Léo Thiercelin
email: thiercelin@lma.cnrs-mrs.fr

Loïc Saint-Aimé

Frédéric Lebon

Aurélien Saulot

The formation of Tribological Surface Transformation (TST), known as white etching layer (WEL) in the wheel rail contact, corresponds to progressive and irreversible transformations on the surface of material due to the interaction of contact between two materials. The presence of WEL, damages the rails and cracks could initiate easily. Understanding the mechanisms of formation of WEL is important to prevent the problem of Rolling Contact Fatigue (RCF) in the rail network. The formation of the WEL is assumed to be due to a cyclic shear loading (at a high-frequency), under hydrostatic pressure associated with a moderate rise in temperature. Based on Transformed Induced Plasticity models and a previous model, a thermo-mechanical model taking into account the hydrostatic pressure, the shear stress and the temperature is presented to predict the formation of the WEL. An internal variable, representing all the steps of evolution of the microstructure leading to the WEL formation is introduced. 2-D finite element simulations of the rail running band (in the longitudinal direction) submitted to a cyclic thermomechanical loading are conducted to identify the model. The results are in good agreement with experimental observations for the size of the transformed zone after a given number of trains. The nonuniform repartition of WEL observed on the running band on trains could be

Thermomechanical modelling of the tribological surface transformations in the railroad network (white eching layer)

explained by local variations of the thermomechanical loading.

Keywords:

Rolling Contact Fatigue (RCF), White Etching Layer (WEL)

Introduction

Due to the increase of rail traffic combined with the rise of speed and loads of trains the rail network is subjected to the occurrence of many Rolling Contact 5 Fatigue (RCF) defects such as squats. Squat defects damage the running band of the rail by the presence of cracks on the surface. In many studies, the squat formation is correlated with the presence of Tribological Surface Transformations (TST) called the White Etching Layer (WEL) [START_REF] Daniel | Final Report on the Rail Squat Project R3-105[END_REF][START_REF] Simon | Tribological characterization of rail squat defects[END_REF][START_REF] Steenbergen | On the mechanism of squat formation on train rails -Part I: Origination[END_REF] for rail steel grades.

Because of its brittleness and incompatibilities with the original material, cracks 10 are observed in the vicinity of WEL either in the interface with the parent phase or within the WEL (Fig. 1). Understanding the mechanism of appearance and the nature of the WEL is then essential to prevent squat formation. Experimental studies have shown that the WEL is a solid-solid phase transformation of an initial pearlitic structure to 15 either a martensitic or a "quasi-martensitic" structure (nano Fe-α grains saturated with C atom). Despite progress in techniques of analysis, the mechanism of WEL formation is still controversial. According to previous researches, WELs are the results of either mechanical [START_REF] Al-Juboori | Squat formation and the occurrence of two distinct classes of white etching layer on the surface of rail steel[END_REF][START_REF] Al-Juboori | Characterisation of White Etching Layers formed on rails subjected to different traffic conditions[END_REF][START_REF] Lojkowski | Nanostructure formation on the surface of railway tracks[END_REF][START_REF] Newcomb | A transmission electron microscopy study of the white-etching layer on a rail head[END_REF], thermal processes [START_REF] Daniel | Final Report on the Rail Squat Project R3-105[END_REF][START_REF] Österle | Investigation of white etching layers on rails by optical microscopy, electron microscopy, X-ray and synchrotron X-ray diffraction[END_REF]9,[START_REF] Wang | Microstructure features on rolling surfaces of railway rails subjected to heavy loading[END_REF] or a combination of both [START_REF] Baumann | Surface modification, corrugation and nanostructure formation of high speed railway tracks[END_REF][START_REF] Wu | Micro and Nanoscale Characterization of Complex Multilayer-Structured White Etching Layer in Rails[END_REF][START_REF] Kumar | Microstructural evolution of white and brown etching layers in pearlitic rail steels[END_REF].

In some cases, due to its martensitic structure and the presence of retained austenite , the authors [START_REF] Daniel | Final Report on the Rail Squat Project R3-105[END_REF][START_REF] Österle | Investigation of white etching layers on rails by optical microscopy, electron microscopy, X-ray and synchrotron X-ray diffraction[END_REF]9,[START_REF] Wang | Microstructure features on rolling surfaces of railway rails subjected to heavy loading[END_REF][START_REF] Wu | Micro and Nanoscale Characterization of Complex Multilayer-Structured White Etching Layer in Rails[END_REF] deduced the WELs are induced by the transformation of the pearlite into austenite followed by a rapid quenching transforming the austenite into the martensite. The presence of retained austenite would be due to an incomplete transformation. The austenitization of the steel 25 is explained by an increase of a flash temperature above 700 °C (austenitization temperature). This temperature rise is generally attributed to friction in the contact zone in cases of braking or accelerating zone [START_REF] Al-Juboori | Squat formation and the occurrence of two distinct classes of white etching layer on the surface of rail steel[END_REF][START_REF] Al-Juboori | Characterisation of White Etching Layers formed on rails subjected to different traffic conditions[END_REF] or after a grinding process [START_REF] Rasmussen | Surface crack formation on rails at grinding induced martensite white etching layers[END_REF]. Moreover, this could be due to the formation of Adiabatic Shear Bands (ASB) generated by an intense plastic deformation [START_REF] Daniel | Final Report on the Rail Squat Project R3-105[END_REF][START_REF] Nakkalil | Formation of adiabatic shear bands in eutectoid steels in high strain rate compression[END_REF]. According

30
to Nakkalil [START_REF] Nakkalil | Formation of adiabatic shear bands in eutectoid steels in high strain rate compression[END_REF], the WEL zones could be linked to a succession of ASBs that would join to form a homogenous layer. However, the contact temperature does not exceed 400 °C in normal condition [START_REF] Baumann | Surface modification, corrugation and nanostructure formation of high speed railway tracks[END_REF][START_REF] Ertz | A comparison of analytical and numerical methods for the calculation of temperatures in wheel-rail contact[END_REF][START_REF] Newcomb | A transmission electron microscopy study of the white-etching layer on a rail head[END_REF][START_REF] Linck | Consequence of contact local kinematics of sliding bodies on the surface temperatures generated[END_REF][START_REF] Zwierczyk | Thermal and Stress Analysis of a Railway Wheel-Rail Rolling-Sliding Contact[END_REF]. In this mechanism the temperature is the driving force and the mechanical stresses facilitate the kinetics of transformation [START_REF] Baumann | Formation of white-etching layers on rail treads[END_REF][START_REF] Al-Juboori | Evolution of rail surface degradation in the tunnel: The role of water on squat growth under service conditions[END_REF][START_REF] Hilliard | Iron-carbon phase diagram isobaric sections of the eutectoid region at 35, 50 and 65 kilobars[END_REF][START_REF] Wu | Laboratory simulation of martensite formation of white etching layer in rail steel[END_REF].

35

Nevertheless, the retained austenite is not present in the whole WELs. Thus, another mechanism induced mechanically has been proposed to explain the formation of a nano Fe-α grains saturated with C atom [START_REF] Baumann | Surface modification, corrugation and nanostructure formation of high speed railway tracks[END_REF][START_REF] Lojkowski | Nanostructure formation on the surface of railway tracks[END_REF][START_REF] Newcomb | A transmission electron microscopy study of the white-etching layer on a rail head[END_REF][START_REF] Simon | Tribological characterization of rail squat defects[END_REF]. Indeed, in the case of cold drawing, Djaziri [START_REF] Djaziri | Deformation-Induced Martensite: A New Paradigm for Exceptional Steels[END_REF] proved the transformation of pearlite into martensite mechanically is possible without a prior austenitization. This mech-

40

anism is based on the severe plastic deformation that creates a high density of dislocations. The dislocations play a key role in the dissolution of cementite, the reduction of grain size and in the C migration [START_REF] Gavriljuk | Decomposition of cementite in pearlitic steel due to plastic deformation[END_REF][START_REF] Languillaume | Cementite dissolution in heavily cold drawn pearlitic steel wires[END_REF][START_REF] Sauvage | Atomic-scale observation and modelling of cementite dissolution in heavily deformed pearlitic steels[END_REF]. In the case of the wheel/rail contact, the rail undergoes high repetitive contact stresses leading also to an accumulation of plastic deformation. The latter induces several 45 microstructural changes until the WEL formation: cementite alignment with the rolling direction [START_REF] Alwahdi | Subsurface microstructural analysis and mechanical properties of pearlitic rail steels in service[END_REF][START_REF] Tyfour | Interaction between Wear and Rolling Contact Fatigue in Pearlitic Rail Steels[END_REF], reduction of interlamellar spacing [START_REF] Dylewski | The tridimensional gradient of microstructure in worn rails -Experimental characterization of plastic deformation accumulated by RCF[END_REF], breaking of cementite lamellaes and reduction of grain size [START_REF] Al-Juboori | Squat formation and the occurrence of two distinct classes of white etching layer on the surface of rail steel[END_REF][START_REF] Al-Juboori | Characterisation of White Etching Layers formed on rails subjected to different traffic conditions[END_REF][START_REF] Lojkowski | Nanostructure formation on the surface of railway tracks[END_REF][START_REF] Newcomb | A transmission electron microscopy study of the white-etching layer on a rail head[END_REF][START_REF] Baumann | Surface modification, corrugation and nanostructure formation of high speed railway tracks[END_REF]. All these steps will facilitate the cementite dissolution and the C migration [START_REF] Ivanisenko | The mechanism of formation of nanostructure and dissolution of cementite in a pearlitic steel 620 during high pressure torsion[END_REF]. In this mechanism the temperature is no longer the driving force of WEL formation the thermo-mechanical loadings history undergone by the running band of rails [START_REF] Kumar | Microstructural evolution of white and brown etching layers in pearlitic rail steels[END_REF][START_REF] Newcomb | A transmission electron microscopy study of the white-etching layer on a rail head[END_REF].

Thus, a thermo-mechanical model is proposed to represent the progressive WEL formation whatever the mechanism of formation. The constitutive equations are mainly based on a previous model [START_REF] Antoni | A combined thermo-mechanical model for Tribological Surface Transformations[END_REF] that is reminded in the Section 2. Then, a description of the improvement of this model with a new yield surface of WEL formation is detailed in Section 3.

Thereafter, the model is implemented into a 2D Finite Element case representing the cumulative passage of trains. A study is done to estimate an order of magnitude of the parameters in wheel-rail contact conditions. Then a study of the variability of the thermomechanical field is realized to see how it could modify the kinetics of the WEL formation.

Thermo-mechanical model

In this section, we recall the thermomechanical model proposed by Antoni's works [START_REF] Antoni | A combined thermo-mechanical model for Tribological Surface Transformations[END_REF][START_REF] Antoni | Return Mapping Algorithms (RMAs) for 625 Two-Yield Surface Thermoviscoplastic Models Using the Consistent Tangent Operator[END_REF]. Thus, all the state variables and all the equations given in this section were presented in Antoni's papers.

Motivations

As observed in the literature, the structure of WELs and their mechanism of formation differ from one study to another. However, from a macroscopic point of view, it is assumed that they have similar mechanical properties and they result from a cyclic thermo-mechanical process. Thus, the WEL formation will be considered as a progressive irreversible transformation of an initial pearlitic structure to a "martensitic-like" structure.

The modelling of the permanent solid-solid phase transformation induced mechanically was done in the past [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases[END_REF][START_REF] Leblond | Mathematical modelling of transformation plasticity in steels II: Coupling with strain hardening phenomena[END_REF][START_REF] Taleb | A micromechanical modeling of the Green-635 wood-Johnson mechanism in transformation induced plasticity[END_REF]. Those models concern materials that could transform from a parent phase to a daughter phase under mechanical stresses where the Von Mises stress is lower than the yield strength. It induces an abnormal plastic flow, called "TRansformed Induced Plasticity" (TRIP).

Using the assumption that the WEL results from a thermomechanical process and that there exists two kinds of plasticity in the process (classical isochoric plasticity and TRIP plasticity), a first phenomenological law of WEL formation was extended by Antoni [START_REF] Antoni | A combined thermo-mechanical model for Tribological Surface Transformations[END_REF].

State Variables

The state variables used in this model are divided into observable variables and internal variables.

Observable variables are the absolute temperature T > 0 and the total strain tensor (with the assumption of small displacements). The strain tensor is decomposed into its spherical part 1 3 T r( )G and its deviatoric part e where G denotes the metric tensor (Eq. ( 1)).

= 1 3 T r( )G + e (1)
Internal variables concern the variables associated with the classical isochoric plasticity and the TRIP plasticity.

For the classical isochoric plasticity, an isotropic hardening is assumed considering v as the isotropic hardening variable and e pc the classical plastic deviatoric strain tensor.

For the TRIP plasticity, an internal variable z, ranging from 0 to 1, is defined. There are several interpretations of this variable from different points of view. From a metallurgical point of view, it corresponds to the proportion of a parent phase (pearlite) that has been transformed into a daughter phase (WEL).

But this approach considers that the transformation is binary and that there exists an equilibrium between the two phases: the pearlitic structure (parent phase) and the WEL (daughter phase). However, the WEL formation is more complex and there are several steps to transform the original pearlite into WEL [START_REF] Ivanisenko | The mechanism of formation of nanostructure and dissolution of cementite in a pearlitic steel 620 during high pressure torsion[END_REF]. Thus z would be more likely an indicator of the different steps of transformation before obtaining the complete transformation. The gradient of the microstructure as observed in the works of Dylewski [START_REF] Dylewski | The tridimensional gradient of microstructure in worn rails -Experimental characterization of plastic deformation accumulated by RCF[END_REF] can be correlated with the gradient of the z value (Fig. 2). The closer z is to 1 the more the structure will approach a complete transformation into a WEL. The secondary variable denotes the TRIP plastic deviatoric strain tensor e pz and corresponds to the plastic flow due to the phase transformation.

TRIP plasticity is accompanied by a density change (ρ martensite = 7800 kg.m -3

and ρ f errite = 7700 kg.m -3 ). So, the total TRIP plastic strain tensor, pz is defined in Eq. ( 2). The parameter κ denotes the density change during the 115 WEL formation.

pz = e pz - 1 3
z κ G (2)

Constitutive equations

The constitutive equations are described below:

σ = -P G + s (3)
P = - 3λ + 2μ 3 tr( ) + z κ -3α(T -T 0 ) (4) s = 2μ(e -e pc -e pz ) ( 5
)
where σ is the Cauchy stress tensor, P is the hydrostatic pressure, s is the stress deviator tensor, (λ, μ) are the Lame coefficient, α the thermal expansion 120 coefficient, T the absolute temperature and T 0 the room temperature.

The heat equation is given by:

ρ i C T T 0 Ṫ -kΔT = -(3λ+2μ)α T T r( ˙ )+s : ( ėpc + ėpz )-hv v+ 1 κ [P -(3λ+2μ)αT ] ż (6)
where " Δ " is the Laplacian. The first term on the right hand side of Eq. ( 6) corresponds to thermo-elasticity, the second one, to TRIP-like plasticity and the classical plasticity and the last one, to the latent heat associated with the 125 solid-solid phase transformation.

The equations of evolution of the internal state variables associated with the TRIP-like process (z and e pz ) and the classical isochoric plasticity process (v and e pc ) are detailed in the following sections.

Equations for the classical isochoric plasticity 130

Cyclic plasticity models for rail steel grades generally introduce a combination of kinematic and isotropic hardening to take into account the ratchetting response [START_REF] Ekberg | An engineering model for prediction of rolling contact fatigue of railway wheels, Fatigue <html ent glyph="@amp;" ascii="&amp[END_REF][START_REF] Meyer | Microstructure and mechanical properties of the running band in a pearlitic rail steel: Comparison between biaxially deformed steel and field samples[END_REF][START_REF] Pandkar | Ratcheting-based microstructuresensitive modeling of the cyclic hardening response of case-hardened bearing steels subject to Rolling Contact Fatigue[END_REF][START_REF] Schleinzer | Residual stress formation during the roller straightening of railway rails[END_REF]. It seems clear that the modelling of the WEL formation requires a cyclic plasticity model but the initial idea of this model was rather to focus on the modelling of the TRIP plasticity. Thus, for a first approach, 135 a simple model of linear isotropic hardening is defined (where h is the linear isotropic modulus) in Eq. [START_REF] Newcomb | A transmission electron microscopy study of the white-etching layer on a rail head[END_REF].

We consider a non-viscous plastic model. However, for the sake of numerical resolutions, this model is approached by a visco-plastic model which is regularized in order to converge to the non-viscous plastic model (Eq. 8 and 9). So, 140 the parameter ξ is chosen small enough to tend towards the plasticity model.

The plastic flow rate is oriented by the stress deviator tensor (Eq. ( 9)).

f pc (σ eq , v) = σ eq -(σ y + hv) (7) Viscoplastic multiplier v = < f pc (σ eq , v) > ξσ y (8) Flow rule ėpc = v 3 2σ eq s (9)
where σ eq is the Von Mises equivalent stress, σ y the classical yield strength, ξ is the characterisitc time of viscous effect associated with respectively the classical plasticity and the TRIP plasticity < . > denotes the Macaulay brackets (< x >= x when x > 0 and < x >= 0 when x ≤ 0).

Equations for the TRIP plasticity

The expression of the yield function of the TRIP plasticity (Eq. ( 10)) is based on the assumption that the WEL transformation is induced by a coupling between a thermal term f therm (T ) with a mechanical term f meca (P, σ eq ).

Whenever the difference between those two terms becomes positive, the WEL transformation is in progress which is numerically expressed by an evolution of the internal variable z. The choice of the formulation of the term f meca (P, σ eq ) and its physical meaning are detailed in Section 3 (Eqs. ( 15) and ( 16)).

Again for the same reasons than the classical plasticity, the model of TRIP plasticity is approached by a visco-plastic model that is regularized. The parameter η must be chosen appropriately in order to converge towards a plastic model (Eq. 11). Similarly to the classical plasticity, the plastic flow rate linked to the WEL formation is also assumed to be oriented by the stress deviator tensor (Eq. ( 9)).

Yield function f pz (P, σ eq , T ) = f therm (T )f meca (P, σ eq ) ( 10)

Viscoplastic multiplier ṗ = < 1 -z > η < f pz (P, σ eq , T ) > H(P ) (11)
Flow rule 1 ėpz = ṗ 3 2σ eq s;

Flow rule 2 ż = κ ṗ [START_REF] Kumar | Microstructural evolution of white and brown etching layers in pearlitic rail steels[END_REF] where H(.) denotes the Heavyside step function (H(x) = 1 when x ≥ 0 and

H(x) = 0 when x ≤ 0).
f therm (T ) is a linear function of the temperature ranging from 0 to 1 for temperatures between the room temperature (T 0 ) and the critical temperature of transformation (T z i ):

165 f therm (T ) = T -T 0 T z i -T 0 (14)
Remarks :

• Eq. ( 11) highlights the fact that phase transformation occurs only in a state of compression (H(P ) > 0) and it ends once z reaches one (< 1-z >).

• Eq. ( 13) is consistent with the condition of the irreversibility of the transformation because ż cannot be negative. 170

A phenomenological yield function of WEL formation

3.1. Formulation of the term f meca (P, σ eq ) of the Eq. ( 10)

Numerous mechanical parameters may be responsible for the formation of WEL. The main factors that can be identified in the literature are the temperature, the hydrostatic pressure [START_REF] Hilliard | Iron-carbon phase diagram isobaric sections of the eutectoid region at 35, 50 and 65 kilobars[END_REF][START_REF] Wu | Laboratory simulation of martensite formation of white etching layer in rail steel[END_REF][START_REF] Al-Juboori | Structural and microstrutural investigation of two distinct classes of white etching layer formation on the rail[END_REF][START_REF] Zhu | Understanding and treatment of squat defects in a railway network[END_REF], shear stresses [START_REF] Ivanisenko | The mechanism of formation of nanostructure and dissolution of cementite in a pearlitic steel 620 during high pressure torsion[END_REF][START_REF] Zhu | Understanding and treatment of squat defects in a railway network[END_REF]42] and the 175 density of dislocations [START_REF] Lojkowski | Nanostructure formation on the surface of railway tracks[END_REF][START_REF] Newcomb | A transmission electron microscopy study of the white-etching layer on a rail head[END_REF][START_REF] Gavriljuk | Decomposition of cementite in pearlitic steel due to plastic deformation[END_REF][START_REF] Ivanisenko | The mechanism of formation of nanostructure and dissolution of cementite in a pearlitic steel 620 during high pressure torsion[END_REF]. Other authors have also suggested dynamic effects such as the rate of plastic deformation [START_REF] Nakkalil | Formation of adiabatic shear bands in eutectoid steels in high strain rate compression[END_REF] or the frequency of stress [START_REF] Newcomb | A transmission electron microscopy study of the white-etching layer on a rail head[END_REF].

The model developed by Antoni [START_REF] Antoni | A combined thermo-mechanical model for Tribological Surface Transformations[END_REF] allowed the qualitative prediction of WEL formation considering only a coupling between the hydrostatic pressure (P ) and the temperature (T ) (Eq. ( 15)). However, by interpolating the data 180 from the literature [START_REF] Hilliard | Iron-carbon phase diagram isobaric sections of the eutectoid region at 35, 50 and 65 kilobars[END_REF][START_REF] Wu | Laboratory simulation of martensite formation of white etching layer in rail steel[END_REF], the hydrostatic pressure reduces linearly the austenitization temperature by only 20 °C/GPa. Thus, under wheel/rail contact conditions (with a hydrostatic pressure in the order of 1 GPa), the austenitizing temperature will be reduced by about 20 °C. So it seems clear that the dependency of the hydrostatic pressure alone is insufficient to predict the WEL (Eq. ( 16) and ( 17)). Then, with this new model the WEL formation would result from a coupling between the hydrostatic pressure and the shear stress which is enhanced by the temperature.

f old = f meca (P ) = e -<P > ω 1 (15)
f new = f meca (P, σ eq ) = < e -<P > ω 1 -e -σ c ω 1 > 1 -e -σ c ω 1 < e -σ eq ω 2 -e -τ (P ) ω 2 > 1 -e -τ (P ) ω 2 (16)
With τ (P ) defined as follow :

τ (P ) = τ c < e -<P > ω 3 -e -σ c ω 3 > 1 -e -σ c ω 3 (17)
For the definition of the model, it is assumed that whatever the values of two variables, there is a value for the third one for which there is necessarily a phase transformation. In other words, it suggests the existence of three critical points:

• a critical temperature T z i • a critical hydrostatic pressure σ c • a critical shear stress τ c
The new yield function f pz new (P, σ eq , T ) is then given as follow:

f pz new (P, σ eq , T ) = T -T 0 T z i -T 0 - < e -<P > ω 1 -e -σ c ω 1 > 1 -e -σ c ω 1 < e -σ eq ω 2 -e -τ (P ) ω 2 > 1 -e -τ (P ) ω 2 (18)
This model takes well into account those critical points. Indeed, considering a thermo-mechanical state with T = T 0 and σ eq = 0, the term f therm is equal o n ure static pressu static pressu ear stress τ τ τ yield function (P, σ P, σ eq eq , T , T ) = ) odel t d to 0 and the new yield function f pz new will be then:

f pz new (P, σ eq = 0, T = T 0 ) = - < e -<P > ω 1 -e -σ c ω 1 > 1 -e -σ c ω 1 (19)
Thus, the only condition leading to a phase transformation will be when P exceeds σ c . With the same reasoning, it is verified that τ c and T z i are also critical transformation points with this formulation.

A 3-D representation

A parametric study of the new yield surface is presented in this section with the goal to develop a strategy to identify the parameters of the model. There are six parameters classified into two groups. The first one concerns the three critical points of transformation (T z i , σ c and τ c ) and the other one is relative to three parameters of sensitivity for the coupling between the temperature, the hydrostatic pressure and the Von Mises equivalent stress (ω 1 , ω 2 and ω 3 ).

A 3-D representation of the yield surface (with T , P and σ eq as variables)

is given by plotting the surface of equation f pz (T, P, σ eq ) = 0 (Fig. 3).

It is possible to modify the curvature of the surface changing the value of one parameter of the parameters ω i (Fig. 3). Each ω i corresponds to the coupling between two variables separately so it is possible to identify them separately. A brief description of the process of identification is then given below.

The parameter ω 1 allows to modify the curvature of the surface in the plane pressure-temperature (Fig. 3). It could be identified by carrying out cyclic compression tests at various temperatures. It will give conditions of pressure and temperature leading to WEL formation or not. Thus, the suitable value of ω 1 will be the one that will modify correctly the curvature of the surface that delimits the conditions of the WEL formation.

The parameter ω 2 allows to modify the curvature of the surface in the plane shear stress-temperature (Fig. 3). It could be identified by carrying out cyclic shear tests at various temperatures.

Finally, The parameter ω 3 allows to modify the curvature of the surface in the plane shear stress-pressure (Fig. 3). It could be identified by conducting If the coupling between the variable is unknown, for lack of experimental results, a linear coupling between the variables is possible taking ω 1 , ω 2 and ω 3 high enough.

A methodology to identify the parameters of the model with the application of the wheel-rail contact

A first parametric study is made to identify and have an order of magnitude of each parameter in the case of the WEL formation in the wheel-rail contact.

This methodology could be applied to any material presenting TSTs formation induced thermo-mechanically. The material modelled concerns a fully pearlitic steel (Carbon atome content of 0.77%) commonly used in the French Railway network (R260 steel grade [START_REF] Steel | Rail steel grades[END_REF]). (see Fe-C diagram under atmospheric pressure [START_REF] Wu | Laboratory simulation of martensite formation of white etching layer in rail steel[END_REF]). For the sake of simplicity, the temperature T z i is fixed to 1000 K. Then, the litterature shows [START_REF] Hilliard | Iron-carbon phase diagram isobaric sections of the eutectoid region at 35, 50 and 65 kilobars[END_REF][START_REF] Wu | Laboratory simulation of martensite formation of white etching layer in rail steel[END_REF] the austenitizing temperature decreases linearly with the hydrostatic pressure of around 20 °C/GPa. Thus, the coupling between the hydrostatic pressure and the temperature is considered as linear, ω 1 is thus arbitrarily chosen very high (100 GP a). Moreover, for the purpose of satisfying the hypothesis of a linear decrease of 20 °C/GPa, the hydrostatic pressure leading to a transformation at T = T 0 is assumed to be equal to

σ c = 30 GP a.
The third critical point concerning the critical shear stress (τ c ) is unfortunately unknown in the literature but some studies showed the effect of the shear stress in facilitating the phase transformation [START_REF] Rittel | The mechanical response of pure iron at high strain rates under dominant shear[END_REF]. A parametric study has to be done to see the sensitivity of this parameter on the yield surface. At this stage, three parameters are identified (T z i , σ c and ω 1 ) and will be fixed for the rest of the study (Table 1).

T z i (K) σ c (GP a) ω 1 (GP a) 1000 30 100
Table 1: Parameters fixed for the rest of the study Thus, a parametric study of the three other parameters (τ c , ω 2 and ω 3 ) is made in the conditions of wheel-rail contact. In that case, the pressure is often around 1 GPa and the temperature in the contact could increase up to 300 • C in normal conditions (600 K). Thus, for the study, a contact pressure of 1 GPa and three temperatures (400 K, 500 K and 600 K) are then considered.

Dynamics contact conditions are experimentally difficult to estimate accurately, especially the tangential component in the real contact zone. Thus, the shear stress that could induce the WEL formation is unknown. So this model makes it possible to determine the critical shear stress σ eq critique which will induce a phase transformation under different rail conditions. Therefore, with fixed temperature and pressure, the yield function f pz new (T, P, σ eq ) is then a variable function depending of the shear stress. So, by plotting the mechanical term f new with the shear stress it is possible to estimate this critical shear stress, i.e the shear stress such that f new is so equal to the thermal term f therm (T ). A parametric study on the parameters ω 2 , ω 3 and τ c is done to see their effect on 280 the value of σ eq critique (Fig. 4, 5 and 6). A general remark on the model is that whatever the values of the parameters, a rise in temperature and/or pressure will lower the value of σ eq critique , making the WEL formation easier to occur. So, the model is coherent with the assumption that the temperature and the pressure facilitate the WEL formation.

Concerning the parametric study, three Tables (Tables 2, 3 and4) are presented to highlight the effect of each parameter. In fact, Table 2 reveals that the lower τ c is, the lower σ eq critique is. The same conclusion is done for ω 2 (Table 3) and ω 3 (Table 4), that denotes respectively the sensitivity of the coupling between the shear stress and the temperature (ω 2 ) and the coupling between the pressure and the shear stress (ω 3 ). Therefore, ω 2 , ω 3 and τ c are three parameters that can modify the curvature (ω 2 , ω 3 ) and the boundaries (τ c ) of the yield function f pz new (T, P, σ eq ). Moreover, they modifiy the value of the critical shear stress σ eq critique necessary to induce the WEL formation. Thus, depending on the contact conditions, there could be WEL formation only if the shear stresses are higher than σ eq critique .

Wheel/rail contact problem (2D-Finite Element case)

Modelling of the wheel-rail contact

For this first study on a real case, simulations of a 2D model of the rail running band in the longitudinal direction submitted to a thermo-mechanical mobile field using the commercial FE software ABAQUS have been carried out (Figs. 7 and8). The material behavior (Sections 2 and 3) has been implemented p σ σ eq eq c cri ri

P (GP a) T (K) ω 2 (GP a) ω 3 (GP a) τ c (GP a) σ eq critique (GP
p p the mechanical problem are coupled. In order to reduce the computation time, a 305 stationary resolution is chosen for the thermal problem. Thus, the two problems are uncoupled into a thermal problem that determines the temperature fields that will feed the mechanical problem.

For the calculation, a square (with a length of 100 mm) modelling the railhead of the rail in the longitudinal direction in 2D is considered (Fig. 7). The 310 mesh size is refined from the surface to five millimeters (that corresponds to the depth where the Von Mises stress is maximum in the rail [START_REF] Zwierczyk | Thermal and Stress Analysis of a Railway Wheel-Rail Rolling-Sliding Contact[END_REF]) with the smallest element size of 10 μm (Fig. 8). The mesh size has been so refined to satisfy the mesh convergence and to have the local thermo-mechanical state in the vicinity of the surface in order to be able to reproduce numerically the WEL formation is then applied to the rail surface. This field is composed of two mechanical fields (along normal and tangentiel directions) and a thermal field (Fig. 8). The normal mechanical field p(x), which follows a theoretical Hertz profile, represents the pressure field exerted by the wheel on the rail (Eq. ( 20)). In 2D, the wheelrail contact is considered lineic with a size of 2a. In real conditions, the contact zone changes because of the plasticity of the material so the size of contact (2a) and the pressure are not constant [START_REF] Sebès | A fast-simplified wheel-rail contact model consistent with perfect plastic materials[END_REF]. Nevertheless, for the calculations the size of contact and the maximal pressure (p 0 ) are considered constant as a first approach. The value of a is then fixed to 5 mm that corresponds to the initial conditions in an elastic case [START_REF] Zwierczyk | Thermal and Stress Analysis of a Railway Wheel-Rail Rolling-Sliding Contact[END_REF]. p 0 denotes the maximal pressure on the contact and is fixed to 1 GPa. The tangential mechanical field q(x) is modelled by the Carter theory (Eq. ( 21)) that is a simple way to model the traction loading of the wheel-rail contact in a first approach (Fig. 9). This theory divides grossly the contact zones into two parts, an adhesion zone that corresponds to the front edge of the wheel and a sliding zone to the back of the wheel. The size of the sliding zone is difficult to evaluate and it depends on the global (wheel slippage on the rail inducing creepage [? ]) and local contact conditions. For the calculation, conditions of rolling with sliding are imposed to represent local sliding during a wheel passage. The adhesion zone and the slipping zone are arbitrarily chosen to be the same size (c = a/2 and d = ac).

The thermal field is modelled by a temperature field. The aim of this study is not to focus on the heat flows induced in the wheel-rail contact but rather 345 to have an idealized case of contact with a simplified thermal field. Thus, a temperature T(x) field is imposed on the contact zone and assumes a uniform temperature T i in the surface (Eq. ( 22)). The temperature is of the order of 450 K [START_REF] Ertz | A comparison of analytical and numerical methods for the calculation of temperatures in wheel-rail contact[END_REF].

p(x) = p 0 1 - x -x 0 a 2 , with x ∈ [x 0 -a, x 0 + a] ( 20
)
Where x is the longitudinal coordinate of the point on the rail, x 0 the coor-350 dinate of the point in the contact center. The goal of this study is to be able to obtain results comparable to the observations of real rails. Based on the observations, three conditions are imposed in order to validate the model: To evaluate the effect of the parameter τ c , defined as the critical shear stress, a linear coupling is defined, between the shear stress and the temperature on one hand (ω 2 = 100 GP a = +∞) and between the shear stress and the hydrostatic pressure on the other hand (ω 3 = 100 GP a = +∞). The higher τ c is, the lower is the depth of the transformed zone. Indeed, by choosing a value of τ c equal to 0.7 GPa the depth is of the transformed zone is of 3.3 mm whereas for a value of 0.75 GPa the depth is reduced to 0.3 mm (Fig. 10).

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ q(x) = q 1 (x) + q 2 (x) q 1 (x) = μ 0 p(x) with x ∈ [x 0 -a, x 0 + a] q 2 (x) = - c a μ 0 p 0 1 - x -x 0 -d c 2 with x ∈ [x 0 + 2c -a, x 0 + a] (21)
T (x) = ⎧ ⎨ ⎩ T i if x ∈ [x 0 -a, x 0 + a] T 0 otherwise (22)
•
The study of the coupling between the shear stress and the temperature is now done (effect of ω 2 ). For this study, τ c is now fixed to 1 GPa in order to have no significative influence of this parameter. A linear coupling between the shear stress and the pressure is once again considered (ω 3 = 100 GP a). The higher ω 2 is, the smaller is the depth of transformed zone. Indeed, by chosing a value of ω 2 equal to 0.4 GPa, the depth is 2 mm whereas for a value of 0.5 GPa the depth of the transformed zone is reduced to 0.5 mm (Fig. 11).

Similar conclusions are drawn concerning the coupling between the shear stresses and the hydrostatic pressure (effect of ω 3 ). In fact, the higher ω 3 is, the smaller the transformed zone is (Fig. 12).

At this stage, there is not a unique set of parameters leading to a transformed zone at most 0.3 mm thick under the classical thermomechanical conditions. Thus, the condition for obtaining a transformed zone of a maximimum thickness of 0.3 mm (condition #1) is satisfied for a multiple set of parameters. The best set of parameters will depend on the sensitivity of each coupling that has to be fed experimentally with dedicated shear test under pressure and temperature controlled (work in progress for a future paper). Because of the lack of experimental data available at this time, an arbitrary set of parameters verifiying the condition #1 is considered and will be fixed for the rest of the study (Table 5).

τ c (GP a) ω 2 (GP a) ω 3 (GP a)
1 100 4

Effects of the variability of the thermomechanical fields

During a train passage, the rail undergoes a cyclic thermomechanical field that is not constant. There are three main variations that are summurized 410 below:

• A variation of the normal forces [START_REF] Aknin | Contribution of railway dynamics codes to the understanding of the metallurgical behavior of the rail tread[END_REF][START_REF] Ishida | The effect of lateral creepage force on rail corrugation on low rail at sharp curves[END_REF][START_REF] Li | Differential wear and plastic deformation as causes of squat at track local stiffness change combined with other track short defects[END_REF][START_REF] Li | An investigation into the causes of squats-Correlation analysis and numerical modeling[END_REF][START_REF] Wild | Microstructure alterations at the surface of a heavily corrugated rail with strong ripple formation[END_REF]: those variations are due to two scales of the contact, from a macroscopic point of view (multibody dynamics, infrastructure) and at the microscale (imperfection of contact, rugosity, corrugation). For this first study, those variations will 415 simply be modelled by a variation of the maximal Hertz pressure (p o +Δp).

• A variation of the tangential forces [START_REF] Busquet | 3D finite element investigation on the plastic flows of rolling contacts-correlation with railhead microstructural observations[END_REF][START_REF] Ishida | Experimental study on rolling contact fatigue from the aspect of residual stress[END_REF][START_REF] Johnson | The Mechanics of Plastic Deformation of Surface and Subsurface Layers in Rolling And Sliding Contact[END_REF][START_REF] Kondo | Cause, increase, diagnosis, countermeasures and elimination of Shinkansen shelling[END_REF]: the tangential forces are often unknown because they depend on the contact conditions that could change with many factors (adhesion, rugosity, climate factors, third

J • • A vari A
are often are ofte could ch uld ch body). The effect of the variation will be numerically driven by a variation of the coefficient of friction (μ o + Δμ).

• A temperature variation [START_REF] Daniel | Final Report on the Rail Squat Project R3-105[END_REF][START_REF] Ertz | A comparison of analytical and numerical methods for the calculation of temperatures in wheel-rail contact[END_REF][START_REF] Ertz | Thermal stresses and shakedown in wheel/rail contact[END_REF][START_REF] Linck | Consequence of contact local kinematics of sliding bodies on the surface temperatures generated[END_REF][START_REF] Nakkalil | Formation of adiabatic shear bands in eutectoid steels in high strain rate compression[END_REF]: the temperature change is due to frictional power dissipation, the plasticity and also, in some specific cases, from electrical breakdown. Depending on the contact conditions, in particular the sliding part there could be a variation of the heat flow and thus variations of the temperature. The change of temperature will then be a variation of the surface temperature imposed (T i + ΔT ).

A parametric study is conducted to see the effect of the variability of the thermomechanical loading on the depth of the transformed zone. Whatever the increase of one of those variables, the thickness of the transformed zone increases (Fig. 13, 14 and 15). The second observation is that when simultaneous variations of several variables are applied, the thickness of the transformed zone increases a lot more than when there is a unique variation (Fig. 16). Thus,

local

Study for cyclic loading

The goal of this section is to determine the number of train producing a WEL zone (quasi complete transformation when the internal variable z is equal A parametric study of the kinetics of the z variable is subsequently carried out to see the effect of the parameter κ and the effect of the variations of the thermomechanical field (Eq. ( 12) and ( 13)).

460

The study of sensitivity to the parameter κ shows that the higher κ is, the faster is the transformation (Fig. 18). Indeed, for a value of κ equal to 200, the number of trains needed to reach a quasi complete transformation is around 90 whereas for κ = 300, the number of trains is reduced to 70.

Then, variations of the thermochanical loading (Δp, Δμ or ΔT ) are consid-465 ered to see their effects on the kinetics of z and the number of trains necessary to obtain a quasi complete WEL transformation with a thickness of a few dozen microns.

Whatever the increase of one of those variables (Δp, Δμ or ΔT ), the WEL transformation is faster than in the classical conditions (Fig. 19, 20, 21 and 22).

470

For example, an increase of 10% of the pressure will decrease the number of the train necessary to have a complete WEL transformation (Fig. 19). Moreover, a combination of simultaneous variations will accelerate the kinetics of WEL formation (Fig. 22).

The second striking feature is that a variation of different variables could crease o ease o friction from 0.3 to 0.4 or an increase of the pressure from 1 GPa to 1.2 GPa, 60 trains are required to reach a quasi complete transformation (Fig. 23). In addition, a simultaneous combination of several variations can also lead to the same number of trains (p = 1.1 GP a, μ = 0.35 and T = 450 K) (Fig. 23). The path to obtain a plain WEL is not unique. This model is a promising tool to better understand and anticipate the formation of WEL in the wheel/rail contact, depending on the thermomechanical conditions applied to the rail. From a broader point of view, this model could be applied to the formation of TST in other areas that would be associated with thermomechanical coupling.

Aknowledgments

This work is part of the multi-disciplinary project MOPHAB, which aims to improve knowledge and understanding of the mechanisms leading to the formation of White Etching Layer in the materials used to construct railways, and to develop numerical models. The authors would like to thank IRT Railenium for their funding support and the industrials partners for their support (RATP, SNCF, British Steel).

The passage to the limit of the function τ (P ) leads to: f pz : lim

τ c →+∞ ω 2 →+∞ τ c ω 2
< e -σ eq ω 2e -τ (P )

ω 2 >
1e -τ (P )

ω 2 = lim ω 2 →+∞
e -σ eq ω 2 = 1

Then the function f meca (P, σ eq , T ) can be simplified: lim

τ c →+∞ ω 2 →+∞ τ c ω 2 f new = < e -<P > ω 1 -e -σ c ω 1 > 1 -e -σ c ω 1
The yield function is then:

f pz new (P, σ eq , T ) = f pz new (P, T ) = T -T 0 T z i -T 0 - < e -<P > ω 1 -e -σ c ω 1 > 1 -e -σ c ω 1 (24)
With the parameters τ c and ω 2 taken large enough a model depending only on the temperature and the hydrostatic pressure is obtained. Only two parameters are to be identified in this case: σ c and ω 1 .

530 By taking a particular value of the parameter σ c (Eq. 25) an equivalent expression of the Antoni's yield surface is obtained (proof below).

h 1 2 3

 123 Linear isotropic hardening modulus ξ, η Characterisitc time of viscous effect associated with respectively the classical plasticity and the TRIP plasticity κ Parameter linked to the volumic change Parameter of sensitivity for the coupling between the temperature T and the hydrostatic pressure P ω Parameter of sensitivity for the coupling between the temperature T and the Von Mises equivalent stress σ eq ω Parameter of sensitivity for the coupling between the hydrostatic pressure P and the Von Mises equivalent stress σ eq

Figure 1 :

 1 Figure 1: Longitudinal rail cross section in worn rail with the presence of cracks on the railroad after WEL formation at the WEL/pearlite interface and within the WEL

50 but

 50 could catalyze the cementite dissolution and the C diffusion.To summarize, the WEL transformation is induced by the accumulation of the train passage and has not a unique mechanism of formation. It depends on 4

  110

Figure 2 :

 2 Figure 2: Schematic view of the tridimensional gradient of microstructure explained by Dylewski [29] interpreted numerically using an internal variable z (blue box)

  new formulation of the mechanical term f meca based on the previous formulation is proposed taking into account a new variable associated with the deviatoric part of the stress tensor (Von Mises equivalent stress σ eq )

  arameter ω ω 2 stress-temper ress-temp ts at va ts at va e The firs The fi τ c τ ) and the o ) and the betwe betwe cyclic shear tests under different hydrostatic pressure conditions.

Figure 3 :

 3 Figure 3: 3-D representation of the yield function f pz (P, σ eq , T ) and effects of the parameters of sensitivity ω 1 , ω 2 and ω 3 on the curvature of the yield surface

  The first step is to identify the three critical points of transformation. The temperature of transformation noted T z i corresponds to the austenitization tem-the case of the pearlitic steel, this temperature is equal to 722 • C

14

 14

Figure 4 :Figure 5 :

 45 Figure 4: Effect of τ c on the critical shear stress of transformation

Figure 6 :

 6 Figure 6: Effect of ω 3 on the critical shear stress of transformation

  315in few dozen of microns from the surface (Fig1).

Figure 7 :

 7 Figure 7: 3D model of the rail and the longitudinal model considered for the study

Figure 8 :

 8 Figure 8: FE model of the rail (2D) with the thermo-mechanical field

Figure 9 :

 9 Figure 9: Carter field: x 0 is the center of the contact zone

Condition # 1 :

 1 In the classical thermomechanical conditions (p o = 1 GP a, 355 μ o = 0.3, T i = 450 K), the transformed zone has a depth of less than 0.3 mm • A variation of the thermomechanical conditions (p o + Δp, μ o + Δμ and or T i + ΔT ) increases the size of the transformed zone (called condition #2) and accelerates the kinetics of the WEL formation compared to classical 360 conditions (called condition #3). The first step is to see the influence of the missing parameters of the model (τ c , ω 2 and ω 3 ) and the thermomechanical field (p o , μ o and T i ) on the size of the transformed zone. A first parametric study (τ c , ω 2 and ω 3 ) after only one cycle is conducted with classical thermomechanical conditions (p o , μ o and 365 T i ). After this step, by chosing a good set of parameters the condition #1 will be validated. Then, using this set of parameters, a study of the sensitivity of thermomechanical fields variations (p o +Δp, μ o +Δμ and or T i +ΔT ) is realized to validate the condition #2).The second step consists in applying a cycling thermomechanical loading 370 in order to see the effects of the last missing parameter (κ) and the thermomechanical fields variations on the kinetics of the WEL formation (condition #3).

4. 2 .

 2 Study for a first cycle 4.2.1. Parametric study of the parameters of the model 375 The parametric study of the model (τ c , ω 2 and ω 3 ) is done in the classical thermomechanical conditions (p o , μ o and T i ). The study is carried out independently for each of the three parameters (τ c , ω 2 and ω 3 ) by fixing the two latter (Fig. 10, 11 and 12). One thermomechanical cycle is applied and corresponds to one wheel pass.

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: Effect of τ c on the depth of the transformed area (with ω 2 = ω 3 = 100 GP a)

  variations of the thermomechanical variables could lead to WEL patches with non uniform size which is exactly what is observed on the worn rails, the condition #2 is thus verified. The next step is to pursue the analysis by now applying cyclic loading to model the progressive WEL formation after a given number of trains.

Figure 13 :

 13 Figure 13: Effect of the contact pressure on the transformed zone (with μ o = 0.3 and T i = 450 K)

Figure 14 :Figure 15 :Figure 16 : 4 (p o = 1 1 )

 141516411 Figure 14: Effect of the temperature on the transformed zone (with P o = 1 GP a and μ o = 0.3)

Figure 17 :

 17 Figure 17: Example of the gradient of transformed zone after 100 trains : P o = 1 GP a, μ o = 0.4 and T i = 450K

475

  lead to both the same kinetics and the same number of trains leading to complete WEL transformation. Indeed, considering an increase of the coefficient of J

Figure 18 :

 18 Figure 18: Effect of κ on the kinetics of the z variable

Figure 19 :Figure 20 :Figure 21 :Figure 22 :Figure 23 :

 1920212223 Figure 19: Effect of the variability of the pressure on the kinetics of the z variable

σ c = ω 1

 1

Table 2 :

 2 Effect of τ c on the critical shear stress σ eq

	a)

Table 3 :

 3 Effect of ω 2 on the critical shear stress σ eq

	eq critique (GP a)

critique P (GP a) T (K) ω 2 (GP a) ω 3 (GP a) τ c (GP a) σ

Table 4 :

 4 Effect of ω 3 on the critical shear stress σ eq

	17

critique as a user material subroutines (UMAT). The resolution of a thermomechanical problem is costly in terms of computational time especially if the thermal and

Table 5 :

 5 Parameters chosen for the WEL model

	22

Appendix

Model independent of σ eq -equivalence with the previous model With this model, it is possible to neglect the shearing effect and to recover an expression of the previous model. To omit the shearing effect in the model, the following values of the parameters must be taken into account:

The passage to the limit of the function τ (P ) gives: lim τ c →+∞ τ (P ) = +∞


Laurent Dupont

Jean-Louis De Bougrenet De

La Tocnaye

Michel Gadonna

Thierry Sansoni

Jean-Louis De Bougrenet De La Tocnaye

Thierry Sanson

Fast liquid crystal end-less polarisation controller for first order PMD compensation

Keywords: Optical fiber transmission, Wave dispersion, Light dispersion, Wave polarization, Optical polarization, Signal convertor, Liquid crystal device

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

CONTROLEUR DE POLARISATION RAPIDE A CRISTAUX LIQUIDES POUR LA COMPENSATION DE PMD DU PREMIER ORDRE

Resume

Cet article expose le principe d'un contr6leur de polarisation rapide, sans fin, pour la compensation dynamique de dispersion de mode de polarisation (PMD). On montre de quelle far;on un tel composant peut-etre utilise comme une lame de phase variable a axe tournant, et les avantages qui en resultent pour /'optimisation des algorithmes de recherche de l 'etat de polarisation. On presente quelques resultats experimentaux obtenus a partir d'une cellule a cristal liquide electrocline, alignee de maniere homeotrope et uti lisee comme lame de phase variable, a axe tournant. On discute ensuite !es peif ormances d'un tel composant et de quelle far;on ii est possible d'ameliorer ses caracteristiques de maniere a repondre aux specifications exigees aujourd'hui par un systeme de compensa tion de PMD. Electro-optic states of polarisation convertors are currently very attractive elements both for transmission systems and optical fibre sensors. This paper focuses here on the first appli cation and more precisely on their uses to mitigate polarisation mode dispersion, in fibre long haul transmission systems. Polarisation mode dispersion presents a major impairment for multi-gigabit systems deployment and its effect on direct detection systems through pulse spreading and depolarisation, results generally in severe performance degradations [1]. In addition, temporal drifts of PMD cause the biggest problems and the greatest challenges since the compensator dynamically must adapt to the changing transmission fibre properties. We restrict here on the first order optical PMD system compensation involving the use of high speed optical polarisation controllers. We briefly discuss the main optical arrangements and the major technologies and then we propose an illustration using high speed liquid crystal materials minimizing the number of control parameters.

II. PRINCIPLE OF FIRST ORDER PMD COMPENSATION

Polarisation mode dispersion results from the fact that the propagation constant for the two eigen-modes of a fibre become distinct during the propagation, when the circular sym metry of the core is broken by stress or geometrically induced birefringence [START_REF] Heisman | Polarisation mode dispersion: fundamentals and impact on optical communi-cation systems[END_REF]. A single uniformly distributed perturbation on a short section of fibre creates two orthogonally polari sed modes along the directions of the stress induced symmetry, called principal states of polarisation. For long optical long fibre span, random polarization mode coupling between each section has to be considered. Polarisation components simultaneously launched along these two principal states arrive at the fibre end section at different times. This is called dif ferential group delay (DGD ). PMD is characterised for a given optical frequency by two time dependent parameters: principal states of polarisation and DGD. These two parameters repre sent the first order polarisation mode dispersion. For long fibre spans, the evolution of PMD with the distance is fairly slow and the random probability density function of DGD is Max wellian [START_REF] Kaminov (l | Polarisation in optical fibre[END_REF]. Therefore, PMD is difficult to assess or measure. Whereas group velocity disper sion is deterministic, PMD characteristics vary in time and frequency. Hence it is better to speak of mitigation than cancellation. The proposed and investigated methods to mitigate PMD can be divided into three main categories: electrical, optical and by use of robust modu-lation formats. We focus here on the optical approach. Most of the practical optical (and also electrical) PMD compensation systems consist of three parts: the compensating element, the error signal, and the control algorithm, the compensating element being made itself of a bire fringent element before the receiver, preceded by a polarisation controller (Figure 1) [START_REF] Francia | Simple dynamic polarisation mode dispersion compensator[END_REF]. A feedback loop featuring a control algorithm optimizes performance by setting the mitigator into one of two possible operating points. The first cancels DGD by setting the polarisation controller so that the line's slow polarisation state coincides with the fast polarisation state of the birefringent element and vice versa. In this case, DGD of the birefringent element should be adjustable. The algorithm is based on degree of polarisation optimisation.

Mithodes optiques pour attinuer la dispersion de polarisation. L'algorithme est base sur /'optimisation de degri de polarisation.

The second acts on principal states of polarisation of the entire path (line fibre and miti gator) to impose that the input polarisation state coincide with one of principal polarisation states . In this case even if the total DGD is not cancelled its effect is reduced. All of these considerations concern only first order PMD, we restrict here.

As displayed Figure 1 optical first order PMD mitigators use a combination of birefringent elements (the PMF and the polarisation controller). One section of polarisation maintaining fibre (PMF) requires one control, the polarisation controller (PC) with a simple algorithm lea ding to potentially fast response, if the other elements are fast enough (interface, Stokes para meters detector and PC). This configuration is of little flexibility to compensate higher order PMD (assuming several section of PMF separated by several PC). Clearly an important issue is a trade-off between flexibility and operation speed. High speed operations are critical to ensure tracking of fast dispersion transients [START_REF] Francia | Simple dynamic polarisation mode dispersion compensator[END_REF]. System speeds depends not only on the com plexity and implementation of the control algorithm but also on the component which gain to be as fast as possible. For instance, the polarisation controller must be able to control polari sation variations as high as 50 revolutions of the polarisation state of the signal around the Poincare sphere within 1 sec.

An another useful feature is that the polarisation convertor, for use in optical commu nications, is required to convert a time-varying arbitrary state of polarisation to another, pos-sibly time-varying arbitrary state of polarisation, without the need to reset the device, i.e. the control must be endless.

III. VARIOUS OPTICAL SOLUTIONS:

Va rious techniques, optical configurations and materials can be used to provide polari sation converters, LiNb03 waveguides, Faraday rotators, rotating wave-plates, and electro optic crystals. We focus here on the most commonly used, both regarding technologies and optical configurations. The most useful way of classifying optical compensators is, howe ver, via the number of degree of freedom (DOF), i.e. the number of elements that are controlled by the control algorithm. The DOF is generally a good estimate of the complexity and hence the cost of the compensator. Basically two degrees of freedom are needed to transform any polarisation state to any each other. However for functionality and techno logy reasons, (e.g. endless control), more parameters are used. This is discussed in the next section.

III.1. Waveguided TE-TM mode converters:

It is basically a pure waveguide approach. To obtain arbitrary polarisation transforma tion, the control of both TE-TM mode conversion and differential phase shift are required. Structures have been reported and realized in which rapidly alternating mode conversions and phase shifts effectively provide a distributed phase shift over the guide [START_REF] Walker | Endless polarisation control using an integrated optic LiNb03[END_REF][START_REF] Walker | Polarisation control for coherent communications[END_REF]. First reali sations have been demonstrated with LiNb03 materials. Three independent voltage controls allow providing a dynamic endless polarisation conversion.

The second approach is based on free-space wave-plate combinations. In this frame, there are three equivalent ways to perform any arbitrary polarisation transform. The first uses three wave-plates with variable electrically-controllable birefringence and fixed optical axes, the second three wave-plates with rotatable axes (typically illustrated by liquid crystals) and fixed birefringence, and the third combining two variable and rotatable wave-plates. Let us now describe rapidly their advantages and limitations.

III.2. Stack of fixed orientations and variable wave-plates:

This configuration has been extensively described [START_REF] Rumbauch | Polarisation control for coherent fibre optic systems using nematic liquid crystal[END_REF]8]. One can demonstrate that arbitrary polarisation transformation can be achieved (see demonstration on Poincare sphere of Figure 2.b ), with only three fractional wave-plates such that the relative orienta tions of these plates are 45° with respect to one another [8]. By changing the applied vol tages, one can control the amount of rotation with respect to each axis and then address any location on the Poincare sphere. Both PLZT (Pb-La-Zr-Ti ceramic) and Liquid crystal can be used. Polarisation control using such a technique needs three parameters and endless control can be achieved but assuming a sophisticated addressing scheme of the three wave-plates in contrast to the next approach which is fundamentally end-less.

A(1)

Stack of rotatable fractional wave-plates:

Three wave-plates are again required but in contrast to the previous case, their birefrin gence is fixed (respectively quarter, half and quarter wave-plates) whereas their principal axis can be rotated (see Figure 3). As previously one can easily demonstrated that any arbitrary polarisation can be addressed. This configuration can be implemented using various techno logies

(PLZT [9), LC [10], LiNb03 [11)).
This technique needs a particular electrode schemes in order to create a rotating electric field. With PLZT it generally uses the four sides of the bulk material (see Figure 4a) and with liquid crystal the electrodes are etched on the glass substrate as depicted in figure 4b.

A vertically aligned (smc layers are parallel to the substrate) LC layer is sandwiched bet ween two glass substrates (see Figure 56). LC molecules incline parallel to the substrate. In this configuration, LC layers work as an, optically uniaxial medium retarder. Its retardation depends on the direction, of the magnitude of the applied field, of the cell gap and the bire fringence value An. Then the principal axis of the rotatable wave-plate depends on the direc tion of the applied field. In order to rotate the electric field vector, in the plane parallel to the substrates, radial electrodes (6, 8 and more) are etched on the glass substrate. Because the cell is thin, it is possible to engrave the electrodes on the glass substrate which authorizes more complex designs. Polarisation control using such a technique (PLZT or LC) needs again three parameters but endless control can be achieved easily in contrast to previously.

III.4. Variable and rotatable wave-plate case

This case is a combination of previous ones. One can demonstrate the same results when using only one cell from which both the axis and the birefringence can be modified. In this case the variable and rotatable wave-plate is controlled by two parameters, the strength and the direction of the applied voltage (for PLZT), the applied voltage value and orientation (for LC). Continuous alteration of these parameters produces a continuous change in polarisation such that the device is able to convert an arbitrary input polarisation into a desired output polarisation but with a significant gain in control parameter and then in speed for the moni toring algorithm. Similarly both PLZT and LC solution have been proposed [12,[START_REF] Dupont | Principle of polarisa tion mode dispersion controller using homeotropic electroclinic liquid crystal confined single mode fibre devi ce[END_REF]. However, to obtain the endless control of the device, at least two variable and rotatable wave-plates are necessary. The phase shift magnitude and the non-blocking behaviour (concerning phase shift) are managed by the monitoring algorithm.

Indirectly the birefringence adjustment can be used to enable a better control in the c) case. When a liquid crystal polarisation controller is made using rotatable fractional wave plates, it is crucial for the plates to be precisely half-wave or quarter-wave as required. Clearly it is difficult to achieve in any conventional way (for instance Lefebvre fibre loops) . The present device has the advantage that the half-wave or quarter-wave plate condition can be achieved very precisely by simply adjusting the magnitude of the drive voltage. In this way, a very flexible and rapid polarisation controller can be manufactured.

Free-space wave-plate solutions have generally higher coupling efficiency to fibre than LiNb03 waveguides and lower polarisation dispersion losses. Among them, liquid crystal solutions present a lot of advantages, such as simplicity, low voltage control, good package, no moving parts, no mechanical fatigue and to be low cost (compared to LiNb03 approaches). However, they suffered for a long time, from a lack of speed, which has been improved considering other liquid crystal or composite materials and/or special electrical addressing schemes, as it will be described in the next section.

IV. PRINCIPLE OF HOMEOTROPIC LIQUID CRYSTAL ENDLESS ROTATOR

The use of electroclinic materials is interesting because providing switching time improvement of a few orders of magnitude with respect to standard nematic materials. The birefringent materials used are smectic A and smectic C* ferroelectric liquid crystal in homeotropic configuration confined between two glass plates [START_REF] De Bougrenet | Complex amplitude modulation using liquid crystal spatial light modulators[END_REF]. In this configura tion, molecular layers are oriented parallel to the substrate. This orientation is obtained by using specific surfactant such as Silane. In the smectic A phase the birefringence is indu ced by the elctric field. The smectic C* exhibits a spontaneous birefringence. In both case, the polar coupling between material and the in-plan electric field induces rotatable wave plate, with or without phase shift control for respectively, smectic A and smectic C* (see Figure 5). With the electroclinic effect, geometrical relations between the principal neutral axis of the index ellipsoid n and the electric fi�ld Eis "ii xi= a Eand I a EI= 8, with a is the elec troclinic coefficient. With smectic C* 8 is constant and the spontaneous polarisation P is parallel to E.

In the case of polarization controller with rotatable fractional wave-plates, smectic C* is a more useful liquid crystal for obvious functionality reasons (the first, is that birefringence is maintained without electric field -due to presence of steady states). In contrast, for a control ler used as a variable and rotatable wave-plate, smectic A is preferable. The orientation of electric field and then of neutral axis is controlled by star -like elec trodes (See figure 6). Typical thickness of liquid crystal are "" 20 and 40 µm for respectively quarter and half wave-plate [START_REF] Dupont | Principle of polarisa tion mode dispersion controller using homeotropic electroclinic liquid crystal confined single mode fibre devi ce[END_REF]. To obtain a large permeation length of electric field within the liquid crystal, thick electrodes are used. The thickness of cupper electrodes is 8 µm and they are realised by Liga* process. Separations between opposite electrode are between 20-30 µm.

V. FIRST EXPERIMENTAL RESULTS: OPTICAL BEHAVIOUR OF SMECTIC A AND SMECTIC C* PLATES:

The polarisation controller should have a short response time (a few lOµs) and should provide endless control. Two liquid crystals have been studied in two different configura tions (smectic A and smectic C*). We determined the optical characteristics of liquid crystals by two ways. First, we apply a sweep frequency of rotating the electric field that determines the rotating response time, second, we apply an electric pulse (0-V) within the liquid crystal that determines the birefringence modulation response time. By rota ting both the in-plane electric field direction and its amplitude, we controlled the polari zation change and then the intensity change through the polarizer (see figure 7). With a uniform rotating frequency f, the signal detected has a frequency four times higher (4/). The optical response gives both the induced birefringence (see figure 7) and the response time.

.39-J'ul-ai 22:17:43

Optical response

Applied voltage

The Jones matrix of the rotatable birefringent plate is:

91 ( e ) ( exp(i Ll <f>/2) 0 ) 91 (-e) o exp(-i Ll <f> 12)
Where Ll<f> is the phase shift of the birefringent material and 91 ( 8) is the rotation matrix of e angle between the neutral axis of the birefringent material and the input polarization. By measuring the amplitude the light passing through the wave-plate by setting it between two polarizers, it is straightforward to calculate the induced phase shift by the formula:

Ll<f> = 2 sin -1 (
Where !�ax is the maximum intensity with polarizer parallel to the input polarizations and !�ax is the maximum of intensity with polarizer crossed w. r. to the input polarizations. Then two types of liquid crystal materials and electrodes implementations have been considered.

V.1. Electro-clinic wave-plate:

The birefringent material used is a smectic A liquid crystal (w415) of the Alkenyl family (provided by the University of Colorado at Boulder) in a homeotropic configuration and confined between two glass plates of 0.7 mm thickness with thin ITO electrodes. This material has a giant electroclinic effect at room temperature (smectic A phase: 23 -32 °C) and the given induced tilt angle is 18° for lOV /µm. The cell thickness of 18 µm is obtained by a joint of cement including 18 µm spacers. The maximum measured phase shift is Li<!>= 0.41 n rad. for high electric field value (I OV /µm) at a 1.55 µm operating wavelength. This low value of phase shift (below the typical index modulation values of smectic A liquid crystals) is the proof of the decrease in electric field magnitude in the thickness of liquid crystal. Therefore, the use of thick electrode is absolutely necessary in this case (see Figure 8). Moreover, the response time is strongly dependent of the temperature: l ms near smectic C* -smectic A phase transition and less than 100 µs near smectic A -Isotropic phase transition.

V.2. Smectic C* wave-plate:

Smectic C* liquid crystal wave-plates have also been studied and realized with thick cup per electrodes [ 151. The liquid crystal used is the zu-4655-100 and shows a good homeotro pic orientation (see Figure 9). The cell thickness is 40µm. The measured phase shift is close to 7t. ln this case, thick electrodes are used to obtain a homogenous response time of liquid crystal in the thickness. The obtained response time is less than 100 µs with an electric field of 1 OV /µm at room temperature ( 150 V between to opposite electrodes). Principe du coup/age optique des lames.

In both smectic cases the driving electronics has been made up by Analog amplifiers that applied phase-shifted voltages on each electrode. Electronics is monitored via a Digital-Analog Converter (National Instrument) driven by a PC. The optical system uses selfoc fibres collimating the optical beam within the polarisation controller aperture (see Figure 10).

Other types of fibres, such as extended mode fibres, can be advantageously used in such a system. Another option consists in using the device in reflection. The main advantage is to reduce the required thickness and to cancel double refraction which can occur with some material and optical configurations. In contrast, for instance, to the nematic rotatable wave plate (rotatable fractional case), where the directors in the bulk are parallel to the glass, under an electric field, with smectic A, the directors make an angle 8 (Figure 5) to the normal resul ting in double refraction under normal incidence. Double refraction also occurs in configura tion (rotatable fractional and rotatable variable cases), where both director orientation and inclination are operating. Reflective devices enable to bypass this problem, but require the additional use of a circulator.

As an illustration here, double refraction results in a beam steering of a few degrees (typically 5-6°), for a beam with a polarisation in the tilt plane. This results in a significant Joss in the output fibre coupling when using single mode fibre (about 3dB). However, this point can be easily mitigated when using beam coupling into an extended mode fibre (typi cally 40 µm).

VI. DISCUSSION AND TRENDS:

In both cases thick electrodes are recommended to obtain a rt phase shift for electro-clinic and to obtain homogenous response time for the smectic C*. In addition, this arrangement prevents generally from residual polarisation dispersion loss sources due to the better homo geneity of the applied field and then liquid crystal orientation. Thermal control with a Pelle tier module is also required in particular for the smectic A wave-plates which have a critical behaviour near the transition temperature. Using smectic C* phases could be an interesting solution for a fractional wave-plate: the birefringence is high and can be weakly adjusted with temperature. Moreover, the wave-plate characteristics remain in the absence of applied electric field. This feature can provide in some case a decisive operational advantage.

Compared to other materials (nematic, PLZT and LinB03), the above solutions have the main advantage of exhibiting a good response time, a good trade-of between low voltage supply and short material thickness, enabling very compact and low loss systems. However, the alignment of such materials needs a particular attention. Therefore, other solutions have been suggested using, for instance, fast nematic liquid crystals [START_REF] Achaya | Fast tunable in-line liquid crystal based polarization controller for reset free operation[END_REF] or nano-PDLC composite liquid crystal materials [START_REF] Matsumoto | Light processing and optical devices using nano-sized droplets of liquid crystal dis persed in polymer[END_REF]. However, if both solutions gain in speed and are easier to manu facture, they require significantly higher voltages (> 20 Vo lts I µm). These high voltage values result generally in severe optical aperture limitations, then making more difficult the device pigtailing, collimation and packaging as well as mitigation of double refraction when it occurs (as mentioned above). This is probably the most serious practical advantage of low voltage solutions such as smectic A and C* solutions over current competitors.

Of course, considerations on device choice cannot be only restricted to material aspects but also to algorithm optimisation. This is the reason why variable and rotatable wave-plates are preferred, because reducing the number of control parameters and then simplifying the algorithm. However, the use of polarisation controllers cannot be restricted only to the frame of PMD compensation, its potential applications include other domains such as polarisation analysis, scrambling and de-multiplexing for which fast response times are often required.

  Mots des : Transmission fibre optique, Dispersion onde, Dispersion optique, Polarisation onde, Polarisation optique, Convertisseur signal, Dispositif cristal liquide.I. Introduction II. Principle offirst order PMD compensation III. Various optical solutions IV. Principle of homeotropic liquid crystal endless rotator V. First experimental results: optical beha viour of smectic A and smectic C* plates VI. Discussion and trends References I. INTRODUCTION

  Photo-diode

FIG. 2 .

 2 FIG. 2. -a) Wave-plates orientation and b) polarisations transformation principle with 3 variable wave-plates. a) L'orientation des lames birefringentes b) Principe de transformation des itats de polarisation avec trois lames birefringentes.

F1G. 3 .FIG. 4 .

 34 FIG. 4.a) Electrodes implementation for a PLZT wave-plate and associated voltages. b) Electrodes implementation on the substrate for liquid crystal wave-plate. a) L'implantation des electrodes pour Les lames PLZT et /es tensions associeesh) L'implantation des electrodes sur le suhstrat pour Les lames a cristaux liquides.

FIG. 5 .

 5 FIG. 5. -Liquid crystal is uniaxial crystal. The index ellipsoid is represented in geometry of a) Srnectic A phase and and b) Srnectic C* phases.

FIG. 6 .

 6 FIG. 6. -Geometry of liquid crystal cell. Geometrie de la cellule cristal liquide.

/

  -il""--.. I\ \ I • /'r \"'-.... I\ I I N..: I y I N..: \ \ \ ..._.. "/ \ I \ ....._, \j 1/ \ / \ \/ \

  FIG. 8. -Decrease of the transverse electric field with thin electrode (a), and thick (b) electrodes. c) View of double thin electrodes. d) cross-section of thick electrodes system. E volution de l'intensiti du champ electrique avec des electrodes fines (a) et epaisses (b). c) Vue en coupe avec des electrodes fines. d) Vue generale des electrodes epaisses.

Acknowledgement

This work has been support by the RNRT COPOLDYN. Authors like to thank Dr. J.P. Hamaide and S. Lanne from Alcatel CRC for valuable discussions about the application and A. Poudoulec from Highwave Optical technology for assembling and pigtailing the wave plates.