), as well as showing a drop in costs (that is, recourse to a doctor) in the case even of relatively limited coinsurance. This reduction in consumption is not linear. It reached 30% with a coinsurance rate of 95% and 20% with a coinsurance rate of 25%. The effect was thus more marked from 0% to 25% than for an increase from 25% to 50% or 95%. The reduction in costs was more evident regarding ambulatory care than for hospital treatment, where coinsurance was less effective. After a medical checkup (visual acuity, tooth decay, blood pressure, and so forth), the study concluded that there was an absence of palpable improvement in the state of health as a result of insurance cover except for the two first income deciles (the least well off).

Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte health-related costs, the others (elective surgery, pregnancy, and so forth) were unwilling to allow themselves to be allocated an insurance policy with a high rate of coinsurance. This "attrition bias" meant that the study measured, above all, the effect of a reduced health cover on a population of healthy people, a factor that skewed the results (see: Nyman 2007). The empirical results of the Rand Corporation study are, in addition, flawed with respect to the health assessments given by doctors during medical examinations (measuring blood pressure for example), while it is now agreed in medical literature that health should be measured based on an assessment provided by patients concerning their own health. A utilitarian approach to care based on quantifiable measurement does not correspond to the outlook concerning their health provided by patients. In addition, health cannot be reduced to short-term measurement and the Rand Corporation study did not take the long-term effects of the deprivation of healthcare into account. Finally, the care available during the 1970s was less effective than it is today, in particular with respect to the treatment of chronic disease, contributing to the effects of care on people's health being underestimated.

Doubts concerning the viability of the Rand Corporation study were then consolidated following numerous studies being in agreement in their placing an emphasis on the negative effects of cost-sharing insurance policies on people's health. The strategy, which involves making patients pay, is based on the belief that nonessential care, which does not contribute to improving the state of an individual's health, should bear the cost. This belief is utterly without foundation. When the patient is made to pay for his or her own healthcare, all care deteriorates, including care that is essential to health. Cost-sharing policies are oblivious of the different types of care and cutting care does not exclude essential care, as is evidenced by a large number of empirical studies carried out in the United States [START_REF] Chandra | Patient Cost-sharing and Hospitalization Offsets in the Elderly[END_REF][START_REF] Trivedi | Increased Ambulatory Care Copayments and Hospitalizations among the Elderly[END_REF]), Canada (Tamblyn et al. 2001), and France [START_REF] Dourgnon | Payer peut nuire à votre santé: une étude de l'impact du renoncement financier aux soins sur l'état de santé[END_REF]. Consequently, making the patient pay is a ludicrous strategy owing to the fact that it leads to an increase in the very costs that it was intended to reduce. Indeed, just as the organized cutting of care causes harm to the health, it also leads to "undesirable events" such as delayed care, treatment being substituted for medication that is better funded but dangerous to the patient's health,4 a more frequent use of emergency services, and a rise in hospitalization, especially in the form of long stays.

These consequences are inconvenient for those involved. They are also counterproductive as far as public policy is concerned since they create new expenditure. The harmful effects of cost-sharing policies on people's health are reflected by an increase in the overall cost of care, as is shown in the more recent studies carried out by the Rand Corporation [START_REF] Mcglynn | Assessing the Appropriateness of Care: How Much is Too Much?[END_REF][START_REF] Goldman | Cutting Drug Co-Payments for Sicker Patients on Cholesterol-Lowering Drugs Could Save a Billion Dollars Every Year[END_REF]): cutting coinsurance on cholesterol-lowering drugs might improve the health of patients at the same time as saving more than a billion dollars in medical costs every year, by increasing patient compliance as far as treatments are concerned, and reducing the risk of hospitalization.

Patient Poverty and the Benefits of Medical Insurance

The harm wrought by cost-sharing policies is all the greater since it is the worst off who are the main victims. In fact, insofar as inequalities in terms of health are above all social inequalities, the sickest are primarily the worst off. The existence of a "social gradient" means that the best off are freer from disease than the worst off [START_REF] Fassin | Inégalités de santé[END_REF][START_REF] Lang | Les inégalités sociales de santé[END_REF]. Moreover, flat-rate payments, coinsurance, and other deductibles mean that it is only the sick-and thus the worst off-that pay. It follows, therefore, that those who are worse off need to visit doctors more, although actual recourse to them is observed more among the better off [START_REF] Devaux | Income-Related Inequalities in Health Service Utilization in 19 OCDE Countries, 200 -2009[END_REF].

This situation, which is widespread in Europe, is particularly marked in France, especially in terms of specialist treatment and prevention. Moreover, a lower instance of visiting a doctor is damaging to people's health and a late appeal for treatment for what is referred to as repair is more expensive for the community as a whole. The causes of the inequality as far as access to care is concerned can largely be attributed to the health cover of the person who is sick [START_REF] Desprès | Le Renoncement aux soins pour raisons financières: une approche socio-anthropologique[END_REF]. Individuals are not insured in accordance with their risk, but their income. Those on the highest incomes and those who have the best positions in the labor market have the most insurance cover [START_REF] Domin | L'Assurance maladie complémentaire, vecteur de la reproduction des inégalités dans l'emploi[END_REF]Batifoulier et al 2010). Those who have the greatest need for a secure access to treatment are also the least protected by their health insurance.

Inequalities in terms of access to healthcare cannot counteract inequalities in the state of health; in fact, they make them worse. In this situation, the populations with the worst state of health are those which suffer most from the effects of cost-sharing policies. By making care more expensive for everyone, and especially for the least well off, who are also the sickest, these policies mean that they have to devote a more significant portion of their income to healthcare and, thus, to sacrifice other items of consumption, some of which are more useful for their health, namely nutrition, hygiene, and accommodation. The deterioration in the state of health of the most vulnerable is a cause of negative externality in society. In addition to the threat posed to the health of individuals themselves, the introduction of financial barriers in terms of access to healthcare may result in a collective risk posed by the spread of diseases and the deterioration of communal well-being, necessitating new public expenditure [START_REF] Bréchat | Solidarité, précarité et handicap social[END_REF][START_REF]Santé et assurance maladie: l'inquiétante dilution des services publics[END_REF]Batifoulier and Parel 2012).

Therefore, rather than being a problem, health insurance is a solution above and beyond an improvement in the state of health. The experiment carried out in the state of Oregon in 2008 is testimony to this. A certain number of individuals won a medical insurance policy through a lottery and it was possible to observe their behavior as a result of the insurance and compare it with that of the "losers"5 [START_REF] Allen | The Oregon Health Insurance Experiment: XVI Evidence from the First Year[END_REF][START_REF] Baicker | The Effects of Medical Coverage: Learning from the Oregon Experiment[END_REF].

As expected, the existence of health cover increased the probability of the access to healthcare for all types of care except emergencies, 6 leading to a 25% increase in costs. The study showed that the state of health improved significantly. Care is, therefore, not unnecessary and answers to a genuine need. The number of days taken off sick diminished and the people who were newly insured declared that, henceforth, and more often than before, a regular place would be reserved for healthcare and the services of a general practitioner.

Having health cover not only has a positive effect on the health, but also leads to better general welfare. This is proved in the experiment by showing that health insurance increases well-being by making the lives of individuals secure so that they are able to honor their debts (a reduction of 40% in the probability of not honoring one's debts) 7 and consume more, whatever their health problems. The fact that medical costs are dealt with minimizes anxiety concerning the future and enables individuals to manage their existence better. This effect is very striking and shows that healthcare is a door into care in general so that people are then able to pay attention to poverty and social exclusion through healthcare. In return, a feeling of well-being and being in control of one's life has an impact on the state of one's health insofar as insecurity fosters inequality, as is demonstrated in Wilkinson's work (2009). Extra health cover, therefore, has a positive effect on the state of the health of the population and is likely to reduce the costs connected with illness over time.

Health insurance, like social insurance in general, contributes to better welfare in the general population. By distributing benefits, which go above and beyond health, a national health service engenders social ties and ensures collective progress. This ambition is a political creation, however, rather than a natural given. By the skilful pretense that health insurance constituted a problem, the economic theory of moral hazard succeeded in eroding the common good as 6. Medicaid health cover increases the likelihood of using external consultation services by 35%, prescribed medicines by 15%, and hospitalization by 30%. Preventive care was higher than for those without cover (a rise of 60% for mammograms, and 20% for cholesterol control). In total, primary healthcare increased by 50%.

7. Outstanding medical debts have become a reality in numerous cases, without an aim. The use of cost-sharing schemes was recommended as a result of the connection between insurance and waste, encouraging the patient to abandon an unjustified consumption of healthcare. This idea leads structurally to greater inequality. Financial barriers to the healthcare system deprive those who need it most of access to healthcare. Necessary healthcare is prevented by the desire to eradicate unnecessary care. It is both a ludicrous and counterproductive strategy since it costs the community dear at the same time as destroying social cohesion.

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It is of singular interest that the whole theory of health-related moral hazard claims allegiance to[START_REF] Arrow | Uncertainty and the Welfare Economics of Medical Care[END_REF], to whom we are indebted for the popularization of the idea of moral risk, but who is opposed to Pauly's view of it and who persisted in his opposition by emphasizing the role of moral judgments: "Pauly's wording suggests that 'rational economic behavior' and 'moral perfidy' are mutually exclusive categories. No doubt Judas Iscariot turned a tidy profit from one of his transactions, but the usual judgment of his behavior is not necessarily wrong"(Arrow 1968, 538). Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte

The empirical study carried out by[START_REF] Pierre | Impact d'une sur assurance santé sur le reste à charge des patients[END_REF] shows that a subscription to a more inclusive form of health insurance is accompanied by an increase in excess fees, since extra insurance enables access to care that was previously too expensive, namely dental and eye treatment. Supplementary insurance does not therefore diminish the financial risk. Neither does it lead to unnecessary expenditure. It enables the satisfaction of a demand that is too expensive without adequate insurance cover. Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte

The negative effect on the health of reducing medicine reimbursement has been observed in France[START_REF] Pichetti | Le Déremboursement des médicaments en France entre 2002 et 2011: éléments d'évaluation[END_REF]. Reduced reimbursement leads to substituting prescribed medicines for the types of treatment that are still reimbursed, causing problems both to public health and the financial efficacy of the measures (substitution for types with better reimbursement). For example, the abandonment of venotonic drugs is reflected in the overprescription of nonsteroidal and antalgic anti-inflammatories. The substitution of mucolytic drugs by bronchodilators and antitussives can be dangerous to an individual's health. Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte

The Oregon Health Plan Standard aimed to provide the worst-off inhabitants, chosen at random, with insurance through an extension of Medicaid: ninety thousand adults on low incomes, who did not fulfil the required criteria for eligibility to Medicaid, volunteered for the lottery for ten thousand available places. Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte

reaching the same level as the United States, where health expenditure is the primary cause of personal financial ruin. Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte Document downloaded from www.cairn-int.info --Batifoulier Philippe -93.19.60.143 -24/05/2015 09h41. © La Découverte