Delphine Aussavy These

Delphine Aussavy

Maitre Sylvain Philippon

Professeur Guillaume Kermouche

Professeur Thierry Grosdidier

Professeur, UTBM François Peyraut

Professeur Ghislain Montavon

Emmanuelle Vincent Charles

NiCrAl-Bentonite, CoNiCrAlY-BN-Polyester and YSZ-Polyester

Keywords: NiCrAl-Bentonite, CoNiCrAlY-BN-Polyester Jiangwei, Caminde, Shushuen

Les grandes connaissances engendrent les grands doutes"

Aristote

Je souhaiterais commencer par remercier les membres de mon jury de thèse ; Monsieur Grosdidier, qui a accepté de présider ce jury de thèse, Messieurs Kermouche et Pilippon qui ont accepté de rapporter mes travaux, merci pour vos remarques et vos conseils, ainsi que pour l'échange très intéressant suite à la présentation de mes travaux.

Je remercie mes directeurs de thèse ; Rodolphe Bolot, Ghislain Montavon ainsi que François Peyraut, merci pour vos conseils et vos critiques qui ont permis d'améliorer ce travail, ainsi que vos mots d'encouragements pendant les périodes de doute.

Un grand merci à Cécile Langlade, qui s'est occupée de coordonner l'organisation de la soutenance alors que chacun des éléments actifs de ces travaux de doctorat sont partis vers de nouveaux horizons.

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emissions should be reduced by 80%. The perceived noise should also decrease by 50%, while reducing operating costs [START_REF]Europe's Vision for Aviation[END_REF].

In 1990, the consortium EIMG (Engine Industry Management Group) was established at the request of the European Commission to represent a single block of all European companies in the aeronautical sector. This consortium allows mutualizing the work on various European technology programs having a better visibility on advanced pre-competitive research, and also ameliorating the exchange of information between the consortium partners.

Several solutions have been identified to increase engine efficiency and reduce fuel consumption, including: reducing the weight of components, increasing the resistance of materials to the environment of a jet engine; improving the engine stability, limiting the peripheral leakage between the rotating blades (the rotor) and the stationary casing (stator).

It is in this context that several projects such as DREAM, SEALCOAT, VITAL, NEWAC or LEMCOTEM were conducted to ensure the technological advances and answer to the objectives of ACARE. The NEWAC and LEMCOTEC projects focused on increasing the thermal efficiency at the engines cores, VITAL and DREAM carried on improving propulsion efficiency and low pressures.

These programs have led to technological innovations that allowed approaching the ACARE's objectives.

This study takes place in the frame of the project E-BREAK (Engine Breakthrough components and subsystems), a program grouping 42 European companies and research laboratories. Previous studies have improved the engine efficiency but in return have induced higher pressure and temperature loads into the engine. The aim of the E-BREAK project is to develop appropriate technologies of subsystems and to adapt them to these new constraints of temperatures and pressures.

To study all the problematic related to subsystems the project was divided into several sub-projects. The thesis takes place in the frame of the SP4, "high temperature materials for breakthrough components", see Figure 0-2. The work involves developing abradable materials and static components which may sustain high temperatures. Abradable materials improve the efficiency of aeroengines by reducing the clearance between the rotor and the stator. For now the application of abradable coatings and thermal barriers are the most appropriate solutions to raise these objectives.

These coatings must be composed of materials able to resist to high temperatures and pressures. The objectives of SP4 are as follows:

• Validation of all new materials, processes and methodologies concerning abradables and thermal barriers.

• Develop and validate specific models to optimize and predict the behavior of coatings during the working phase of the devices.

• Create a database on the state of the art related to abradable systems developed by thermal spraying.

In the framework of the thesis, I was particularly involved in workpackages WP4.2 and WP4.3

(with MTU and Turbomeca companies as leader partners, respectively) with the aim of determining experimentally and numerically the extrinsic properties of several materials with different microstructures. However, the final objectives of these two workpackages are different. The final objective of WP4.2 was to conduct a numerical model to reproduce abradability tests on different abradable coatings. For this, it is mandatory to provide information about the extrinsic properties of abradable coatings, which is the aim of this thesis work. These properties will serve to feed the final numerical model of abradability test elaborated by WP4.2 partners. On the other hand, the final objective of WP4.3 was to dimension modified thermal barrier coatings for use as abradable coating in the HP turbine. The job of the thesis work is to determine the extrinsic properties of several modified TBC materials with different microstructures obtained with the same powder, in order to differentiate their performances for abradable application in the HP turbine. YSZ-polyester is a solution for modified thermal barriers including abradability performances and is studied to replace existing thermal barriers presently used as clearance seal [START_REF] Rhys-Jones | Thermally sprayed coating systems for surface protection and clearance control applications in aero engines[END_REF][3][4] in the hightemperature High Pressure Turbine (HP) Turbine. To conduct this study in the best conditions, a number of deliverable has been provided to the project partners.

• State of the art and the establishment of means of characterization protocols of the structure of materials

Introduction

To better understand how to achieve the objectives set by ACARE (performance improvement, reduction of NOx and CO 2 emissions and decrease of the specific fuel consumption) the description of the general operating principle of turboreactor is required. The jet engine is the propulsion system of aircrafts, and is composed of various sub-systems. The air enters through the intake section and is accelerated before leaving the engine through the exhaust nozzle. A linear momentum is then created at the back of the engine providing the engine thrust. At admission, the air is circulated by the fan (consisting of blades) and is then compressed through the compressor. Once in the combustion chamber, compressed air is mixed with fuel to initiate the combustion, burn the mixture and expand the combustion gases. Before exiting the turboreactor engine through the exhaust nozzle which allows accelerating the air and induces the thrust of the engine, the air passes through a turbine, composed of fixed vanes and moving blades. Part of the power induced by the turbine will be recovered and redistributed to operate under the different components (compressor, blower, rotor shaft ...).

Today the majority of civil aviation park is equipped with turbofan engines, Figure 0 The primary flow undergo the entire thermodynamic Joule-Brayton cycle, as described on Figure 0-5, and this produces the energy required to operate the engine. The stages of the thermodynamic cycle are as follows:

• Reversible adiabatic compression (from 1 to 2)

• Irreversible isobaric combustion (2 to 3)

• A reversible isobaric relaxation (from 3 to 4)

• A reversible isobaric cooling (4 to 1) The thermal efficiency  th,ideal of the Joule-Brayton cycle is given by 𝜂 𝑡ℎ,𝑖𝑑𝑒𝑎𝑙 = 1 -

𝑇 4 𝑇 3 = 1 -( 𝑃 4 𝑃 3 ) 𝑘-1 𝑘 (I.1)
Where k, is an adiabatic index whereas T i and P i (i varying from 1 to 4) are respectively the Joule-Brayton cycle temperatures and pressures at various stages of the cycle represented on Figure 012345.

From Figure 0-5 and equation 1, increasing the admission temperature (T3) of the gas into the turbine by keeping a constant cooling should improve the system efficiency. However, the admission temperature is limited by materials properties.

Moreover, the efficiency of the turbine is decreased by various losses taking place within the turbofan. Due to the non-adiabatic aspects in the compressor and turbine, heat losses occur. There is also mechanical friction losses occurring between turbine components. Friction and thermal processes dissipate the kinetic energy and reduce efficiency. There is a functional clearance between the rotor and the stator for limiting the mechanical friction, providing leakages in the compressor and turbine. It follows that the entire quantity of air entering through the inlet does not participate in the energy transformation process. The effectiveness of the turboengine is then decreased and the mechanical work capacity is again reduced, and the thrust produced is therefore lower. These leakages are found to have other adverse effects such as increased fuel consumption or instability of compression [4,[START_REF] Wiseman | An investigation of life extending control techniques for gas turbine engines[END_REF].

To reduce these losses, one solution (which is the one studied here) is to reduce the peripheral clearances. However it is not possible to reduce to zero the clearance during the mechanical design of the turbofan due to manufacturing tolerances and possible movement of the casing during the commissioning of the rotor. Indeed, the expansion coefficients of the materials used as well as the high centrifugal forces acting on the rotating parts of the turbine engine would induce shocks among the various components. The application of mechanical seals usually used in piston compressors and engines is not an option here [START_REF] Dorfman | Thermal Spray Technology Growth in Gas Turbine Applications[END_REF].

The solution mentioned was therefore to create seals that would allow blades to widen a road in the coating by removing the material if they come in contact. This property, providing preferentially wear of the coating instead of the blade, is called abradability. The interest of the abradability property is to preserve the blades which have a more complex technical nature and would cost more to maintain or change. Additionally, it is mandatory to protect the optimized aerodynamic shape of the blade.

The use of abradable materials reduces as much as possible the clearance between the blades and the casing while ensuring their preferred wear in the case where the blades encounter the coating. The abradability of a material can be quantified by its facility to be wear during a frictional contact with another part. Nevertheless, this abradability property is antagonistic with the thermal and mechanical properties necessary for the survival of a material in the environment of high temperature and highpressure jet engine. These types of coatings must have good abradability, but it is also necessary to have a good erosion resistance, a good resistance to high temperature oxidation and also to sustain mechanical loads at high temperatures during long times [START_REF] Hajmrle | New Abradable Seals for Industrial Gas Turbines, Thermal Spray[END_REF]. It is therefore mandatory to determine the coatings providing the best compromise between abradability, mechanical property and thermal resistance. The range of temperature and pressure within a turbojet is very large. It is therefore necessary to use materials whose thermal and mechanical properties are different to adapt them to the various operating conditions occurring in different section parts. To reach this goal, a wide range of different abradable family is available but each type of them has its own advantages and disadvantages. Typically, these families are classified by the nature of the materials that make up the coating and the abradable seal is formed according to the scheme below:

 A metal or ceramic matrix providing to the coating its thermal and mechanical properties  A lubricating phase or brittle phase facilitating the pull-out of the coating material by a shearing effect. That limits the size of debris in the event that the blade breaks the coating. It also limits the transfer of the coating material to the blade  Sometimes a polymer phase is added, e.g. AlSi-Polyester abradable. This polymer phase will hence the abradability performance and limit the adhesive transfer between the blade and the coating. For abradable coating applied to temperature range higher than 350°C, this phase polymer must be removed after a heat treatment, e.g. CoNiCrAlY-BN-Polyester, YSZ-Polyester. That allows creating porosity within the deposit in order to restrict the stresses propagation during interaction of the blade onto the coating.

However, the chemical composition is not the only variable affecting the coating properties because two coatings having the same chemical nature may have highly different microstructures after the spraying operation and therefore possibly different macroscopic properties. The various microstructural objects as the rate, the size, the shape of the porosities and their connectivity will affect the coating properties. The scale at which the material is studied also turns out to be important and will be later detailed in the manuscript.

The studied materials, defined under the European project, are abradables taking place in the high These different abradable coatings are manufactured by the Thermal Spraying process [START_REF] Herman | Plasma-Sprayed Coatings[END_REF].

Thermal spray processes involve creating a high temperature and high velocity jet in order to spray the powder material (directly injected in the jet) onto the surface to coat. Powder particles are melted, accelerated and sprayed. The crashing particles flatten on the surface and form the coating which thus consists of a network of lamellae resulting from the crushing of individual molten droplets. This process can manufacture different facies of microstructures.

Facies

In order to obtain the best resulting properties of the coatings, it is possible to select the most appropriate process parameters for modifying the jet physicochemical (plasma or flame) properties as well as the conditions of insertion of the powder. The selected parameters allow modifying the particles momentum and temperature while the physical state of the droplet at the time of impact on the substrate determines the shape of the lamellae (Figure 01234567) and the condition of the contact interface between each particle [START_REF] Fauchais | Knowledge concerning splat formation: An invited review[END_REF]. That is why diverse and varied coating facies can be obtained. Abradable coatings obtained by thermal spraying are strongly heterogeneous materials, as well concerning the chemical composition as the microstructure. These characteristics will have strong influences on their mechanical and thermal properties.

The erosion, abradability or thermal cycling tests can be used to establish the macroscopic properties of the coatings. However, these tests are expensive and time-consuming, with a low availability, and do not provide directly the material behavior laws. For these reasons, it is useful to study the intrinsic mechanical properties of abradables coatings. Nevertheless, abradable materials are difficult to characterize by conventional tests in the mechanical or thermal framework. The forming of the coating does not actually guaranty standardized specimen for the tests and does not always fulfill the criteria required in terms of thickness. Another issue is that it is difficult to delete the substrate on which the coating is deposited. It is then necessary to use tests taking into account the influence of the substrate on the measured properties [START_REF] Steinke | Process Design and Monitoring for Plasma Sprayed Abradable Coatings[END_REF][START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF], or find ways to remove it [START_REF] Johnston | Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings[END_REF][START_REF] Kuroda | The quenching stress in thermally sprayed coatings[END_REF][START_REF] Baïz | Etude expérimentale du contact aube/abradable : contribution à la caractérisation mécanique des matériaux abradables et de leur interaction dynamique sur banc rotatif avec une aube[END_REF].

Many analytical models have been developed for determining the mechanical and thermal properties of coatings, such as the Maxwell, Batchelor Hill, O'Brien, Hadley, Hashin and Shtrikman or

Bergman models [START_REF] Kaviany | Principles of Heat Transfer in Porous Media[END_REF][START_REF] Prat | Transferts de chaleur dans les milieux poreux Conduction, convection, rayonnement, Techniques de l'ingénieur Transferts thermiques[END_REF][START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomecanique de matériaux réfractaires modèles[END_REF][START_REF] Hashin | Note on a variational approach to the theory of composite elastic materials[END_REF][START_REF] Hashin | The Elastic Moduli of Heterogeneous Materials[END_REF][START_REF] Hashin | Analysis of Composite Materials-A Survey[END_REF][START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF]. Nevertheless, the use of such models needs to make assumptions about the structure of the material and are not fully representative of their true microstructure. Particularly, they do not take into account the shape and the size of the porosities and the connectivity between them. To link the microscopic behavior of the matter to its counterpart at the macro-scale of the coating, it is necessary to use numerical approaches based on the organization of the structure of the material. Since the pioneer works of Duvaut [START_REF] Duvaut | Analyse fonctionnelle et mécanique des milieux continus -Application à l'étude des matériaux composites élastiques à structure périodique -Homogénisation[END_REF], Bensoussan et al. [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF], Sanchez-Palencia [START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF] and Suquet [START_REF] Suquet | Plasticité et homogénéisation (in english : "Plasticity and homogenization[END_REF], the homogenization theory has raised a growing interest among the scientific community and has proven its efficiency for modeling heterogeneous composites made of a large amount of repeated unit cells called REV (Representative Elementary Volume) but also named a RVE (Representative Volume Element). It should however be mentioned that the homogenization method fails to deal with dynamic calculation if the characteristic wavelength is smaller than the size of the heterogeneity [START_REF] Peyraut | Calcul et visualisation de l'amortissement d'une onde harmonique dans un multicouche viscoélastique[END_REF].

Nevertheless, one of the important benefits is that this method can be extended to heterogeneous media presenting a non-periodic geometric organization by selecting a suitable set of boundary conditions related to the microscopic problem [START_REF] Carazo | Effective properties of nodular castiron: A multi-scale computational approach[END_REF]. Even if the choice of the REV in this situation is still questionable, it can be defined as the minimum volume of material whose behavior is equivalent to the homogeneous fictitious material and keeping invariant the material properties if the volume is increased [START_REF] Fernandino | Determination of effective elastic properties of ferritic ductile cast iron by computational homogenization, micrographs and microindentation tests[END_REF]. In order to determine the limit defined by this invariance property, it is required to use actual micrographs of the microstructures and to perform several computations on different REV corresponding to different magnifications. An interesting and detailed discussion on the size of the REV for random composites can be found in [START_REF] Kanit | Determination of the size of the representative volume element for random composites : statistical and numerical approach[END_REF].

In a complementary manner to the homogenization method, which is based on the asymptotic expansion of the displacement field, the micro-dilatation theory developed by [START_REF] Nunziato | A nonlinear theory of elastic materials with voids[END_REF] for modeling elastic porous media [START_REF] Nunziato | A nonlinear theory of elastic materials with voids[END_REF] must also be mentioned. In the same spirit but earlier, the Cosserat brothers have developed the notion of micro-rotation related to the independent rotation of each particle at the micro-scale length [START_REF] Cosserat | Théorie des corps déformables[END_REF][START_REF] Cosserat | Note sur la théorie de l'action euclidienne[END_REF]. More details on the recent potentiality of these two theories can be accessed in [START_REF] Jeong | Etude sur la modélisation multiéchelles et multidisciplinaires des matériaux[END_REF].

Several previous studies have tried to determine the properties of coatings by modeling real or artificial microstructures thanks to the finite element software OOF which is dedicated to the analysis of microstructures (Z. Wang, A. Kulkarni et al. [START_REF] Kulkarni | Advanced microstructural characterization of plasma-sprayed zirconia coatings over extended length scales[END_REF][START_REF] Kulkarni | Processing effects on porosityproperty correlations in plasma sprayed yttria-stabilized zirconia coatings[END_REF][START_REF] Kulkarni | Comprehensive microstructural characterization and predictive property modeling of plasma-sprayed zirconia coatings[END_REF][START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF], R. Bolot et al. [START_REF] Bolot | Predicting the Thermal Conductivity of AlSi/Polyester Abradable Coatings: Effects of the Numerical Method[END_REF], M.E. Cunningham et al.

[43], K. Bakker [START_REF] Bakker | Using the finite element method to compute the influence of complex porosity and inclusion structures on the thermal and electrical conductivity[END_REF] and S. Grandjean et al. [START_REF] Grandjean | Numerical calculations of the thermal conductivity of porous ceramics based on micrographs[END_REF]). In most cases, a simplified mesh was necessary because of the limitations of computer tools in terms of CPU and RAM. All these studies demonstrate the influence of the microstructure on thermal and mechanical properties of the coatings. To account for this influence in the modeling process, a multi-scale approach was developed and is presented and discussed in the third part of this manuscript. In this approach, finite element calculations are performed directly on images of real coating microstructures to determine their thermal and mechanical properties. The mesh is built pixel by pixel, each pixel becoming an element and matching with a chemical phase constituting the coating. However, the general equivalent behavior law was not really determined because the calculations were only performed in simple test case such as the uniaxial tensile one. That means that the approach that is commonly practiced in the homogenization theory

where the mean values of the stress and strain fields are integrated over the RVE and linked together by the equivalent elastic properties, were not used. Nevertheless, this thesis work is more focused on experimental aspects than numerical issues and the numerical part that was developed can be considered as a first insight on the application of a simplified homogenization approach to abradable coatings. More precisely, the objective of this thesis work is to determine an approach to analyze and understand the link between the microstructures and the properties of the coatings as illustrated by The expected properties of the different families of abradable coatings are presented in chapter 1.

This chapter aims to draw up an inventory of their mechanical and thermal characteristics. The various tests for their characterization and their limitations, as well as the manufacture processes, will be introduced and the influence of the process on the microstructure features of the coating will be discussed. Studies concerning the influence of those microstructural objects onto the coating properties will be listed [START_REF] Kulkarni | Comprehensive microstructural characterization and predictive property modeling of plasma-sprayed zirconia coatings[END_REF][START_REF] Azarmi | Young's modulus measurement and study of the relationship between mechanical properties and microstructure of air plasma sprayed alloy 625[END_REF][START_REF] Bengtsson | Characterization of microstructural defects in plasma-sprayed thermal barrier coatings[END_REF][START_REF] Bolelli | Microstructure-based thermo-mechanical modelling of thermal spray coatings[END_REF][START_REF] Johnson | Effective modulus and coefficient of thermal expansion of Ni-YSZ porous cermets[END_REF][START_REF] Laurent | Etude microstructurale fine de revêtements céramiques de type barrièrre thermique; incidence sur la résistance thermomécanique de ces revêtements[END_REF][START_REF] Sesso | Effect of Plasma Spray Parameters on Thermal Barrier Coating Formation and Microstructural Properties[END_REF][START_REF] Öksüz | Microstructure and wear properties of plasma-sprayed aluminum-silicon-polyester coatings[END_REF][START_REF] Sevostianov | Plasma-sprayed ceramic coatings: anisotropic elastic and conductive properties in relation to the microstructure; cross-property correlations[END_REF] as well as studies that collected data on these materials [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF][START_REF] Johnston | Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings[END_REF][START_REF] Borel | The wear mechanisms occurring in abradable seals of gas turbines[END_REF][START_REF] Chappel | Gas Path Blade Tip Seals: Abradable Seal Material Testing at Utility Gas and Steam Turbine Operating Conditions[END_REF][START_REF] Clegg | NiCrAl/bentonite thermal spray powder for high temperature abradable seals[END_REF][START_REF] Peyraut | Numerical Simulation of the HR15Y Hardness Test of Abradable Thermally Sprayed Coatings[END_REF][START_REF] Ma | Investigation of abradable seal coating performance using scratch testing[END_REF][START_REF] Sporer | Processing and Properties of Advanced Ceramic Abradable Coatings, Thermal Spray[END_REF][START_REF] Stringer | High speed wear testing of an abradable coating[END_REF][START_REF] Yi | Friction and wear behaviour and abradability of abradable seal coating[END_REF][START_REF] Yi | Erosion wear behaviour and model of abradable seal coating[END_REF]. Finally, different modeling strategies already used in the literature to characterize abradable composite materials are presented [START_REF] Bolot | Predicting the Thermal Conductivity of AlSi/Polyester Abradable Coatings: Effects of the Numerical Method[END_REF][START_REF] Peyraut | Numerical Simulation of the HR15Y Hardness Test of Abradable Thermally Sprayed Coatings[END_REF][START_REF] Bolot | Application of FEM for the Estimation of Thermo-Mechanical Properties of Plasma Sprayed Composite Coatings[END_REF][START_REF] Bolot | A Two-Dimensional Heat Transfer Model for Thermal Barrier Coating Average Thermal Conductivity Computation[END_REF][START_REF] Seichepine | Numerical simulation of the thermo-mechanical behaviour of thermally sprayed abradable coatings[END_REF][START_REF] Seichepine | Mechanical modelling of highly heterogeneous thermally sprayed abradable coatings, International Thermal Spray[END_REF][START_REF] Peyraut | Finite element modeling of abradable materials -Identification of plastic parameters and issues on minimum hardness against coating's thickness[END_REF][START_REF] Faraoun | Modelling route for abradable coatings[END_REF][START_REF] Faraoun | Improvement of thermally sprayed abradable coating by microstructure control[END_REF][START_REF] Langer | OOF: an image-based finite-element analysis of material microstructures[END_REF][START_REF] Bartsch | Simulating Thermal Response of Eb-Pvd Thermal Barrier Coating Microstructures[END_REF][START_REF] Tan | Modeling thermal conductivity of thermal spray coatings: comparing predictions to experiments[END_REF].

Chapter 2 will review the various experimental equipments and methods required for the measurement of thermal and mechanical properties of the coatings investigated. It will also present the protocols used to describe the microstructures of various coatings under the same conditions and will introduce the methodology to perform modeling calculations of the coating properties.

Chapter 3 presents in more details the microstructures examined and the preliminary studies required for feeding the numerical model. Finally, experimental and numerical results obtained on these different microstructures are analyzed in Chapter 4. This section will be broken down into three main areas:

 The first will describe in more details each studied microstructure. This will highlight the necessity of using several scales at the microstructural description of these coatings.

 The second will highlight the influence of the microstructure on experimental and calculated properties. It will discuss issues concerning the information to implement as model data.

 The third axis will discuss the results in the light of the different scales used. In particular, it will compare the calculated and experimentally measured properties of each of the microstructures.

The conclusions of these three axes will be grouped and discussed in a synthesis that will arise on the general conclusion of this work.

The aim of this chapter is to provide more information concerning the abradable coating problematic, to describe the process used to manufacture the abradable coating and to focus on a methodology to elaborate numerical and analytical modeling methods which allow estimating the material extrinsic (i.e., macroscopic) properties.

The use of abradable materials reduces as much as possible the clearance between the blades and the casing while ensuring their preferred wear in the case where the blades encounter the coating [START_REF] Rhys-Jones | Thermally sprayed coating systems for surface protection and clearance control applications in aero engines[END_REF], [4]. The abradability of a material can be quantified by its facility to be wear in a frictional contact with another part. Nevertheless, this abradability property is antagonistic with the thermal and mechanical properties required for the survival of a material in the high temperature and high-pressure environment of a jet engine. These types of coatings must have a good abradability, but it is also necessary to have a good resistance to erosion, to high temperature oxidation and also to sustain mechanical loads at high temperatures during long times [START_REF] Hajmrle | New Abradable Seals for Industrial Gas Turbines, Thermal Spray[END_REF]. It is therefore mandatory to determine the coatings providing the best compromise between abradability, mechanical properties and resistance to the high temperature environment.

It is therefore necessary to use materials whose thermal and mechanical properties are adapted to the various operating conditions occurring. However, the chemical composition is not the only variable affecting the coating properties because two coatings having the same chemical nature may have highly different microstructures after the spraying operation and therefore possibly different macroscopic properties [START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Karthikeyan | Developing empirical relationships to estimate porosity and Young's modulus of plasma sprayed YSZ coatings[END_REF][START_REF] Friis | Influence of particle in-flight characteristics on the microstructure of atmospheric plasma sprayed yttria stabilized ZrO2[END_REF]. The various microstructural objects such as the rate, the size, the shape of the porosities and their connectivity will affect the coating extrinsic properties. The scale at which the material is studied also turns out be important and will be later detailed in the manuscript.

The elaboration process to manufacture the abradable coating has a strong influence on their microstructures. Several studies [START_REF] Fauchais | Knowledge concerning splat formation: An invited review[END_REF][START_REF] Fauchais | Quo vadis thermal spraying?[END_REF][START_REF] Vardelle | Spray parameters and particle behavior relationships during plasma spraying[END_REF] highlight the influence of these process parameters onto coating morphologies and then onto their thermal and mechanical properties [START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Karthikeyan | Developing empirical relationships to estimate porosity and Young's modulus of plasma sprayed YSZ coatings[END_REF][START_REF] Friis | Influence of particle in-flight characteristics on the microstructure of atmospheric plasma sprayed yttria stabilized ZrO2[END_REF]. For a same abradable coating nature, several facies can be obtained, due to the various microstructural features, i.e. oxides, porosity, burnt polymer, intra and inter lamellar cracks and unmelted particles [START_REF] Herman | Plasma-Sprayed Coatings[END_REF][START_REF] Fauchais | Plasma Spraying: From Plasma Generation to Coating Structure, Advances in Heat Transfer[END_REF].

Discrimination is then necessary to determine the coating having the most appropriate in-service performances for a given application.

The material nature and coating microstructure permit to vary strongly the coating properties. It is mandatory to determine which combination allows providing the desired properties. But abradable dimensioning is problematic due to the poorly capitalized experience on abradable coatings. Solutions have to be found. The first part of this chapter will define what an abradable is, what its main functionalities are, what are its expected characteristics and properties, and which information is important to dimension them. The importance of obtaining intrinsic and extrinsic properties of abradable coatings will be highlighted. Several techniques to determine abradable coating performance will be presented with their advantages and drawbacks.

The erosion, abradability or thermal cycling tests establish the macroscopic properties of the coatings. However, they are expensive and time-consuming, with a low availability, and do not provide directly the material behavior laws. For these reasons, it is useful to study the intrinsic mechanical properties of abradables coatings. Nevertheless, abradable materials are difficult to characterize by conventional tests in the mechanical or thermal framework. The forming of the coating does not actually guaranty standardized specimen for the tests and does not always fulfill the criteria required in terms of thickness. Another issue is to measure experimentally coating properties without take into account its substrate. The last part of this chapter focuses on a methodology that has become widespread. The aim is to elaborate numerical and analytical modeling methods to obtain material properties from element details concerning their microstructure.

Many analytical models have been developed for determining the mechanical and thermal properties of the coating, [START_REF] Kaviany | Principles of Heat Transfer in Porous Media[END_REF][START_REF] Prat | Transferts de chaleur dans les milieux poreux Conduction, convection, rayonnement, Techniques de l'ingénieur Transferts thermiques[END_REF][START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomecanique de matériaux réfractaires modèles[END_REF][START_REF] Hashin | Note on a variational approach to the theory of composite elastic materials[END_REF][START_REF] Hashin | The Elastic Moduli of Heterogeneous Materials[END_REF][START_REF] Hashin | Analysis of Composite Materials-A Survey[END_REF][START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF]. Nevertheless, assumptions about the structure of the material are mandatory which is not fully representative of their true microstructure. Numerical approaches based on the organization of the structure of the material are required to link the microscopic behavior of the matter to its counterpart at the macro-scale of the coating, .The homogenization theory [START_REF] Duvaut | Analyse fonctionnelle et mécanique des milieux continus -Application à l'étude des matériaux composites élastiques à structure périodique -Homogénisation[END_REF][START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF][START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF][START_REF] Suquet | Plasticité et homogénéisation (in english : "Plasticity and homogenization[END_REF], has proven its efficiency for modeling heterogeneous composites made of a large amount of repeated unit cells called REV (Representative Elementary Volume). One of the important benefits is that this method can be extended to heterogeneous media presenting a non-periodic geometric organization by selecting a suitable set of boundary conditions related to the microscopic problem [START_REF] Carazo | Effective properties of nodular castiron: A multi-scale computational approach[END_REF].

Several previous studies have tried to determine the properties of coatings by modeling real or artificial microstructures thanks to the finite element software OOF which is dedicated to the analysis of microstructures (Z. Wang, A. Kulkarni et al. [START_REF] Kulkarni | Advanced microstructural characterization of plasma-sprayed zirconia coatings over extended length scales[END_REF][START_REF] Kulkarni | Processing effects on porosityproperty correlations in plasma sprayed yttria-stabilized zirconia coatings[END_REF][START_REF] Kulkarni | Comprehensive microstructural characterization and predictive property modeling of plasma-sprayed zirconia coatings[END_REF][START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF], R. Bolot et al. [START_REF] Bolot | Predicting the Thermal Conductivity of AlSi/Polyester Abradable Coatings: Effects of the Numerical Method[END_REF], M.E. Cunningham et al.

[43], K. Bakker [START_REF] Bakker | Using the finite element method to compute the influence of complex porosity and inclusion structures on the thermal and electrical conductivity[END_REF] and S. Grandjean et al. [START_REF] Grandjean | Numerical calculations of the thermal conductivity of porous ceramics based on micrographs[END_REF]). All these studies demonstrate the influence of the microstructure on the thermal and mechanical properties of the coatings. To account for this influence in the modeling process, a multi-scale approach was developed and is presented and discussed in the third part of this manuscript. In this approach, finite element calculations are performed directly on images of real coatings microstructures to determine their thermal and mechanical properties.

I Abradable composite coatings

As explained in the introduction, some air leakages and combustion gases leakages occur respectively into the compressor and the turbine parts of a turbo engines. Those leakages drastically reduce the engine efficiency. This point is of a major interest for the aeronautical industry because fuel represents more than half of direct operating costs [START_REF] Pritchard | Abradable Coatings used for Gas Path Seals in Turbine Engines[END_REF]. That is why even a fuel saving of 1% onto an engine applied to the entire aeroengines park represents a real increase of profit.

Leakages bypass flow through the space between the rotating blade tips and the stationary housing (stator). An appropriate sealing will permit to reduce these spaces and therefore to improve the engine efficiency but knowing that this gap cannot be totally reduced to zero. Moreover, due to the thermal expansion and high centrifugal forces acting on the rotating components, the application of mechanical seals (usually used for piston engines and compressors) is an irrelevant solution [START_REF] Dorfman | Thermal Spray Technology Growth in Gas Turbine Applications[END_REF]. That is the reason why abradable coating solution has been introduced into turbo engine since years 1960 [START_REF] Sporer | Novel Ceramic Abradable Coatings With Enhanced Performance[END_REF].

This solution is an easy way to reduce gas leakages occurring in clearance part of the engine.

I.1. Role of Abradable seals

The technical solution to control and reduce the leakages and consequently to increase the engine efficiency in the range of 1 to 5 % is the application of abradable seals [4,[START_REF] Ghasripoor | Abradables improve gas turbine efficiency[END_REF][START_REF] Rhys-Jones | The use of thermally sprayed coatings for compressor and turbine applications in aero engines[END_REF]. Abradability is the material ability to be pull out during a rubbing contact with another component [START_REF] Hajmrle | New Abradable Seals for Industrial Gas Turbines, Thermal Spray[END_REF].

Indeed, in order to be the most efficient, blades tips must be the closest possible from the sealing coating. This solution will ensure that the engine working fluids flow over the blade aerofoil surfaces with the minimum loss between blade tips. But due to the operating temperature in the turbine, blades can expand and touch this coating, leading to blades and coating damages. A mean of achieving the smallest possible clearance is to allow the blade to machine a path into the shroud [START_REF] Rajendran | Gas turbine coatings -An overview[END_REF]. Abradables seal systems are applied to the surface of the stator (Figure 1-2) and allow a blade tip to freely incur into the shroud. The coating thickness can reach several millimeters and acts like a sacrificial layer. This will thereby reduce the tip gap to a minimum. Ideally, the abradable must not damage the incurring component, i.e. the blade, and must be perfectly smooth to reach this aim. The main interest of using abradable seals is therefore to offer a technical solution to keep undamaged the blade components which are the most complex and expensive parts to change or to repair [START_REF] Rhys-Jones | Thermally sprayed coating systems for surface protection and clearance control applications in aero engines[END_REF].

I.2. Expected properties of abradables

Abradables are applied throughout the hot gas turbine and/or compressor sections, meaning that they are covering a wide temperature range and all the blade tip velocity range. But a unique abradable coating is not able to account for all operating conditions met with all the compressor and turbine parts. As the gas temperature increases, only few materials might survive in such an environment, strongly reducing the range of the available coatings. Each range of temperature and blade tip velocity requires in fact a specific abradable coating. Figure 1-3 gives an insight on the different abradable coating families usually selected depending on the engine temperature range. To allow their utilization, the abradable seal must have suitable mechanical and thermal properties to survive in such harsh environment but must also fulfill other properties criteria than good abradability, namely:

 Resistance to erosion because turbine high-speed gas can contain solid particles moving with a high velocity.

 Resistance to corrosion and oxidation promoted by the high temperatures and pressures.

 Capability to avoid debris released during a rub because these debris can damage the components.

 Thermo shock resistance because the engine go through large temperatures cycles  Minimum affinity for the substrate material to avoid interdiffusion of the coating material into the carter.

Unfortunately, as shown on Figure 1-4, the coating characteristic in terms of abradability is opposite to the one in terms of erosion resistance. A compromise between these two opposite trends has therefore to be found.

I.3. Abradable composition

Abradables are composite materials made up of a ceramic or a metallic matrix with a lubricant or/and porosity former phases. The matrix furnishes to the abradable seal its mechanical and thermal properties while the lubricant and the porosity former phases provide its capacity to be removed by the blade without damaging or wearing it.

The matrix material must be resistant, in terms of oxidation and corrosion, to the hottest engine stage in which it will be utilized. The main materials used to cover the entire engine range are matrices composed of aluminum, MCrAlY and Yttrium stabilized zirconia. To obtain a clean cut with a minimum of material transfers from the coating to the blade, the fractures occurring during blade contact must be micrometric. The coating matrix voids allow transferring the energy from the blade incursion to the metal matrix by giving priority to inter-particle bond fractures [START_REF] Faraoun | Improvement of thermally sprayed abradable coating by microstructure control[END_REF].

The functionality of the lubricant agent, entrapped into the matrix, is to act as a dislocator to reduce the forces required for the expulsion of particles. In this way, the energy needed for removing material during the abrading process is reduced and the size of the debris will be limited. The lubricant agent also presents the interest of minimizing the coating transfers to the blade [START_REF] Faraoun | Improvement of thermally sprayed abradable coating by microstructure control[END_REF]. A polymer phases is codeposited with the dual purpose of reducing the stresses inside the coating and, when burnt out, of providing crack sources for particles release. In other words, the polymer is added to induce porosity in the abradable coating and to reduce stresses propagation during an interaction between the blade and the coating. For example, in the case of a CoNiCrAlY-hBN-polyester powder, the CoNiCrAlY is the metal matrix, hBN (also called white graphite) is the release agent and the polyester once burnt permits the occurrence of porosity. It must be additionally underlined that the hBN agent does not induce corrosion when in contact with aluminum and has a high oxidation temperature.

Abradable seal properties are of course determined by their chemical composition as previously explained, but also by their microstructure. Indeed, two coatings with the same chemical composition, but with strongly different microstructures, will not behave in the same manner under the same abradable wear, high temperature and high pressure conditions. Several studies have highlighted the influence of coating microstructure on the effective thermomechanical properties and on the global behavior. It is for example reported in [START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Karthikeyan | Developing empirical relationships to estimate porosity and Young's modulus of plasma sprayed YSZ coatings[END_REF][START_REF] Friis | Influence of particle in-flight characteristics on the microstructure of atmospheric plasma sprayed yttria stabilized ZrO2[END_REF] that pores and cracks network directly interfere on the heat flow through the coating leading to a lower thermal conductivity of the coating.

To conclude this section, one can say that the selection of the appropriate abradable system does not only depend on the chemical composition of the powder to be sprayed but also on the operating conditions and on the blade characteristics. Depending on these conditions, the different available abradable coatings can be classified into generic families as described in the next section.

I.4. Abradable families

Because the temperature and pressure ranges are very large into a turbo engine, the application of several materials with different mechanical and thermal properties is necessary to account for the operating conditions of the different engine section. Several abradable classes exist, with their own benefits and disadvantages which determine their engine part application [START_REF] Giannozzi | On the potential of metal and ceramic based abradables in turbine seal applications[END_REF]. They are described in the following.

I.4.1. AlSi-polyester

Into low-pressure and low-temperature compressors, the most common abradable coating elaborated by atmospheric plasma spraying belongs to the AlSi-polyester family. They present a low resistance to shear stress providing good performance in terms of abradability. Those coatings can be dense but their operating temperature cannot be higher than 325°C and depends on the polymer nature.

This abradable family is recognized to provide the best performance in terms of abradability and low blade wear.

I.4.2. Metal matrix with solid lubricant addition

For high-temperature HP Compressors, the abradable family generally used is made of a metal matrix including solid lubricant addition. For working with temperature higher than 450°C, it is however necessary to spray a coating free of polymer. The solution is to remove the polymer phase or to use graphite or clay for the lubricant phase and to select an elaboration process giving a very high porosity level into the coating. Two of the studied coatings in this work come from this kind of abradable family, namely the NiCrAl-Clay and the MCrAlY-polyester abradables.

The first one, NiCrAl-Clay must be highly porous to fulfill its abradable role. The NiCrAl phase will provide to the coating its oxidation resistant properties. The friable clay will permit to shear easily the coating, to reduce the sizes of fragments induced by the cutting and to minimize the matter transfer between the blades and the coating.

A MCrAlY matrix resistant to hot gases erosion and oxidation composes the second studied coating. To confer abradability performance to the coating, the presence of lubricating and/or dislocating phases is required as well as a high porosity level (between 35 to 45%). The choice of the lubricating phase, hBN most of the time, is limited by its behavior under high temperatures. Porosity is induced in the coating by the presence of a polyester phase, which will be removed after spraying by application of a thermal treatment. This will conduct to remove and replace the polymer phase by pores.

I.4.3. Modified thermal barrier coating

To deal with high-pressure and high-temperature environment, the family of abradable commonly called the modified thermal barrier coatings is used. Thermal barrier coatings are composed of three different layers to reduce the metallic substrate temperature.

The first one in contact with the casing is a MCrAlY metal to protect the metallic substrate against hot corrosion and oxidation and to reduce the thermal expansion coefficient mismatch between the substrate and the top of the coating. The second one is a YSZ dense layer used as an insulator.

The last one is a technical oxides ceramic layer having good thermal insulating properties, a high expansion coefficient and is phase-stable at high temperatures. This ceramic layer has therefore the capability to resist to high-temperatures gases. The stabilization of Zirconia with Yttria oxide improves the high-temperature thermal lifecycle but this abradable ceramic is highly abrasive.

Polymer phases were thus added in order to improve the abradability properties of the layer. The polymer phase is removed after thermal treatment and gives place to porosities. Additionally, the structure of the ceramic material can be submicronic (i.e., large ceramic particles composed of smallest ones) in order to improve the abradability and decrease the size of the debris. We remind that the properties expected for an abradable coating are generally ranked in the following growing order of importance: good erosion resistance, good thermal cycle resistance and then abradability performance. Unfortunately, until now, no satisfactory abradable solutions exist for the highest operation temperature (around 1000°C) and coating spallation have been already observed under thermal cycling [4].

I.5. Abradable performances indicator

The principal characteristics to evaluate the in-service performances of abradable coatings are related to their microstructure description, their level of porosity, their apparent hardness, their thermal shock resistance, their erosion resistance to particles and finally, of course, their abradability performance [START_REF] Sporer | Novel Ceramic Abradable Coatings With Enhanced Performance[END_REF]. All these characteristics can be classified in two categories as shown on Figure 1-5 in order to well separate the most important ones, namely the abradability property, from the others.

The experimental and numerical procedures commonly used to determine these characteristics are presented in the forthcoming sections. The rig test is composed of a blade mounted on a rotating disc and made of the same material as the one used in real conditions, and of a shroud segment specimen coated with the abradable system to be studied. A stepper motor is used to force the contact between the coated specimen and the dummy blade. Rigs are equipped with a heating device to vary the shroud surface temperature and create a thermal gradient similar to the case of a real engine. The blade tip velocity can be varied as well as the incursion rate to test abradability performances under various conditions. Several sensors may be used to collect information of interest such as the coating and blade temperatures, the radial and tangential forces, the sensor distance to the blade tip, the disk speed and so on. These tests can give additional information about the nature and quality of wear occurring during the blade-coating contact, but also about the presence or not of material transfer which may occur in between the coating and the blade tip. The surface of the coating after cutting is also observed to determine the quality of the material abradability as well as the wear of the blade (Figure 12345678). Several studies have used this device to investigate the wear mechanisms occurring during rub in between the blade and the coating. Laverty [START_REF] Laverty | Rub Energetics of Compressor Blade Tip Seals[END_REF], Stringer [START_REF] Stringer | High speed wear testing of an abradable coating[END_REF] and Borel [START_REF] Borel | The wear mechanisms occurring in abradable seals of gas turbines[END_REF] have for example demonstrated the good relevancy of abradable wear obtained with abradable rig tests and those obtained during in-service engine conditions. These studies allowed to draw a wear map of abradable coatings and to discover all the wear mechanisms occurring during rub-in (e.g., cutting off, adhesive transfers, abrasion, grooving, densification, erosion, thermal shock…). This experimental approach thus provides useful results concerning the abradable performances but also contains some limitations listed below:

1. The repetition of tests is difficult due to the size and complexity of the test. Many parameters have to be implemented in order to reproduce turbine conditions, i.e. temperature at the rear face of the segment, blade tip velocity, wear track length, incursion rate and depth, blade height discrepancy, surface temperature of the abradable coating, wear track roughness, vibration data, wear mechanisms. Due to this, those tests are time consuming and costly.

2. Many abradable rigs exist with a similar approach: see Figure 1-6 and Figure 1-7 and also refer to the Praxair abradability rig test [START_REF] Taylor | Multilayer Abradable Seal Coating System, Heat Treating and Surface Engineering[END_REF], the Vac aero Hot Abradability Testing [START_REF] Vac Aero | Abradable Coatings used for Gas Path Seals in Turbine Engines Available[END_REF] and a numerous of house made test-rigs (Bounazef [90], Borel [START_REF] Borel | The wear mechanisms occurring in abradable seals of gas turbines[END_REF], Laverty [START_REF] Laverty | Rub Energetics of Compressor Blade Tip Seals[END_REF] and Stringer [START_REF] Stringer | High speed wear testing of an abradable coating[END_REF] for example). However, up to now, it does not exist a normalized test. Therefore the results obtained cannot be compared if they are not coming from the same abradable rig test and conducted in the same conditions. I.5.2. The HR15Y hardness test: an alternative to the abradability rig test

Considering the drawbacks of the rig tests listed in the previous section, alternative solutions have been studied to estimate the abradability performances [START_REF] Baïz | Etude expérimentale du contact aube/abradable : contribution à la caractérisation mécanique des matériaux abradables et de leur interaction dynamique sur banc rotatif avec une aube[END_REF][START_REF] Cuny | Caractérisation locale des couples de matériaux mis en jeu lors du contact rotor/stator dans une turbomachine[END_REF]. One of them consists in taking advantage of the correlation between coating hardness and abradability performances [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF][START_REF] Yi | Friction and wear behaviour and abradability of abradable seal coating[END_REF] . To determine the hardness of high porosity coatings, the applied method must have an averaging effect and be able to measure very low hardness. The Rockwell superficial hardness test (HR15Y) is appropriate to account for these two constraints and is employed for measuring the hardness of abradable coatings. The HR15Y test uses a preloading of 29 N to reduce the influence of the surface preparation; otherwise, a slight surface preparation is required (Figure 123456789). X. Ma and A. Matthews [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF] have shown a correlation between the Young modulus, the HR15Y hardness and the abradability of the coating. However, since these authors have only obtained an estimation of the abradability performances, the mechanical properties must not be used alone to determine the coating abradability. This correlation has been previously noticed by M. Yi [START_REF] Yi | Friction and wear behaviour and abradability of abradable seal coating[END_REF] who evaluated the abradability performances and hardness of several abradable coatings used in intermediate temperature conditions. He demonstrated a decreasing trend of the abradability with the increase of the coatings hardness, which does not seem surprising. However, M. Yi also showed that coatings with similar hardness values can show a highly different abradability behavior.

However, the hardness test can only give a first trend of abradability performances because the conditions are very far from those of a real engine. Additionally, a particular attention must be given to the thickness of the sample because a high variability of hardness against thickness has been already reported for real materials [START_REF] Lim | Structures and properties of BON and multilayered TiN/BON thin films prepared by PAMOCVD method[END_REF]. However, it has been demonstrated in [START_REF] Peyraut | Finite element modeling of abradable materials -Identification of plastic parameters and issues on minimum hardness against coating's thickness[END_REF] that a minimum hardness, invariant with thickness, can be reached, and that this minimum hardness is related to the boundary conditions applied to the substrate. Some other basic materials testing have been considered as an alternative: scratch, impact or pin on disc tests [START_REF] Ma | Investigation of abradable seal coating performance using scratch testing[END_REF][START_REF] Yi | Friction and wear behaviour and abradability of abradable seal coating[END_REF][START_REF] Sutter | Dynamic analysis of the interaction between an abradable material and a titanium alloy[END_REF]. They offer the advantage of both instrumentation and repetition under controlled condition. Unfortunately those test are far from real engine working condition.

I.5.3. Secondary tests

Secondary tests are necessary to validate the use of abradable coatings in real working conditions because abradable materials must fulfill other criteria than a good abradability. Erosion wear and thermal shock resistance are for example important properties to determine the coating lifetime [START_REF] Dorfman | Recent Development of High Temperature Thermal Spray Abradable Coatings, Ceramics Japan[END_REF][START_REF] Wilson | Thermally sprayed abradable coating technology for sealing in gas turbines[END_REF]. In practice, abradable seals must actually offer an average life of approximately 10,000 hours before replacement.

The main erosion wear mechanism is due to erosive particles and high-speed gas flow. The erosion resistance test consists in the determination of the material loss provided by solid particle impingement erosion. As shown in Figure 1-10, a diversity of test devices exists to determine the coating erosion resistance to impinging particles.

In house erosion test -M. Yi et al. [START_REF] Yi | Erosion wear behaviour and model of abradable seal coating[END_REF] General Electric (GE) company erosion test During a long time, the erosion wear mechanisms of abradable materials were unknown. Due to the high porosity of such coatings and to the presence of non-metallic phases, the erosion wear rules actually differ from conventional ones. To investigate several kinds of intermediate temperature abradable coatings, M. Yi et al. have realized a homemade test machine [START_REF] Yi | Friction and wear behaviour and abradability of abradable seal coating[END_REF]. However, the tests are often conducted according to the General Electric (GE) specification E50TF121CL-A [START_REF] Sporer | Processing and Properties of Advanced Ceramic Abradable Coatings, Thermal Spray[END_REF][START_REF] Rajendran | Gas turbine coatings -An overview[END_REF][START_REF] Hajmrle | Abradable Seals for Gas Turbines and Other Rotary Equipment[END_REF], even if an ASTM G76-13 standard test method exists to determine the erosion resistance of abradable coatings [START_REF]Standard Test Method for Conducting Erosion Tests by Solid Particle Impingement Using Gas Jets[END_REF]. The principle of those two standard methods, the ASTM and the GE ones, are very similar, but they are conducted with quite different parameters. First, the GE test can be conducted at high temperature whereas the ASTM test run at room temperature only [START_REF] Smith | High-temperature erosion testing standard and round robin testing[END_REF]. The other differences concern; the standoff distance (ASTM: 10 mm, GE: 100 mm), the impingement angle (ASTM: 90°, GE: 20°) and the powder feed rate (ASTM: 2 g.min -1 , GE: 300 g.min -1 ). However, the GE specification is the one mostly applied in the aeronautic industry and is the one considered in this manuscript. The GE number represents the time required to erode a coating thickness of 25.4 μm (1/1000 of inch). Thus, the best erosion resistance corresponds to the highest values of the GE erosion number. The erosion resistance required depends on the coating application and on its environment during engine operating service.

Complementary to the erosion test described above, the thermal shock resistance must also be considered because of the large temperature fluctuation existing in aero engines [START_REF] Schütze | Standardisation of Thermal Cycling Exposure Testing[END_REF]. The highest the temperature variations occurring in the engine are, the stronger the materials damages are. The thermal shock test consists in subjecting the coating and the substrate to a rapid heating, then a soak period at constant temperature and finally a fast cooling during a short time. This test is conducted to determine the behavior of a material submitted to large temperature variations and to predict the in-service lifetime of the component. Indeed, under high temperature cycling, the material physical characteristics change due to the thermomechanical stresses induced by alternating high-and lowtemperature extremes. This test is also interesting to observe the behavior of two different materials interconnected. Due to the large stresses induced by the dilatation mismatch of the two materials the thermal shock testing can cause failures. In the compressor and turbine, materials are exposed to extremely high fluctuations of temperature, so knowing their behavior is important.

The transition between temperature extremes occurs very rapidly, with a variation speed greater than 15°C per minute. Equipments with single or multiple chambers are typically used to perform thermal shock testing. When using a single chamber thermal shock equipment, the samples remain in one chamber, and the chamber air temperature is rapidly cooled and heated. Some equipment uses separate hot and cold chambers with an elevator mechanism that moves the samples in two or more chambers. This procedure is thus quite complex with many parameters [START_REF] Buessem | Thermal Shock Testing[END_REF], i.e. temperature range, number of cycles, sample size and shape, number of material layers, mass of the samples, number of samples, airflow around the samples, dwell time at each temperature level, and also minimum rates of temperature changes. This diversity of the parameters may induce a difficult interpretation of the results. The upper and lower temperature levels must be carefully determined. The temperature limits must be chosen to avoid exceeding the material limits of the sample and being representative of the real operating conditions. Temperatures that exceed the melting point of any material contained in the sample would likely result in invalid test failures. The erosion and thermal shock tests provide a first idea of the lifetime of coatings and permit to understand the coating behavior under critical conditions. They are conducted in laboratories and not in real turbine environment conditions. Indeed, from a cost and time-consuming point of view, measurements on such materials in real working conditions are difficult to plan.

I.5.4. Abradable extrinsic properties in the literature

Most of the tests previously presented do not provide information about intrinsic properties of abradable coatings, that is to say the thermal and mechanical properties measured at the scale of the components. This issue is of major interest because intrinsic properties are linked to the extrinsic ones used to dimension the abradable coating. This link between the micro and macro scales will be discussed later.

Determining extrinsic properties such as thermal diffusivity, thermal conductivity or the Young Modulus, should allow a better understanding of the abradable behavior and facilitate their dimensioning by creating a database in function of their chemical compositions and microstructures.

Extrinsic properties correspond to homogenized properties and are measured at the macro scale level.

They are strongly dependent upon: 1) the scale at which they are measured, 2) the manner the samples have been manufactured, 3) the measurement techniques that has been selected. The thermal history of the material during processing is also an influent parameter on the measured properties. For example, the intrinsic properties of polyester measured using the nanoindentation technique can be different when measured in the starting material (that is to say in the feedstock powder) or in the coating (that is to say when embedded in the coating and after having being thermally processed in a high-temperature flow).

A wide category of experimental tests allows the determination of extrinsic properties:

1. The Laser Flash analysis allows quantifying the thermal diffusivity, the heat capacity and then calculating the thermal conductivity [START_REF] Degiovanni | Conductivité et diffusivité thermique des solides, Techniques de l'ingénieur Mesure des grandeurs thermophysiques[END_REF][START_REF] Houlbert | Measurement of thermal diffusivity and anisotropy of plasma-sprayed coatings[END_REF].

2. The dilatometer test provides the thermal expansion coefficient [START_REF] Murat | Dilatométrie[END_REF].

3. The tensile test gives the Young modulus in one direction.

4.

Resonance frequency damping analysis provides information about the elastic properties, i.e.

Young's modulus and shear modulus.

Unfortunately, very few information related to the extrinsic properties of abradable coatings are available in the literature, particularly the thermal ones as shown in Table 1

-1.
Moreover, a scattering of the measured data is sometimes observed with for example a variation of 100% (from 2 to 4 GPa) of the Young modulus of AlSi graphite [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF]. In fact, most of the time, conventional tests (pull off, shear, peel or indentation tests) describe partially the behavior of the coating/substrate system, leading to an inaccurate evaluation of the coating properties. Additionally, due to the porous microstructure, the adhesion tests also induce inaccuracies because the glue can penetrate into the coating and increase its adhesion. However, concerning the structural material properties, Johnston has developed a repeatable method to produce free standing abradable test specimen [START_REF] Johnston | Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings[END_REF]. This under patent method [106] allowed him providing the static and cyclic tensile stresses and determining the Young modulus, the tensile strength and the strain to failure of three different abradable coatings (i.e., AlSi-hBN, NiCrAl-Bentonite-hBN and NiCrAl-Bentonite). AlSi-hBN 17.22 [START_REF] Johnston | Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings[END_REF] x x AlSi-graphite 2 -4 [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF] x x

NiCrAl-Bentonite-hBN 0.749 [START_REF] Johnston | Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings[END_REF] x x

NiCrAl-Bentonite 1.395 [START_REF] Johnston | Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings[END_REF] x 18.3 [START_REF] Taylors | Evaluation of NiCrAl-Bentonite (LCA-314) Coatings by AST on Trent 800 Stage 6 Compressor Rings after Flight Exposure[END_REF] Ni-graphite 3 -4.5 [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF] x x YSZ-polyester 76 [START_REF] Gupta | Design of next generation thermal barrier coatings -Experiments and modelling[END_REF] 0.64-1.08 [START_REF] Curry | Design of Thermal Barrier Coating Systems[END_REF] x

AlSi-polyester 1.5 -2.5 [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF] x x

I.6. Elaboration processes of abradable materials

The main processes used to manufacture abradable coatings are plasma and flame spraying [3,[START_REF] Dorfman | Thermal Spray Technology Growth in Gas Turbine Applications[END_REF].

These processes allow manufacturing a wide range of composite coatings with a large variety of operating process parameters. They particularly allow spraying thick layers, i.e. from a few micrometers to several millimeters, with several natures of materials, even the most refractory ones.

The adjustment of the spray parameters allows acting on several factors to obtain the desired coating characteristics.

One other benefit of the plasma and flame spraying processes is the possibility to elaborate coatings on a large variety of surface geometries, even the more complex ones. The implementation of coatings in aeronautic technologies sometimes requires multilayer deposits, e.g. modified Thermal

Barrier Coating (TBC), to confer supplementary functions to the initial material, under very high thermal and pressure environment. The thermal spray technology, well adapted to manufacture these kind of multilayer coatings, is presented in details in the next section.

II Thermal spraying -Elaboration process

II.1. Process description

As explained previously, thermal spraying is the main process used in industry to manufacture abradable coatings. Thermal spraying has been invented in 1909 by Shoop [START_REF] Audisio | Revêtement et traitements de surface. Fonctionnalités, durabilité[END_REF]. The aim is to generate a high temperature high velocity jet using a gas and a source of heat. A material in powder, wire or rod form is then injected into this jet, providing droplets in molten or plastic state. The droplets are accelerated and sprayed toward the surface to coat. They flatten, solidify and cool down onto the substrate. The resulting microstructural architecture of the coating is a network of intermingled lamellae, characteristic of the thermal spray process. Compared to other coating processes like chemical/physical vapor deposition or electroplating, and due to its high enthalpy, the thermal spraying process can allow very high coating deposition rates to obtain thick coatings.

The thermal spray process is also a versatile technique which allows operating with a large variety of gas temperatures and velocities. This will permit to coat a wide range of materials of different natures. Depending on the feedstock material properties, it can confer a variety of properties to the material surface, such as corrosion or oxidation resistance, hardness increase, thermal insulation or electrical properties. Moreover, surface preparation techniques, such as degreasing, sanding or grit blasting allow providing a good bond strength of the coating on the substrate. This versatility is useful for satisfying the numerous characteristics (abradability, thermal shock resistance and erosion resistance capability) expected for an abradable coating. The coating properties depend in fact on three main families of parameters that can be adapted to satisfy the desired applications. These families of parameters are (1) the physical and chemical characteristics of the gas jet (plasma gas or combustion gas mixture) and its interaction with the environment, (2) the powder characteristics in terms of shape, diameter size and nature, as well as the injection parameters and (3) the lamellae formation process.

Characteristic of the gas jet, injection parameters and choice of powder can be also adapted to satisfy the desired applications.

The thermal spraying process regroups several techniques with their own characteristics [START_REF] Proner | Revêtements par projection thermique, Techniques de l'ingénieur Traitements de surface des métaux par voie sèche et en milieu fondu[END_REF]. The main differences between them concern the energy sources. The processes can be divided in two main categories, i.e. combustion process or electric discharge source [3]. These two categories can themselves be divided in sub-categories as shown on Figure 1-12. Only two of these sub-categories have been used in this work to manufacture abradable coatings: flame spraying and atmospheric plasma spraying (APS) processes. Those are the processes commonly used to manufacture abradable coatings considered in the frame of this work, to obtain the desired microstructures. Their principle and differences are schematized on Figure 1-13 and will be detailed in the forthcoming sections.

II.1.1. Combustion-Powder Flame Spray

The combustion flame spray of powder was the first thermal spraying process developed and patented in 1911 by shoop [START_REF] Audisio | Revêtement et traitements de surface. Fonctionnalités, durabilité[END_REF][START_REF] Sokolov | Contribution au développement de la projection thermique à très faible pression[END_REF]. The flame is generated by the combustion of fuel in air or pure oxygen. The flame velocity is in the range 50-100 m/s and the maximum flame temperature can reach 3100°C for a combustion in pure oxygen. The material in a powder form is transported by a carrier gas and injected into the flame. A secondary gas such as inert gas or compressed air can be used to "atomize" the particles (subdivide them into smaller particles). The particle velocity in the flame can be in the range 30-70 m.s -1 .

This process allows reaching relative densities of the coatings from 85 to 98%. The void level is high and gives microstructure made of coarse splats. This process is mostly used to produce coatings adapted for basic wear, corrosion resistance and abradable or also clearance control applications.

II.1.2. Atmospheric Plasma Spray

Atmospheric plasma spraying, created in 1951 by the company plasmadyne [START_REF] Audisio | Revêtement et traitements de surface. Fonctionnalités, durabilité[END_REF], consists in the generation of a plasma. The plasma is a matter state similar to the gaseous state, i.e. a diluted and disordered material but with charged particles, i.e. ions and electrons, however, their proportions induce a neutral matter state, named the plasma [113]. An high frequency discharge is used to initiate the electric arc between an anode (most of the time in copper with insert of sintered tungsten) and a cathode (thoriated tungsten). A DC generator providing the plasma jet then maintains this arc. This DC arc heats the working plasma gas mixture (most of the time composed of Argon, Hydrogen, and/or Helium) up to a very high temperature, thus providing a dissociation (for H 2 ) and a ionization of the elements. A high expansion also occurs providing the formation of a plasma jet [START_REF] Proner | Revêtements par projection thermique, Techniques de l'ingénieur Traitements de surface des métaux par voie sèche et en milieu fondu[END_REF]. The plasma jet has a very high temperature in the range 12 000-20 000°C at the torch exit, whereas the corresponding velocity is in between 600 m.s -1 when working gas is pure Argon, and can reach a velocity up to 2200 m.s -1 with Argon/Hydrogen working gas mixture. A carrier gas will transport the material (in powder form) and will inject it into the core of the plasma where particles will reach a velocity between 100 and 500 m.s -1 .

Atmospheric plasma spraying (APS) often produces coatings with higher density and bond strength in comparison with the flame spray process due to a higher degree of melting and a higher velocity of the particles. APS coatings also present a lower contamination state than flame spray coatings.

Moreover, this method allows producing ceramic and other refractory materials coatings.

To conclude, thermal spraying processes allows forming new material coatings having very different extrinsic properties, which is perfect for the manufacture of abradables

II.2. Coating formation process

To understand how process parameters influence the coating microstructure, it is necessary to study the coating formation process. The plasma jet allows the melting and acceleration of the sprayed particles. During their impact onto the substrate, the droplets crash and flatten to form lamellae. The construction of the coating is thus achieved by a succession of layers formed by the spreading, flattening and solidification of the melted particles onto the substrate [START_REF] Fauchais | Quo vadis thermal spraying?[END_REF][START_REF] Fauchais | Plasma Spraying: From Plasma Generation to Coating Structure, Advances in Heat Transfer[END_REF]. A network of intermingled lamellae also called "splats" therefore builds up the coating. The shape of these lamellae influences the coating microstructure.

The formation of a coating from particles results from the thermal and kinetic energy transfers between the enthalpy source and the particles [3,[START_REF] Fauchais | Plasma spray: Study of the coating generation[END_REF]. The kinetic energy is transmitted to the particles by friction between the high velocity jet and the particles. These transfers depend on the size and on the density of the particles as well as on the density, the velocity, and the viscosity of the jet.

The thermal energy transfers into the plasma jet are due to two main mechanisms, namely convection and conduction. Convective transfers act on the surface of the particles whereas conductive transfers operate inside the solid (or molten) particle. This can lead sometimes to particles having a solid core and a surface at the vaporization temperature of the material (for refractory material or ceramics for example). This induces disk shaped lamellae, partially fingered.

The major factors influencing the shape of the lamellae after flattening of the particles are their powder nature, powder size, and also their temperature and velocity prior to the impact [START_REF] Mcpherson | A review of microstructure and properties of plasma sprayed ceramic coatings[END_REF][START_REF] Fantassi | Influence of the velocity of plasma-sprayed particles on splat formation[END_REF]. Some of these factors, i.e. powder nature, powder size, are linked to the feed material and to the substrate surface preparation but some others, i.e. particle temperature and velocity are more dependent on the process parameters. Indeed, process parameters act on the plasma jet characteristics and then on energy transfers and subsequent interactions with the powder particles. For example, when the plasma enthalpy increases, thermal transfers from the plasma to the particles increase, so that the fraction of melted particles also increases. From these interactions, particles impact onto the substrate with a given velocity and temperature. It results in different shapes of the splats depending on the velocity and temperature (Figure 1 -14). Indeed, a rise in the particle temperature provides a decrease in the dynamic viscosity of the material. Together with a higher particle velocity, it results in a higher degree of flattening. A higher degree of flattening corresponds to a decrease in the splat thickness and to a larger area of the splat surface in contact with the underlying material. That may lead to a higher cooling velocity after solidification. A lot of experimental and numerical studies have been conducted

to understand the splat formation process and a wide number of them have been indexed by Fauchais et al. [START_REF] Fauchais | Knowledge concerning splat formation: An invited review[END_REF].

If the particles stay longer in the hottest part of the plasma jet, their average temperature increase [START_REF] Bisson | Influence of hydrogen on the microstructure of plasma-sprayed yttria-stabilized zirconia coatings[END_REF]. However, for a same physical state of the plasma jet, changing the particles velocity change also their dwell time in the jet and influence their temperature, for example a too high velocity of particles reduce their particles heat. If the thermal transfers are too low to melt a particle, the solid particle may bounce onto the substrate. On the contrary, a well molten liquid particle with a high velocity can fragment during its impact on the substrate, leading to an increase of the porosity into coating. The lamellae forms, and interactions between them, thus depend on the physical state of the particles prior to their impact [START_REF] Fauchais | Knowledge concerning splat formation: An invited review[END_REF][START_REF] Fauchais | Quo vadis thermal spraying?[END_REF][START_REF] Vardelle | Spray parameters and particle behavior relationships during plasma spraying[END_REF]. Additionally, the nature of the interface between individual lamellae, and the internal structure of the lamellae produced by their rapid cooling and solidification, strongly influence the microstructure of plasma sprayed coatings. Coating in-service properties depend on about 50-60 spraying parameters that are all dependent from each other. Several process parameters are important to control the plasma properties and then the coating elaboration. Fisher [START_REF] Fisher | Variables Influencing the Characteristics of Plasma-Sprayed Coatings[END_REF] has listed numerous variables influencing the plasma jet characteristics. Some of the most relevant are the flowrate of the working gases, the electric power, the angle and velocity of the powder injection, and the spraying distance. This variety of parameters permits a large flexibility to design the appropriate coating microstructure.

It is also possible to change the gas flow rate or to use a feedstock material with a different morphology. These processes use different kinds of powders manufactured for abradable application.

The manufacture of powders is based on:

 The use of a release agent encapsulating the individual particles of the matrix, e.g. NiCrAl-Bentonite.

 The mix of different powders selected in a wide range of materials providing different types of properties, e.g. abradability and erosion resistance for CoNiCrAlY-BN-Polyester.

 The process to form the powder, e.g. agglomeration to form YSE-Polyester powder.

However, European partners fixed the powders from the early stage of this thesis work so that only the process parameters were therefore adjusted.

II.3. Microstructure influence on coating properties

Composite coatings elaborations lead to complex materials and microstructures. Understanding how the microstructure influences the coating properties is a key point for improving the coating performances; namely reducing the thermal conductivity of thermal barrier coatings, limit the cracks and stresses propagation in abradable seals and enhancing the abradability performances. Several studies already highlighted the influence of the coating microstructure on their effective thermomechanical properties and on their global behavior. For example, it has been proved in [START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Karthikeyan | Developing empirical relationships to estimate porosity and Young's modulus of plasma sprayed YSZ coatings[END_REF][START_REF] Friis | Influence of particle in-flight characteristics on the microstructure of atmospheric plasma sprayed yttria stabilized ZrO2[END_REF] that pores and cracks networks directly interfere the heat flow through the coating and provide a lower apparent thermal conductivity of the coating.

Abradable seal properties are sensitive to their chemical composition but also to their microstructure. Indeed, two chemically identical coatings with strongly different microstructures will not behave similarly under abradable wear, high temperature and high pressure. given application, it is therefore important to understand interactions between the coating microstructures and properties.

Composite coatings investigated here are used in the turbofan environment and are submitted to high temperatures (up to 1800 °C), high pressure oxidation and erosion wear due to debris engulfed in the airstream [119]. They are made up of a ceramic or a metallic matrix with a lubricant and/or porosity former phase. Varying the plasma spraying operating parameters permits to implement different types of features in the coating microstructure, i.e. phases, pores, cracks. Those features are interesting for several reasons;

1. The network of voids and cracks is beneficial for abradable performances and limits the stress propagation [START_REF] Fauchais | Plasma Spraying: From Plasma Generation to Coating Structure, Advances in Heat Transfer[END_REF].

2. Inter and intra-lamellar cracks play a huge role in the decrease of the coating thermal conductivity, and subsequently on the thermal insulation [START_REF] Boire-Lavigne | The relationship between the microstructure and thermal diffusivity of plasma-sprayed tungsten coatings[END_REF], which is important in the case of high temperature abradables for example.

T. Steinke et al. [START_REF] Steinke | Process Design and Monitoring for Plasma Sprayed Abradable Coatings[END_REF] have drawn some primary conclusions about the effect of plasma process parameters on abradable coating performances. To increase the coating hardness, the erosion resistance and the deposition efficiency, the ideal parameters were a low argon flow rate, a high current intensity and a low spray distance. Of course, the selection of the process parameters depends on the nature, thermal and mechanical intrinsic properties of the considered material. D.

Sporer et al. [START_REF] Sporer | Processing and Properties of Advanced Ceramic Abradable Coatings, Thermal Spray[END_REF] obtained the same conclusions: to increase the coating hardness and erosion resistance, the selected parameters allow increasing the plasma power and providing a higher temperature of the plasma, the polymer phase of the powder was then burnt in-flight leading to a very low presence of the polymer phase in the coating. The coating microstructure itself is strongly influenced by the elaboration process [START_REF] Mcpherson | A review of microstructure and properties of plasma sprayed ceramic coatings[END_REF]. As represented on Figure 1-15, thermal spray processes permit to obtain coatings with a variety of features that are described below, namely oxides, porosity, polymer phase, intra and inter lamellar cracks and unmelted particles:

 Oxides: Oxides are produced due to interactions between the hot particles and air entrained in the plasma jet. This air entrainment leads to an oxidation of the surface of particles and then to the presence of oxide inclusions into the coating. The presence of these oxides can be beneficial in increasing the coating hardness for example.

However, they can also increase the brittleness of the coatings. In practice, this microstructure feature can be interesting for wear resistance or thermal insulation applications. However, a too high oxide level may also have negative effects on the lamellae cohesion.

 Porosity: The porosity network directly interfere the heat flow through the coating providing a lower thermal conductivity [START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Karthikeyan | Developing empirical relationships to estimate porosity and Young's modulus of plasma sprayed YSZ coatings[END_REF][START_REF] Friis | Influence of particle in-flight characteristics on the microstructure of atmospheric plasma sprayed yttria stabilized ZrO2[END_REF]. The voids network can be described by the pore level but also by their size, their shape and their distribution.

The presence of large porosities can improve the thermal-shock resistance [START_REF] Scrivani | Thermal Fatigue Behavior of Thick and Porous Thermal Barrier Coatings Systems[END_REF] whereas finely distributed porosities tend to enhance the thermal shock resistance of brittle materials [START_REF] Scrivani | Thermal Fatigue Behavior of Thick and Porous Thermal Barrier Coatings Systems[END_REF][START_REF] Novinski | Modified zirconia abradable seal coating for high temperature gas turbine applications[END_REF]. For example, porosities and cracks can improve the mechanical compliance of Thermal Barrier Coatings systems, increasing their thermomechanical compatibility during cyclic thermal exposure between room temperature and an operating temperature over 1000 • C [START_REF] Herman | Survivability of thermal barrier coatings[END_REF][START_REF] Vaßen | Correlation between spraying conditions and microcrack density and their influence on thermal cycling life of thermal barrier coatings[END_REF]. The pores size and morphology also have an influence on brittle fracture of the coating and on electrical and thermal properties [START_REF] Antoine | Quantification de l'architecture poreuse de dépôts finement structures (sub-micromètre -nanomètre) de zircone yttriée réalisés par projection plasma de suspension[END_REF][126][127].

Besides, some studies have shown the non-proportionality existing between the Young's modulus and the void rate as well as the non-proportionality between the thermal conductivity and the void rate [START_REF] Arai | Effect of Thermal Spray Condition on Characterization of Thermal Barrier Coating, Thermal Spray[END_REF]129]. For example, Arai et al. [START_REF] Arai | Effect of Thermal Spray Condition on Characterization of Thermal Barrier Coating, Thermal Spray[END_REF] have

shown that the contribution of the pore level is secondary for inelastic deformation processes. Other microstructural features existing in the coating also influence the evolution of the Young's modulus and of the thermal conductivity. Several studies highlighted the influence of the coating microstructure (not only that of the void rate) on the coating effective thermomechanical properties and on its global behavior [START_REF] Kulkarni | Comprehensive microstructural characterization and predictive property modeling of plasma-sprayed zirconia coatings[END_REF][START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Azarmi | Young's modulus measurement and study of the relationship between mechanical properties and microstructure of air plasma sprayed alloy 625[END_REF][START_REF] Bengtsson | Characterization of microstructural defects in plasma-sprayed thermal barrier coatings[END_REF][START_REF] Bolelli | Microstructure-based thermo-mechanical modelling of thermal spray coatings[END_REF][START_REF] Johnson | Effective modulus and coefficient of thermal expansion of Ni-YSZ porous cermets[END_REF][START_REF] Laurent | Etude microstructurale fine de revêtements céramiques de type barrièrre thermique; incidence sur la résistance thermomécanique de ces revêtements[END_REF][START_REF] Sesso | Effect of Plasma Spray Parameters on Thermal Barrier Coating Formation and Microstructural Properties[END_REF][START_REF] Öksüz | Microstructure and wear properties of plasma-sprayed aluminum-silicon-polyester coatings[END_REF][START_REF] Sevostianov | Plasma-sprayed ceramic coatings: anisotropic elastic and conductive properties in relation to the microstructure; cross-property correlations[END_REF].

 Polymer phase: in most cases, the polymer phase present in coating is just a porosity former, meaning that the polymer is often removed by pyrolysis. The spray parameters must be adapted to the polymer content and size desired. The purpose is to reduce stresses inside the coating and provide crack sources for the particles release.

In other words, the polymer is added to induce porosity in the abradable coating and to reduce the propagation of stresses.

 Inter-lamellar cracks: they are due to the imperfect contact occurring at the lamellae interface, which are especially related to sliding due to the lamellae contraction after solidification. Increasing horizontal microcracks can result in a reduction of the thermal shock lifetime [START_REF] Vassen | Influence of Microstructure on the Thermal Cycling Performance of Thermal Barrier Coatings, Thermal Spray[END_REF] and their propagation close to the substrate/coating interface can cause a spalling and failure of the coating, and can strongly affect the coating cohesion.

 Intra-lamellar cracks: these micro-cracks are due to residual stresses occurring when the particles are cooled down quickly after impacting the substrate surface (or the surface of the previously deposited coating layers) [START_REF] Arai | Effect of Thermal Spray Condition on Characterization of Thermal Barrier Coating, Thermal Spray[END_REF]. In these conditions, if a sliding is not possible, a fracture of the lamellae may occur, especially for ceramic splats. Cracks perpendicular to the substrate/coating interface decrease the stiffness of the coating and thus improve its thermal shock resistance [START_REF] Friis | Influence of particle in-flight characteristics on the microstructure of atmospheric plasma sprayed yttria stabilized ZrO2[END_REF]. They also improve the insulating ability of the coating.

 Unmelted particles: they are due to insufficient thermal exchanges between the plasma and the particle, inducing a partial melting of the particle. Unmelted particles tend to decrease the quality of the contact between splats, leading to a decrease of the cohesion between lamellae and to the occurrence of a premature cracking, delamination or spalling. However, unmelted particles are poorly oxidized, their presence is desired for abradable performance specially for AlSi/polyester abradable.

To highlight the influence of microstructure on coating performance, one of the most well-known coating application is thermal barrier coatings (TBCs). It must be outlined that the performance of TBCs depends on the process parameters used to manufacture the deposit [START_REF] Sesso | Effect of Plasma Spray Parameters on Thermal Barrier Coating Formation and Microstructural Properties[END_REF][START_REF] Friis | Influence of particle in-flight characteristics on the microstructure of atmospheric plasma sprayed yttria stabilized ZrO2[END_REF]127,[START_REF] Markocsan | Effect of Thermal Aging on Microstructure and Functional Properties of Zirconia-Base Thermal Barrier Coatings[END_REF][START_REF] Marple | Yttria-Stabilized Zirconia Thermal Barriers Sprayed Using N[2]-H[2] and Ar-H[2] Plasmas : Influence of Processing and Heat Treatment on Coating Properties[END_REF][START_REF] Prystay | Structure of plasma-sprayed zirconia coatings tailored by controlling the temperature and velocity of the sprayed particles[END_REF] knowing that porosity is an important attribute of TBCs. It is actually the main contributing factor to lower the effective thermal conductivity of TBCs, and for being able to deal with higher operating temperatures of turbine engine [START_REF] Vassen | Influence of Microstructure on the Thermal Cycling Performance of Thermal Barrier Coatings, Thermal Spray[END_REF][START_REF] Jadhav | Low-thermalconductivity plasma-sprayed thermal barrier coatings with engineered microstructures[END_REF][START_REF] Wang | A new structural model of effective thermal conductivity for heterogeneous materials with co-continuous phases[END_REF].

II.4. Residual stresses

The thermal spray process induces residual stresses in coating. Residual stresses are due to quenching of individual lamellae, expansion mismatch between the coating and substrate materials, temperature gradients and phase changes of materials taking place during thermal spraying (flattening/solidification and cooling processes). Among these phenomena, one can retain two main contributions. The first one is the rapid cooling of lamellae causing a thermal contraction of splats. A tensile stress appears from this contraction by the presence of the underlying material (substrate or other splats). The second contribution comes from the dilatation mismatch between the coating and substrate materials, during cooling down of the system to the ambient temperature.

In practice, during the final cooling of the substrate/coating system to ambient temperature, a thermal contraction of the materials occurs and generates stresses by the dilatation mismatch of the two materials (i.e., difference between the thermal expansion coefficients of the coating and substrate materials). Microstructures features (micro or macro cracks, yielding or creep, as well as plastic deformation for metals and alloys) are relaxation mechanisms of these stresses [START_REF] Kuroda | Quenching stress in plasma sprayed coatings and its correlation with the deposit microstructure[END_REF][START_REF] Clyne | Residual Stresses in Thermal Spray Coatings and Their Effect on Interfacial Adhesion: A Review of Recent Work[END_REF]. Depending on the relative thermal expansion coefficients of the two materials, these "thermal stresses" can be compressive in the coating (𝛼 𝑠 > 𝛼 𝑐 ) or tensile (𝛼 𝑠 < 𝛼 𝑐 ), so that thermal stresses can sometimes contribute to offset quenching stresses and sometimes contributes to increase the final stress level.

Application of preheating of the substrate may thus sometimes be useful in order to decrease the tensile stress level in the coating layer (due to quenching of lamellae) or even provide a final compressive state of stresses in the coating: it is particularly the case for ceramics that exhibit most of the time a low expansion coefficient in comparison with most common metal substrates. On the contrary, the use of efficient cooling devices during thermal spray may also be useful for some other applications: one can mention the case of AlSi/polyester abradable coatings that often exhibit a high dilatation coefficient.

Several studies have highlighted the influence of these residual stresses on the coating properties.

For example, G.G. Stoney et al. [START_REF] Stoney | The Tension of Metallic Films Deposited by Electrolysis[END_REF] have established equations to predict residual stresses in thin coating. Timoshenko [START_REF] Timoshenko | Analysis of bi-metel thermostats[END_REF] has developed an analytical model of elastic thermal stresses in a two-layer system. Other studies have also been conducted on multilayer systems and on metal/ceramic interlayer [START_REF] Hsueh | Residual stresses in thermal barrier coatings: effects of interface asperity curvature/height and oxide thickness[END_REF][START_REF] Hsueh | Modeling of Thermal Stresses in Joining Two Layers with Multi-and Graded Interlayers[END_REF][START_REF] Williamson | Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part I. Model description and geometrical effects[END_REF]. However, in the context of abradable materials, the question of residual stress has received only little attention even if one can mention the work of R.E. Johnston on this subject [START_REF] Johnston | The Sensitivity of Abradable Coating Residual Stresses to Varying Material Properties[END_REF].

Nevertheless, this issue will not be addressed in this manuscript due to lack of experience capitalized on abradable coating.

II.5. Abradable problematic

As described previously, usual tests applied to abradable materials can be uneasy to conduct. The determination of the abradability or erosion properties in real or laboratory conditions are indeed time and cost consuming (sample preparation, availability of the testing bench). Alternative tests have been used these last years to establish correlations with the abradability performances by means of statistical functions, models, stereological analysis or analytical studies [START_REF] Ma | Evaluation of abradable seal coating mechanical properties[END_REF][START_REF] Ma | Investigation of abradable seal coating performance using scratch testing[END_REF]. However, the experience on abradable materials is poorly capitalized. For this reason, dimensioning of abradable coatings is still a difficult task.

Moreover, thermal spraying processes allow manufacturing different coating microstructures with the same chemical composition. These coatings are designed for working in the same turbo-reactor section and in the same operating conditions. It is then mandatory to find solutions to discriminate these different microstructures. The solution adopted in this thesis work, and developed in the forthcoming chapters, was to realize a first discrimination of the most interesting coatings by comparing the influence of their microstructures on extrinsic (i.e., macroscale) thermomechanical properties. The coating morphological characteristics were described and discretized using image analysis and stereological protocols. A feasibility study was then conducted to determine whether the abradable coating properties can be obtained by 2D numerical models. The corresponding extrinsic properties should allow dimensioning abradable coatings but also feeding numerical models to reproduce abradability tests. These tests concern the abradable coatings developed in the framework of the European E-BREAK program.

III Modeling of thermal and mechanical properties

As discussed previously, the coating microstructure dramatically affects the coating properties.

Several studies were therefore conducted the last decade to understand these strong interactions [START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Grandjean | Numerical calculations of the thermal conductivity of porous ceramics based on micrographs[END_REF]. Mastering these interactions is actually a strategic issue in order to improve efficiently the properties of coatings for aero-engine applications.

The literature regroups numerous works trying to predict the extrinsic behavior of multi-phase materials by using information on their phases [START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomecanique de matériaux réfractaires modèles[END_REF][START_REF] Carazo | Effective properties of nodular castiron: A multi-scale computational approach[END_REF][START_REF] Bolot | Predicting the Thermal Conductivity of AlSi/Polyester Abradable Coatings: Effects of the Numerical Method[END_REF][START_REF] Grandjean | Numerical calculations of the thermal conductivity of porous ceramics based on micrographs[END_REF][START_REF] Azarmi | Young's modulus measurement and study of the relationship between mechanical properties and microstructure of air plasma sprayed alloy 625[END_REF][START_REF] Johnson | Effective modulus and coefficient of thermal expansion of Ni-YSZ porous cermets[END_REF][START_REF] Bolot | Application of FEM for the Estimation of Thermo-Mechanical Properties of Plasma Sprayed Composite Coatings[END_REF][START_REF] Bolot | A Two-Dimensional Heat Transfer Model for Thermal Barrier Coating Average Thermal Conductivity Computation[END_REF]144,[START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF]. The extrinsic properties correspond to homogenized properties determined at the macro-scale of the coating and deduced from the intrinsic properties at the micro-scale level of the coating components. Even if the homogenization theory has originally emerged to predict the behavior of periodically organized composites [START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF], the theory also applies to randomly arranged materials [START_REF] Carazo | Effective properties of nodular castiron: A multi-scale computational approach[END_REF][START_REF] Fernandino | Determination of effective elastic properties of ferritic ductile cast iron by computational homogenization, micrographs and microindentation tests[END_REF].

In this section, the basic principles of the homogenization theory are first recalled, simplified approaches based on analytical models are introduced and the potential of numerical methods is discussed.

III.1. Homogenization theories

Abradable coatings being composite heterogeneous structures, their thermomechanical behavior can be observed at different scales. At the macroscopic one, the thermomechanical behavior is described at the global scale of the working conditions of the coating. At the microscopic scale, the thermomechanical behavior is examined at the local level of the phases. To connect these two scales by general equivalent behavior laws, one way is to perform a finite element computation by considering all micro-structural features in the mesh. Unfortunately this would require billions of elements to respect the micro-geometry and current computers are not able to support such a mesh in terms of memory size and computation time.

To link the macro and micro scales, one other solution is to replace the composite by an artificial homogeneous medium having a macroscopic average physical behavior corresponding to the initial heterogeneous medium. In this manner, the heterogeneous medium will be homogenized into an equivalent homogeneous structure suitable for a finite element analysis. A prior knowledge of the thermomechanical properties of all the phases of the heterogeneous structure is of course mandatory.

Unfortunately, in the case of abradable materials, it is sometimes uneasy to get reliable information about the phase properties from the powders manufacturer. That can unfortunately lead to uncertainty on the estimation of the homogenized properties. In fact, the properties of a composite material depend upon:

1. the inclusion nature and properties, 2. the inclusion repartition, that is to say a random or a periodic organization, 3. the inclusion geometry, in terms of shape and size, 4. the interface nature between the different inclusions.

In conventional homogenized models, interfaces are assumed continuous, meaning that stresses and displacements are perfectly transmitted from one phase to another. This assumption could be questionable but was adopted to pay more attention on the best magnification to apply on micrographs in order to obtain what is commonly called a REV (Representative Elementary Volume),or RVE.

Actually, to apply the homogenization procedure, it is necessary to preliminary define this REV on which the computation will be performed (Figure 12345678910111213141516). It must be large enough compared to the size of heterogeneities in order to correctly represent the averaged behavior of the whole structure, but small enough compared to the structure size to allow a reasonable computation time. As claimed by F.D. Carazo et al. [START_REF] Carazo | Effective properties of nodular castiron: A multi-scale computational approach[END_REF]: "… there are numerous definitions of RVE in the literature [START_REF] Hashin | Analysis of Composite Materials-A Survey[END_REF][START_REF] Drugan | A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites[END_REF][START_REF] Evesque | Fluctuations, Correlation and Representative Elementary Volume (REV) in Granular Materials[END_REF][START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF][149][START_REF] Trusov | The theory of constitutive relations: Part I[END_REF][START_REF] Van Mier | Fracture Processes of Concrete[END_REF] but there is a general agreement that a RVE can be considered as the minimum volume of material whose behavior is equivalent to a volume of a homogeneous fictitious material". For heterogeneous medium with a periodic organization, the REV reduces to a repeated basic cell [START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF] while, for random heterogeneous medium, all the REV portions, even if they are not superimposed, should have a similar macroscopic behavior in a statistical manner. In fact, as outlined by D.O. Fernandino et al. [START_REF] Fernandino | Determination of effective elastic properties of ferritic ductile cast iron by computational homogenization, micrographs and microindentation tests[END_REF], there are three different ways to generate the micro-scale features of the REV: 1) Assume that it is periodically located but this is definitely not the case for abradable materials; 2) Generate artificially the REV by means of computer algorithms; 3) Extract the REV from actual micrographs. In this thesis work, the third way was considered because numerous micrographs are available through the E-BREAK European program. However, the choice of the best magnification level of these micrographs

is still an open question debated for example in [START_REF] Fernandino | Determination of effective elastic properties of ferritic ductile cast iron by computational homogenization, micrographs and microindentation tests[END_REF] and also discussed in the third chapter of this manuscript. Following all above remarks, the REV must satisfy the following requirements:

 The average properties must not evolve when the REV dimensions increase.

 The average properties resulting from the use of the REV with the homogenization procedure must be representative of the macroscopic behavior of the whole structure.

 If several thermal and mechanical macroscopic properties are needed, they must all be taken into consideration to adjust the REV.

 The average REV behavior must not depend on the boundary conditions applied to the whole structure.

However, the achievement of all these requirements can lead to very large REV which is not desirable [START_REF] Forest | Milieux continus généralisés et matériaux hétérogènes[END_REF] specially with composite having large microstructure feature size. To avoid this, an alternative is to conduct a statistical approach where several Elementary Volumes (EV) of smaller dimensions are used. The effective behavior of the considered heterogeneous media will then be estimated by the statistical average of the apparent behavior on these elementary volumes [START_REF] Kanit | Determination of the size of the representative volume element for random composites : statistical and numerical approach[END_REF]. The REV is then the combination between the statistical approach and the EV chosen.

To summarize this introductive section on the homogenization theory, even if different approaches exist, a common core of methodology can be described as follows and as related by F. Di Paola [START_REF] Paola | Modélisation multi-échelles du comportement thermo-mécanique de composites à renforts sphériques (in english: Multiscale modeling of thermo-mechanical behavior of composite with spherical reinforcements[END_REF]:

 Select the appropriate Representative Elementary Volume.

 Operate the localization process to link the micro and the macro scales.

 Apply the homogenization procedure to deduct the apparent properties.

III.2. Analytical models

Even if analytical models are not able to give accurate predictions of the thermal and structural macroscopic properties, they are useful to provide lower and upper bounds of these properties. By providing a first estimation of the average value linked to the microstructures, these bounds permit particularly to check the appropriateness of more accurate computational approaches. Moreover, the estimation provided by analytical models is sufficient in situations where only a global trend is needed.

However, the number of analytical models developed in the literature is important so that all models cannot be presented in this manuscript. Some of them are described in the PhD dissertations of M. Gupta [START_REF] Gupta | Design of Thermal Barrier Coatings: A Modelling Approach[END_REF], N. Tessier-Doyen [START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomecanique de matériaux réfractaires modèles[END_REF] and J. Qiao [START_REF] Qiao | Modélisation des propriétés thermomécaniques effectives de dépôts élaborés par projection thermique[END_REF]. To name a few among the most wellknown, we can mention the contributions of R. Hill [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF], Z. Hashin et al. [START_REF] Hashin | Note on a variational approach to the theory of composite elastic materials[END_REF][START_REF] Hashin | The Elastic Moduli of Heterogeneous Materials[END_REF], A. Degiovanni Azarmi et al. [START_REF] Azarmi | Young's modulus measurement and study of the relationship between mechanical properties and microstructure of air plasma sprayed alloy 625[END_REF]. This list is non-exhaustive and the purpose is only to show the wide variety of possibilities and assumptions which have been already taken in this domain. All the references mentioned above are examined with more details in the next section.

III.2.1. Two phases model

Voigt and Reuss have established simple in-series and in-parallel models yielding lower and upper bounds of the thermal conductivity of a two-phase composite structure [START_REF] Voigt | Theoretische Studien über die Elasticitätsverhältnisse der Krystalle[END_REF][START_REF] Reuss | Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle[END_REF]. This composite structure consists in alternative layers of two phases as shown on Figure 1-17. R. Hill adapted the Voigt and Reuss equations to a more general case including n phases [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF], see equation (I.2).

However, this model gives very large lower and upper limits around the thermal properties and does not take into account the material morphology (phase size and repartition).

The equivalent thermal conductivity к 𝑒𝑞 is expressed as follows, the left-hand side describing the in-series model and the right-hand side the in-parallel one:

к 𝑒𝑞 = 1 ∑ 𝑓 𝑖 к 𝑖 𝑛 𝑖=1
or:

к 𝑒𝑞 = ∑ 𝑓 𝑖 × к 𝑖 𝑛 𝑖=1 (I.2)
where f i corresponds to the volumetric fraction of each phase and к i is the intrinsic thermal conductivity of each phase.

In-series (lower limit) In-parallel (upper limit) The same in-parallel or in-series equations as those given by (I.2) can also be used for other thermal and mechanical properties as the Young modulus E, the shear modulus G and the bulk modulus K. However, for calculating the Poisson ratio ν and the coefficient of thermal expansion α, equation (I.2) must be adapted [START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomecanique de matériaux réfractaires modèles[END_REF][START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF][START_REF] Hill | A self-consistent mechanics of composite materials[END_REF]. In the case of a two-phase model, the adaptation takes the form: Hashin et al. in 1961 [22, 23] and R. Hill in 1965 [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF] determined equations to obtain bounds of the macroscopic Young modulus of a two-phase solid composite with spherical inclusions encapsulated into an infinite homogeneous matrix. However, these equations only consider the relative volume of the two phases and none of the geometry aspects of the phases was taken into account, except when one phase consists in aligned continuous fibers. In fact, as reported in the PhD dissertation of Tessier-Doyen [START_REF] Tessier-Doyen | Etude expérimentale et numérique du comportement thermomecanique de matériaux réfractaires modèles[END_REF], by calculating the upper and lower limits of the bulk and shear modulus, it is possible to deduce the upper and lower bounds of the Young modulus, of the Poisson ratio, of the thermal conductivity and of the thermal coefficient of expansion.

𝜈 𝑒𝑞 = 𝜈 1 ×𝑓 1 ×𝐸 2 +𝜈 2 ×𝑓 2 ×𝐸 1 𝐸 2 ×𝑓 1 +𝐸 1 ×𝑓 2 𝛼 𝑒𝑞 = 𝛼 1 ×𝑓 1 ×𝐾 1 +𝛼 2 ×𝑓 2 ×𝐾 2 𝑓 1 ×𝐾 1 + 𝑓 2 ×𝐾 2 (I.3) Z.
Maxwell works described by F. Cernuschi et al. in [156] correspond to a two-phase model of spheres dispersed into a continuous matrix. This model is applicable only for a very low level of the second phase (below 10%) and leads to the following expression of the effective thermal conductivity к eff of the coating:

к eff = к m × 2κ m + κ 2 -2𝑓 2 (κ m -κ 2 ) 2κ m + κ 2 + 𝑓 2 (κ m -κ 2 ) (I.4)
where f 2 corresponds to the volume fraction of the dispersed sphere phase and κ m and κ 2 are the thermal conductivity of the continuum media and of the dispersed sphere phase, respectively.

Bruggeman [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF] extended the Maxwell model by assuming that the dispersed sphere embedded into the continuous matrix could have an infinite range of radius values. It is then possible to remove the volumetric fraction limitation from the Maxwell equation (I.4) which can be replaced by [156]:

1 -𝑓 2 = ( к 𝑒𝑓𝑓 к 𝑚 - к 2 к 𝑚 ) ( к 𝑒𝑓𝑓 к 𝑚 ) 1 3 (1- к 2 к 𝑚 ) (I.5)
where f 2 corresponds to the volume fraction of the dispersed sphere phase. It is observed that equation (I.5) does not provide к eff explicitly. An iterative computational approach is thus required to determine the effective thermal conductivity from the non-linear equation (I.5).

M.E. Cunningham et al. [START_REF] Cunningham | Heat conduction in spheres packed in an infinite regular cubical array[END_REF] performed an analytical study of a material in function of its inclusion arrangement. The 3D analytical solution of the temperature was applied unit cell by unit cell in a domain made up of unit cubic elements containing a sphere located at the center of each cube. To take into account the material porosity, a corrective factor was applied. However, this method is only valid for materials with repetitive microstructural inclusion.

J. Wang et al. [START_REF] Wang | A new structural model of effective thermal conductivity for heterogeneous materials with co-continuous phases[END_REF] developed a theory for composites having multiple continuous phases by combining two different conventional models (namely the Voigt-Russ [START_REF] Voigt | Theoretische Studien über die Elasticitätsverhältnisse der Krystalle[END_REF][START_REF] Russ | Practical Stereology[END_REF] and the Maxwell

к eff = к m к 2 к m (1-f 2 1 3 )+f 2 1 3 -f 2 1 3 к 2 к m (1-f 2 1
3 )+f 2

к eff = к s 2 (√ 8к p к s + 1 -1) (I.7)
Where к s and к p are defined by:

к s = 1 ∑ f i к i n i=1 ; к p = ∑ к i f i n i=1
(I.8)

III.2.2.
Porous materials and thermal spray coatings models All analytical models described in the previous section do not take into account the effect of porosity although this aspect is of major interest for abradable coatings [4]. It is therefore necessary to establish specific equations by making assumptions on the thermal conductivity of pores. This conductivity is often considered as very low and the radiative contribution is generally neglected [START_REF] Prat | Transferts de chaleur dans les milieux poreux Conduction, convection, rayonnement, Techniques de l'ingénieur Transferts thermiques[END_REF][START_REF] Daïan | Equilibre et transferts en milieux poreux[END_REF]. This simplifies the models and the main results related to these simplifications are reminded on Table 1-3 where f p represents the volume fraction of pores.

Models

Effective conductivity

Voigt and Russ [START_REF] Voigt | Theoretische Studien über die Elasticitätsverhältnisse der Krystalle[END_REF][START_REF] Reuss | Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle[END_REF] 0 ≤ к eff ≤ (1 -f 𝑝 ) × к m with к 𝑝 assume as negligible

Maxwell [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] к eff = к m (1 -3 2 f 𝑝 )

Bruggeman [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF]

к eff = к m (1 -f 𝑝 ) 3 2
Bruggeman with spheroidal dispersion [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF] к

𝑒𝑓𝑓 = к 𝑚 (1 -𝑓 𝑝 ) 𝑋 ; X = 1-cos φ 2 F + cos φ 2 F
where F is the shape factor Figure 1-18 and φ the angle between the revolution axis of the spheroid and the non perturbed heat flux. However, the models described on Table 1-3 are valid only for isolated pores while it is well known that the thermal and mechanical properties decrease with the number of pores [START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF][START_REF] Sevostianov | Plasma-sprayed ceramic coatings: anisotropic elastic and conductive properties in relation to the microstructure; cross-property correlations[END_REF][START_REF] Antoine | Quantification de l'architecture poreuse de dépôts finement structures (sub-micromètre -nanomètre) de zircone yttriée réalisés par projection plasma de suspension[END_REF]. To overcome this drawback, the Sprigg's equation [START_REF] Spriggs | Expression for Effect of Porosity on Elastic Modulus of Polycrystalline Refractory Materials, Particularly Aluminum Oxide[END_REF] is commonly used for determining the Young modulus of porous material with a uniform distribution of pores:

E eff = E m exp(-b f p ) (I.9)
Where exp stands for the exponential function and b is a constant parameter equal to 5.16.

B. Budiansky has given linear laws [START_REF] Budiansky | On the elastic moduli of some heterogeneous materials[END_REF] to determine the upper limits of the shear and bulk modulus:

G eff + = G bulk (1 - 15(1-ν bulk ) 1-5ν bulk ) f p ; K eff + = K bulk (1 - 3(1-ν bulk ) 2(1-2ν bulk )
) f p (I.10)

Where G bulk and K bulk respectively represent the shear and bulk modulus and  bulk is the Poisson ratio of the dense material. All these material properties are related to the bulk material, that is to say the fully dense matrix material.

From the Budiansky work, J.C. Glandus [START_REF] Glandus | Rupture Fragile et Résistance aux Chocs Thermiques de Céramiques à Usage Mécanique[END_REF] expressed the Young modulus with respect to the Poisson ratio of the porous material :

E eff + = E bulk (1 - 3(1-ν)(1-2ν)+(9+5ν) 2(7-5ν bulk )(1-2ν bulk ) ) f p (I.11) R.
McPherson [START_REF] Mcpherson | A model for the thermal conductivity of plasma-sprayed ceramic coatings[END_REF] proposed one of the first analytical model adapted to plasma sprayed coatings.

The coatings were considered having alternative region of good and poor contact between lamellae.

The microstructure consists in lamellae having a thickness δ and a fraction f bulk and the apparent contact is assumed to be perfect. The effective conductivity corresponding to these microstructural parameters is given by: к eff = к m 2δf bulk πr (I.12)

Where r represents the radius of each individual contact area.

Several other authors [120, 121[46, 156, 160] have improved this model by integrating the oxide layers resistance or by characterizing the deposit lamellae structure.

I. Sevostianov and M. Kachanov [START_REF] Sevostianov | Plasma-sprayed ceramic coatings: anisotropic elastic and conductive properties in relation to the microstructure; cross-property correlations[END_REF]159] have proposed to link the macroscopic thermal conductivity and Young modulus to a coating microstructure composed of two families of ellipses representing inter and intra lamellar cracks. The thermal conductivity and the Young modulus perpendicular to the substrate are then expressed in function of a component representing the crack density tensor in the direction perpendicular to the substrate. This component integrates as well cracks densities of both families as well as their orientation.

Hasselman (as reported by Dutton et al. [START_REF] Dutton | Effect of heat treatment on the thermal conductivity of plasma-sprayed thermal barrier coatings[END_REF]) studied the effect of cracks having various orientation onto the coating thermal conductivity. The cracks shape was considered ellipsoidal and flat with the minor axis length c close to zero:

κ eff = κ bulk (1 + ( 2f p π ) ( b 𝑐 )) -1 (I.15)
Where b represents the length of major axis.

The conclusion was that cracks perpendicular to the thermal flow direction offer the maximum of thermal insulation whereas cracks parallel to the thermal gradient do not have any effect. More generally, the use of analytical models is not fully representative of the coating microstructure even if it is a timeless methodology to obtain quickly the properties of composite materials. Indeed, they do not take into account the real shape and size of the porosity, and the connectivity between the different pores. Most of analytical models found in the literature and presented previously are only adapted to two-phases or simply organized multi-phase media while the complexity of abradable materials comes from the combination of a random multi-phase organization with porosity. This is why an analytical approach of abradable materials is very limited considering their microstructural complexity. Nowadays, the most competitive approach to achieve accurate predictions of abradable coatings properties is in fact the numerical one, even if this approach is time consuming. Anyway, the constant evolution of computer tools technology allows reducing significantly the time necessary for modeling with this kind of approach. The next section presents very recent advances in developing efficient numerical tools specifically adapted to abradable coatings.

III.3. Numerical models

Several previous studies [START_REF] Bakker | Using the finite element method to compute the influence of complex porosity and inclusion structures on the thermal and electrical conductivity[END_REF][START_REF] Grandjean | Numerical calculations of the thermal conductivity of porous ceramics based on micrographs[END_REF][START_REF] Bolot | Application of FEM for the Estimation of Thermo-Mechanical Properties of Plasma Sprayed Composite Coatings[END_REF][START_REF] Seichepine | Mechanical modelling of highly heterogeneous thermally sprayed abradable coatings, International Thermal Spray[END_REF][START_REF] Faraoun | Modelling route for abradable coatings[END_REF]170,171] K. Bakker [START_REF] Bakker | Using the finite element method to compute the influence of complex porosity and inclusion structures on the thermal and electrical conductivity[END_REF] applied the Finite Element Method on a 2D image corresponding to a cross section of a microstructure to take into account the shape, the orientation and the distribution of the dispersed phases onto the coating thermal conductivity. Bakker particularly discussed the misrepresentation of 3D microstructures by 2D microstructure images. Due to the limited computation tool capacity, avoiding using 3D FE models, Bakker established a relationship between the thermal conductivity к 2𝐷 obtained from the 2D FE model and the 3D actual thermal conductivity к 3𝐷 of the microstructure. However, this relationship given below is suitable only for spherical inclusions:

к 𝑝 к 𝑚 = 1-((1-( к 𝑝 к 𝑚 ))𝑓 𝑝 )- к 2𝐷 к 𝑏𝑢𝑙𝑘 1-((1-( к 𝑝 к 𝑚 ))𝑓 𝑝 )- к 3𝐷 к 𝑏𝑢𝑙𝑘 (I.16)
Where к bulk is the thermal conductivity of the fully dense matrix. based on color differences to create the mesh elements over the images, and then assigning to the several features some intrinsic properties to perform virtual experiments [START_REF] Langer | OOF: an image-based finite-element analysis of material microstructures[END_REF]. From these previous steps, it is possible to determine the macroscopic properties but also to measure and visualize the stresses. Two approaches for representing the coating morphology were studied by means of OOF by the authors of the papers [START_REF] Kulkarni | Advanced microstructural characterization of plasma-sprayed zirconia coatings over extended length scales[END_REF][START_REF] Kulkarni | Processing effects on porosityproperty correlations in plasma sprayed yttria-stabilized zirconia coatings[END_REF][START_REF] Kulkarni | Comprehensive microstructural characterization and predictive property modeling of plasma-sprayed zirconia coatings[END_REF][START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF]. The first one consisted in quantifying YSZ coating microstructure by Small-Angle Neutron Scattering (SANS). Then an artificial microstructure was rebuilt by using the previous average information on pore size, shape, orientation and distribution. The second approach consists in using real coating microstructure images obtained by Scanning Electron Microscopy (SEM). SEM approach gives a realistic simulation of pore morphology in a local region for coatings which have been thermal treated after the spraying whereas the SANS based approach provides a consistent global averaging of the pore morphology and distribution over the entire coating. A good accordance was found between predicted values and experimental values for the thermally treated coatings but not for the as-sprayed ones. The predicted values are actually higher than the measured data due to the lack of information on the splat boundaries and pores size distribution. Anyway, even if the splat boundaries affect the thermal conductivity, it is not taken into account via SEM images.

The lamellae contact is better after thermal treatment, which explains why the simulated results are better when they are compared to the measurements coming from a thermally treated coating rather than from an as-sprayed one.  For both codes, decreasing the element size increases the accuracy of results.

 FEM gives higher values than FDM.

 FDM is less memory consuming than FEM for the largest models.

R. Bolot et al. [START_REF] Bolot | A Two-Dimensional Heat Transfer Model for Thermal Barrier Coating Average Thermal Conductivity Computation[END_REF] developed since 2004 an in-house finite difference code to estimate the effective thermal conductivity of porous YPSZ coatings and to determine the influence of the pore architecture. The heat transfer modeling is computed directly onto SEM images of the coating cross section, each pixel of the binary picture being considered as a grid cell. The procedure applied in this work allows decreasing the required number of iterations and the CPU time. More recently, in 2009, R. Bolot et al. [START_REF] Bolot | Predicting the Thermal Conductivity of AlSi/Polyester Abradable Coatings: Effects of the Numerical Method[END_REF] worked on the prediction of the thermal conductivity of AlSi/polyester abradable coatings. The modeling was conducted directly onto cross-section images of the microstructure by use of two different numerical codes (OOF and TS2C), respectively based on the finite element and the finite difference methods. The lattice structure was determined by the pixels color. The results show differences due to computational aspects, such as a nodal calculation approach with OOF and a cell center formation with TS2C. Moreover, as previously mentioned by Bartsch, the finite element code OOF requires much more RAM in comparison with TS2C (similar to TBCTool). Additionally, the calculation time required with TS2C was of a few minutes only for a standard image (1024x768 pixels), which is very reasonable. The authors found some discrepancy between experimental results (measured by the laser flash technique) and numerical predictions, proving that this technique needs further investigations. More precisely, the modeling results were higher than the experimental ones for YSZ coatings while it was the contrary in the case of Molybdenum coatings. However, image analysis appeared to be an effective tool for assessing the thermal conductivity of thermal spray coatings.

S. Grandjean et al. [START_REF] Grandjean | Numerical calculations of the thermal conductivity of porous ceramics based on micrographs[END_REF] compared numerical and analytical results obtained on micrographs corresponding to coatings of tin oxide samples with pore volume fractions from 10% to 50%.

Preliminarily, the accuracy of the procedure has been studied on two simple geometries, i.e. an isolated pore inclusion represented by a sphere into a solid matrix represented by a cube and an isolated cylinder into a cube. Up to 20% of pore volume fraction, the values obtained by the analytical expressions of Maxwell-Eucken and Landauer [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] and by numerical simulations performed with the finite element ABAQUS software are in a good agreement. However, beyond 20%, an increasing deviation was noted between these two methodologies. In fact, numerical methodologies do consider the connectivity of porosity network, contrary to analytical models. But the 2D numerical approach was found to underestimate the thermal conductivity of real three-dimensional structures.

A.D. Jadhav et al. [START_REF] Jadhav | Low-thermalconductivity plasma-sprayed thermal barrier coatings with engineered microstructures[END_REF] compared the OOF code with analytical models to determine the thermal conductivities of APS (Atmospheric Plasma Spraying) and SPPS (Solution Precursor Plasma Spray) coatings. Both of these two calculations were also compared to experimental results measured by Laser flash analysis. It appeared that the use of the OOF code was more accurate than the analytical model.

III.3.2. Thesis orientation

As seen in the previous section, analytical models are easy to use but numerical modeling based on microstructure images is more faithful for a realistic representation of the actual microstructure of the material. Numerical methods are increasingly popular because of their reliability even if, most of the time, lattice simplification concerning the microstructure and the mesh was necessary to overcome the computer limitations in terms of memory size and computation time. Nevertheless, due to new computer generations, computational times are shortened, allowing working with a better resolution of images without simplifying the microstructure.

The approach considered here is to conduct a modeling methodology directly on real coating microstructure images to calculate their mechanical and thermal properties. Meshing was performed pixel by pixel to avoid simplification of the microstructure. Each pixel color corresponds to a material phase, image analysis and stereological protocols allowing to differentiate all the phases and features of the microstructure. The code used is the multipurpose commercial software ANSYS which allows structural and thermal finite element simulations. The details of the procedure methodology will be given in chapter 2.

IV Conclusion -Thesis strategy

Abradable materials are strongly heterogeneous composite structures with several phases including porosity. The process used to manufacture the abradable material is an important factor influencing the microstructure organization by inducing several kinds of features. This chapter highlighted the influence of the material chemical composition and microstructure on the coating extrinsic properties:

it is thus mandatory to manage efficiently the spray process. The coating extrinsic properties allow discriminating the microstructures and therefore constitute an excellent indicator for dimensioning abradable coatings. In the framework of the E-BREAK European project, the extrinsic properties are also needed for feeding the models of the abradability test bench.

Two solutions are available to determine the extrinsic properties of abradable coatings. The first is the application of experimental testing. However, a large number of experiments is generally required in order to characterize the various manufactured microstructures. Additionally, some of the required tests may be long and tedious, or may also be expensive. Moreover, their availability may also be a difficulty. Finally, most of them are far from real conditions, and standard test specimens are sometimes difficult to manufacture. For these reasons, the use of modeling raised a growing interest among scientists and engineers during the last decade. Modeling constitutes a low cost, easy-to-use and complementary solution for testing and estimation of extrinsic properties of abradable coatings.

Modeling may also be used in order to decrease the number of experiments required after validation.

However, it is still an open research field, the main difficulty coming from the realistic representation of the multi-scale interactions with various random phases and porosities.

The aim of this work is to characterize three types of abradable coatings in the framework of the E-BREAK European project (7 th Framework programme). These coatings are applied in different parts of turboengines and their good use requires the knowledge of different thermomechanical properties.

Experimental and numerical tests have to be conducted and a specific modeling method has to be developed. The works realized in the framework of this PhD try to reach these objectives and are presented in the three forthcoming chapters.

Processes used to manufacture the coatings and to prepare the samples (in order to characterize their experimental properties and observe their microstructure) are described in Chapter 2. Moreover the protocols used for image analysis of the sample and the methodology applied to conduct the modeling works are also presented in chapter 2. Since continuity of the displacement fields was assumed at the boundary between phases, interactions at interfaces of phases are not represented in a perfect manner. This point constitutes a trail to further improve the models developed in the frame of this work. Techniques applied to determine the coating properties are also presented.

Chapter 3 describes the works conducted to obtain the information required for feeding the modeling methodology. It concerns the microstructure description of the three coatings, but also the choice of their Representative Elementary Volume and the determination of intrinsic properties of the phases. Our contribution was incorporated within the framework of two working groups of the E-BREAK European project with their own strategies. Among the three coatings studied different microstructures have been implemented in accordance with the subproject strategies, the CoNiCrAlY-BN-polyester coating has the merit to oppose two different microstructures in order to represent the extreme behaviors that could be obtained with this material. On contrary to the CoNiCrAlY-BNpolyester, the prepared NiCrAl-bentonite coatings have relatively similar microstructures, but these slightly different microstructures even induce different properties. This chapter will also briefly outline the materials used. More details about their composition and microstructural features will be given in the chapter 3.

The process elaboration and parameters applied to coat the abradables layers will be discuss here as well as the technics applied to determine the mechanical and thermal properties of coatings.

Then, the coating preparation surface protocols, the image analysis and image description protocols will be describes.

To finish the details about the conventional numerical routine to calculated the coating properties will be explained.

All the results obtained concerning the image analysis or the studies conducted to develop this methodology will be detailed in chapter 3. To finish in chapter 4 all the results concerning the experimental measurement and the numerical calculation will be discussed.

I Manufacturing process

I.1. Presentation of Materials

Three different coating natures were studied. The two first coatings may be used up to 600°C and 750°C respectively (compressor section) and are composed of a metal matrix with a lubricant phase and/or porosity former (polymer phase). The third coating is composed of a ceramic matrix with a polymer phase used as porosity former: this material may be used up to higher temperatures, thus allowing its use in the turbine section. Oerlikon-metco group (Switzerland) manufactures all powders used in this frame. Information about the powder materials (i.e., reference name, composition, size range and manufacturing process) is listed in Table 2-1. The powder manufacturing process is an important variable which influences the coating quality and morphology. It determines the powder composition, the shape and type of the particles, the powder and coating densities, and the specific of the coating, which governs the ability of a material to absorb heat. These variables influence interactions between the thermal spray flame and the powder.

I.2. Presentation of the coating elaboration processes

Thermal spray enables mixing and co-deposition of materials with very different properties such as ceramics and polymers. Details concerning the manufacturing processes of the coatings are listed in Switzerland). The HOSP™ process (manufacturing process of the powder) allows providing hollow spheres and combines the advantages of pre-alloyed, fused and crushed powders together with the free flowing, consistent shape of spray dried powders. This material is used as the thermal insulating interlayer in a three-layer ceramic abradable system, consisting of a thermal sprayed bond coat and an abradable top coat. The last layer is the abradable top coat made of Zirconia-polyester ceramic. The polymer phase is added in the spray powder to impart voids inside the coating.

The coating service properties depend on about 50-60 spraying parameters [START_REF] Fauchais | Quo vadis thermal spraying?[END_REF][START_REF] Vardelle | Spray parameters and particle behavior relationships during plasma spraying[END_REF] among which several are important vectors to control the plasma properties and the coating elaboration process. One can mention for exemple:

 The flow rates and nature of the working gases  Electric power (adjustable thanks to the electric arc current intensity)

 Powder injection (adjustable thanks to carrier gaz and powder flow rates)

 Spraying distance

For confidentiality reasons, parameters used for the coating elaboration are not listed in this manuscript. Two NiCrAl-bentonite coatings with different microstructures were elaborated for this study: one is more porous than the other (pore levels of 40 and 35% respectively). Two CoNiCrAlY-hBN-polyester coatings were also elaborated, but with larger differences; one may be qualified as porous (pore level of about 30%) whereas the other one can be qualified as very porous (≈50%) with a highly connected void network. In practice, the APS process offers much more latency in the thermal spray parameters in comparison with flame spraying, so that larger differences can be obtained on the coating microstructures for M2043 coatings in comparison with M314 (NiCrAl-Bentonite) ones.

YSZ-polyester coatings with different microstructures were manufactured with different plasma torches (e.g., F4, 9MB and SINPLEX torches). However, due to confidentiality aspects, again the operating parameters are not reported here.

The approach used for YSZ-polyester coatings was the following one:

(i) identification in the literature of the most influent APS parameters on the coating morphology [START_REF] Bisson | Influence of hydrogen on the microstructure of plasma-sprayed yttria-stabilized zirconia coatings[END_REF][START_REF] Marple | Yttria-Stabilized Zirconia Thermal Barriers Sprayed Using N[2]-H[2] and Ar-H[2] Plasmas : Influence of Processing and Heat Treatment on Coating Properties[END_REF]177,[START_REF] Vaidya | Process maps for plasma spraying of yttria-stabilized zirconia: An integrated approach to design, optimization and reliability[END_REF],

(ii) determination of a reference spray parameter set for each gun: some of the reference parameters were provided by Oerlikon-Metco for thermal spraying of YSZ-polyester powder with their equipment, some others were provided by Turbomeca (SAFRAN group, Tarnos) (iii) identification in the literature of limit values of the parameters, (iv) determination of a matrix of tests in order to vary the selected parameters one by one, to allow the discrimination of their influence on the coatings microstructure and to compare the manufactured coatings to the reference as listed in Table 2- Five different microstructures of YSZ-polyester coatings will be considered in this manuscript (although many more coatings were manufactured). One is the reference coating (porosity level of about 8%) elaborated with 9MB torch, the second one is a dense coating (porosity level of about 3%, a part of this void is due to coarse pores), the third one is a more porous coating (about 12%), another one corresponds to a coating with a coarse microstructure very similar to the reference one (≈8%) and the last one is a dense coating without coarse pores were the void ratio is about 3%. Other microstructures (manufactured with other guns for example) will be given in the annexes. Before spraying of the three different abradable coatings, the surface of the substrates was degreased (Ethanol) and grit-blasted (α Alumina grits of 250 µm, average size, at a pressure of 3x10 5

Pa) to ensure the best possible coating adhesion. Then, samples were installed on a rotating substrate holder. During spraying, the samples were cooled by air jets located on both sides of the assembly.

The samples were fixed on a stainless steel 304 holder.

It is important that no layering takes place during the deposition of metal-polymer abradables. The layering consists in successive distinct layers with variable ratio of polymer and metal phases. This phenomenon can occur in the plasma due to the very different properties of those two phases (the metal particles are heavier than the polymer ones, so that their penetration in the plasma jet is higher, giving rise to a segregation of the particles). The use of a high spray distance can favor this layering.

II Experimental protocols to determine coating properties

II.1. Thermal properties

Due to the high temperature in the compressor and turbine sections of aero-engines, using thermal insulating materials is important. However, this solution is efficient when engines surface is cooled.

The materials considered for abradable applications are composites consisting of materials with different natures and composition… The complexity of their composition and microstructure induces the necessity to study and understand their behavior and response to high temperature environments.

The thermal conductivity and thermal diffusivity of materials are the main properties useful to quantify heat diffusion. Macroscopic properties of such multi-material structures are difficult to quantify using conventional methods (experimentally), indeed only some of them can offer this possibility such as laser flash.

Diffusivity measurements have been performed for this study. However for confidentiality aspect, the values measured cannot be given and must be normalized. Only a light qualitative comparison between coating of same nature can be made. For this reason the description of the experimental measure and their results will be given in annex.

II.1.1. Laser flash measurements

The most popular method used for measuring the thermal diffusivity is the flash method. It has the advantage of being fast while providing data with good accuracy and reproducibility. Today this method is very useful to estimate the thermal diffusivity, the thermal conductivity and the specific heat even for composite materials. In the ideal case, the material is homogeneous, isotropic and opaque with adiabatic boundaries. The pulse duration is very short, the pulse absorption at the front surface is geometrically uniform and heat losses are negligible. The temperature of the sample is then given by the Parker equation [START_REF] Degiovanni | Conductivité et diffusivité thermique des solides, Techniques de l'ingénieur Mesure des grandeurs thermophysiques[END_REF][START_REF] Hay | Mesure de la diffusivité thermique par la méthode flash[END_REF][START_REF] Thermitus | Laser Flash Method for testing Spherical specimens[END_REF]:

𝑇(𝑥, 𝑡) = 𝑄 ρ𝐶𝑒 (1 + 2 ∑ cos ( 𝑛𝜋 𝑒 𝑥) ∞ 𝑛=1 exp ( -𝑛 2 𝜋 2 𝑎𝑡 𝑒 2 )) = 𝑇 𝑙𝑖𝑚 𝑓( 𝑥𝑎𝑡 𝑒 ′ 𝑒 2 ) (II-2)
𝑄 ̂ is the excitation energy, laser pulse per unit area (J.m-²), ρ is the material density (kg.m -3 ), Cp is the specific heat of the material (J.kg -1 .°C -1 ), e is the thickness of the sample (m), stands for time (s)

and T for temperature (°C). In addition, x corresponds to the direction of the depth of the sample (i.e., along the thickness). For the rear surface flash analysis, the thermogram obtained represents the temperature at x=e (i.e., rear face of the sample).

From the previous equation, II-2, it is possible to determine II-3. Indeed, by considering the hypothesis that boundaries are adiabatic, the temperature of the rear face rises a maximum level and should stay at this value indefinitely. The thermal diffusivity can thus be determined by considering the time 𝑡 1/2 (s) at which the half of the maximum temperature increase is reached:

𝑎 = 0.138786 𝑒² 𝑡 1/2 (II-3)
This model is very simple because only one measure is required (𝑡 1/2 ). However, one problem is the underestimation of several factors. Indeed, the heat losses from the sample are not taken into account, as well as the non-uniformity of the pulse or also the finite size of the width of the laser point.

Nevertheless, most of the time, measurements are non-ideal cases. Several models were thus suggested to enhance the Parker method and in particular to account for heat losses. These methods can be classified in one of the six following categories: modification of the Parker method, use of partial time or temperature ratio, apparent thermal diffusivity, least squares minimization, partial temporal moment and logarithmic transform [START_REF] Thermitus | Laser Flash Method for testing Spherical specimens[END_REF].

The model used for the measurements performed in the frame of this study is that of Clark and

Taylor: this model is in the category of the "use of partial times ratio" and permits to take into account With Cp corresponding to the specific heat (J.kg -1 .°C -1 ), m is the mass of the sample (kg) and T the temperature difference (°C).

The study of the thermogram provides the thermal diffusivity of the considered material and the specific heat can be calculated. It is then possible to calculated de thermal conductivity by the following relationship:

к=𝜌. 𝑎. 𝐶 𝑝 (II-8)

In which к is the thermal conductivity (W.m -1 .K -1 ), ρ is the density of the sample (kg.m -3 ) and a is the thermal diffusivity (m².s -1 ). Due to the complexity of erosion or abradability tests, some other evaluation tests may provide trends concerning the performances of abradables. For example, a lot of studies have shown the relationship existing between HR15Y Hardness and abradability [START_REF] Chappel | Gas Path Blade Tip Seals: Abradable Seal Material Testing at Utility Gas and Steam Turbine Operating Conditions[END_REF][START_REF] Peyraut | Numerical Simulation of the HR15Y Hardness Test of Abradable Thermally Sprayed Coatings[END_REF][START_REF] Peyraut | Finite element modeling of abradable materials -Identification of plastic parameters and issues on minimum hardness against coating's thickness[END_REF]: the highest is the abradability of a coating, the lowest is the hardness. However, as it will be discuss later in this manuscript this relationship is not always obvious.

Hardness testing provides simplest and cheapest normalized measures. However, a special hardness measurement method is required for abradables because of their high void level: this method is the Rockwell superficial hardness. It consists in applying a spherical indent with constant load on the coating surface and to measure the penetration depth. The load applied on the material surface is lower than for conventional Rockwell hardness measurements. In principle, the load applied is comprised

e = hp-hi (II-9)

The Hardness value depends upon this last penetration depth of the indenter. The hardness scale range is 0 to 100, 100 is the highest hardness.

Fifteen measurements were conducted for each coating, following ASTM E18 standard [181].

Indents were randomly located at the surface of the samples. Finally, the hardness values provided in this report are the average of those fifteen measurements.

This test is usually used as quality control test of abradable coatings and to compare coatings with each other. This mechanical property is experimentally measured in quasi-static conditions. No information is provided concerning the dynamic behavior. However, some studies demonstrated that different coatings could have the same HR15Y Hardness without showing the same abradable behavior [START_REF] Yi | Friction and wear behaviour and abradability of abradable seal coating[END_REF][START_REF] Yi | Erosion wear behaviour and model of abradable seal coating[END_REF]. Hence, HR15Y cannot be the only criteria to determine the quality or abradability of coatings. This observation has been confirmed in the frame of this study. Indeed several ceramic abradable coatings with different microstructures have been studied (see annexes section) but HR15Y

did not allow discriminating these coatings. Peyraut et al highlighted the influence of the coating thickness on HR15Y hardness: it was shown that for coatings with a thickness lower than 3 mm, the hardness decreases with the increase of the coating thickness. However for coatings with a thickness larger than 3 mm, the hardness tend to increase with the thickness and then reaches a constant value [START_REF] Peyraut | Finite element modeling of abradable materials -Identification of plastic parameters and issues on minimum hardness against coating's thickness[END_REF]. The influence of the substrate on the measurement can also be an issue.

II.2.2. Erosion tests

Erosion wear is caused by the impact of solid or liquid particles on the coating surface and several wear mechanisms are involved, which are controlled by the particle material, the angle of impingement, the impact velocity, and the particle size. Those tests were performed by Oerlikon-Metco in Switzerland.

The General Electric erosion bench test is standardized using GE specifications (specifications E50TF121-S2). Erosion wear results from the impact of fused and crushed alumina erodent particles on the surface of the sample at room temperature. The angle of impingement is 20°, the pressure of the feeding air is 1.75-2.25 bar, the particles mass flowrate is 300g.min -1 , and the total mass of erodent particles is about 600g (i.e., maximum duration of 2 minutes). The erosion wear is evaluated thanks to the measure of the coating thickness variation before and after the test for a specific time and compared to a known reference (Lexan). The erosion resistances provided in this report correspond to the average of those three measurements. This test has been performed on YSZ-polyester abradable coatings only (more sensitive to the erosion problematic) due to the EBREAK project expectations.

II.2.3. Nano indentation

This technique consists in determining the local mechanical properties of materials. It can be applied as well to massive materials and coatings or thin films. The aim is to measure the penetration depth of a Berkovich indent to obtain the elastoplastic properties of the material. The indenter is initially in contact with the sample surface. A normal load is applied and progressively increased with a given speed (load cycle) until a maximum normal force is reached. The load is then decreased, with the same loading speed, until partial or total relaxation of the material. To avoid any contribution of the substrate, the maximum penetration depth of the indenter in the sample must not exceed 10% of the sample thickness. This procedure is repeated several times at different locations on the sample.

During the entire test, the indenter position is controlled by a differential capacitive sensor. This type of device is very sensitive and accurate. The load applied is very low (a few tens of milli-Newton) as well as the penetration depth (a few micrometers) to avoid any influence of the substrate during the measurement.

For each load cycle, the force applied is drawn versus the indenter position, see Figure 2 in which F is the load, F m is the maximum load, h is the penetration depth measured during the test, h m is the maximum depth, h p is the final residual depth after complete unload, and m is a constant depending on the indenter geometry.

From those data is it then possible to determine the Stiffness S which graphically corresponds to the slope of the tangent to the unload curve and which is established by differentiating equation II-11.

S=(

𝑑𝐹 𝑑ℎ ) 𝑚𝑎𝑥 = 𝑚 × 𝐹 𝑚 × (ℎ 𝑚 -ℎ 𝑝 ) -1

II-12

The intersection between this tangent and the abscissa axis allows determining h r to calculate the contact depth h C . Finally the calculated contact depth h C and the calculated projected contact area A p (hc) can be determined by the following equations:

ℎ 𝑟 = ℎ 𝑚 - 𝐹 𝑚 𝑆 (II-13) ℎ 𝑐 = ℎ 𝑚 -𝜀(ℎ 𝑚 -ℎ 𝑟 ) (II-14) 𝐴 𝑝(ℎ 𝑐 ) = 𝐶 0 × ℎ 𝑐 2 (II-15)
in which C 0 depends on the indenter geometry (C0=24.5 for a Berkovich indenter).

Knowing the stiffness and the projected contact area, it is then possible to calculate the reduced elastic modulus (Eq. 2-16) and then the instrumented elastic modulus (Eq. 2-18).

𝐸 𝑟 = 𝑆√𝜋 2𝛽 √ 𝐴 𝑝(ℎ 𝑐 ) (II-16
)

With E r is the reduced elastic modulus, and β is the shape factor of the indenter extremity (circular =1, triangular =1.034, square = 1.012).

Finally the instrumented Hardness (H IT ) and the instrumented Elastic Modulus (E IT ) are determined by following equations:

𝐻 𝐼𝑇 = 𝐹 𝑚 𝐴 𝑝(ℎ 𝑐 )
(II-17)

𝐸 𝐼𝑇 = (1-𝜈 2 ) 1 𝐸 𝑟 - (1-𝜈 𝐼 2 )
𝐸 𝐼

(II-18)

In which E I is the elastic modulus of the indenter, ν is the Poisson's ratio of the material and ν I is the Poisson's ratio of the identer. Those measurements were conducted by CSM Instruments, Peseux, Switzerland. Figure 2-13 shows a photo and schematic view of the device. The measurements were performed on polished cross sections of the coatings to observe and target all the coating phases individually. Fifteen indents were conducted for each phase.

Some of the test parameters must be defined depending on the size of the phase entities. Indeed it is important to determine the maximal size of the indenter to use in order to minimize the lateral influence on the measurement performed on the considered phase entities. For the phases with a large size such as CoNiCrAlY and NiCrAl, the criteria applied for the selection of the indent size represent maximally 1/7 of the phase entities sizes. Some uncertainties persist mainly due to the unknown size of the phase entities in the direction of the sample depth. In addition, the different phases may influence each other measurement.

II.2.4. Resonance frequency damping analysis

This test allows determining the elastic properties of the materials by applying continuous resonance frequency excitations. This test is a non-destructive dynamic measure. The excitations are applied with an impact tool and equations adapted for rectangular or cylindrical samples permit the calculation of E, G, and ν. Details required to conduct the measurements are the shape, density and stiffness of the sample, and the natural vibration frequencies of a sample.

The vibrations are produced by gently hitting the sample with a small projectile in the sample center. The resulting vibration is picked up with a captor to be analyzed. The impulse excitation can be performed using a small hammer or an automated tapping device. Several captors, e.g. piezoelectric sensor, microphone, laser-vibrometer or accelerometer can be used to detect the vibrations in the sample. A microphone or a laser-vibrometer can be used in order to avoid any contact with the sample, hence improving the results. Laser-vibrometers also allow measuring signals with extreme frequencies.

The captor records the vibration signal during all the solicitation. The signal is then send to the RFDA software and the initial resonant frequency is calculated by using a Fast Fourier Transform (FFT). This information is then used to rebuild the vibration signal. The software modifies this rebuilt signal by using an iterative process in order to match with the measured vibration signal. The resonance frequency of the sample corresponds to the one of the rebuilt signal displayed by the software.

To determine the Young modulus, the resonance frequency must be determined by the longitudinal mode of vibrations as represented in Figure 2 (II-20)

with: f t : torsion frequency, A and B : correction coefficients.

E G

a brittle behavior. Typically, there is no distinct elastic limit. The material fails shortly after reaching the ultimate tensile strength or shows a large drop of stress till rupture. Unfortunately, the stress-straincurves cannot be presented in the manuscript due to the confidentiality reason.

The testing at higher temperatures showed early rupture of the test samples during clamping or during heating up of the test facility. Tests without stress ("free hanging samples") showed that the samples get deformed under high temperature. It was concluded that there are slight residual stresses in the samples due to spraying. The samples were heated and treated before testing at higher temperature for stress relief. A stress relief temperature was thus performed at the testing temperature for ½ hour in Argon atmosphere Thermal sprayed coatings are formed by the flattening of particles and by the solidification of the subsequent lamellae (named splats). This type of architecture is thus formed of a network of intermingled splats with a significant amount of pores. In general, the pore network itself is composed of globular pores (stacking defects occurring during the thermal spray process), as well as intra-and inter-lamellar pores (thin cracks due to, respectively, incomplete contacts between flattened splats and stress relaxation). These microstructural features appear on micrographs representing a coating crosssection:

The simple use of the void rate to describe the microstructure of a coating does not permit to give any information on the pore network. The network formed by the different coating phases can be described more precisely by different ways, such as their content, size, shape, distribution, orientation, and connectivity, Figure 2-16. This network has an influence on the coating effective (macroscopic)

properties. Indeed two different coatings with the same overall void rate but different pore networks, can have different behaviors during mechanical or abradable tests. Very often, each feature can be discriminated by different grey levels (thresholding during image analysis). The different phases embedded in coatings can be qualitatively and quantitatively described.

For this, image analysis and stereological protocols must be implemented following the methodology explained below.

III.1. Preparation protocols of the samples

Changes in the coating microstructure have been identified and related to the variation in the thermal spray operating parameters. Several steps of preparation are required to obtain samples suitable for the discrimination of microstructural features, i.e. sample cutting, impregnation and polishing. However, several damages can occur during this preparation and induce modifications in the coating microstructure. For this reason, it is mandatory to determine the most appropriate protocol for the preparation of the samples in order to be as representative (and reproducible) as possible and to limit artifacts. Several studies reported general rules and advices concerning the preparation of plasma sprayed coating samples suitable for the capture of micrographs [START_REF] Sauer | Accepted Practices of Thermal Spray Technology: Mettallographic Preparation and Testing of Thermal Spray Coatings: Sectionning[END_REF][START_REF] Puerta | Accepted Practices of Thermal Spray Technology: The Preparation and Evaluation of Thermal Spray Coatings: Fine Grinding & Polishing[END_REF][START_REF] Puerta | Accepted Practices of Thermal Spray Technology: The Preparation and Evaluation of Thermal Spray Coatings: Mounting[END_REF][START_REF] Puerta | Accepted Practices of Thermal Spray Technology: The Preparation and Evaluation of Thermal Spray Coatings: Grinding[END_REF]. Table 2-6 displays the different preparation steps, the artifacts which may appear during each step and the methodology used for the surface preparation.

The first step of the sample preparation consists in cutting a piece of the coating. Inappropriate sectioning could induce delamination, localized cracking in the coating structure and sometimes separation of the coating from the substrate. The second step is the mounting of the cut samples.

Several methods of mounting exist but not all of them are appropriate to abradable coatings which exhibit usually a highly porous structure. Inappropriate mounting may induce pull-out defects during the polishing step, Figure 2-16. A polishing recipe must be established in accordance with the coating nature in order to reveal the coating microstructure with a minimum amount of artifacts (scratches, pull-outs…). For those reasons, it is mandatory to determine the most appropriate protocol for the sample preparation for each abradable family. For each sample of the same abradable family the same preparation method will be applied in order to be as representative (and reproducible) as possible and to limit artifacts due to the operator. Most of the time, the first step of polishing consists in quickly remove a certain depth of material to reach a zone free of damages due to sectioning. This step is named the grinding step. Because of the use of a slow transverse feed speed during cutting, this grinding step is not mandatory and the number of polishing steps (and time) could be reduced. Indeed it was observed that increasing the polishing time and the number of steps induces a higher depths of coating removed for those abradable structures. However it was observed that the resin impregnation was less efficient at deepest coating thicknesses. Due to the less efficient sample mounting, damages occurred. For those reasons it was chosen to work with a low cutting velocity and to limit the number of polishing steps. A semiautomatic polishing is performed to reduce damages induced by the operator. The polishing recipe used is given in Table 2-7: sample preparation protocol for thermal sprayed NiCrAl-bentonite, CoNiCrAlY-BN-polyester and YSZ-polyester samples. To finish, the samples were cleaned with distilled water and alcohol, and then dried in an oven at 80°C during 10 minutes.

III.2. Observation of the coating microstructures

Coatings were observed by Optical Microscopy (OM) and Scanning Electronic Microscopy (SEM).

Each method has its own advantages and disadvantages. Indeed OM permits to have an overview of the coating and to observe the distribution of coarse voids in the entire coating. However it is difficult to discriminate the small dimension microstructure details. The use of SEM permits to observe all microstructure features of the coating. However the biggest void distribution may be misrepresented at high magnification level. The use of the two acquisition techniques is thus a good way to have a good description of the coating microstructure.

Optical microscopy allows observing and interpreting the macro and microstructure of a material.

After polishing of the surface, the different microstructural features can be observed at low magnification level, e.g. continuity defects, granular structure…..Optical microscopy permits to observe the phase details of the microstructure, i.e. dimensions, shape, surface fractions, and so on…

The contrast changes are due to several reflecting power of the different phases. Amorphous or semi crystalline polymer phases are observed in black. The limitation of optical microscopy is the low reflecting power and the low field depth.

Acquisition of images must always be performed in the same conditions to be representative and to allow comparing different coatings between each other.

The Optical Microscopy (OM) device used in the present work is an EPIPHOT-TME from Nikon, Japan with a SONY camera. Some of the abradable coatings studied in this manuscript could not be correctly described using observations by OM. For example, the bentonite phase was misrepresented.

Scanning electronic microscopy allows observing more precisely the surface topography. An electron beam scans the sample surface and interacts with the sample matter. Interaction between the electron beam and the sample material gives birth to electron or X-ray emissions.

In fact, depending on the SEM mode, three major types of emissions may be detected to provide the image of the sample:

 Secondary electrons (SE) are due to the loss of energy of primary electrons of the beam during the choc on the atoms of the material surface. These secondary electrons have a low energy and come from the superficial layers near the sample surface. That is why they are sensitive to the surface relief and provide information on the sample topography, i.e.

voids, cracks…

 Back scattered electron (BSE) are due to quasi elastic interaction between the primary electron beam and atoms in the core of the sample. The direction of primary electrons is then modified with a very low loss of energy. Back scattered electrons come from deeper layers of the sample and are sensitive to the atomic number of the material. They give information on the phase composition of the coating, due to different contrasts during image acquisition.

 X-rays are due to the inelastic interaction of electrons with the sample. They give information about the chemical composition of the sample. There are usually considered during Energy Dispersive Spectrometry (EDS) measurements.

The device used in the frame of this work is a SEM with field emission gun (FEG-SEM), FESEM 5800LV, JEOL Corporation, Japan. The FEG [START_REF] Erdman | Multispectral imaging in an FEG-SEM[END_REF] allow producing a very narrow beam of precisely controlled high-energy electrons. This type of device provides a higher spatial resolution (a few nanometers), a better reliability, a higher signal-to-noise ratio, and a longer life than previous technologies. The maximum spatial resolution of images is about 100 nm. An X-Ray detector allows using EDS technology to characterize and analyze the surface composition (qualitatively and quantitatively).

The microstructure of the coatings was observed at high resolution, but also at low magnification level for coatings misrepresented by OM. The surface and cross-section of powders were also observed. The cross-section microstructures of the coatings have been observed mostly in the BSE (back-scattered electrons) mode. Parameters of the electronic column have to be the same in terms of voltage (V=15 kV) and working distance (WD=10 mm) in order to collect data in same conditions [126].

A gold metallizing of the sample surface was applied (with the device SCD005/CEA035, Baltec) to improve the electric conductivity of the samples for observations by SEM.

III.3. Image analysis

The description of the coating microstructure can be performed quantitatively and qualitatively by the use of image analysis software implementing stereological tools. In order to be representative and to compare all microstructures, the samples must be prepared and analyzed in the same conditions.

Image analysis protocols were implemented using the Image J Software (http://rsbweb.nih.gov). The process consists of image acquisition by FESEM (BSE mode) and OM. Concerning the image analysis software, some automatic macro can be implemented and adapted to the coating nature.

One of the first steps of the method consists in the discrimination of all phases (materials) of the coating. Images are constituted of pixels with different grey levels. As explained in the section about the observation of the coating microstructure, different ranges of grey level correspond to the different phases of the coating. For example, in This image can then be segmented to create a ternary image (i.e., containing three different grey levels only), this step is called thresholding. The grey level of all pixels corresponding to the voids (initial range 0-S1) it set to a single value (here 0). The grey level of all pixels corresponding to the matrix (range S2-255) is set to 255 and so on (third color for pixels in the intermediate range S1-S2).

The same methodology can be applied to CoNiCrAlY-BN-polyester and YSZ-polyester coatings. After segmentation an "open" command can be applied to the image to remove residual isolated black pixels and reduce noise. This "open" command consists in applying an "erosion" command followed by a "dilate" command. The "erode" command removes a layer of pixels starting from the

(Voids) (Bentonite Phase) (Metal matrix)
contour of the feature whereas the "dilate" command adds a layer of pixels starting from the contour of the feature. The thinnest "black" details (i.e., pores) of the image are thus deleted with this method.

III.4. Representative Elementary Volume (REV)

Extracting quantitative data from a coating microstructure requires, in order to avoid biases, the capture of images representative of the whole structure, that is to say to consider a Representative Elementary Volume (REV). The REV (corresponding to the coating area) allow observing all microstructural details of the coating. When selecting two areas located in different places in the coating, extracted data (such as the phase contents for example) must remain very similar if the REV condition is respected. Usually, in periodic media, the REV corresponds to the selection of a volume element which can be repeated several times to rebuild the whole structure [START_REF] Bornert | Chapitre 1: changement d'échelle : motivation et méthodologie, Mécanique et Ingénierie des Matériaux -Homogénéisation en mécanique des matériaux 1 -Matériaux aléatoires élastiques et milieux périodiques[END_REF]. In random media, such as plasma sprayed coatings, two different areas selected in the coating could never be superimposed even if the corresponding macroscopic behavior is the same. To determine the appropriate magnification level, the evolution of calculated effective properties must be analyzed considering several REV of growing sizes. Some studies have shown the limitation of this methodology. To satisfy all the conditions to consider a REV can lead to a too large REV [START_REF] Forest | Milieux continus généralisés et matériaux hétéogènes[END_REF], that is why a statistically approach is mandatory: the size of the smallest element is considered but the study is conducted on several images. This assumes that the average behavior determined from those images is equal to that of the overall heterogeneous media [START_REF] Kanit | Determination of the size of the representative volume element for random composites : statiscal and numerical approach[END_REF]. For those reasons, a statistical analyze is mandatory to determine the REV. Several criteria can be used to quantify the void size, e.g. equivalent diameter, caliper diameter, area, perimeter, etc. Due to the presence of strongly different void sizes and shapes in such microstructures, it is hence mandatory to identify a common descriptor to compare one coating to another. The void equivalent diameter, D eq(S) , appeared, after several attempts, as the most relevant criterion. It permits to consider voids whatever their forms and is defined as follows:

Deq(S) = 2√ 𝑆 𝑒𝑞 𝜋 (II-24)
where, S eq , represents the surface area of voids.

The determination of the distribution of the surface equivalent diameter is based on different steps: D eq ≤1.5 µm 1.5 < D eq ≤10 µm 10 < D eq ≤ 𝟐0 µm 20 < D eq ≤ 𝟒0 µm 40 µm <D eq

IV Modeling methodology

Modeling is an alternative method to estimate the materials properties, especially for porous and heterogeneous materials, as explained in chapter 1. This method has been largely developed those last years and permits to take into account the influence of the microstructure features on the material properties. Calculations can be performed directly on micrographs captured by Scanning Electron Microscopy or Optical Microscopy. For this reason, images have to be as reliable as possible to represent correctly the coating microstructure.

The prediction of the thermal conductivity and of the Young's modulus was conducted by finite element analysis (with ANSYS Software) directly on domains resulting from the discretization of captured SEM micrographs. The ANSYS version was release 14.5. An interface called

TGA_TO_ANSYS and developed at IRTES-LERMPS laboratory (University of Technology of

Belfort-Montbéliard, France) was used to provide directly an ANSYS Parametric Design Language (APDL) script from a picture in TGA format (Truevision Targa). The mesh corresponds to the pixels of the considered micrograph (i.e., use of a square element for each pixel). The computer was equipped with 8 CPU (Intel® Core™ i7 CPU 960 3.2 GHz) and 48 Go RAM.

IV.1. Modeling Steps

The modeling routines can be divided in three different steps represented in Figure 2 The first step consists in generating an image of the cross-section of the coating corresponding to the whole micrograph, and to generate directly a meshing corresponding to the pixels of the micrograph. Therefore, square elements are considered (one for each pixel). One or more thresholds must be considered to obtain a discrimination of the different materials of the coating. Different materials are defined within the ANSYS script (the number of materials corresponds to the number of phases in the coating). Each pixel becomes an element for which a material is assigned depending on its grey level (i.e., phase). The input data needed to conduct the modeling routines are listed below:

 Methodology to discriminate the different phases: applied by image analysis thresholding.

 Implementation of the properties of the different materials. Researches in the literature have been considered to determine these properties.

 Boundary conditions, representing different virtual loadings, are then applied on the discretized domain; e.g., a vertical displacement is imposed to the nodes pertaining to the top line to obtain the effective Young's modulus in the vertical direction, and a temperature difference between the top and bottom lines is imposed to obtain the through thickness effective thermal conductivity. The output data of the calculation permits to provide the properties by considering conventional behavior laws.

IV.2. Input Data

The model must be feed by several input data. One of the first steps of the method is to discriminate all phases of the coating. The images are constituted of pixels with different grey levels.

The methodology to discriminate the microstructures is the same as that described in the section about "Image analysis". Then the model requires the mechanical and thermal intrinsic properties of the different phases of the coating, i.e. 8YSZ (dense or powder type structure), CoNiCrAlY, oxides (in the CoNICrAlY-BN-Polyester coating), NiCrAl, bentonite, voids filled with air.  Intrinsic properties of materials corresponding to the properties of a single phase of the different materials with a given composition. Usually two methodologies are implemented to measure these properties experimentally:

o Data collected from measures considering "as dense as possible" and "as pure as possible" material samples at the macroscale (i.e., a few cubic millimeters, at least),

o Data collected at the nanoscale using nanoprobes (i.e., Young modulus calculated from nano-indentation tests using the Oliver-Pharr model [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation techniques[END_REF])

 Effective properties of coatings corresponding to "homogenized" properties measured at the macro or micro scales.

Those data will hence strongly depend upon 1) the scale at which the considered property has been measured, 2) the manner the samples have been manufactured, 3) the test that has been selected. Also, the thermal history of the material during its elaboration is a parameter influencing the measured properties. For example, the intrinsic properties of polyester measured using nano-indentation can be different for the initial state of the material (that is to say the feedstock powder material) or in the coating (that is to say when embedded in the coating and after processing by the high-temperature flow during the spray process).

The Young's modulus, thermal conductivity and Poisson's ratio were determined according to the literature data and sometimes by nano-indentation and are listed in Table 2-10 and Table 2-12 ( [START_REF] Antou | Amélioration de revêtements barrières thermiques par un procédé de refusion laser in situ utilisant un laser à diodes[END_REF][193][START_REF] Bassani | Hydrostatic Lubrication[END_REF]). In the ideal case, the input data set to the different elements correspond to the bulk properties of each phases.

For YSZ, CoNiCrAlY, and NiCrAl phases, these data were not difficult to found. For example for YSZ, the thermal conductivity data found in literature vary between 2.5 and 3 W.m -1 .K -1 . The values of the Young's modulus vary in between 200 and 220 GPa. For this example, the dispersion is thus quite low for both properties. It is thus possible to use an average value of properties found in the literature to implement it as input data in the model.

On contrary, important details were missing (i.e., real composition and elaboration) to determine the bulk properties of bentonite and BN phases. The dispersion of data found in literature is listed in Table 2-10. To determine the properties of voids, some hypothesis were conducted as follows:

 voids are fulfilling with dry air  the pressure in the pores is considered as equal to the atmospheric pressure  for the modeling the voids are associated to a compressible material (ν=0.33) [START_REF] Coussot | Rhéophysique la matière dans tous ses Etats[END_REF]. However the sensitivity of the value of ν on the calculation is almost null.

 Bulk modulus E= 1.10 5 Pa [START_REF] Bassani | Hydrostatic Lubrication[END_REF]. [START_REF] Johnson | Effective modulus and coefficient of thermal expansion of Ni-YSZ porous cermets[END_REF][START_REF] Antou | Amélioration de revêtements barrières thermiques par un procédé de refusion laser in situ utilisant un laser à diodes[END_REF][START_REF] Ingel | Lattice Parameters and Density for Y2O3-Stabilized ZrO2[END_REF] 1.10 -4 ** -ν (-) 0.3 [START_REF] Antou | Amélioration de revêtements barrières thermiques par un procédé de refusion laser in situ utilisant un laser à diodes[END_REF] 0.33 ** 0,3 [START_REF] Saeidi | Mechanical Properties and Microstructure of VPS and HVOF CoNiCrAlY Coatings[END_REF] 0,3 [START_REF] Saeidi | Mechanical Properties and Microstructure of VPS and HVOF CoNiCrAlY Coatings[END_REF] к (W.m -1 .K -1 ) 2.5 -3 [START_REF] Dutton | Effect of heat treatment on the thermal conductivity of plasma-sprayed thermal barrier coatings[END_REF][START_REF] Antou | Amélioration de revêtements barrières thermiques par un procédé de refusion laser in situ utilisant un laser à diodes[END_REF][START_REF] Poulain | Etude de la conductivité thermique de revêtements céramiques à base de zirconerelation avec la composition, la microstructure et la morphologie[END_REF] 0.025 [193] 25 [199] 15 [200,[START_REF] Verdy | High heat flux thermal cycling of multi layered deposit using a HVOF Gun[END_REF] ** Hypothesis.

The bulk mechanical properties of CoNiCrAlY and NiCrAl phases were not found in the literature.

However, their extrinsic properties in coatings elaborated by thermal spraying can be found in the literature [START_REF] Johnston | Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings[END_REF][START_REF] Saeidi | Mechanical Properties and Microstructure of VPS and HVOF CoNiCrAlY Coatings[END_REF]202,[START_REF] Waki | Development of Lateral Compression Method of Circular Tube Thin Coating for Mechanical Properties of Plasma Sprayed CoNiCrAlY[END_REF].

For CoNiCrAlY-BN-polyester coatings, some previous studies have shown that an intermediate phase corresponding to a mix of BN and aluminum oxide is formed during the plasma spray process [START_REF] Giannozzi | On the potential of metal and ceramic based abradables in turbine seal applications[END_REF]. A lack of information exists concerning the composition and properties of this new phase.

Concerning Al 2 O 3 , depending on the fraction of impurities and depending of the presence of gamma or alpha phase, the corresponding Young's modulus may vary between 200 and 400 GPa, whereas the thermal conductivity can vary between 10 and 35 W.m -1 .K -1 . Concerning bentonite, the corresponding properties are different at the grain scale or at the scale of a compressed powder. The bentonite composition is also important to determine the properties.

Bentonite is mostly constituted by montmorillonite. Two main types of bentonite exist: i.e., rich sodium bentonite and rich calcium bentonite. However, there is no precise information about the composition of bentonite contained in the powder.

There is also a lack of information concerning the elaboration process and phases of materials contained in the thermal spray powders, i.e. with the hBN powder. Indeed, depending on the process elaboration of hBN powder for example, the corresponding bulk properties may vary in between 10 and 75 GPa for the Young's modulus and in the range 15-80 W.m -1 .K -1 for thermal conductivity. Most of the time, hBN powders are densified by Hot Pressing (HP) and Hot Isostatic Pressing (HIP)

techniques, which induce respectively anisotropic or isotropic behaviors of the bulk material [START_REF] Lipp | Hexagonal boron nitride: Fabrication, properties and applications[END_REF].

From this, the properties of hBN found in the literature could depend on the considered load direction.

Those properties also depend of the B 2 O 2 oxide content.

Data provided for Aluminum oxide (Al 2 O 3 ) in Table 2-11 correspond to purity in the range 60-99.8%. For a high purity alpha-alumina, the Young's modulus E is in the range 380-400 GPa, the Poisson's ratio ν is in the range 0.22-0.25 and the thermal conductivity  in the range 30-40 W.m -1 .K -1

[208].

Concerning Al 2 O 3 and hBN phases, the thermal conductivities and Young's modulus found in the literature have an important dispersion, which cannot be neglected. Another point is the huge difference between the properties of those two phases. An estimation of the composition of the phase contained in the coatings is thus mandatory.

The solution applied to estimate the Young's modulus of thoses phases existing in the coatings was to conduct nano-indentation measurements, However, estimation of the thermal conductivity of Al 2 O 3 and bentonite is more difficult, since no experimental solution was found. In these conditions, it was decided to use the average thermal conductivity of data found in the literature. Nevertheless, it is just an assumption, so that it is thus important to determine the influence of the considered thermal conductivity of the phases on the calculated thermal conductivity of the coatings. For all these reasons it is important to obtain the maximum of information about the nature of the powder, its composition, the corresponding elaboration process, and also the same information for each of the different constituents. Those details allow considering the most relevant input properties as possible. However, those data are not always indicated by the powder manufacturer, especially in the case of composite powders such as the present ones.

Due to the assumptions made concerning the properties of some phases, it is important to study the influence of the input data on the calculated coating properties, and to determine if this influence is significant. This study is detailed in chapter 3, "results" section. The determination of the representative volume is also an important step to conduct before the modeling. The methodology for this is explained in section III.4 Representative Elementary Volume (REV).

IV.3. Routines of the model

This paragraph describes the step 2, see figure 2-29, of the modelling method depicted in figure 2-28.

Figure 2-29: Conventional modeling methodology

The "conventional" modeling consists in the calculation of the material properties by FEA. This implies the use of images presenting an area representative of the elementary volume of the structure e.g. Figure 2-30, and to apply directly a meshing pixel by pixel (each pixel becomes an element). After discrimination of the different phases in the coating, allowing affectation of a material number to each element, and after setting the properties of each material, the modeling may be conducted. Boundary conditions are imposed to represent virtual loadings applied on the discretized domain.

Several temperature, displacement, strain or stress loading conditions can be applied. However, stress and strain loading is never applied on abradable microstructure. Indeed applying a stress is problematic if there is voids on the image limits, an important stress can induce a huge deformation and misrepresent the coating properties.

To conduct the modeling, some assumptions are made:  Voids are filled with dry stationary air at atmospheric pressure and supposed as compressible. This assumption permits to neglect convective effects so that the only heat transfer process in pores is thermal conduction.


 The load applied is continuous at the interface of the phases and voids  Knudsen effects were not taken into account. Some studies have shown that in the case of plasma sprayed coatings, the pore size may have an influence on the convective and radiation heat transfers [221].

To determine the Young's modulus of the coating a tensile test is simulated. An imposed mechanical strain is considered (imposed displacements to nodes pertaining to lines). The corresponding displacements must be sufficiently small to remain in the elastic domain. The boundary conditions applied to compute the effective Young's modulus for the through-thickness direction are the following ones:

 an imposed through-thickness mechanical strain ε=10 -4 is applied on the upper edge of the image (the displacement of nodes of this edge are set to 10 -4 x the image thickness) The longitudinal effective thermal conductivity can also be calculated with the same methodology but by applying inverse boundary conditions. Several other virtual loadings can be applied to determine other properties such as the Poisson's ratio: some of them are listed in annexes [Aussavy itsc 2015 [START_REF] Aussavy | YSZ-Polyester Abradable Coatings Manufactured by APS[END_REF]]. This methodology can also be applied to determine the properties of the coatings at high temperature. However, input data (high temperature properties of all coating phases) are required, which is not always the case.


The use of micrographs allows representing precisely the microstructure of a coating and to take into account the influence of microstructural objects in calculating the coating properties. This method directly uses real images of the coatings, and has a good reliability therefore. However the reliability of the representation of the coating microstructure is critical for a representative modeling analysis, and to get results in accordance with experimental measurements. That is why the resolution of micrographs must be high enough to reveal a maximum of microstructural details. For this, SEM imaging is often the most suitable device. However, this tool always requires working under the same conditions in order to be as reproducible as possible, and to compare the coatings one with each other.

The surface preparation of samples, and the treatments of images may cause artifacts. These artifacts may induce differences between computed and experimental results, but allow a qualitative comparison between the various manufactured coatings if a reliable protocol is well developed and applied to each coating.

With this modeling methodology, it is however difficult to take some phenomena into account. For example, the influence of bad lamellae contacts is not considered by working directly from SEM images. However these bad contacts may have a significant influence on both thermal and mechanical properties of coatings. The heat treatment sometimes applied on as-sprayed coatings may improve the contact quality between splats and reduce the differences between the computed and experimental results.

Chapter 3. : Model input data and computational domain

The methodology applied for numerical modeling has been presented in Chapter 2. As described in Figure 3-2, this methodology can be divided into three distinct steps: input data, numerical modeling and data processing. The final objective (step 3 of Figure 3-2), is to determine the extrinsic properties of the coating. For this a modeling methodology is applied directly from images of its microstructure (step 2, of Figure 3-2). Input information are mandatory for this (step 1 of Figure 3-2). The present chapter will focus on the first step of the methodology; i.e., the input data required for modeling and the definition of the computational domain. Three main conditions have to be determined: i) the selection of an appropriate magnification level in order to fulfill the requirement of the representative elementary volume (REV) of the coating microstructures, ii) the appropriate discrimination of the coatings phases, and iii) on the selection of intrinsic phase properties as input data.

To determine the coating effective properties directly from images of their microstructure the representative elementary volume is commonly used [START_REF] Carazo | Effective properties of nodular castiron: A multi-scale computational approach[END_REF][START_REF] Fernandino | Determination of effective elastic properties of ferritic ductile cast iron by computational homogenization, micrographs and microindentation tests[END_REF][START_REF] Paola | Modélisation multi-échelles du comportement thermo-mécanique de composites à renforts sphériques (in english: Multiscale modeling of thermo-mechanical behavior of composite with spherical reinforcements[END_REF]. Indeed choosing an accurate REV is mandatory to describe the coating microstructure. The selection of the appropriate magnification level will depend on the microstructural features size and shape. The REV must be large enough compared to the size of the heterogeneities in order to correctly represent the averaged behavior of the whole structure but small enough compared to the structure size. At the REV, all the random images representing the heterogeneous medium should have a similar macroscopic behavior in a statistical manner. The protocol to determine REV of the studied coating is described in chapter 2, and in this chapter 3 the REV the three abradable coating will be given.

Once the REV is determined it will be possible to analyze more precisely the coating microstructure. The appropriate acquisition tools have to be chosen in function of the objective, which is to obtain the most realistic representation of the microstructure. Then, a software is used to extract the microstructure information. Combined with stereological protocols [START_REF] Russ | Practical Stereology[END_REF] it is possible to quantify the microstructural features. In function of all the different features and phases existing into the coating, their images must be transformed into binary or ternary images before to conduct the modeling methodology. To do this, image analyses protocols can be applied, and will be explained in this chapter 3.

To finish, all the coating features and phases will play a role on the extrinsic coating properties. As described in chapter 2, the modeling method consists in using the intrinsic bulk properties of all the coating phases and features to calculate the extrinsic coating properties. In this chapter 3 the methodology to determine the phases and features properties is given. They are determined mostly by literature research but also by nanoindented measurements [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation techniques[END_REF].

Three abradable coatings having different chemical composition have been considered; i.e., NiCrAl-Bentonite, CoNiCrAlY-BN-polyester and YSZ-polyester.

I Selection of an appropriate magnification level

Plasma spraying is the manufacturing process implemented to build those coatings. Consequently, they are made by the stacking of individual lamellae resulting from the impact, spreading and solidification of impinging particles. Depending upon the characteristics of those particles upon impact, in terms of momentum, molten state and viscosity, impact angle, etc., and of the substrate, in terms of composition, temperature, etc., mostly, various features will be identified in the coatings structure: flattened lamellae, voids resulting from stacking defects, cracks resulting from stress relaxation locally, oxides, etc. As a consequence, coatings exhibit heterogeneous composite structures [START_REF] Herman | Plasma-Sprayed Coatings[END_REF] interesting for abradability application [START_REF] Faraoun | Improvement of thermally sprayed abradable coating by microstructure control[END_REF][START_REF] Novinski | Modified zirconia abradable seal coating for high temperature gas turbine applications[END_REF]. Those features can be quantified in terms of nature, size, spatial distribution, connectivity, etc. from which will depend the coating extrinsic thermomechanical properties [START_REF] Mcpherson | A review of microstructure and properties of plasma sprayed ceramic coatings[END_REF][START_REF] Poulain | Etude de la conductivité thermique de revêtements céramiques à base de zirconerelation avec la composition, la microstructure et la morphologie[END_REF]. For modeling such properties, the computational domain, limited in dimensions, has to be representative of the whole coating structure, that is to say has to represent the variety of features and their characteristics. In other terms, the computational domain needs to correspond, at least, to the representative elementary volume (REV) which can be defined as the minimum volume sample from which a given parameter becomes independent of the size of the sample. Considering 2D modeling, this definition can be extended to the required minimum surface.

On SEM or OM images, the REV will depend upon the selected magnification to capture them. For this reason, the determination of the REV is mandatory for each considered coating structure. At first, those structures will be depicted qualitatively. At second, the statistical methodology used to determine REV for each coating structure will be presented.

I.1. Microstructure description of the coatings

At first, polished cross-sections of sprayed coatings have been observed at different magnification levels, implementing OM and SEM, with the objective of identifying the main structural features. A common description can be made for the three materials. The brightest colors on microstructures represent the metallic or the ceramic matrices of the coatings. The darkest colors correspond to voids, which exhibit different characteristic sizes. For NiCrAl-Bentonite coating structure, this network is highly connected and is due on the one hand to the manufacturing process implemented (i.e., flame spraying) and on the other hand to the initial feedstock powder architecture (i.e., a core made of bentonite surrounded by the metallic matrix). For CoNiCrAlY-BN-polyester coating structure, the darkest areas correspond to the cavities formed by the pore former (polyester) which decomposes during a post treatment at high temperature (in the 500°C range for 3 hour). For YSZ-polyester coating structure, the darkest areas correspond to coarse voids induced by the suppression of the polymer phase by applying a thermal treatment after the spraying process. Meanwhile those features are located randomly in the coatings, their spatial distribution can be typified as "homogeneous" at this magnification level.

OM presents several limitations however which impede the description of those coatings. On the one hand, the reflectance of some features is too low to be detected. This is the case of the bentonite phase embedded in NiCrAl-bentonite coatings. For this reason, it does not contrast enough with the voids. On the other hand, the limited resolution of this observation technique does not allow observing the finest structural details, such as cracks and delaminations. Those observations have been hence complemented with SEM observations. I.1.2. Observation at high magnification level by SEM At a higher magnification level (×500), specific features can be also observed in YSZ coatings: grey features, which will be discussed into more details hereafter (feature 7, Figure 34) and voids, of characteristic dimensions in the 1 -10 µm size range1 .

As explain in chapter 2, the determination of the equivalent diameter distribution is based on different steps:

ii) measurement of the surface area of each void which is realized for the whole data set if those grey phases are due to a morphological variation with the matrix and the BSE mode permits to reveal areas of different chemical composition by contrast variation. EDS and XRD measurement have been conducted also to get a better insight on those phases composition, transformation or impurities.

Those measurements do not reveal anything unexpected concerning the coating composition.

Therefore, there is no phase transformation occurring during the spray process, or in case of partial phase transformation, it is not significant enough (in volume) to be detected.

The high magnification SEM view (secondary electron mode), Figure 3-6 a), reveals a structure made of an agglomeration of nano-particles of 50 nm in diameter, average value, and nano-voids. Figure 3456b) shows the surface of the powder before thermal spraying. The surface is composed of several grains of different size and shape. Some circular grains have been measured less than 0.05 µm as the small grain which composed the coating grey phases. The structure represented in Figure 3-6 a) is very similar to the one of the starting feedstock material, as displayed in Figure 3456). Therefore, it is concluded that those grey features correspond to the core of feedstock particles that have been incompletely processed (i.e., melted) during their flight in the plasma jet and that remain embedded in the coating structure. Complementary analyses performed on those coatings by XRD did not show any identification of new phases compared to the ones already identified in the starting feedstock, whereas those grey areas in the coatings represent a volume fraction of about 20%, average value, well above the limit of detection of phases conventionally admitted with XRD techniques. This result constitutes supplementary evidence that those features are formed by embedded unmelted cores of feedstock particles that have not been processed. The coating microstructure at the two magnification level presented in Figure 3-4 does not permit to observe this "nano-porosity". A third magnification level is necessary in order to describe even more precisely those microstructures, Figure 3456. Some similar microstructures have been observed in previous studies. Curry et al. [START_REF] Curry | Next Generation Thermal Barrier Coatings for the Gas Turbine Industry[END_REF] observed this fine granular structure on coating elaborated with YSZ Homogenized Oven Spheroidized Powder (HOSP). powder [START_REF] Friis | Influence of particle in-flight characteristics on the microstructure of atmospheric plasma sprayed yttria stabilized ZrO2[END_REF][START_REF] Bisson | Influence of hydrogen on the microstructure of plasma-sprayed yttria-stabilized zirconia coatings[END_REF][START_REF] Marple | Yttria-Stabilized Zirconia Thermal Barriers Sprayed Using N[2]-H[2] and Ar-H[2] Plasmas : Influence of Processing and Heat Treatment on Coating Properties[END_REF][START_REF] Antou | Amélioration de revêtements barrières thermiques par un procédé de refusion laser in situ utilisant un laser à diodes[END_REF][START_REF] Myoung | Microstructure design and mechanical properties of thermal barrier coatings with layered top and bond coats[END_REF]. The network of inter and intra lamellar cracks is widely depicted.

FOCUS : Microstructure of Modified TBC

Some globular porosities are noticeable, their size approach the 10 µm diameter, but strongly depends of the initial powder size [START_REF] Kulkarni | Processing effects on porosityproperty correlations in plasma sprayed yttria-stabilized zirconia coatings[END_REF].

In comparison, the microstructure with the porosity former, at the same magnification level reveal much higher porosities size with irregular shapes. At this magnification level, the inter and intralamellar cracks network is not as much depicted than YSZ coating. To observe this network it is necessary to observe the microstructure at a higher magnification levels. This conclusion has been made also by Curry et al. [START_REF] Gupta | Design of next generation thermal barrier coatings -Experiments and modelling[END_REF], which used higher magnification level to observe the effect of voids networks on such structure. Moreover, the grey features corresponding to the feature 7 into Figure 3-6, phase do not exist in a typical YSZ coating and are certainly due to the powder composition and its particles form induced by the powder elaboration process.

I.1.3. Conclusion and notion of material complexity

Three coatings with significantly different chemical compositions and microstructures have been depicted. It appeared that those structures have to be observed at several magnifications (i.e., scales) to identify all features forming them. The number of scales varies depending upon the considered coating composition. For example a minimum two different magnification levels are required to describe CoNiCrAlY-BN-Polyester and YSZ-polyester coatings. This is the reason why an index of complexity has been defined in order to typify the considered structures. It comprises two indicators: the number of phases embedded in the coatings and the number of scales needed to observe them. Table 3-2 displays the index of complexity for the three studied coatings. [note 3] YSZ, voids, greys features. random selection, and to plot it versus the number of analyzed images, Figure 3456789. This corresponds to a stabilized variability of the calculated void content. This stabilized variability is intrinsic to the coating structure and not related to biases due to the REV condition which would not be fulfilled.

The study of these curves for the different magnification levels reveals that increasing the magnification level permits to observe more precisely the details of the microstructure but requires increasing the number of images.

For each magnification level, four different curves were plotted and were obtained by randomly sorting the values. When the four curves tend toward the same asymptote, it is possible to determine the minimum number of images necessary to be representative. 3-3 reveal that increasing the magnification level permits to observe more precisely the details of the microstructure but induces increases the number of images required to provide a representative value, and increases the variability. Due to the presence of large voids, the use of a high magnification level induces larger differences of the phase content measurement from an image to another. For these reasons, for YSZ-Polyester coating, it was chosen to work with a magnification level of ×150 (1 pixel corresponds to 0.93 µm) which represents the best compromise.

Coating set Magnification level variability of void content Number of images

YSZ-Polyester

Reference coating ×150 0.12 12 ×200 0.18 13 ×300 0.24 17

The second step consists in the determination of the number of images, at the selected magnification level, required to be representative of the whole structure. The same approach as the one described previously was followed and several properties, i.e. E, κ and voids rate, were calculated on 15 different images of coatings.

I.2.2. Second step, Image number selection

To determine the number of images in order to have statistically significant results the previous procedure describe with the void ratio have been applied to use also the Young's modulus and the Thermal conductivity as discriminative features. In this example, the minimum number of images required to reach a representative area is pointed out by a black arrow on Figure 3-11. The estimation of the Young's modulus and the calculation of the thermal conductivity require 7 and 8 images respectively. The determination of the porosity rate determined previously requires a minimum of 12 images to be representative. Table 3-4 regroups the conclusion of the study for properties studied onto YSZ-polyester coatings.

Coating set Properties Number of images

Abradable YSZ-polyester

Reference coating

Void content 12

Young's modulus 7

Thermal conductivity 8

The number of images chosen from these results in order to have statistically representative results

is 15 images at a magnification level of ×150 by SEM. Determining the void rate of those coating requires a higher number of images than for determining the Young's modulus and thermal conductivity.

I.2.3. Conclusions

This procedure has been applied to the different coatings nature in order to determine the appropriate number of images and magnification level to be in RVE of the coatings. The main conclusions of the study are displayed in Table 3-5.

II Phase discrimination

As explain in chapter 2, image analysis requires several steps. The first is related to the images capture protocol. The second one corresponds to the conversion of grey scale images into binary or ternary ones. For this, a threshold filter, based on the contrast (i.e., on the grey color of the feature) is applied. The third one aims at identifying all features in the coatings and at extracting geometrical data related to them. Finally, those raw data are statistically studied.

II.1. Detailed protocol

NiCrAlY-Bentonite coating structure has been selected to detail here the image analysis protocol that has been optimized, owing the fact that it has been implemented, systematically 2 , for each coating.

The main objective is to have available a robust and reproducible protocol for processing the coating structures, whatever the index of complexity. Figure 3-11 displays the generic flowchart of such a protocol.

Grey color SEM images are encoded in 8 bits, that is to say that they contain 2 8 = 256 levels of grey, where, arbitrarily, the 0 value corresponds to the black color and the 255 value to the white one.

Meanwhile they were kept constant to the maximum possible extend, the SEM parameters for capturing the images can differ with time. Consequently, the distribution of grey levels can evolve, for the same coating, from one capture to another. This is why the first step is to normalize the distribution of grey levels, by arbitrarily setting to 0 the darkest grey level of the original image and to 255 the lightest one. Each feature in the picture is spatially determined by a range of grey colors. A threshold, based on pixel color (contrast) over a given range of grey colors allows in a second time discriminating each type of features. The resulting image comprises artifacts, such as isolated pixels, which, obviously, are not representative of the considered feature. The opening of the image allows hence in a third time to delete from the image those artifacts. The resulting binary image can be now analyzed and data related to the considered features extracted. The same procedure is applied for each type of features identified in the coating structure. In a second time, the collection of binary images are added (Boolean logic) together to reconstruct the whole coating structure. This time, each feature is spatially referenced in the image. Figure 3456789101112displays an example of this reconstruction step that has been automated in a dedicated subroutine. Each feature in the picture is spatially determined by a range of grey colors. For two-phase coatings (e.g., voids embedded in a matrix), the tangents on both sides of a peak are determined. The intersection of a tangent with the background, corresponding indeed to an inflection point, corresponds to the threshold value applied to the image (i.e., values framed in red in Figure 3-14).

For three-phase coatings (e.g., voids and particles embedded in a matrix), the median value of the plateau between two successive peaks has been selected (i.e., values framed in green Figure 3 -14).

II.2. Some specific issues encountered in image processing and implemented solution

An issue in discriminating features by color (contrast) thresholding of such structures is the presence of fine cracks, which, most of the time, exhibit a large range of colors. This is especially the case of cracks in YSZ coating images. This is due to the depth of field of SEM which is much higher than the one of OM. Consequently, reflections are the sides of the cracks change their contrasts, making more difficult their discrimination from other features, such as unmelted cores of particles, appearing in the same color. After multiple attempts, a dedicated protocol has been determined, as displayed in Figure 3 subroutine: the initial one, the one with only the largest features and their exact shape and size (unlike when opening an image) and one with only the smallest features and cracks. The smallest feature and cracks can be then considered and transformed into black pixels to be added (Boolean logic) to the image depicting the globular voids in view of generating a picture of the void network, whatever the shape of the voids (i.e., globular of cracks).

Initial network

Step 1

Step 2

Generated images

II.3. A few results

The aforementioned protocols have been applied to each nature of coatings. Results, expressed in terms of ratios of each feature, are displayed hereafter:

 the two microstructures considered for the NiCrAl-Bentonite coating, corresponding to two sets of spray operating parameters;

 the two microstructures considered for the CoNiCrAlY-BN-polyester coating, corresponding to two sets of spray operating parameters;

 an example of five microstructures for the YSZ-polyester coating, knowing that coatings have been manufactured implementing two plasma spray torches (i.e., Sulzer F4 and Sulzer 9MB) operated with a large range of spray operating parameters. One coating corresponds to the reference coating elaborated with the 9MB torch. Another show a similar microstructure than the reference coating, a third is more porous, there is also a less porous coating and finely a dense coating with no coarse porosities.

NiCrAl-Bentonite

NiCrAl+Bentonite ratio: 65% ± 5%  NiCrAl ratio: 37% ± 3%  Bentonite ratio: 28% ± 4% Voids ratio: 35% ± 5% NiCrAl+Bentonite ratio: 60% ± 5%  NiCrAl ratio: 35% ± 4%  Bentonite ratio: 25% ± 3% Voids ratio: 40% ± 4% CoNiCrAlY-BN-Polyester -Elaborated by APS with F4-torch 8 and 6 mm nozzle diameter respectively.

CoNiCrAlY+BN ratio : 70% ± 2%  CoNiCrAlY ratio: 57% ± 2%  Grey phase ratio: 13% ± 1% Voids ratio: 30% ± 2% CoNiCrAlY+BN ratio : 46% ± 2%  CoNiCrAlY ratio: 39% ± 2%  Grey phase ratio: 7% ± 1% Voids ratio: 54% ± 1%

YSZ-Polyester -Elaborated by APS with 9MB-torch

Reference coating

YSZ+grey phase ratio : 92% ± 2%

 YSZ ratio: 75% ± 2%  Grey phase ratio: 18% ± 1% Voids ratio: 8% ± 2%

Similar than reference

YSZ+grey phase ratio : 92% ± 1%  YSZ ratio: 72% ± 1%  Grey phase ratio: 19% ± 1% Voids ratio: 8% ± 1%

More porous than reference

YSZ+grey phase ratio : 88% ± 3%  YSZ ratio: 70% ± 3%  Grey phase ratio: 18% ± 1% Voids ratio: 12% ± 2%

Less porous than reference

YSZ-grey phase ratio : 97% ± 1%  YSZ ratio: 79% ± 1%  Grey phase ratio: 18% ± 1% Voids ratio: 3% ± 1%

Dense coating with no coarse porosities

YSZ-grey phase ratio : 97% ± 31%  YSZ ratio: 76% ± 3%  Grey phase ratio: 21% ± 1% Voids ratio: 3% ± 1%

III Intrinsic properties of phases

For modeling the apparent properties of coatings from domains discretized from their microstructure, intrinsic thermomechanical properties of features / phases are used as input data. In this work, the considered features / phases are: 8YSZ, CoNiCrAlY, Al 2 O 3 -BN oxide, NiCrAl, Bentonite, air filling up the voids.

An intensive review of data available in databases and in the literature has been undertaken: most of the intrinsic properties used come from such an origin.

A drawback associated to data from the literature can be a weak consideration of specificities induced by the specific processing of the materials by the high-energy flow (i.e., flame or thermal plasma). For example, bentonite is colloidal hydrated clay. Depending upon its hydration rate, the elastic property of the material can vary in a large extend. No information is available regarding the change in the hydration rate of bentonite after having being processed by the high-energy flow. This is why some additional experimental tests, implementing mostly nanoindentation tests, have been carried-out to validate or determine additional elastic properties of features.

III.1. Intrinsic properties from databases available in the literature

Multiple studies have been conducted along the years to determine the thermomechanical properties of YSZ. This material exhibits a low thermal conductivity and high mechanical performances (toughness, etc.), especially at elevated temperatures (i.e., above 600°C).

Gadag et al. [START_REF] Gadag | Thermo-elastic properties of dense YSZ and porous Ni-ZrO2 monolithic and isotropic materials[END_REF] reviewed those studies, especially the ones published by Pace et al, Buckley et al, Seluc et al, and Kandil et al (reference [4] and [START_REF] Stöver | Directions of the development of thermal barrier coatings in energy applications[END_REF] to [START_REF] Dorfman | Thermal Spray Technology Growth in Gas Turbine Applications[END_REF] into [START_REF] Gadag | Thermo-elastic properties of dense YSZ and porous Ni-ZrO2 monolithic and isotropic materials[END_REF]). They completed this review by additional data related to isotropic single crystal YSZ and cubic polycrystalline structure of YSZ.

Most of properties have been determine at room temperature and some other at high temperature (i.e., from 20 to 900°C order).

Antou [START_REF] Antou | Amélioration de revêtements barrières thermiques par un procédé de refusion laser in situ utilisant un laser à diodes[END_REF], in his Ph.D. thesis typescript has also listed several studies providing mechanical and Youngblood et al. [START_REF] Youngblood | Thermal diffusivity of partially and fully stabilized (yttria) zirconia single crystals[END_REF], Poulain [START_REF] Poulain | Etude de la conductivité thermique de revêtements céramiques à base de zirconerelation avec la composition, la microstructure et la morphologie[END_REF] or Ibegazene [START_REF] Hocine | Etude microstructurale de la stabilité thermique de phases métastables dans les systêmes à base de ZrO2, HfO2, et d'oxydes de terres rares : application aux barrières thermiques[END_REF], determined also Young's modulus, thermal conductivity and diffusivity of dense polycrystalline or monocrystalline YSZ. A selection of those properties is listed in Table 3

-9.
Concerning the intrinsic properties of CoNiCrAlY and NiCrAl phases collected in the literature, they mostly concern properties determined directly onto sprayed coatings. For example, Saeidi has work on as-sprayed and annealed High Velocity Oxygen Fuel (HVOF) and Vacuum Plasma Spray (VPS) MCrAlY coatings [START_REF] Saeidi | Microstructure, oxidation & mechanical properties of as-sprayed and annealed HVOF & VPS CoNiCrAIY coatings[END_REF]. The measurements were conducted by a conventional Dynamic Mechanical Analyzer (DMA) 3-point bend test onto free standing coating (coatings were detached previously from the substrate). As-sprayed Young's modulus varies between 100 and 130 GPa, depending upon the composition of the MCrAlY coating and the considered manufacturing process.

Other authors have developed specific technics to determine mechanicals properties of MCrAlY free standing, such as Waki et al. [202,[START_REF] Waki | Development of Lateral Compression Method of Circular Tube Thin Coating for Mechanical Properties of Plasma Sprayed CoNiCrAlY[END_REF] implementing a lateral compression method and Johnston [START_REF] Johnston | Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings[END_REF] implementing tensile test applied on coating specimen elaborated into dissolvable molds. Waki et al. made measurement onto as sprayed CoNiCrAlY coatings elaborated by VPS, HVOF and APS and measured Young's moduli varying between 10 to 110 GPa, depending upon the manufacturing process and the voids content into the coatings. Johnston has made measurement onto NiCrAl-Bentonite abradable coating and measure a Young's modulus of 1.39GPa ± 0.17GPa. This value will be interesting to compare with calculated and experimental values related in chapter 4 of the manuscript.

Tamarin [START_REF] Tamarin | Protective Coatings for Turbine Blades[END_REF] Taylor et al. [START_REF] Taylor | Thermal expansion of MCrAlY alloys[END_REF][START_REF] Taylor | Dilatometer studies of NiCrAlY coatings[END_REF] studied the coefficient of thermal expansion (CTE) of APS MCrAlY coating having various composition. It varies in a very large extend (i.e., from 13.10 -6 to 17.10 -6 °C-1 ), depending upon the composition of the material.

ν (-) 0.3 [START_REF] Ingel | Initial Characterization of Partially Stabilized HfO2 Single Crystals[END_REF][START_REF] Radovic | Thermophysical properties of YSZ and Ni-YSZ as a function oftemperature and porosity[END_REF] 0.3 [START_REF] Saeidi | Microstructure, oxidation & mechanical properties of as-sprayed and annealed HVOF & VPS CoNiCrAIY coatings[END_REF]239] 0.3 [START_REF] Saeidi | Microstructure, oxidation & mechanical properties of as-sprayed and annealed HVOF & VPS CoNiCrAIY coatings[END_REF]239] κ (W.m -1 .K -1 ) 3-2.5 [START_REF] Jadhav | Low-thermalconductivity plasma-sprayed thermal barrier coatings with engineered microstructures[END_REF][START_REF] Dutton | Effect of heat treatment on the thermal conductivity of plasma-sprayed thermal barrier coatings[END_REF][START_REF] Antou | Amélioration de revêtements barrières thermiques par un procédé de refusion laser in situ utilisant un laser à diodes[END_REF][START_REF] Poulain | Etude de la conductivité thermique de revêtements céramiques à base de zirconerelation avec la composition, la microstructure et la morphologie[END_REF][START_REF] Wu | Low-Thermal-Conductivity Rare-Earth Zirconates for Potential Thermal-Barrier-Coating Applications[END_REF] 25 [199] 15 [200,[START_REF] Verdy | High heat flux thermal cycling of multi layered deposit using a HVOF Gun[END_REF] The mechanical properties of MCrAlY phases found in the literature relate to phases already elaborate either by thermal spraying or via other methods. So there is a limitation to implement these properties in the model for two main reasons:

 The composition of MCrAlY phases found in the literature is different from those studied here.

 A voids network has been created during the elaboration process and affects the value of properties measured.

For this reason, one has performed nanoindentation measures onto NiCrAl and CoNiCrAlY phases of the coating elaborated on this work to obtain the mechanical properties values corresponding to their real chemical composition. Values are given in Table 3-10. Concerning void properties, some assumption have been made in order to determine some thermomechanical properties. The hypotheses are given in Table 3-11 as well as the properties used as input values of voids to conduct the modelling methodology.  For the modeling, the voids are assumed to behave as a compressible material (ν=0.33) [START_REF] Coussot | Rhéophysique la matière dans tous ses Etats[END_REF]  Bulk modulus K a = 𝐸 3-6𝜈 = 10 5 Pa [START_REF] Bassani | Hydrostatic Lubrication[END_REF]. If ν=0.33 then K a =E= 10 5 Pa. ν (-) 0.

*Hypothesis.

III.2. Issues in the identification of intrinsic properties of some features and implemented alternative solutions

Issues in the identification in the literature of intrinsic properties of some features / phases, bentonite, Al 2 O 3 and BN, have been faced.

Regarding bentonite, colloidal hydrated clay, its properties varies in a very large extend, depending upon, among other parameters, the hydration amount of the clay. Bentonite Young's modulus varying between 0.02 [START_REF] Tang | Thermo-mechanical behaviour of a compacted swelling clay[END_REF] and 14 GPa [START_REF] Bathija | Elastic Properties of Clays[END_REF] and thermal conductivity varying between 0.2 [START_REF] Tang | A study on the thermal conductivity of compacted bentonites[END_REF] and 1.4

W.m -1 .K -1 [START_REF] Hicks | Role of Bentonite in Determination of Thermal Limits on Geological Disposal Facility Design[END_REF] have been found. No information is currently available regarding the hydration rate of the bentonite embedded with NiCrAl in feedstock particles. Moreover, as already said, those materials are processed by the high-energy flow, experiencing thermal treatments that may lead to a variation in the hydration rate of bentonite in the feedstock and in the coating.

Regarding Al 2 O 3 , its formation results from the preferential in-flight oxidation of Al during the processing of NiCrAl or CoNiCrAlY particles. The kinetics of such an oxidation reaction is very fast (in the 1 ms order) and other materials can oxidize / react also at the same time, such as BN, leading to the formation of composite BN-Al 2 O 3 materials about which no relevant information exists.

Depending upon their purity, intrinsic Young's moduli varying between 200 [212] and 400 [START_REF] Auerkari | Mechanical and physical properties of engineering alumina ceramics[END_REF] GPa for Al 2 O 3 and between 25 [START_REF] Eichler | Boron nitride (BN) and BN composites for high-temperature applications[END_REF] and 80 GPa [START_REF] Lipp | Hexagonal boron nitride: Fabrication, properties and applications[END_REF] for hBN are found in the literature.

Therefore, nanoindentation tests performed on thermal spray coatings cross-sections have been implemented to experimentally determine the elastic property of the Al 2 O 3 -BN composite feature, appearing as grey areas (feature 6) in Figure 3-4 together with Bentonite features. Results are displayed in Table 3-11. Experimental determination of the intrinsic thermal conductivities of those features is impossible, since no local test considering a representative volume of a few µm 3 are available. This is why it was decided, as an alternative, to assess the sensitivity of the apparent thermal conductivity of such coatings in regard with a variation of the considered intrinsic thermal conductivity of the Al 2 O 3 -BN composite compound. For that, it was assumed that the thermal conductivity could vary between 10 and 90 W.m -1 .K -1 (other intrinsic properties regarding other features being depicted in Table 3 -13).

Results are depicted in Figure 3-17. From those results, one can conclude that:

 a variation of +80 W.m -1 .K -1 (i.e., from 10 to 90 W.m -1 .K -1 ) in the thermal conductivity of the Al 2 O 3 -BN composite compound leads to a variation of +3.5 (i.e., from 7.4 to 10.9) in the calculated apparent thermal conductivity of the coating, representing a relative variation of about 30%. Such a variation can be considered, at this stage of the work, as acceptable and could be integrated as a deviation in the calculated apparent thermal properties ;

 The higher the selected intrinsic thermal conductivity, the lesser is its influence on the calculated coating apparent thermal conductivity, as illustrated in Table 3-13.

Table 3-13: Influence of the variation in the selected intrinsic thermal conductivity of the Al2O3-BN compound on the calculated coating apparent thermal conductivity.

Variation in input data (intrinsic property of Al 2 O 3 -BN composite compound) Δ input Δ output

Variation in output data (calculated coating apparent thermal conductivity)

Between 70 to 90 W.m -1 .K -1 20 W.m -1 .K -1 0.3 Between 10.6 to 10.9

Between 10 to 30 W.m -1 .K -1 20 W.m -1 .K -1 1.4 Between 7.4 to 8.8

For the modeling work which will be describe in chapter 4, it was chosen to selected an intrinsic thermal conductivity of 50 W.m -1 .K -1 as input data for the Al 2 O 3 -BN composite compound in CoNiCrAlY-BN-Polyester coatings. Moreover, such a value represents the median value of data identified in the literature.

The same study has been conducted for determining the influence of the intrinsic thermal conductivity of the bentonite phase as input data onto the calculated coating apparent thermal conductivity of NiCrAl-bentonite coatings as output data. The input data was varied between 0.2 and 2 W.m -1 .K -1 , that is to say a variation of 90% over the considered range. The normalized output data varied accordingly between 0.44 and 1.03, that is to say a variation of 57% over the considered range.

It was also chosen to select as input data a median value of 1.4 W.m -1 .K -1 and integrate in the deviation of calculated values this effect.

I.1. Conclusion

Table 3-13 synthesizes intrinsic thermomechanical properties of features and phases identified in the coatings and considered in the modeling work depicted in chapter 4 of this typescript

* Experimental nanoindentation measurements ** Assumption

II Conclusion

The description of the microstructures of selected coatings highlighted the presence of several structural features having different size and shapes. An index of complexity has been hence defined and applied. It appears hence that the less complex structure is NiCrAlY+Bentonite, the most complex one is YSZ+polyester.

Besides, the Representative Elementary Volume for each coating has been determined and, for a given magnification, the number of images to account with has been determined (Table 34).

In parallel, a reproducible protocol for image processing has been established. It allows systematically processing images in view of discriminating the different features, without statistic biases due to the processing protocol (Figures 3-11 and 3-16).

Finally, intrinsic properties of features / phases identified in the coatings have been determined, either by selecting them in databases or by measuring them experimentally (Tableau 3-13).

Chapter 4. Results and discussions -

Improvement of the modeling method

As discussed more in details in Chapter 1, the coating microstructure dramatically affects the effective thermomechanical properties. Mastering this interaction is actually a strategic issue in order to improve efficiently the properties of coatings for aeroengine applications. Analytical models, numerical models and homogenization theories have been developed to predict the extrinsic behavior of multi-phase materials by using information on their phase.

The majority of analytical models are easy to use and represent a timeless methodology to obtain quickly the properties of composite materials, but are not fully representative of the coating microstructure. Indeed, they do not take into account the real shape and size of the porosities, and the connectivity between the different pores. Most of analytical models found in the literature and presented previously in Chapter 1 are only adapted to two-phases or simply organized multi-phase media while the complexity of abradable materials comes from the combination of a random multiphase organization with porosity. For this reason, an analytical approach of the properties of abradable materials seems very limited considering their microstructural complexity.

Nowadays, the most competitive approach to achieve accurate predictions of abradable coating properties is in fact the numerical one. Modeling based on micrographs is more faithful for a realistic representation of the actual microstructure of the material. Numerical methods are increasingly popular because of their reliability even if, most of the time, lattice simplification concerning the microstructure and the mesh is necessary to overcome the computer limitations in terms of memory size and computational time. Nevertheless, due to new computer generations, computational times are shortened, allowing working with a better resolution of images without simplifying the microstructure.

The literature regroups a wide number of works trying to predict the extrinsic behavior of multiphase materials by using information on their phase. The extrinsic properties correspond to homogenized values determined at the macro-scale of the coating and deduced from the intrinsic properties of the coating components at the micro-scale. Even if homogenization theories have originally emerged to predict the behavior of periodically organized composites [START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF], they are also applied to randomly arranged materials [START_REF] Carazo | Effective properties of nodular castiron: A multi-scale computational approach[END_REF][START_REF] Fernandino | Determination of effective elastic properties of ferritic ductile cast iron by computational homogenization, micrographs and microindentation tests[END_REF]. Nevertheless, this thesis work is more focused on experimental aspects than numerical issues and the numerical part can be considered as a first insight on the application of a simplified homogenization approach of abradable coatings.

The approach considered here is to conduct a modeling methodology directly on real coating microstructure images to estimate their mechanical and thermal properties. The meshing is performed pixel by pixel to avoid any simplification of the microstructure. Each pixel color corresponds to a material phase.

Chapter 3 presented data concerning the input information required by the model, i.e. concept of representative elementary volume, phases discrimination, intrinsic properties of the phases. As describes by Figure 3-2, this point corresponds to the first step of the modeling method. Chapter 4 relates to the modeling step of the methodology (i.e., step 2 represented on Figure 3-2) and will allow comparing the calculated effective properties (i.e., step 3 on Figure 3-2) with coating experimental extrinsic properties. The 2 nd step of the modeling methodology consists in applying the conventional modeling method describes in chapter 2, to all microstructures studied. Then, the results obtained will be compared to experimental properties measured for those same coatings. From the first analyses, the conclusions will lead to an improvement of the method by considering several scales. The results obtained will then be compared to experimental results and to the results obtained with the initial conventional modeling methodology (i.e., single scale calculations). Due to the differences of strategy adopted by the two sub-working groups of the European project E-BREAK, experimental properties measured on the different abradable coatings are not the same and are listed in Table 4-1.

For NiCrAl-Bentonite coatings, the standard deviation on the Young's modulus measured by RFDA is higher than the one on the Young's modulus measured by tensile test. However, to compare the variability of measurements for both techniques, a comparison of the coefficient of variation is required. The coefficient of variation (i.e., defined as the percentage ratio of the standard deviation to the average value) is convenient to compare properties of coating of same nature having widely different means. Here, the coefficient of variation of the Young's modulus is in between 17% and 33 % for microstructure A and B respectively in the case of tensile test measures, and between 25% and 40 % by considering RFDA results. The variability of the Young's modulus measurements for tensile tests and RFDA are thus of the same order for each coating.

I.1.2. CoNiCrAlY-BN-polyester coatings

Two strongly different coatings were considered for the CoNiCrAlY-BN-polyester abradable material (manufactured with two different sets of parameters), which shows a higher complexity than NiCrAl-Bentonite. The less porous coating will be called C and the most porous one will be called D, see The coefficient of variation of HR15Y hardness represents between 2 and 6 % of the average value which is quite low (the data are hence consistent). Except for the last coating (i.e., the densest one), those data do not allow us to establish a behaviour law to predict the coating hardness in function of the porosity rate and microstructure description of the coating.

The HR15Y hardness measured on the reference is the lowest value, whatever is the microstructure of the other coatings. Indeed, as shown on Figure 4-5, no correlation can be found between the HR15Y hardness and coating void content or erosion resistance. This observation has been extended to other YSZ-Polyester microstructures, and the conclusion remains the same. The HR15Y test, widely used to characterize abradable coatings used in the compressor section, is thus not adapted for ceramic-based abradable materials since it is not sensitive enough. One explanation could be the low plastic deformation of the ceramic matrix which leads to the propagation of cracks during the test and influences the measures between one coating and another. A second one could be the very low penetration of the indenter during the preliminary charge of the HR15Y test (see chapter 2 for the test description). As explain in chapter 1 and 3, a preliminary charge is applied on the coating to remove all the effects of surface irregularities. The hardness is given by the difference between the penetration deph of the second load and the preliminary load.

The hardness of the ceramic matrix is very high, meaning that the penetration of the indenter is low (during both preliminary charge and second charge). This does not permit to remove the effects of surface irregularities and to observe an effect of the microstructure.

On contrary, the results of erosion tests are in accordance with the microstructure description. The densest coatings show the highest erosion resistances, whereas the coatings corresponding to the most porous microstructures have the lowest erosion resistances. Those microstructural differences and the resulting variation of the erosion resistance are explained by the differences in the process parameters applied to manufacture the YSZ-Polyester coatings. Indeed, the resulting particles temperature and velocity conditions at their impact onto the substrate were different considering the different sets of parameters.

I.2. Calculated Young's modulus by "conventional" 1-scale modeling

The modeling method described in Chapter 2 was applied to the different coatings and the results calculated are listed in the following tables: Table 4-5, Table 4-6 and Table 4- ) is lower than the one corresponding to coating A. However, considering their standard deviation, it is possible to say that the calculated Young's modulus is of the same order for the two coatings. Microstructural differences between the two coatings do not induce strong differences in the calculated properties. This result is explained by the few differences observed when the two coatings were described in chapter 3. The standard deviations on the calculated Young's modulus are very high in both cases. Their relative standard deviations are 83% and 67% for coatings A and B respectively. This can be explained by the porous network, which is strongly connected. With 2D images of the microstructure, the porosity network between one image and another is quite different inducing this high standard deviation. This is illustrated by the following Figure 4-6, showing to images of the same coating (i.e. coating A). The Young's moduli calculated is lower than the one of the metal matrix (i.e., 150 GPa). For the less porous coating (coating C), the Young's modulus is higher than that of NiCrAl-bentonite coatings.

The calculated Young's modulus of the most porous coating (microstructure D), is much lower than the one calculated for coating C. This difference is quite important and is explained by the large difference in the porosity rate of about 20% between the two coatings (i.e., void rates of 30% and 50%

for coatings C and D respectively). Moreover the variability of the Young's modulus calculated for coating D is very high. The variability is of 25% for coating C, which is already an important value, and increases up to 200% in the case of coating D. This can be explained by the high porosity rate of about 50% for coating D, which induces a misrepresentation of the matrix and voids connectivity for 2D images of the microstructure. This contributes to decrease strongly the calculated Young's modulus since the connectivity of the metal phase becomes zero (i.e., no continuous white path through the vertical direction).

I.2.3. YSZ-polyester coatings

For the YSZ-polyester coatings, the applied strategy was a little different than for the two previous coatings. To observe if the model permits to take into account the influence of the microstructure on the calculated effective properties, it was chosen to perform the calculations for coatings manufactured with the 9MB torch for different sets of thermal spray parameters. More precisely, only five of them, described in chapter 3, were considered in this manuscript. An assumption was made concerning the dark grey areas observed on the YSZ-polyester coating micrographs (described in Chapter 3). It was concluded that those grey features correspond to the core of feedstock particles that have been incompletely processed. In the following results, the dark grey areas were considered as similar to the matrix phase. The calculated Young's moduli were normalized from the reference coating and are listed in Table 4-7. The Young's moduli calculated with the modeling method were different from one microstructure to another.

On the contrary, the densest coatings were obtained for sets of parameters providing an increase of the specific enthalpy of the plasma, compared to the reference set. In that case, the particle temperatures and velocities at the moment of the particle impact on the substrate were higher. Those sets of parameters provide a decrease of the coating void size and content.

For the coatings having coarse porosities, this modeling method permits to discriminate the coatings manufactured with the different sets of parameters. Indeed the less porous coatings have the highest calculated Young's modulus. On the contrary, the sets corresponding to the lowest calculated Young's modulus correspond to the most porous microstructures. In conclusion, the parameters that permit to increase the Young's modulus and erosion resistance are those that allow increasing both the temperature and velocity of the particles. These parameters influence strongly the coating microstructure compared to the reference one.

However, for coatings with similar porosity rates, some little differences exist between the corresponding calculated Young's modulus. The porosity rate is not the only microstructural feature playing a relevant role on the coating properties. Some differences (i.e. corresponding to void size, shape, repartition, orientation and connectivity) must exist between those two coatings. The relative difference between experimental and calculated data will be calculated by applying the following equation:

I.3. Comparison of experimental and calculated properties

I.3.1. NiCrAl-Bentonite coatings
relative difference Δ (%) = ( 𝑣𝑎𝑙𝑢𝑒 𝑐𝑎𝑙𝑐 -𝑣𝑎𝑙𝑢𝑒 𝑒𝑥𝑝 )×100 𝑣𝑎𝑙𝑢𝑒 𝑒𝑥𝑝 (IV.1)
Value exp is the experimental data and the value calc corresponds to the calculated properties.

Coating A, the less porous one among NiCrAl-Bentonite coatings (Figure 4-7, on the left) shows difference between the calculated and measured Young's modulus (their relative difference is equal to Δ=-56%). The calculated young's modulus is lower than the experimental one. However, considering the standard deviation of both calculated and measured data, the Young's moduli are in the same range.

The graph corresponding to coating B, the most porous among NiCrAl-Bentonite coatings, Figure 4-7, on the right) shows a small difference between the calculated and measured Young's modulus, i.e.

the measured properties are smaller than the calculated ones and Δ=+32%. However, considering the standard deviations of both values, the results seem to match quite well. Comparing coatings A and B, some differences exist between the measured Young's moduli. However, this difference is much smaller for the calculated data which are very similar, especially when considering the standard deviation. In fact, although the two coatings present different experimental properties (including hardness), the coating microstructures seem not so different, so that the calculated data are also not different. For the microstructure presenting coarse porosities, the experimental erosion resistances and the calculated Young's moduli allow discriminating the coatings performances from their microstructure.

A porous coating provides a lower cohesion, thus decreasing the erosion resistance, and gives rise to a decrease of the elastic modulus. Figure 4-9 confirms this information.

The most porous coating has a lower erosion resistance and a lower Young's modulus in comparison with the reference coating. The coating with a similar microstructure in comparison with the reference one shows equivalent mechanical properties. However, their properties are not equal, which is certainly due to microstructural details that differ between those two coatings. Indeed, the voids induce a low cohesion of the coating; intra-lamellar cracks decrease the cohesion at the scale of the particles forming the coating microstructure. Two coatings with a similar overall porosity level can exhibit differences between the measured erosion resistance and the calculated Young's modulus, if the inter and intra lamellar crack network differs from one microstructure to the other. As expected, the less porous coating shows the highest erosion resistance. Beyond the microstructures having the coarse voids, the less porous coating shows the highest calculated Young's modulus. Nevertheless, the very dense coating with no coarse porosities shows the highest erosion resistance but a quite low calculated Young's modulus. This result is explained later in this manuscript.

II Thermal properties

Experimental method

Grade C (less porous) Grade D (most porous)

Laser flash analysis 1.35± 0.14 1.00 ± 0.02

The experimental thermal conductivity of CoNiCrAlY-BN-polyester samples are also low compared to the thermal conductivity of the matrix (i.e., about 25 W.m -1 .K -1 according to the literature

Numerical method

Grade C (less porous) Grade D (most porous) к through-thickness 10.0± 5.0 0.33 ± 0.25

The calculated through-thickness thermal conductivities of coatings C and D are lower than that of the CoNiCrAlY matrix (i.e., 25 W.m -1 .K -1 , according to the literature [199]. The calculated thermal conductivity for the most porous coating (namely D) is much lower than the one calculated for coating C. This difference is very high and can be explained by (1) the large difference in the void rate between the two microstructures (i.e., variation of about 20 % that is to say void rate of 30% for coating C against 50% for coating D), and (2) the difference in the connectivity of the void network.

According to these calculations, coating D is still considered as a good insulator, whereas coating C seems much less efficient for insulating applications. However the standard deviation is much higher for coating D: the coefficient of variation corresponds to 50% for coating C, which is already quite high, and increases up to 76% for coating D. This can be explained by the high porosity level of about 50% for microstructure D. Indeed, this high porosity level induces a misrepresentation of the connected matrix with 2D images of the microstructure, which contributes to decrease the calculated thermal conductivity. In other words, the probability to find a continuous path through the metal matrix is higher for a real 3D structure than for 2D slides of this 3D structure.

II.2.3. YSZ-polyester coatings

Concerning the third material, the same coatings as those described previously in chapter 3 have been used to estimate the thermal conductivity of YSZ-polyester coatings. An assumption was made to consider the dark gray phases having the same properties than matrix phase. The calculated thermal conductivity has been normalized from that of the reference coating. The calculated results are listed in Table 4-12. The results obtained with the modeling method were different from one coating to another and are represented in Figure 4-10. The observations are in accordance with those made for mechanical properties: the model permits to discriminate the different coating microstructures as well for the Young's modulus than for the thermal conductivity for the coating having coarse porosities. Indeed the comparison between Figure 4-10 and Table 4-12 shows that the less porous coatings have the highest thermal conductivity whereas the lowest thermal conductivities correspond to the most porous coatings.

The coating showing a similar microstructure in comparison with the reference one shows a similar thermal conductivity. However, the thermal conductivity calculated for the densest coating is surprising since a higher thermal conductivity was expected compared to other coatings. At this stage of the study no good interpretation on these surprising results can be made.. Due to the previous hypotheses listed overhead, (consideration of the grey phase as similar to the matrix phase), the thermal conductivity is certainly overestimated. Coating A, the less porous among NiCrAl-Bentonite coatings, show a reasonable relative difference between the calculated and experimental data, i.e. Δ=-22%: the calculated thermal conductivity is a little lower that the measured one. Moreover, when considering the standard deviation, one can confirm that the calculated and experimental thermal conductivities are in the same range. The graph concerning coating B even shows a smaller difference between the calculated and measured thermal conductivity: the calculated thermal conductivity is lower than the measured one with a relative difference of Δ=-10%. Moreover, considering the standard deviations, the results seem to match very well. However, the standard deviations of the calculated thermal conductivities are quite high.

II.3. Comparison of experimental and calculated properties

Comparing coatings A and B, the same conclusion than for the mechanical properties can be made, i.e. some reasonable differences are noticed between the computed and experimental thermal conductivities.

Coating C shows much larger discrepancies between experimental and calculated data. In particular, the calculated thermal conductivity is much higher than the measured one and the relative difference between the two values is very high, i.e. Δ=+640%. The difference between experimental and calculated data is a little lower for coating D (i.e., the most porous CoNiCrAlY-BN-polyester coating). The relative difference reaches -70% which remains high (but lower than the relative difference concerning coating C). Considering the standard deviation, it reaches 20% and 76% of the average calculated thermal conductivity for coatings C and D respectively.

Even by considering the standard deviations, the calculated properties for those two coatings (C and D) do not correspond to the measured ones. As explained previously this may be due to a wrong choice concerning the magnification level, inducing a misrepresentation of the real coating microstructure. Moreover, for the thermal conductivity, these discrepancies could also be due to the assumption made for the input properties applied for the modeling methodology (see Chapter 3).

Unfortunately, due to the difference of strategy existing between the two "work-packages" of the E-BREAK European project, thermal conductivity measurements were not performed for the third material (i.e., YSZ-polyester coatings used as abradable in the turbine section) so that no correlation can be made between calculated and experimental properties for this material.

III Analysis of mechanical and thermal properties

III.1. General observation

In thermal spraying, the intrinsic properties of the sprayed materials may be affected during the deposition process (i.e., total or partial melting, resolidification, formation of cracks, etc.).

According to experimental measurements, the Young's modulus and thermal conductivity of NiCrAl-Bentonite coating A (the less porous with a void rate of 35%) are higher than those measured for coating B (the most porous with a void rate of 40%). On contrary, the computed Young's modulus and thermal conductivity are very similar for these two coatings: this results is obviously obtained due to the small differences observed concerning the coating microstructures for the two sets of parameters (the micrographs are not so different for coatings A and B so that the calculated properties are similar).

Concretely, the calculated thermal conductivities are slightly lower than experimental measurements for coatings A and B. However, the calculated Young's modulus of coating B is a little higher than the experimental value, and inversely for coating A.

However, considering the standard deviation, the calculated and experimental results are in the same range, Figure 4-8 and Figure 4-11. the calculated and experimental properties are finally in quite good accordance and confirm that the thermal conductivity is low and the module also. Moreover, experimental measures performed on this material also indicate that the coatings behave like a brittle material under tensile solicitations (i.e., rapid failure after the elastic limit is reached). As depicted in chapter 2, those conclusion have been made by the partner IFAM-DD, and the stress and strain curves cannot be given in the manuscript.

The conclusions for CoNiCrAlY-BN-polyester coatings are quite different. According to the micrographs, coatings C and D are very different one from the other. Coating C is much less porous than coating D and its void network is much less connected. For more details the overall void rate of coating D reaches 50% whereas that of coating C is 30% only. The behavior of this material is strongly different regarding to experimental and calculated results. For example, the experimental Young's modulus and thermal conductivity are higher than the calculated data for coating C.

Considering the calculated properties the same tendency can be observed for both the thermal conductivity and the Young's modulus, with strong differences between the calculated Young's modulus and thermal conductivity for the two coatings (due to the large differences between the 2 microstructures). Nevertheless, experimental and calculated properties do not match at all even when considering the standard deviations. According to experimental data, the two coatings seem to have good thermal insulation performances. On contrary, the calculated data do not reveal as well this result for coating C, for which the thermal conductivity is quite high in comparison with that of coating D.

Concerning YSZ-polyester coatings, a coherent variation of experimental erosion resistance in function of the coating microstructure can be observed (i.e. evolution of the erosion resistance with porosity). This tendency is also valid for the calculated properties of the microstructure having coarse voids. Indeed one of the result can seem incoherent (i.e., the Young's modulus and thermal conductivity calculated for the densest coating without coarse porosities are lower than those calculated for other YSZ-polyester coatings). The modeling methodology allows discriminating the different coatings, even for those which seem similar to the reference coating. However, even if the results seem in accordance with the evolution of the microstructure, the use of a single magnification level certainly induces the same problem as that described for the CoNiCrAlY-BN-polyester coatings.

Indeed, YSZ-polyester coatings as CoNiCrAlY-BN-polyester coatings show higher index of complexity (see chapter 3 for details). Due to the coarse porosity network provided by the polymer contained in the powder, it becomes difficult to find an adequate magnification level to study these coatings. Indeed, the use of a high magnification level permits to see the microstructure details but does not allow providing an accurate representation of the coarse porosity distribution. On the contrary, the application of a low magnification level permits to have an accurate representation of the coarse porosity network but the microstructure details can then not be observed with sufficient precision. Finally the initial magnification level chosen for the previous calculated data was a compromise to observe as well coarse and small features, but this choice induces an underestimation of the influence of the smallest features on the coating properties, due to the network of small microcracks that can be observed at high magnification level only. Unfortunately there are no experimental results for the Young's modulus and thermal conductivity to confirm those observations for the YSZ-polyester coatings.

III.2. Issues and drawbacks

It was explained in Chapter 3 that modeling was implemented for two different NiCrAl-Bentonite coatings produced with two sets of thermal spray parameters. It was explained in chapter 1 that the material properties depend on the coating microstructure so that two coatings with similar chemical compositions can show different effective properties due to different microstructures. Here, experimental results illustrate these differences of properties in function of the microstructure.

However, the calculated results do not show clear differences between those two coatings (namely coatings A and B). On contrary, the calculated properties for CoNiCrAlY-BN-polyester and YSZpolyester materials are strongly different for the different coatings (manufactured with different sets of parameters). This result clearly illustrates the role of the microstructure: whereas coatings A and B

show similar microstructures (according to the micrographs), the coatings obtained for CoNiCrAlY-BN-polyester and YSZ-polyester materials are strongly different depending on the considered set of parameters (still according to the micrographs). For this reason, the model is not able to predict differences between the properties of coatings A and B, whereas it was possible for other materials. In other words, the model is able to predict differences between coatings if their microstructures are different: it is not the case for coatings A and B so that the computed properties are quite similar for these 2 coatings. For this reason, CoNiCrAlY-BN-polyester coatings show strong differences in their properties according to both experimental and calculated data (the microstructures are much more different between coatings C and D than between coatings A and B). The YSZ-polyester coatings also show strong differences of microstructure depending on the thermal spray parameter set.

It can be noticed that the fractions of the matrix and porosity are quite similar for both NiCrAl-Bentonite coatings (namely coatings A and B). Precisely, the porosity and the matrix both represent between 35 and 40 % (volume fraction estimated from 2D micrographs), with a widely connected void network. With this kind of composition, it seems difficult to observe obvious differences for the calculated properties. The standard deviations of the calculated properties are high. The modeling method is based on calculations performed on two-dimension micrographs, which explains easily the difference with experimental results. For CoNiCrAlY-BN-polyester coatings, the standard deviations of the calculated properties are also high and strongly different between calculated and experimental properties. Besides, the comparison of the experimental and calculated properties of CoNiCrAlY-BNpolyester coatings leads to different interpretation for microstructure D and C. Indeed, image of microstructure D used to calculated thermal conductivity and Young's modulus, show a zero connectivity of the matrix from the bottom to the top of the image inducing low values comparing to experimental ones. On contrary, for microstructure C, the calculated Young modulus and thermal conductivity are higher than experimental ones and this can be due to a misrepresentation of the details of the microstructure. It seems that at this magnification level, the details of the microstructure are misrepresented and the model does not take into account their influence on the coating properties.

However, due to the important standard deviation of the calculated properties, it seems that at this magnification level, the largest features of the microstructure as well as the connected porosity network are also misrepresented. The void distribution and their connectivity in the coatings are not really well considered. Using a higher magnification level, should resolve the problem of the misrepresented details, but on the contrary would increase the misrepresentation of the biggest microstructural features, leading to a higher standard deviation of the calculated properties. Using a lower magnification level, should resolve the problem of the misrepresentation of the biggest microstructural features (the largest globular pores) and decrease the standard deviation of the calculated properties, but on the contrary would increase the misrepresentation of the small-scale microstructural details (thin cracks), inducing an overestimation of the calculated properties. In both cases, the change of the magnification level is hence not a good way to improve the modeling method.

Other options have to be considered.

Considering the similar size of the microstructural features between CoNiCrAlY-BN-polyester and YSZ-polyester coatings, the same interpretation can certainly be made. Even if experimental measurements of the Young's modulus are not available for YSZ-polyester coatings, it is possible to assume that the initial choice of the magnification level is certainly not well adapted for applying the model on these types of microstructures. This assumption is confirmed by the large standard deviation of the calculated properties.

III.3. Solution

The solution submitted in this manuscript is to apply the modeling method at two different magnification levels; one representing correctly the microstructure details and the second one representing the global void network distribution in the coating.

This method will consist in taking into account the influence of smallest microstructural features on the properties of the coating matrix (i.e., NiCrAl, CoNiCrAlY and YSZ) thanks to the first magnification level (the highest one). The so-calculated matrix properties will then be taken as input properties of the matrix for a second series of calculations performed on micrographs at a lower magnification level. This lower magnification level will be chosen to observe the coating at a larger scale, and probably over the total coating thickness. This second magnification level will thus allow taking into account the influence of the largest microstructural features on the coating properties.

Obviously, the results obtained with this two-scale modeling method will be compared to the results obtained previously with the more conventional method consisting in applying a single magnification level.

IV 2-scale modeling method

The description of the microstructures in chapter 3 and the first results provided in this chapter highlight the necessity to improve the modeling methodology described in chapter 2. Indeed, the description of the microstructure for each coating nature (each material) shows increasing levels of complexity for NiCrAl-Bentonite, CoNiCrAlY-BN-polyester and YSZ-polyester materials respectively. Due to the large range of the size of the microstructural features, several magnification levels were required to observe the coatings as precisely as possible, as described in Table 3-2 .

Concerning NiCrAl-Bentonite coatings, a single scale was considered to observe all details of the coating microstructure (i.e., the two phases, the void network and the cracks). This single scale was considered as suitable because of the powder type (chemically clad powder) and thermal spray process (flame): in fact, the initial structure of the powder is still present in the coatings (bentonite particles within a metal matrix with large pores) and the metal matrix does not contain many cracks. It is certainly for this reason that the measured and calculated properties are in quite good accordance for this material. Concerning CoNiCrAlY-BN-polyester and YSZ-polyester materials, several scales were required to describe correctly the shape and the size of the smallest microstructural features (i.e., such as thin cracks in the metal or ceramic matrix) and the distribution of the largest ones (i.e., such as large globular voids for example). Those remarks could explain the differences noticed between experimental and calculated properties. Polyester microstructures. Indeed, the use of a high magnification level permits to see the microstructure details but does not allow providing an accurate representation of the coarse porosity distribution. On the contrary, the application of a low magnification level permits to have an accurate representation of the coarse porosity network but the microstructure details cannot be observed with sufficient precision. Those features (small-scale details) have an influence on the coating properties and have to be taken into account more precisely during the modeling procedure. To determine the influence of all microstructural features on the coating properties, the use of calculations performed on micrographs captured at two different magnification levels could be a solution. The improvement of the modeling methodology will be described here and will be named "2-scales modeling" approach.

Most of the input information studied in chapter 3 are still suitable (e.g., values of the intrinsic properties of the phases). Some others have to be conducted again (e.g., such as the representative elementary volume). However, the methodology to determine the properties is still the same as that described before. The first stage of the new modeling method consists in calculating the properties of the matrix (metal alloy or ceramic phase) of the abradable coating. Micrographs of the matrix have to be captured at a high magnification level. This allows considering the influence of microstructural details of the matrix on its properties (i.e., smaller coarse porosities + thin cracks network). The properties of the matrix calculated with this first stage of the modeling are then used as input properties for the second stage of the modeling method. The coating micrographs used for this second stage must be captured at a much lower magnification level in order to observe the coating in its overall. This allows taking into account the influence of the largest pores network on the coating properties.

IV.1. 2-scale modeling methodology

For example, considering the case of YSZ-polyester coatings, calculations performed on micrographs with a high magnification level allows taking into account the influence of small-scale structural details with a resolution limit of 0.11 µm for Figure 4-14. Thus, the first stage of the model consists in performing calculations of the properties on micrographs zooming in the YSZ phase in order to take into account the influence of the network of inter and intra lamellar voids. For this first stage, the input data used for the matrix phase and voids correspond to the bulk properties of YSZ and voids (see . The modeling methodology is conducted as described in chapter 2. The same strain and boundary conditions are applied to the boundary edges of the considered coating micrograph. Figure 4-16 is a schematization of the boundary conditions applied to the coating micrographs to determine the corresponding Young's modulus by applying the 2-scale modeling method. The strain is imposed at the top edge boundary (by imposing Uy=10 -4 .NY.DP (1) for the nodes of this edge), whereas the vertical component of the displacement of points pertaining to the bottom edge is set to zero (as for the horizontal component of the displacement for points pertaining to the left edge). A quite similar procedure is conducted to calculate the coating thermal conductivity, except that temperatures are imposed instead of displacements, and that the left and right edges are both free for the thermal calculations.

(

NY represents the number of pixels along the vertical direction and DP represents the size of each pixel, so that the product NY.DP represents the thickness of the micrograph (its vertical dimension). Two different magnification levels are used to determine the properties of each coating. For this, it is mandatory to determine the representative elementary volume for those two magnification levels.

To do this, the same methodology than the one described in chapter 2 and 3 is conducted:

 Determination of the suitable magnification level by studying the variability of the void content at three different magnification levels.

 Selection of the number of images by studying the variability of mechanical and thermal

properties at the magnification level determined before For the 2-scale modeling method, this procedure must be applied twice, firstly for the highest magnification level. Three different magnification levels are hence chosen in order to observe only the matrix phase of the microstructure (the largest voids must be avoided when capturing the different pictures). Secondly, the same procedure is applied to select the low magnification level. Different magnification levels that allow observing the global coating are studied. An example of the porosity variability is given at the two magnification levels of the 2-scale modeling method in Figure 4-17.

60 magnification levels, but also allows avoiding the necessity to rework too much the images, see From the entire study about determination of the representative elementary volume, all magnification levels and number of images have been determined and the results obtained are provided in Table 4-15 below. For YSZ-polyester, ten images captured by SEM were considered at a similar magnification level of ×800, and seven images captured by SEM were considered at a magnification level of ×80. These magnification levels and number of images will thus be retained for the 2-scale modeling method.

IV.2.2. Input properties

Another problematic detailed in chapter 3 was the choice of the input properties to consider in the modeling methodology. Some of the properties found in the literature indicate a large range of possible values depending on the material elaboration or real composition, which are sometimes unknown information. The solution applied in chapter three was to evaluate the influence of the input thermal conductivity of the phases on the calculated coating effective property. This principle has been applied to NiCrAl-Bentonite coatings but also to CoNiCrAlY-BN-polyester ones. The conclusion was that the influence of the variation of the input data (thermal conductivity of bentonite and BN-oxide phases) was low on the calculated extrinsic thermal conductivity of the coating. This conclusion is valid for the conventional modeling method but has to be checked for the 2-scale modeling process.

The same procedure as that described in chapter 3 was thus applied but by using the two magnification levels. The input properties of each phase in the different coatings are given in Table 3-14, except for the properties of the grey phase, which were varied in between 5 W.m -1 .K -1 to 90 W.m -1 .K -1 . *nanoindentation measurements, ** Hypothesis.

The extrinsic properties of the coating calculated with this variation of the input properties are plotted on Figure 4-19 and compared to the results calculated with the conventional modeling method.

The value are normalized by the smallest thermal conduction calculated. is changed. Whatever the modeling method applied, the variation of the input thermal conductivity has the same influence on the calculated effective thermal conductivity. As described in details in chapter 3, it is possible to conclude that this influence is not significant comparing to the gap existing between the input thermal properties applied here (i.e. the input thermal conductivity is in the range 20-90 W.m -1 .K -1 ). Indeed, a variation of the input thermal conductivity of the grey phase from 20 W.m - 1 .K -1 to 90 W.m -1 .K -1 induces a variation from 2.7 to 3.8 of the computed effective thermal conductivity, considering the conventional modeling, and from 1.7 to 2.7 when considering the 2-scale model. However, for a lower value of the input thermal conductivity of the grey phases, the calculated effective thermal conductivity strongly decreases from 1.7 to 1.0 W.m -1 .K -1 . The range existing for the computed effective conductivity is quite low, considering the high range of value applied as the input thermal conductivity of the grey phase. However, the influence has to be considered in the standard deviation.

IV.3. Conclusion

An application of the 2-scale model is a way to circumvent modeling issues described for CoNiCrAlY-BN-polyester and YSZ-polyester coatings. The method is similar to the "conventional" one described in chapter two, but is applied on images captured at two different magnification levels.

Considering NiCrAl-bentonite coatings, the results obtained with the "conventional" modeling method are satisfactory and a good match was obtained between computed and experimental data.

However, the standard deviation of the calculated properties is quite high and one can note the difficulty to discriminate coatings A and B (manufactured with two sets of parameters) by the corresponding computed properties, due to small differences in the corresponding micrographs. In such conditions, it is obvious that applying the 2-scale modeling method could allow improving the computed properties and reducing the standard deviation of the calculated properties. Moreover it is difficult to consider two different magnification levels due to the difficulty to zoom in the matrix phase only. Indeed, due to the elaboration process of the powder (chemical cladding of the bentonite particles), the metallic matrix encloses the bentonite particles, so that zooming in the matrix phase would imply to work with a very high magnification level. In these circumstances, the corresponding high magnification level would provide a zoom on a few splats of NiCrAl only, Figure 4-20, which does not give relevant information on the matrix microstructure, but only on isolated lamellae forming the matrix. For this kind of materials, the only way to improve the results and to discriminate those highly porous microstructures would be to perform 3-D calculations on 3-D images of the microstructure. On contrary, it is much easier to zoom in the matrix phase for CoNiCrAlY-BNpolyester and YSZ-polyester coatings, and the corresponding matrices reveal much information about the embedded microstructural features. Some difference between those two microstructures can be observed at the two magnifications level. The difference of voids network have been highlighted with the use of the conventional model, but here it is also possible to observe the difference of microstructure existing between the matrix of those two coatings.

V.1.1. Mechanical properties calculated with 2-scale modeling The Young's modulus was calculated using the 2-scale modeling strategy for the two CoNiCrAlY-BN-polyester coatings (i.e., previously referenced C and D). In addition, the calculations were performed longitudinally (i.e., in the coating plane) and transversally to the coating structure (i.e., in the direction of the thickness of the coating), in order to investigate the anisotropy of the coatings. The Only the Young's moduli calculated longitudinally will be compared to the experimental results.

Indeed tensile tests and RFDA provide the Young's modulus in the longitudinal direction of the coating. The mismatch between experimental and computed Young's moduli corresponding to coating C is strongly decreased when using the 2-scale modeling method as shown in The results differences between the two numerical approach and the experimental results will be discussed in part V.I.3 of this chapter.

A quite strong anisotropy of the calculated Young's modulus can be noticed for both coatings C and D. This anisotropy seems even more pronounced for coating C (i.e., the Young's modulus calculated in the through-thickness direction is 10% lower than that computed for the in-plane direction).

V.1.2. Calculated thermal conductivity with the 2-scale modeling The relative difference between the calculated and experimental data was evaluated and is provided in Table 4-21. The relative difference between the measured and computed data is decreased when using the 2scale modeling method, but this decrease is not sufficient to get data in accordance with experimental data.

Considering coating D, the conclusions are quite different. Indeed, the calculated thermal conductivity is higher for the 2-scale model (in comparison with conventional modeling) so that it becomes closer to the experimental value. However the calculated thermal conductivity remains lower than the experimental one (i.e., 0.56 for 2-scale modeling method against 0.33 for the conventional model to be compared to the reference value of 1.0 measured experimentally). Those results are represented in Figure 4-24. Another difference between the two modeling methodology is the strong decrease of the standard deviation of the computed thermal conductivity. The coefficient of variation is decreased from 75%

for the conventional modeling to 9% for the 2-scale modeling. On contrary with the case of the Young's modulus, the thermal conductivity measurements are much more reproducible for this microstructure.

For the both microstructures, the calculated anisotropy of the thermal conductivity is quite low.

The difference in the thermal conductivity calculated for the longitudinal and transverse directions is almost inexistent.

V.1.3. Discussion

Applying the 2-scale modeling methodology to compute the Young's modulus of coating C really allows taking in better consideration the influence of several microstructure features on the coating properties. The 2-scale modeling allows being more representative of the real coating microstructure, which provides better results in comparison with experimental measurements of the Young's modulus.

The computed Young's modulus is quite close to experimental data.

On contrary a limitation of the modeling method seems to be reached for coating D, the most porous one (50%) with a highly connected porosity network, as well for the conventional and 2-scale methods. The calculated Young's modulus is much different from the experimental measurement, and remains quite similar with both modeling methods. This result is certainly correlated to the misrepresentation of the connectivity of the CoNiCrAlY phase when using 2D images of the microstructure for coating D. The influence of the porosity distribution on the computed Young's modulus is then overestimated. The only improvement that may be mentioned for the 2-scale

V.2. YSZ-polyester coatings

V.2.1. Young's modulus calculated with the 2-scale modeling The Young's modulus was estimated on the basis of calculations performed on the five YSZpolyester coatings with different microstructures. The calculations were computed with the 2-scale modelling method for the longitudinal and transverse directions. The results were normalized in function of the transverse (i.e., through-thickness) Young's modulus calculated for the reference coating with the conventional modeling method. The results are listed in Table 4-22. This normalization allows comparing as well the influence of the coating microstructure between one coating and another, and the influence of the 2-scale modeling compared to the conventional modeling method. The evolution of the Young's modulus is in accordance with that of the coating microstructure.

Indeed, a decrease of the porosity level provides an increase of the calculated Young's modulus as well for the transverse and longitudinal directions. The Young's moduli calculated with the 2-scale modeling are much lower than those calculated by conventional modeling, except for the densest coating with no coarse porosities. Indeed, for this coating (the last one) the Young's modulus calculated is quite similar whatever the modeling methodology. Due to the absence of coarse porosities, the size range of microstructural features is thus reduced. In fact, there is no need to apply the 2-scale modeling method to determine this coating. The only interest of using the 2-scale modeling for this coating is the application of the same methodology for all coatings, in order to compare all the calculated values between each other. To improve the estimation of the coating properties of such coating (the last one), only one magnification level is enough, but at a lower scale to take into account more precisely the influence of the microstructural features.

The Young's moduli calculated longitudinally and transversally are compared on Figure 4-25 and in Table 4-22. To conclude, the Young's moduli computed for both directions are sensitive to the porosity network and evolve as expected with the microstructure evolution. However the computed values are different for each direction, which indicates that YSZ-polyester coatings show an anisotropic effective Young's modulus.

V.2.2. Thermal conductivities calculated by 2-scale modeling

The effective thermal conductivities of YSZ-polyester coatings were also calculated and normalized in function of the transverse thermal conductivity calculated with the conventional modeling method for the reference coating. The results are listed in Table 4-12 and are represented on Figure 4-26. As expected, the thermal conductivity increases for decreasing porosity levels, as well for the longitudinal as transverse directions. The evolution tendency of the thermal conductivity with the coating microstructure is the same for both modeling methods. The 2-scale modeling provides a decrease of the calculated thermal conductivity compared to the conventional modeling method. It can be noticed that the most porous coatings are more sensitive to the application of the 2-scale modeling.

Indeed, the corresponding difference between the thermal conductivity calculated with the conventional and the 2-scale methods is more important in comparison with the results computed for the less porous coatings. Concerning the densest coating without any coarse porosity, the calculated thermal conductivity is quite similar whatever the considered modeling method, as it was already observed for the Young's modulus.

The thermal conductivities calculated longitudinally and transversally are compared on Figure 4-26 and in Table 4-23. To conclude, the thermal conductivities computed for both directions are sensitive to the porosity network and evolve as expected with the coating microstructure. On contrary to the Young's modulus, the differences between the longitudinal and transverse effective thermal conductivities are lower. The thermal conductivities computed for both directions are quite similar, which indicates that YSZ-polyester coatings exhibit a low anisotropy in terms of thermal conductivity.

Even if there were no experimental measurements performed for the YSZ-polyester coatings in the frame of this work, some data are reported in the literature for similar coatings. Those experimental results are listed in Table 4-24.

V.2.3. Grey phase issue

The previous results calculated with the conventional modeling method were computed with estimated input properties of the grey phase. One can remember that it was chosen to consider the same properties as those of the matrix for the grey phase (i.e., non-melted part of the powder with a structure composed of small agglomerated particles). However, it could also as well have been chosen

to implement properties closer to the porosity properties (or intermediate properties which corresponds more certainly to the reality).

One arising question is: what is the influence of the input properties considered for the grey phase on the coating effective properties?

Figure 4-27 shows the calculated transverse thermal conductivities and Young's moduli computed with both modeling methods for the reference coating. As expected, considering that the grey phase has the same properties as the porosity, provides a strong decrease of the calculated coating properties. For the two calculated properties (Young's modulus and thermal conductivity), the use of the 2scale modeling method provides lower extrinsic properties than the use of the conventional model (1scale modeling). This decrease may be explained by the greater consideration of the crack network when considering the 2-scale modeling. Indeed this second method permits to take more precisely into account the microstructure details of the coating. The properties calculated with the 2-scale modeling method are affected by the microstructure of the ceramic phase during the first stage of the method.

Moreover, the properties are also affected, in the second stage of the method, by the contribution of the amount of porosity in the coatings. The use of the conventional modeling method induces that details like the smallest pores or intra and inter-lamellar cracks are misrepresented and their effect on the Young's modulus and on the thermal conductivity are underestimated. The calculated upper and lower limits provide a large range of possible values for the coating properties. Indeed, for the single scale modeling, the thermal conductivity calculated can differ of more or less 2 W.m -1 .K -1 and more or less 100 GPa for the Young's modulus. For the 2-scale modeling, the thermal conductivity calculated can differ of more or less 1 W.m -1 .K -1 and more or less 30 GPa for the Young's modulus. This solution (i.e., estimation of an upper and lower limit) permits to compare the different coatings, but due to the large range of the data, no clear idea of the real effective coating properties can be obtained.

To determine the real properties of the grey phase, nanoindentation may be a good way for the Young's modulus. However the question of the thermal conductivity seems more delicate.

Nevertheless, the range is not no large concerning thermal conductivity, which indicates a lower sensibility.

V.2.4. Discussion

The conventional and 2-scale modeling methods allow discriminating the different coatings one from each other. Even in the absence of experimental measurements for those coatings, the results obtained with the 2-scale modeling allow obtaining thermal conductivities in the same range as experimental data found in the literature for coatings of similar nature.

A tendency can be observed for both the thermal conductivities and the Young's moduli which both decrease with the increase of the porosity level. However, for two coatings with a same porosity level the computed thermal conductivities and Young's modulus are not perfectly similar due to the complex coating architecture manufactured with the thermal spraying process. Indeed, as well the voids size, shape, orientation and distribution influence the calculation of the properties. To really obtain some correlation between the calculated properties and the coating microstructure, all those information on the coating phases and pores are required and play a role. For this, image analysis protocols have been developed in other studies [126,[START_REF] Russ | Practical Stereology[END_REF][START_REF] Antou | Quantifying Thermal Spray Coating Architecture by Stereological Protocols: Part II. Key Points to be Addressed[END_REF][245][START_REF] Pons | Morphological analysis of pharmaceutical powders[END_REF][START_REF] Pons | Particle morphology: from visualisation to measurement[END_REF] and should be adapted for this kind of coatings.

For all coatings, except the densest one with no coarse voids, the 2-scale modeling method gives rise to a decrease of the calculated extrinsic Young's modulus and thermal conductivity. It also strongly decreases the corresponding standard deviations. However, for the densest coating with no coarse voids, only few differences between the extrinsic Young's modulus and thermal conductivity calculated with the conventional and the 2-scale modeling are observed. It appears that the voids size do not justify the application of 2-scale modeling to provide representative data. Indeed, as there is no coarse void in this coating, application of a low magnification level to observe their influence on the global structure of the coating is not mandatory. However, in order to compare several coatings of a single material it is better to use the same modeling methodology.

From this observation it may be interesting to determine a criteria on microstructural features allowing estimating the interest (or not) to use the 2-scale model. For this the equivalent "surface diameter" of voids may be used. In chapter 2, an image analysis method was described to quantify the voids size. The equivalent diameter of voids, Deq(S) is a common descriptor chosen to compare a coating to another. It permits to consider voids whatever their shapes. The several classes determined are reminded in Table 4-25. versus their equivalent diameter. This allow comparing the repartition of each void class between the three coating. The void rate of each class will be also given in Table 4-26, which allow representing the predominant voids class in each coating. Those two information will allow determining a limit criterion concerning the utility of 2-scale modeling. The densest coating with no coarse voids is mainly constituted of Fine and Medium voids size. There The difference between the three coatings, concerns mainly the area fraction represented by the coarse and very coarse voids. Indeed, the most porous coating is composed of coarse and very coarse pores, but also of fine pores. From the previous results, it was shown that 2-scale modeling is mandatory to provide results taking into account all sizes of the microstructural features. For the less porous coating, 2-scale modeling is also mandatory. Even if the coarse and very coarse voids are fewer than for the most porous coating, they still exist and the use of conventional 1-scale modeling limits the information on the entire porous network of this coating. However, concerning the densest coating, most of the voids are in the finest to medium size range. In this case, there are no coarse voids, and 2scale modeling is not really mandatory to obtain results taking into account all the features existing in the coating.

Size class of voids

The minimum equivalent diameter of voids measured for all coatings is about 0.7 µm (corresponding to one isolated pixel), which corresponds to 1 pixel. In this example, the densest coating with no coarse porosity may be considered as a limit of the interest in the use of 2-scale modelling. For this coating, there are few voids with an equivalent diameter higher than 30 µm (only one pore was higher than 30µm on the evaluation of pore size on 15 images), which is forty times higher than the minimum void size measured. It is possible to consider this value as a limit criterion of the interest of the 2-scale modeling method. The use of the 2-scale model is thus required for coatings with a maximum size of voids higher than forty times the minimal void size.

V.3. Conclusion about the use of the 2-scale modeling methodology

To conclude concerning the results calculated with the 2-scale modeling, the calculated effective properties (Young's modulus and thermal conductivity) are lower than the ones calculated with the conventional modeling method whatever the coating material. As expected, the influence of the smallest microstructural features is better taken into account with 2-scale modeling and the difference of properties calculated between one image and another (of a same coating) is lower (lower variability of the computed results). This methodology is more appropriate to calculate the thermal conduction and Young's modulus of the real abradable coating microstructure than the conventional modeling method.

Concerning the CoNiCrAlY-BN-polyester material, the computed Young's moduli are in better accordance with experimental results than the calculated thermal conductivities. However, due to the assumption made concerning the thermal conductivity to consider for the unknown grey phase embedded in the coatings, it is not really possible to conclude about the good or bad relevancy of the thermal conductivity results. However, concerning YSZ-polyester coatings, the calculated thermal conductivities are in the same range as the experimental thermal conductivity measured by Laser Flash analysis found in the literature for quite similar coatings.

From the results obtained for the YSZ-polyester coatings it was possible to determine a criterion concerning the microstructural features. This criterion will be use to estimate the interest (or not) of the 2-scale model for a given coating: according to this criterion, the interest of the 2-scale model is effective for coatings with a maximum void size forty times higher than the minimal void size, in terms of equivalent diameter. extrinsic properties. The Young's modulus and thermal conductivity for different constitution of YSZ and voids constituting the grey phases will be calculated with equations (VI-2) and (VI-3), and implemented in the model as input data for the dark grey phase areas.

The Young's modulus and thermal conductivity corresponding to different volume fractions of nano-YSZ grains and nano-voids can thus be estimated using these relationships and implemented in the model as input data for the dark grey phase areas. In practice, analytical equations different from those described in chapter 1 were applied. As shown by equation VI-2, the "like-parallel" model suggested for example by Voigt [START_REF] Voigt | Theoretische Studien über die Elasticitätsverhältnisse der Krystalle[END_REF] or Reuss [START_REF] Reuss | Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle[END_REF] was considered for the Young's modulus.

Eeff=x E1+y E2 (VI-2) (IV-3)

E
in which κ 1 is the thermal conductivity of continuous medium (YSZ nano grains) and κ 2 is the thermal conductivity of the dispersed spherical phase (the nano voids) having a volume fraction y.

However it is possible to assume that using those analytical equations will induce an overestimation of the properties of the grey-phase areas. The results are reported in Table 4-27.

VI.3. Influence of the composition of the dark grey phase areas on the computed output extrinsic properties.

As explained, the intrinsic properties of the grey phase areas are unknown. The influence of the properties of the grey-phases areas on the computed results has to be evaluated. It was thus chosen to conduct the modeling by changing the input properties of this phase according to data listed in Table 4-27. The corresponding calculations were performed for the reference coating and are represented on According to the normalized results, the calculated extrinsic Young's modulus of the coating is in the range 18-115 and the computed effective thermal conductivity is in the range 0.9 to 2.2 depending on the considered input data. The choice of the input data considered for this grey phase has hence a major influence on the calculated extrinsic properties of the coating.

VI.4. Discussion

Considering a third scale to determine the properties of the grey phase areas is not a good solution due to artefacts occurring during the sample preparation. Indeed, during the capture of images for a high magnification level the grey phase area will be misrepresented because of the defect.

For these coatings, in order to consider the influence of the coating microstructure without knowing the real properties of the grey phase, a capping of the calculated properties could be a better solution.

From the observation of the structure of grey-phase areas, it is possible to say that those phases are composed of spherical nano-grains of YSZ and nano-voids in quite similar volume fractions. To discriminate the different coatings, it is mandatory to calculate the extrinsic properties by implementing data of Table 4-27 for volume fractions of 1: 50% YSZ + 50% void, 2: 100% YSZ + 0% void and 3: 0% YSZ + 100% void. To discriminate the coatings, the properties calculated for those three volume fractions must be compared.

VII Conclusion

Three different abradable materials, NiCrAl-Bentonite, CoNiCrAlY-BN-polyester and YSZpolyester were studied both experimentally and numerically. The Young's moduli and thermal conductivities of different coatings (of these three materials) were estimated with two different modeling methods: the first model may be considered as conventional (calculations performed of micrographs of coating cross-sections at a single magnification level), whereas the second model may be considered as more advanced (based on calculations performed on images captured at two different magnification levels of the same coating). In this chapter, the relevancy of using one or the other modeling method was discussed.

For NiCrAl-Bentonite coatings which may be considered as the less complex structure among the three abradable natures, conventional modeling was found to provide a good accordance between the measured and calculated Young's modulus and thermal conductivities. However, the standard deviation was high for both calculated properties, and there was no possible discrimination between the two coatings (more or less porous in principle) by comparing the corresponding calculated properties, on contrary with experimental results. This result is related to the micrographs obtained for the two coatings, which were not sufficiently different, thus resulting in quite similar computed properties for the two coatings. Using 2-scale modeling could improve the consistency of the calculated properties by decreasing the standard deviation. However, due to the coating microstructure (resulting from the powder manufacturing process, i.e., chemical cladding of the bentonite particles), using a high magnification level to zoom in the matrix of the coating does not provide relevant information.

For CoNiCrAlY-BN-polyester coatings, the modeling method was found to be much more discriminant between the two coatings, for both the Young's modulus and the thermal conductivity.

However, the conclusion of the modeling method differs for the two coatings (more or less porous again but with larger differences), and depends on the considered property. For the less porous coating (porosity level of about 30%), the use of the 2-scale modeling method allowed providing a Young's modulus in accordance with experimental data (but a higher computed thermal conductivity). On contrary, for the most porous coating, the calculated Young's modulus and thermal conductivity are much lower than experimental data. For the most porous coating, the observed discrepancies are related to the presence of continuous cracks throughout opposite edges of the micrographs at the considered magnification level. Nevertheless, it was difficult to conclude concerning the use of the modeling method to determine the thermal conductivity, due to the presence of an unknown phase in the coatings (with unknown properties): the assumption made for the intrinsic properties of this phase was found to be sensitive on the results.

Tensile test

No data available because the test was performed by IFAM, partners of E-Break project. However for tensile tests, removing the coating substrate is mandatory which does not guarantee the success of the sample preparation. RFDA Moreover, the file generated for the modeling test can be used again to calculate other coating properties and to conduct calculations for the other direction. Only the boundary conditions applied to the picture are modified and it is not necessary to conduct again the three previous steps of coating preparation and image analysis.

When and why using the 2-scale modeling methodology?

The most important interest in the use of 2-scale modeling is a complex microstructure of the coating with large differences in between the range size of the different microstructural features inducing a difficulty to represent all of them with a single magnification level Using a 2-scale modeling allows taking into consideration the influence of all those microstructural features on the coating extrinsic properties, and leads to more realistic calculated properties, compared to using the conventional 1-scale modeling method. First the most important step is to determine an appropriate method for coating surface preparation, as describing in chapter 2. To choose the most appropriate preparation, the main interrogation concerns the sensitivity of the coating. Indeed, it will determine the choice of the cutting wheel, the cutting velocity, the type and methodology for the mounting as well as the polishing method. An inadequate methodology will induce microstructural degradation and will lead to a misrepresentation of the real microstructure. different magnification levels. One will be a high magnification level to represents the matrix details and the second one will be a low magnification level representing the global coating microstructure.

Then the modeling method is applied as described by the protocols details in chapter 2 for the conventional model. Concerning the 2-scale modeling method the protocols is detailed in chapter 4.

But before that, the intrinsic properties of each phase existing in the coating has to be determines, by literature research or experimental measurement on bulk material corresponding to the phase nature.

One main conclusion of this work is the following one: If the material complexity increases, the number of scale mandatory to describe the real microstructure increases. That is why, in this work a "multi-scale" modelling approach have been developed to determine abradable coating properties. This modeling method has been validated through comparison of the calculated values with those obtained experimentally. Table 4-29 lists the properties whose values are in agreement between the experimental measurements and numerical modeling.

YSZ-Polyester

Value found in literature 0.64 <κ< 1.31 W.m -1 .K -1 were in the same range that value calculated by the 2-scale modeling. However due to confidential aspect the real values cannot be given in this manuscript but were varying in between 0.7 and 1.90 W.m - 1 .K -1 from the most porous to the denser coating

The calculated thermal properties of YSZ-polyester coatings were in the same range that one's found in the literature but no experimental test were conducted on those microstructures. Young's modulus will be determine by RFDA and compare with the calculated values. But the modeling methodology allow differentiate the coating microstructure between each other.

One major difficulty of the modeling method is the material database properties. The typical example illustrating this problematic is the bentonite phases. Its bulk properties are difficult to determine only by literature research because its properties strongly depends on its real composition.

As well illustrated with the YSZ-polyester coating, this modeling method allows discriminating the coating performances among layers manufactured by different parameters. Moreover, a comparison between time mandatory to apply all the modeling process and time mandatory to conduct all the experimental measure has been made in the conclusion of chapter 4 and are listed in Table 4-28 .

There is no significant difference in term of time mandatory but, the significant interest concerns the by Halim Al Baida and presented the 20 th November 2015. Some experimental test such as, impact, indentation, ultrasound propagation and Archimedes' principle measures, have been applied on an AlSi-Polyester coating. Those results were used to propose a law of static and dynamic behavior in compression of the abradable coating. The static behavior law obtained, combined with experimental results were used to obtain a simulation of HR15Y hardness test. A comparison of the simulated values and the HR15Y experimental results has been made to validate the law of behavior. The results seem to be in good accordance between experimental and numerical HR15Y hardness. These results must be deepened.

On medium term basis, two perspectives can be considered:

 Behavior laws determined by the above mentioned method may be implemented in the model. This would allow calculating the properties in elastic-plastic field. The test to determine behavior law could be also improved to be conducted at different temperatures.

 One other perspective of this work could be to applied 3-Dimensions images modeling in order to allow determining by modelling the coating properties even if the coating porosity rate is equal or higher than 50%. Previous works to calculate the thermal conductivity of microstructures on the 3-dimensional images have been made in the laboratory (Jianghao Qiao thesis [START_REF] Qiao | Modélisation des propriétés thermomécaniques effectives de dépôts élaborés par projection thermique[END_REF]). It should be interesting to apply it to abradable coating to solve the problematic concerning tri-dimensional limitation with 2D modelling. However the resolution of tomographic tools may not permit to observe all the microstructural details and the time required to conduct the modeling is long. The model is already implemented to conduct the 3D modeling of thermal conductivity on abradable structures. One question is, is this would be only a simple adaptation of the modeling methodology? Or it would be mandatory to develop again the entire methodology?

On long term basis, two perspectives can be considered:

 Modeling the mechanical properties in 3-dimensions. The same issues than for determining the thermal conductivity must be taken into account. Contrary to the thermal conductivity the entire 3D model have to be implemented to determine mechanical properties of abradable coatings.

 Determining numerically the breaking point of the material. The problematic is to model the interface of a strongly heterogeneous microstructure by FEM. This should require knowing the properties of the interface (metallic/metallic or metallic/ceramic). For NiCrAl-Bentonite coatings some differences can be observed between the more porous and the less porous coating. Indeed the less porous coating shows higher displacement under the increase of temperature than the most porous coating. The coefficient of dilatation of the less porous coating is then higher.

Annex n°4-CoNiCrAlY-BN modelling of several thermal and mechanicals properties

The properties results presented here have been normalized by a same random coefficient. Those results have been performed with the high scale of the 2-scale modelling method on the less porous CoNiCrAlY-BN-Polyester coating. The results are an average of the properties calculates on 15 images of the coating. At the high magnification level, representing the CoNiCrAlY-BN matrix, the coefficient of thermal expansion is similar in both directions. A surprising result is the little difference of the coefficient of Poisson calculated in the both directions. Indeed the result expected was a similar value. However this difference is very low. Concerning the thermal conduction and the Young modulus, the values calculated show that this coating as an anisotropic behaviour.

To perform the calculation on the lowest scale of the 2-scale methodology, it was necessary to implement in the Ansys script the bulk modulus of each phase. It was not possible to realize this work before the end of the thesis.

Figure 0 - 1 :

 01 Figure 0-1: E-BREAK Objectives and organization. ................................................................................................

Figure 0 - 2 :

 02 Figure 0-2: Work Package description and zoom onto work package thesis context. ..........................................

Figure 0 - 3 :

 03 Figure 0-3: Abradable coating localization into Turbojet [5]. ...............................................................................

Figure 0 - 4 :

 04 Figure 0-4: Illustration of turbofan engines [7]. ....................................................................................................

Figure 0 - 5 :

 05 Figure 0-5 : Diagram of the Joule-Brayton cycle [8]. .............................................................................................

Figure 0 - 6 :

 06 Figure 0-6 : Operating temperature range of the several abradable families. .....................................................

Figure 2 - 4 :

 24 Figure 2-4: Surface of the YSZ-polyester powder (left-top), zoom on the surface of a single particle (left-bottom) and cross-section of the powder (right) observed by SEM. ...................................................................................

Figure 2 - 5 :

 25 Figure 2-5: Schematic of Laser flash analysis methodology (left), zoom on the sample interaction. ...................

Figure 2 - 6 :

 26 Figure 2-6: Characteristic thermogram obtained by flash method .......................................................................

Figure 2 - 7 :

 27 Figure 2-7: Photo of the experimental device (FlashLine TM 3000, Anter Corporation) ..........................................

Figure 2 - 8 :

 28 Figure 2-8: Pictures of the sample dimensions for laser flash analysis .................................................................

Figure 2 - 9 :

 29 Figure 2-9: Durometer type NR3D from Ernst .......................................................................................................

Figure 2 - 10 :

 210 Figure 2-10: Steps of Rockwell superficial hardness (figure modified from Peyraut et al [57]) ............................

Figure 2 -

 2 Figure 2-11: GE Erosion test bench (left) Erosion Sample (right) ..........................................................................

Figure 2 - 12 :

 212 Figure 2-12: Typical Load/displacement curves ....................................................................................................

Figure 2 - 13 :

 213 Figure 2-13: Photography of the NHT device of CSM instruments (left), indentation system graph (right). ........

Figure 2 -

 2 Figure 2-14 RFDA test to determine the Young modulus (Top) and the shear modulus (Bottom) ........................

Figure 2 - 15 :

 215 Figure 2-15: Tensil test specimen ..........................................................................................................................

Figure 2 - 16 :

 216 Figure 2-16: Schematic representation of different type pore networks with different attributes .......................

Figure 2 - 17 :

 217 Figure 2-17: Example of pull out damages occurring during sample preparation ................................................

Figure 2 -

 2 Figure 2-18: micro-sectioning process and figure taken from TSS Committee on Accepted Practices J.P. Sauer, « Accepted Practices of Thermal Spray Technology» [183] .....................................................................................

Figure 2 - 19 :

 219 Figure 2-19: Pictures of the mounting process ......................................................................................................

Figure 2 - 20 :

 220 Figure 2-20: Cross-section micrograph of a NiCrAl-bentonite coating (top-left), zoom in the coating microstructure to observe the grey level of pixels (top-right), equalized histogram of the micrograph (bottomleft), histogram of the micrograph after double thresholding (bottom-right) ....................................................

Figure 2 - 21 :

 221 Figure 2-21: Example of the effect of morphological operators of Image J software. Figure coming from Image-J guide. ..................................................................................................................................................................

Figure 2 - 22 :

 222 Figure 2-22: Selection of REV for periodic structure (left), representation of a random structure (right). .........

Figure 2 - 23 :

 223 Figure 2-23: Overview of a typical YSZ-polyester coating structure (Optical microscopy). .................................

Figure 2 - 24 :

 224 Figure 2-24: Magnified view of a typical YSZ-polyester coating structure (FESEM). ...........................................

Figure 2 - 25 :

 225 Figure 2-25: Very high zoom into the coating microstructure.............................................................................

Figure 2 - 26 :

 226 Figure 2-26: Void content variability in function of the number of images to determine a REV [191] ...............

Figure 2 - 27 :

 227 Figure 2-27: Distribution of the void fraction in the coating versus their surface equivalent diameter..............

Figure 2 - 28 :

 228 Figure 2-28: Steps of the modeling routines .......................................................................................................

Figure 2 - 29 :

 229 Figure 2-29: Conventional modeling methodology .............................................................................................

Figure 2 - 30 :

 230 Figure 2-30: Example of SM 2640 coating micrographs used for modeling........................................................

Figure 2 - 31 :

 231 Figure 2-31: Description of the boundary conditions representing a vertical displacement load applied on the computational domain (image) to predict the through-thickness effective Young's modulus of the coating. ...

Figure 2 - 32 :

 232 Figure 2-32: Boundary conditions to estimate the longitudinal effective Young's modulus of the coating ........

Figure 2 - 33 :

 233 Figure 2-33: Boundary conditions representing a thermal difference load applied on the computational domain (image) to predict the coating effective thermal conductivity in the through-thickness direction. ....................

Figure 3 - 1 :

 31 Figure 3-1: Summary of chapter 3. ......................................................................................................................

Figure 3 - 2 :

 32 Figure 3-2: Scheme of the numerical modeling methodology. The present chapter refers to the first step. ......

Figure 3 - 3 :

 33 Figure 3-3: Microstructure of NiCrAl-Bentonite, CoNiCrAlY-BN-polyester, and YSZ-polyester by OM (magnification: ×25). ...........................................................................................................................................

Figure 3 - 4 :

 34 Figure 3-4: Abradable coatings microstructure captured by SEM (at two different magnification levels: low level corresponds to ×150 and high level corresponds to ×350, x800 and x500 for NiCrAl-Bentonite, CoNiCrAlY-BN-Polyester and YSZ coating respectively). .............................................................................................................

Figure 3 - 5 :

 35 Figure 3-5: EDS analyses on CoNiCrAlY-BN-polyester coating.............................................................................

Figure 3 - 6 :

 36 Figure 3-6: Image taken by SEM in SE mode. a) zoom onto the grey phase of the coating. b) zoom onto the powder surface. ...................................................................................................................................................

Figure 3 -

 3 Figure 3-7: a) typical morphology of an APS YSZ coating. b) Morphology of an YSE-Polyester coating .............

Figure 3 - 8 :

 38 Figure 3-8: Microstructures of the same YSZ-Polyester coating at several magnification levels ........................

Figure 3 - 9 :

 39 Figure 3-9: YSZ-Polyester coating -Variability of the coating void content versus the number of images at different magnifications. .....................................................................................................................................

Figure 3 - 10 :

 310 Figure 3-10: Voids variability at a magnification level of ×150 on YSZ-microstructure.......................................

Figure 3 -

 3 Figure 3-11: YSZ-Polyester coating -Variability of the coating Young's modulus or thermal conductivity versus the number of images at different magnifications. ............................................................................................

Figure 3 - 12 :

 312 Figure 3-12: Image processing protocol applied on SEM images representative of the coating structure. As a demonstrative purpose, the protocol is applied here on the Bentonite phase embedded in a NiCrAl-Bentonite coating structure. ................................................................................................................................................

Figure 3 - 13 :

 313 Figure 3-13: Reconstruction step of coating structures by adding binary image representative of each identified feature. As a demonstrative purpose, the protocol is applied here on the Bentonite phase and voids embedded in a NiCrAl-Bentonite coating structure. The complementary feature, in white, represents the NiCrAl matrix. .

Figure 3 - 14 :

 314 Figure 3-14: Example of a distribution of grey levels in a captured image of a coating structure where each type of feature is spatially represented by pixel colors covering a range of values. ...................................................

Figure 3 - 15 :

 315 Figure 3-15: Illustration of image transformation procedure applied for YSZ-polyester coating. ......................

Figure 3 - 16 :

 316 Figure 3-16: Schematization of the several steps of the homemade program ...................................................

Figure 3 - 17 :

 317 Figure 3-17: Influence of input thermal conductivity of the grey phase (CoNiCrAlY-BN-polyester coating) onto the extrinsic coatings properties ...............................................................................................................................

Figure 4 - 1 :

 41 Figure 4-1: Summary of chapter 4. ......................................................................................................................

Figure 4 - 2 :

 42 Figure 4-2: Schematization of the numerical modeling methodology. ...............................................................

Figure 4 - 3 :

 43 Figure 4-3: Microstructures of two NiCrAl-Bentonite coatings, A) the less porous and B) the most porous. ......

Figure 4 - 4 :

 44 Figure 4-4: CoNiCrAlY-BN-polyester coating microstructure, C) the less porous and D) the most porous ..........

Figure 4 - 5 :

 45 Figure 4-5: HR15Y and erosion resistance normalized values function of YSZ-Polyester microstructure. ...........

Figure

  Figure 4-6: Representation of the porosity network difference between one image and another from the same

Figure 4 - 9 :

 49 Figure 4-9: Calculated through-thickness Young's modulus and measured erosion resistance normalized from the reference YSZ-Polyester coating. ...................................................................................................................

Figure 4 - 10 :

 410 Figure 4-10: Calculated through-thickness thermal conductivity for YSZ-polyester coatings and normalized from the reference coating. .........................................................................................................................................

Figure 4 - 11 :

 411 Figure 4-11: Comparison of the experimental and calculated through-thickness normalized thermal conductivity of NiCrAl-Bentonite coatings. ..............................................................................................................................

Figure 4 - 12 :

 412 Figure 4-12: Comparison of experimental and calculated through-thickness normalized thermal conductivity of CoNiCrAlY-BN-polyester coatings. Right: Coating C and D. Left: Zoom on Coating D values. .............................

Figure 4 - 13 :

 413 Figure 4-13: Schematization of the two-scale modeling methodology. Top: general protocol. Bottom: zoom in the second step. ..................................................................................................................................................

Figure 4 - 14 :

 414 Figure 4-14: View of the microstructure of a coating, zoom in the YSZ phase of the coating. ............................

Figure 4 - 15 :

 415 Figure 4-15: Global view of the microstructure of a coating. ..............................................................................

Figure 4 -

 4 Figure 4-16: schematization of the 2-scale modeling method used to calulated the coating in plane Young's modulus ...............................................................................................................................................................

Figure 4 - 17 :

 417 Figure 4-17: Study of the porosity variability at high magnification (Top) and low magnification (Bottom). ....

Figure 4 - 18 :

 418 Figure 4-18: Left: coating at a magnification level of x80. Right: coating at a magnification level of x70, showing presence of the edges of the SEM detector. ........................................................................................................

Figure 4 - 19 :

 419 Figure 4-19: Influence of the input thermal conductivity of the grey phase (CoNiCrAlY-BN-polyester coating C) on the extrinsic thermal conductivity of the coatings -2-scale modeling method. .................................................

Figure

  Figure 4-20: Up: a smaller magnification level (x600) is not sufficient to observe only the NiCrAl matrix. Down:

Figure 0 - 1

 01 Figure 0-1 summarizes these different tasks.

Figure 0 - 1 :

 01 Figure 0-1: E-BREAK Objectives and organization.

Figure 0 - 2 :

 02 Figure 0-2: Work Package description and zoom onto work package thesis context. In this context, three different abradable materials are studied; NiCrAl-Bentonite, CoNiCrAlY-BN-Polyester and YSZ-polyester. Each of these materials is applied in different sectors of aeroengines and therefore are able to survive to specific temperature constraints, see Figure 0-3. NiCrAl-Bentonite and CoNiCrAlY-BN-Polyester are used in the high-temperature High Pressure Compressor (HPC) stages.

Figure 0 - 3 :

 03 Figure 0-3: Abradable coating localization into Turbojet [5].

  -4. The secondary flow is subjected to a compression before joining the main flow at the nozzle outlet. This secondary flow provides most of the thrust (above 90%) [6].

Figure 0 - 4 :

 04 Figure 0-4: Illustration of turbofan engines [7].

Figure 0 - 5 :

 05 Figure 0-5 : Diagram of the Joule-Brayton cycle [8].

  temperature high-pressure compressor and high-pressure turbine. Each of them has different operating temperature ranges and is applied to different sectors of the engine, see Figure 0-6. Their selections are based on the preferences of the aeronautic engine suppliers. They are listed in the Table 0-1.

Figure 0 - 6 :

 06 Figure 0-6 : Operating temperature range of the several abradable families.

Figure 0 - 7 :

 07 Figure 0-7: Several lamellae shapes after droplets crushing onto the substrate [13].

Figure 0 - 8 Figure 0 - 8 :

 0808 Figure 0-8 highlights the influence of parameters on the microstructure with a same chemical composition of the coating, here YSZ-polyester. Starting with the same powder, three microstructures were obtained, from the most (left) to the less (right) porous by adjusting the thermal and kinetic transfers between the plasma and the particles.

Figure 0 - 9 .

 09 Figure 0-9. It will be particularly demonstrated that the relationships between the process parameters, the structure of the developed coatings and the macroscopic properties of the coating are the key elements to optimize, design and manufacture abradable seals. To address this issue, the thesis manuscript was organized as schematically shown on Figure 0-10.

Figure 0 - 9 :Figure 0 - 10 :

 09010 Figure 0-9: Integration of the thesis work into the process-microstructure-properties triptych link.

Figure 1 - 2 :

 12 Figure 1-2: Abradable application into turbo engine, general localization (left), zoom localization (right).

Figure 1 - 3 :

 13 Figure 1-3: Abradable coatings applied in each engine part, function of exposition temperature. Picture modified from [4].

Figure 1 - 4 :

 14 Figure 1-4: Erosion resistance VS abradability performance function of ultimate tensile strength, from Douglas E. Chupp from [83].

Figure 1 - 5 :

 15 Figure 1-5: Properties of interest concerning abradable coating description.I.5.1. Main propertiesAbradability is the main property to examine because this is of major interest for the suppliers of jet engines. It therefore constitutes the ultimate goal of this PhD dissertation. The experimental set-up dedicated to the determination of the abradability performance of coatings is the abradable rig-test[START_REF] Sporer | Processing and Properties of Advanced Ceramic Abradable Coatings, Thermal Spray[END_REF][START_REF] Oerlikon-Metco | Improve efficiency and reduce emissions with high pressure turbine abradable coatings for industrial gas turbines[END_REF]. It consists in a single blade (plus an opposite blade that does not enter into contact with the abradable material) whose the tip is rubbing against the coated casing (Figure1-6 and Figure1-7).

Figure 1 - 6 :

 16 Figure 1-6: High temperature Abradability rig test from Oerlikon-Metco [59, 85].

Figure 1 - 7 :

 17 Figure 1-7: High speed rig-test: HSR-ALSTOM IPEK [86].

Figure 1 - 8 :

 18 Figure 1-8: Abradability wear marks observed on coatings -Measurements realized with the high-speed rig-test from HSR-ALSTOM IPEK (Rapperswil, CH) in the frame of the E-BREAK European project.

Figure 1 - 9 :

 19 Figure 1-9: HR15Y test schematization (modified from Peyraut et al. [67]).

[ 95 ]

 95 Nova Swiss company erosion test[START_REF] Swiss | [END_REF]

Figure 1 - 10 :

 110 Figure 1-10: Examples of erosion test devices.

Figure 1 -

 1 11 is a schematization of one example of thermal cycling test.

Figure 1 - 11 :

 111 Figure 1-11: Thermal cycling test schematization [102].

Figure 1 - 12 :

 112 Figure 1-12: Thermal Spray techniques in function of the energy sources.

Figure 1 - 13 :

 113 Figure 1-13: Schematization of the flame/plasma process (left) and coating formation process (right).

Figure 1 - 14 :

 114 Figure 1-14: Example of different splat shapes obtained by Fukumoto [13].

Figure 1 - 15 :

 115 Figure 1-15: Plasma coating microstructure features [12].

Figure 1 - 16 :

 116 Figure 1-16: Schematization of a heterogeneous structure, REV and equivalent homogeneous structure.

[ 103 ]

 103 , F. Cernuschi et al. [156], M.E. Cunningham et al. [43], J. Wang et al. [135], B. Budiansky [157], R. McPherson [158], I. Sevostianov and M. Kachanov [53, 159], R. Dutton et al. [160] and F.

Figure 1 - 17 :

 117 Figure 1-17: In series and in parallel representation of a two-phase model [103, 161, 162].

[ 25 ]

 25 ones) with the Effective Medium Theory (EMT) introduced by Landauer[START_REF] Landauer | The Electrical Resistance of Binary Metallic Mixtures[END_REF]. The Effective Medium Theory assumes a microstructure with a random distribution of all phases. It leads to the following equation:

1 3 (

 13 I.6) Resulting from the combination of the Voigt-Russ and Maxwell models with the EMT theory (equation (I.6)), the equation established by J. Wang et al. is:

Figure 1 - 18 :

 118 Figure 1-18: Shape factor F as a function of the axial ratio a/c of the spheroid [156]

F

  .Cernuschi et al. [156] have reported the influence of the pore shape and size on the coating thermal properties by using the Maxwell and Bruggeman equations. Pores were assimilated to oblate ellipses with a revolution axis a and the same major and minor axis b=c. F. Cernuschi et al. have also elaborated iterative equations in order to describe more precisely the porosity network and all the contribution due to the globular pores and intra/inter lamellar cracks. However, this increases significantly the complexity of the related analytical model. F. Azarmi et al. [46] have compared several analytical model (Spriggs' equation, Hashin-Hasselman and Zhao models) for determining the Young modulus of alloy 625 coatings. Differences between theoretical and experimental results were significant. Azarmi et al. explained this by the idealization of the microstructure when using analytical models.

  allowed estimating the coating properties by modeling real or artificial microstructures of these coatings, most of the time by meshing directly the real or artificial images with Finite Element (FE) or Finite Difference (FD) codes. The numerical model is built directly onto the real or artificial images. These studies are listed below and their own advantages and drawbacks are discussed. III.3.1. FE or FD Modelling T. Nakamura et al. [170] worked on microstructures generated with a randomly distribution of pores with several sizes and shapes. By the use of finite element modeling, they investigated the influence of the pore size, shape and orientation on mechanical properties of ceramic coating elaborated by plasma spraying. Their work allows giving an idea of the relationship between the pores nature and coating properties. But this methodology is not representative of real microstructures because the microstructure are artificially built.

  Z. Wang and A. Kulkarni et al.[START_REF] Kulkarni | Advanced microstructural characterization of plasma-sprayed zirconia coatings over extended length scales[END_REF][START_REF] Kulkarni | Processing effects on porosityproperty correlations in plasma sprayed yttria-stabilized zirconia coatings[END_REF][START_REF] Kulkarni | Comprehensive microstructural characterization and predictive property modeling of plasma-sprayed zirconia coatings[END_REF][START_REF] Wang | Effects of pores and interfaces on effective properties of plasma sprayed zirconia coatings[END_REF] have calculated the Young modulus and the thermal conductivity with the OOF code. OOF is a public domain Oriented Object Finite element software developed by the « National Institute of Standards and Technology » (NIST) of the United States of America in the years 1997 [172]. It has been created to help material scientists determining macroscopic properties of materials. It consists in reading an image to create an adaptive meshing

  J.M. Dorvaux et al. [171] determined the thermal conductivity of porous coatings with a finite difference model. The authors particularly developed a code named TBCTool to study the contribution of the pores size and shape onto the coating properties. This methodology allows taking into account the very complex microstructure of TBC coatings by performing directly computations onto binary SEM images of a cross section of the real microstructure of the coating. The authors concluded that cracks perpendicular to the heat flow are the major heat insulation features of the structure. Bartsch et al. [71] compared results obtained with the TBCTool code developed by the French Aerospace Lab. ONERA (Office National d'Etudes et de Recherche Aérospatiale)[171] and the OOF code developed by the NIST (National Institute of Standards and Technology). The first code is based on the Finite Difference Method (FDM) while the second one uses the Finite Element Method (FEM). These codes were used to determine the thermal conductivity on binary images onto TBC microstructures elaborated by EB-PVD (Electron Beam -Physical Vapor Deposition). The main conclusions are:

  H.I. Faraoun et al. and J.L. Seichepine et al.[START_REF] Seichepine | Numerical simulation of the thermo-mechanical behaviour of thermally sprayed abradable coatings[END_REF][START_REF] Seichepine | Mechanical modelling of highly heterogeneous thermally sprayed abradable coatings, International Thermal Spray[END_REF][START_REF] Faraoun | Modelling route for abradable coatings[END_REF] worked together several years to conduct FE modeling onto micrographs of abradable coatings. They investigated AlSi-polyester, AlSi-hBN and NiCrAl-Bentonite coatings to determine numerically their thermal and mechanical properties. The microstructures were composed of coarse pores distributed randomly into the matrix. The software used was OOF, and to decrease the computation time, the coating microstructure have been simplified by transforming the pores by equivalent ellipses with the same surface by means of the LEG software (Lermps Ellipse Generator). This software has been developed by the Lermps laboratory of the UTBM. However, to take into account the weakening of intrinsic properties caused by plasma spraying, a corrective coefficient was applied on intrinsic phases properties. Y. Tan et al.[START_REF] Tan | Modeling thermal conductivity of thermal spray coatings: comparing predictions to experiments[END_REF] developed a similar method than Bolot in 2006, but combining image and finite element analysis. This code was applied to determine the thermal conductivity of YSZ and Molybdenum thermally sprayed coatings (several techniques were used to manufacture the coatings).

Figure 2 - 2 :

 22 Figure 2-2: Surface of the NiCrAl-Bentonite powder (left) and powder cross section (right) observed by SEM. Micrographs taken from Oerlikon-metco material product data sheet [174].Concerning the NiCrAl-Bentonite powder, the clay particles are chemically cladded thus providing bentonite particles (core) encapsulated in the NiCrAl alloy. Bentonite particles are easily entrapped into the coating with a regular distribution (all bentonite particles are in a metal matrix). This provides a binder free composite with no segregation occurring during transport, storage and spraying. Powder particles are represented on Figure2-2. The surface of the powder is rounded and its shape irregular.The right picture (cross-section of the powder) shows a metallic fine layer encapsulating a porous bentonite core.

Figure 2 - 3 :Figure 2 - 4 :

 2324 Figure 2-3: Surface of CoNiCrAlY-BN-polyester powder (left) and cross-section of the metal phase particles (right) observed by SEMThe YSZ-polyester powder is agglomerated. Agglomeration consists in bringing together two materials of different compositions. This permits to obtain a good distribution of individual phases in the coating. YSZ presents excellent resistances to high temperature environments, high pressure oxidation and erosion wear. Nevertheless, it exhibits a poor abradability and is abrasive for superalloy blades. The use of additional elements, such as polyester, is hence mandatory to improve the coating abradability. In practice, the polyester phase, acting as a pore former, is removed by thermal treatment at 450°C for 8 hours (see M2460 product data sheet referenced as Oerlikon Matco DSMTS-0014.3) after spraying, thus leading to the formation of macro voids in the coating.

3 .

 3 The following parameters were varied in the range of the lowest and highest values: electric arc current intensity (A), Argon volume flow rate (slpm), Hydrogen ratio (%vol), spray distance (mm) and number of powder dispenser injection ports.

F4 torch Test 1 :Test 6 :Test 2 :Test 7 :Test 3 :Test 8 :Test 4 :Test 9 :Test 5 :Test 10 :Test 1 : 8 :Test 2 :Test 9 :Test 3 :Test 10 :Test 4 :Test 11 :Test 5 :Test 12 :Test 6 :Test 13 :Test 7 :

 1627384951018293104115126137 Reference (Ref) Decrease in the Hydrogen volume fraction (H-= H ref -25%) Decrease in the arc current Intensity (I-= I ref -10%) Increase in the Hydrogen volume fraction (H+ = H ref + 50% ) Increase in the arc current Intensity (I+ = I ref + 20%) Decrease in the spraying distance Dspray-= Dspray ref -20%) Decrease in the Argon plasma gas flowrate (Ar -= Ar ref -20%) Increase in the spraying distance (Dspray+ = Dspray ref + 20%) Increase in the Argon plasma gas flowrate (Ar + = Ar ref + 20%) Decrease in the powder feed rate (dpowder-= dpowder ref -50%) 9MB torch Reference (Ref) Test Decrease in the spraying distance (Dspray-= Dspray ref -15%) Decrease in the arc current Intensity (I-= I ref -20%) Increase in the spraying distance (Dspray+ = Dspray ref + 5%) Increase in the arc current Intensity (I+ = I ref + 25%) Decrease in the powder feed rate (dpowder-= dpowder ref -50%) Decrease in the Argon plasma gas flowrate (Ar -= Ar ref -10%) Decrease of the displacement velocity of the torch (kinetic -= kinetic ref -25%) Increase in the Argon plasma gas flowrate (Ar + = Ar ref + 10%) Decrease of the displacement velocity of the torch and Increase in arc current Intensity (kinetic -= kinetic ref -25%) (I+ = I ref + 25%) Decrease in the Hydrogen volume fraction (H-= H ref -25%) Decrease of the displacement velocity of the torch, Increase in the arc current Intensity and Decrease in the spraying distance (Dspray-= Dspray ref -15%) (I+ = I ref + 25%) (kinetic -= kinetic ref -25%) Increase in the Hydrogen volume fraction (H+ = H ref + 50% )

Figure 2 - 5 :Figure 2 - 6 :

 2526 Figure 2-5: Schematic of Laser flash analysis methodology (left), zoom on the sample interaction.This unsteady method consists in subjecting the front surface of a sample to a high intensity short duration energy pulse (flash), Figure2-5. The generated heat flow propagates by conduction through the sample thickness and reaches the rear face of the sample after a short time. The temperature is measured versus time at the rear face of the sample (thermogram, e.g. Figure 2-6). Due to the flash excitation, all characteristic frequencies of the material are simultaneously excited. The analysis is fast but sensitive to noises. The environment temperature is kept constant during each experimentation. The study of the thermogram provides the thermal diffusivity of the considered material. It is then possible to calculated de thermal conductivity by the following relationship: к=ρ.a.Cp (II-1)

2 (II- 5 )

 25 the radiative losses from the sample surfaces. Clark and Taylor examined the thermogram at different points before the maximum temperature rise was reached and determined the setup ratio at different points of the heating curve. Then they developed a correction factor K r and computed it to correct the Parker's equation (Eq.2-3) to give II-4 and II-5:𝑡 𝑥 is the time required to reach xof the maximum temperature rise, and 𝑎 x is the thermal diffusivity corresponding to a temperature rise to x% of its maximum. Here, 𝑎 0.5 is the one determine by the simplified Parker equation II-3.The specific heat measured by laser flash analysis consists in making a comparison of the temperature rise between the sample and a reference. The temperature rise is recorded during the diffusivity measurement and both samples are tested under the same conditions. It is then possible to obtain the heat capacity with the following equation (II-6, II-7 and II-8):Q= absorbed energy= (mCpΔT) ref= (mCpΔT)sample (

Figure 2 - 7 :

 27 Figure 2-7: Photo of the experimental device (FlashLine TM 3000, Anter Corporation)These tests were performed during a period spent at the Institute of Plasma Physics of Prague (IPP). Figure2-7shows the thermal diffusivity device by method flash available at IPP (FlashLine TM

Figure 2 -

 2 [START_REF] Aainsqatsi | [END_REF] shows the thermal diffusivity device by method flash available at IPP (FlashLine TM 3000, Anter Corporation). The samples were in an oven, it was possible to change the environment temperature. The temperature range of the measurements is from room temperature up to 1000°C. The samples were 10mm squares with a thickness of 1.5 mm, Figure2-8. The measurements were repeated on three samples for each kind of coatings. The experimental thermal diffusivities, thermal conductivities and specific heat, provided in this report, are the average of the three measurements.

Figure 2 - 8 :

 28 Figure 2-8: Pictures of the sample dimensions for laser flash analysis Due to the translucency of the materials, a thin layer of colloidal graphite was sprayed on the surface of the samples. The front face becomes opaque for the radiation of the excitation source and the back face emissivity is increased and facilitates the IR detection. The measurements were conducted at room temperature and high temperature, Table 2-4.

  between 60 kgf (689 N) and 150 kgf (1471 N) for conventional Rockwell tests. However, the load applied is comprised between 15kgf (147 N) and 45 kgf (441 N) for superficial Rockwell tests. It permits to test abradable materials which could be strongly damaged under conventional conditions of typical hardness tests. The device used for those tests is a Durometer from Ernst (SOMECO, 6 avenue Charles de Gaulle; 93421 VILLEPINTE Cedex, FRANCE), type NR3D, Figure 2-9.

Figure 2 - 9 :

 29 Figure 2-9: Durometer type NR3D from ErnstThis test is widely used by manufactures of abradable coatings. It involves indenting the material with a hardened steel ball indenter (1/2 in. diameter). The measure is performed in 3 successive steps.A preliminary load of 3 kgf (i.e., 29 N) is first applied. This preliminary load permits to remove any effects of surface irregularities due to the sample preparation. Then an additional load of 15 kgf (i.e., 147 N) is applied and increases further the penetration depth.

Figure 2 - 10 :

 210 Figure 2-10: Steps of Rockwell superficial hardness (figure modified from Peyraut et al [57])Finally, the additional load is removed while the preliminary one is still applied. The depth of indentation is automatically determined. The Rockwell measure is given by e on Figure2-10 and by II-9.

Figure 2 - 11 :

 211 Figure 2-11: GE Erosion test bench (left) Erosion Sample (right)The deepest point and its exact position on the sample is determined for the reference Lexan material. The Lexan sample and the specimens to be tested have to be positioned similarly and at the same height. The resistance erosion value, so-called E normal is then calculated by II-10 as an average of three tests:

  -12. Finally, those curves allow obtaining some mechanical characteristics of the material, such as the local hardness and Young modulus by applying analytical models. The model considered here has been developed by Oliver and Pharr[START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation techniques[END_REF] and is the most popular in the nanoindentation domain.The two mechanical properties usually measured with this method are the local Hardness (H it ) and local Young Modulus (E it ) by using the standard NF EN ISO 14577.

Figure 2 - 12 :

 212 Figure 2-12: Typical Load/displacement curves The first step of the Oliver-Pharr method consists in fitting the unload portion of the curve with the following law:

Figure 2 - 13 :

 213 Figure 2-13: Photography of the NHT device of CSM instruments (left), indentation system graph (right).

- 14 .

 14 To determine the shear modulus, the resonance frequency must be determined by the torsion mode of vibration, Figure2-14. It is then possible to calculate E and G with the following equations:

Figure 2 - 21 )

 221 Figure 2-14 RFDA test to determine the Young modulus (Top) and the shear modulus (Bottom)

Figure 2 - 15 :

 215 Figure 2-15: Tensil test specimen

Figure 2 - 16 :

 216 Figure 2-16: Schematic representation of different type pore networks with different attributes

Figure 2 - 17 :

 217 Figure 2-17: Example of pull out damages occurring during sample preparation

Figure 2 - 19 :

 219 Figure 2-19: Pictures of the mounting process

Force

Figure 2 -

 2 [START_REF] Prat | Transferts de chaleur dans les milieux poreux Conduction, convection, rayonnement, Techniques de l'ingénieur Transferts thermiques[END_REF], it may observe that the coating has clearly three different phases. The white pixels correspond to the metallic NiCrAl phase, the grey pixels are associated to the bentonite phase and the black pixels to voids. The grey level histogram of the image is represented below, Figure2-20. It corresponds to the distribution (number) of the pixels in function of the grey level (256 grey level possibilities). This histogram contains peaks and valleys. The peaks correspond to the most common brightness values of the pixels. On contrary, the valleys indicate that those brightness values are less common. The darkest grey level corresponds to the value 0 (black color) whereas the highest grey level corresponds to the value 255 (white color). Three different peaks can be observed corresponding to the three phases of the coating. The peak near from the 0 value corresponds to voids, whereas the peak near from the value 255 corresponds to the metal matrix. The intermediate peak corresponds to the bentonite phase. For some materials image having a low contrast the histogram can be modified to enhance it. Equalization may be performed to adjust the distribution of intensities on the histogram. This allows covering the whole range of gray levels, and permits to enhance the visibility of some details.

Figure 2 - 20 :

 220 Figure 2-20: Cross-section micrograph of a NiCrAl-bentonite coating (top-left), zoom in the coating microstructure to observe the grey level of pixels (top-right), equalized histogram of the micrograph (bottom-left), histogram of the micrograph after double thresholding (bottom-right)

Figure 2 - 21 :

 221 Figure 2-21: Example of the effect of morphological operators of Image J software. Figure coming from Image-J guide.

Figure 2 - 22 :Figure 2 -

 2222 Figure 2-22: Selection of REV for periodic structure (left), representation of a random structure (right).The REV is directly related to the magnification at which the structure is studied. Indeed an inadequate magnification level could misrepresent the continuity (or discontinuity) of the different phases in the coating. The use of a too high magnification level may overestimate the void content in the coating on one image and the contrary on a second one. On the contrary, the use of a too low magnification level does not permit to observe the microstructural details. For example,

Figure 2 - 23 :

 223 Figure 2-23: Overview of a typical YSZpolyester coating structure (Optical microscopy).

Figure 2 - 24 :

 224 Figure 2-24: Magnified view of a typical YSZ-polyester coating structure (FESEM).

Figure 2 - 25 :

 225 Figure 2-25: Very high zoom into the coating microstructureTo determine this representative elementary volume, it is mandatory to study the variability (𝝈/𝝁) of the various properties calculated from the coating images to get statistically significant results.

  measurement of the surface area of each void which is realized for the whole data set (15 images) of each coating, (ii) voids in contact with the image borders are discarded from the distribution to avoid statistical biases, (iii) calculation of the void equivalent diameter of each void by Eq. 2-24, (iv) the surface area fraction of voids is then plotted versus their equivalent diameter in Figure 2-27, and (v) five different size classes are then identified from this graph (vi) for each class, void content is determined.

Figure 2 - 27 :

 227 Figure 2-27: Distribution of the void fraction in the coating versus their surface equivalent diameter.The first class represents the so-called very fine voids with an equivalent diameter smaller than 1.5 µm. The second class is the so-called fine voids with an average diameter between 1.5 and 10 µm. The third class is the so-called medium voids with equivalent diameter ranging from 10 to 20 µm. The

  -28. The first one concerns implementation of input data required (such as the properties of the phases) before conducting the modeling (2 nd step). The last step concerns printing output data (results of the calculation) allowing calculating the extrinsic properties of the coating (3 rd step).

Figure 2 - 28 :

 228 Figure 2-28: Steps of the modeling routines

Figure 2 - 30 :

 230 Figure 2-30: Example of SM 2640 coating micrographs used for modeling.

  The media is continuous  Each phase is considered as a pure substance, the properties applied to a phase are homogenous in all the phase. No mass transfer is considered at phases interfaces, no phase change or chemical phenomena are taken into account  Each phases has a linear elastic behavior. Intergranular friction are not taken into account in our modeling.

  left and bottom boundaries are assumed to be perfectly embedded (the normal component of the displacement of the concerned nodes is imposed to 0),  the right edge is free of movement This calculation allows computing the effective Young's modulus in the through-thickness direction.

Figure 2 - 31 :Fy:

 231 Figure 2-31: Description of the boundary conditions representing a vertical displacement load applied on the computational domain (image) to predict the through-thickness effective Young's modulus of the coating.Determination of the coating extrinsic Young's modulus is then possible by considering the two dimensional Hook's law. It is then possible to consider relationships between stress and strains for a "plane deformation" calculation.

Figure 2 - 32 : 28 )

 23228 Figure 2-32: Boundary conditions to estimate the longitudinal effective Young's modulus of the coating From equation 2-25 and boundary conditions as shown on Figure 2-32, it is possible to apply equation 2-27 to calculate the longitudinal effective Young's modulus:

Figure 2 - 33 :

 233 Figure 2-33: Boundary conditions representing a thermal difference load applied on the computational domain (image) to predict the coating effective thermal conductivity in the through-thickness direction.

Figure 3 - 2 :

 32 Figure 3-2: Scheme of the numerical modeling methodology. The present chapter refers to the first step.

I. 1 . 1 .Figure 3 - 3 :

 1133 Figure 3-3 displays the microstructures of the three coating captured on polished cross-sections by OM at a low magnification level (×25).

Figure 3 -Figure 3 - 4 )

 334 Figure 3-4 displays each coatings microstructure captured by SEM at two different magnification levels: ×150 and ×350, x800, x500 for the highest magnification level representation of NiCrAl-Bentonite, CoNiCrAlY-BN-Polyester and YSZ-Polyester respectively. The coating matrices appear in white and correspond to NiCrAl, CoNiCrAlY and YSZ, respectively. Features #2, #3 and # 4 in Figure 3-4 are identical for the three coatings. They correspond to small voids due to stacking defects during the spraying process, to inter lamellar cracks due to incomplete contacts in between flattened splats and to intra lamellar cracks due to stress relaxation, respectively.

( 15 images

 15 ) of each coating, iii) voids in contact with image borders are discarded from the distribution to avoid statistical biases, iv) calculation of the void equivalent diameter of each void, v) the surface area fraction of voids versus their equivalent diameter is then plotted and vi) for YSZ-Polyester coating, five different size-classes have been then then identified from this graph. The first size-class represents so-called very fine size voids with an equivalent diameter smaller than 1.5 μm. The second class corresponds to so-called fine size voids with average diameters ranging from 1.5 to 10 μm. The third class corresponds to so-called medium size voids with equivalent diameter ranging from 10 to 20 μm. The fourth class corresponds to so-called coarse voids with equivalent diameters ranging from 20 to 40 μm. The last class is related to so-called very coarse voids with equivalent diameters larger than 40 μm. The total void content of each coating is calculated by summing the average void contents of each void size class and results were compared from one coating to another.

Figure 3 - 4 :

 34 Figure 3-4: Abradable coatings microstructure captured by SEM (at two different magnification levels: low level corresponds to ×150 and high level corresponds to ×350, x800 and x500 for NiCrAl-Bentonite, CoNiCrAlY-BN-Polyester and YSZ coating respectively).

Figure 3 - 6 :

 36 Figure 3-6: Image taken by SEM in SE mode. a) zoom onto the grey phase of the coating. b) zoom onto the powder surface.

Figure 3 -Figure 3 - 7 :

 337 Figure 3-7 compared the microstructure of a modified TBC (studied in this manuscript) with conventional TBC. For a similar magnification level, it is possible to observe important difference between YSZ conventional coating (Metco 204 powder feedstock) and YSE-polyester coating (Metco 2460 powder feedstock).

Figure 3 -

 3 10, represents those four curves at a magnification level of x150.

Figure 3 - 9 :

 39 Figure 3-9: YSZ-Polyester coating -Variability of the coating void content versus the number of images at different magnifications.

Figure 3 -

 3 Figure 3-10 allows determining, for a given magnification, the number of required images to fulfill the REV condition. This corresponds to a stabilized variability of the calculated void content. This stabilized variability is intrinsic to the coating structure and not related to biases due to the REV condition which is not fulfilled. The number of considered images is 15 under those conditions.

Figure 3 - 10 :

 310 Figure 3-10: Voids variability at a magnification level of ×150 on YSZ-microstructure

Figure 3 -

 3 11 corresponds to Young's modulus variability and thermal conductivity variability versus number of analyzed images.

Figure 3 -

 3 Figure 3-11: YSZ-Polyester coating -Variability of the coating Young's modulus or thermal conductivity versus the number of images at different magnifications.

Figure 3 - 12 :

 312 Figure 3-12: Image processing protocol applied on SEM images representative of the coating structure. As a demonstrative purpose, the protocol is applied here on the Bentonite phase embedded in a NiCrAl-Bentonite coating structure.

Figure 3 - 13 :Figure 3 - 14 :

 313314 Figure 3-13: Reconstruction step of coating structures by adding binary image representative of each identified feature. As a demonstrative purpose, the protocol is applied here on the Bentonite phase and voids embedded in a NiCrAl-Bentonite coating structure. The complementary feature, in white, represents the NiCrAl matrix.

Figure 3 -

 3 14 displays an example of the grey levels distribution where the globular voids are represented by pixels with colors varying mostly between 0 and 15, the bentonite particles mostly between 55 and 100 and the NiCrAl matrix mostly between 200 and 255. The threshold protocol aims at defining contrast limits for each feature.

- 16 .

 16 It is based on two different processing of a same image and on the recombination of features representing the voids (globular voids and cracks) in a unique image. A dedicated subroutine has been systematically implemented for processing YSZ coating images.

Figure 3 - 15 :

 315 Figure 3-15: Illustration of image transformation procedure applied for YSZ-polyester coating.

Figure 3 - 16 :

 316 Figure 3-16: Schematization of the several steps of the homemade program

  thermal properties of YSZ for various amounts of Y 2 O 3 . Hasselamn et al. [227], Ingel et al. [228],

  has listed properties of arc plasma spraying MCrAlY of several compositions at different temperatures. The technics used to determine mechanical properties of flat and cylindrical cast samples were: 3-point bend test, 4 -point bend test, tensile tests, dynamic mechanical analysis and indentation. Young's modulus varies in function of MCrAlY compositions and measurement temperature. At room temperature, NiCrAlY Young's modulus varies in between 157 and 178 Gpa, and CoNiCrAlY Young's modulus varies in between 182 and 207 Gpa. The thermal intrinsic properties collected in the literature concerns APS NiCrAl and CoNiCrAlY coatings studied, by Brandt et al. [200] and Patterson et al. [199], respectively. Thermal conductivity at room temperature was 15 and 25 W.m -1 .K -1 , respectively.

Table 3 - 12 :

 312 Nanoindentation Young's modulus values of Bentonite and grey phases of CoNiCrAlYpurity of Al 2 O 3 and hBN phase, the intrinsic thermal conductivities varying between 10 [212] and 35 [210] W.m -1 .K -1 for Al 2 O 3 and between 15 [215] and 90 W.m -1 .K -1 [210] for hBN are found in the literature.

Figure 3 - 17 :

 317 Figure 3-17: Influence of input thermal conductivity of the grey phase (CoNiCrAlY-BN-polyester coating) onto the extrinsic coatings properties

Figure 4 - 2 :

 42 Figure 4-2: Schematization of the numerical modeling methodology.

- 4 .Figure 4 - 4 :

 444 Figure 4-4: CoNiCrAlY-BN-polyester coating microstructure, C) the less porous and D) the most porous For confidentiality reasons, the results are normalized again. As for coatings A and B, the Young's modulus is normalized by considering the experimental value measured by RFDA for the most porous coating.

Figure 4 - 5 :

 45 Figure 4-5: HR15Y and erosion resistance normalized values function of YSZ-Polyester microstructure.

  7 for respectively the NiCrAl-Bentonite, CoNiCrAlY-BN-Polyester and YSZ-Polyester materials. Only the Young's modulus calculated in the in plane direction of the coating will be compared with experimental results. Indeed tensile tests and RFDA provide the Young's modulus along the in plane direction of the coating. The calculated Young's modulus are normalized by the experimental value measured by RFDA for the more porous coating for confidentiality reasons. I.2.1. NiCrAl-bentonite coatings

Figure 4 - 6 :

 46 Figure 4-6: Representation of the porosity network difference between one image and another from the same coating.

Figure 4 - 7 and

 47 Figure 4-7 and Figure 4-8 allow comparing the experimental Young's modulus with the calculated one. The corresponding standard deviations are also considered. The blue points onto the graph represent experimental values whereas the calculated values are represented in red.

Figure 4 - 7 :

 47 Figure 4-7: Comparison of the experimental and calculated in plane normalized Young's modulus of NiCrAl-Bentonite coatings.

Figure 4 - 8 :

 48 Figure 4-8: Comparison of the experimental and calculated in plane normalized Young's modulus of CoNiCrAlY-BN-polyester coatings. Coating C, the less porous one among CoNiCrAlY-BN-polyester coatings (Figure 4-8, on the left)shows strong differences between experimental and calculated data. The calculated Young's modulus is 2.3 time higher than the experimental one. The relative difference between the two values is very high, i.e. Δ=+140%.

Figure 4 - 9 :

 49 Figure 4-9: Calculated through-thickness Young's modulus and measured erosion resistance normalized from the reference YSZ-Polyester coating.

II. 1 .

 1 Experimental values Thermal conductivities of NiCrAl-Bentonite and the CoNiCrAlY-BN-polyester coatings have been measured by Laser flash analysis. The provided data represent an average of 3 different measures on a same sample. Those tests have been performed with the facilities of the Institute of Plasma Physics located in Prague. The results are given in the following tables (

Figure 4 - 10 :

 410 Figure 4-10: Calculated through-thickness thermal conductivity for YSZ-polyester coatings and normalized from the reference coating.

Figure 4 -

 4 Figure 4-11 and Figure 4-12 allow comparing experimental thermal conductivity data with calculated ones. The standard deviation is also considered as well as the relative difference (computed from equation VI-1) between experimental and calculated data. The experimental thermal conductivity is represented by the red dotted line and the standard deviation by the red rectangle. The calculated thermal conductivity and the corresponding standard deviation are represented in blue on the graph.

Figure 4 - 11 :

 411 Figure 4-11: Comparison of the experimental and calculated through-thickness normalized thermal conductivity of NiCrAl-Bentonite coatings.

Figure 4 - 12 :

 412 Figure 4-12: Comparison of experimental and calculated through-thickness normalized thermal conductivity of CoNiCrAlY-BN-polyester coatings. Right: Coating C and D. Left: Zoom on Coating D values.

Figure 4 -

 4 Figure4-13 represents the several steps of the model methodology. As described in chapter 2, the first step consist in choosing all the input information required to feed the model. And the third step consists in considering the output results (i.e., thermal fluxes or forces) to calculate the extrinsic coating properties. The second step concerns the calculation itself, and is the one which has to evolve compared to the "conventional" modeling method. It will be divided in two stages as described in Figure 4-13.

Figure 4 - 13 :

 413 Figure 4-13: Schematization of the two-scale modeling methodology. Top: general protocol. Bottom: zoom in the second step.

Figure 4 - 14 :

 414 Figure 4-14: View of the microstructure of a coating, zoom in the YSZ phase of the coating.

Figure 4 - 15 :

 415 Figure 4-15: Global view of the microstructure of a coating.During the second stage of the modeling procedure, calculations performed on micrographs with a low magnification level, Figure4-13, with a resolution limit of 1.5 µm, allow considering the global microstructure of the coating. The input data of the phase properties are those previously calculated for the high magnification level micrographs (first stage): this corresponds to a so-called "homogenization" process. For his second stage, the properties applied for voids still correspond to those listed in Figure4-30.

Figure 4 - 16 :

 416 Figure 4-16: schematization of the 2-scale modeling method used to calulated the coating in plane Young's modulus

Figure 4

 4 Figure 4-18.

Figure 4 - 18 :

 418 Figure 4-18: Left: coating at a magnification level of x80. Right: coating at a magnification level of x70, showing presence of the edges of the SEM detector.

Figure 4 - 19 :

 419 Figure 4-19: Influence of the input thermal conductivity of the grey phase (CoNiCrAlY-BN-polyester coating C) on the extrinsic thermal conductivity of the coatings -2-scale modeling method.

Figure 4 -

 4 Figure 4-19 shows a similar behavior of the calculated thermal conductivity vs. the input thermal conductivity of the phases for the two methods. Only the input thermal conductivity of the grey phase

Figure 4 - 20 :

 420 Figure 4-20: Up: a smaller magnification level (x600) is not sufficient to observe only the NiCrAl matrix. Down: Zoom in the NiCrAl matrix to avoid bentonite particles, the issue is that this low magnification level only represents a single lamellae splat.

Young's moduli

  of CoNiCrAlY-BN-polyester coatings C and D are listed in Table 4-18 and represented in Figure 4-21 and Figure 4-22.

Figure 4 -

 4 Figure 4-21, considering the standard deviation of both experimental and computed values of the Young's modulus, the two values are in the same range.

Figure 4 - 21 :

 421 Figure 4-21: Comparison between the experimental and calculated Young's modulus (longitudinal direction) for coating C (CoNiCrAlY-BN-polyester coating, normalized values).

Figure 4 - 22 :

 422 Figure 4-22: Comparison of the experimental and calculated Young's modulus (longitudinal direction) for coating D (i.e., CoNiCrAlY-BN-polyester material, normalized values).The only difference between the two modeling methods applied to coating D is the strong decrease of the standard deviation of the Young's modulus for the 2-scale model. The coefficient of variation is decreased from 200% for the conventional method to 67% for the 2-scale model. Even if this coefficient of variation remains high, it is in the same range in comparison with experimental results which shows a coefficient of variation of the Young's modulus of 77% for coating D. It seems thus that the reproducibility of the results in not good for this coating, for both experimental and computed results.

Figure 4 - 23 :

 423 Figure 4-23: Comparison of the measured transverse thermal conductivity with the values computed with the two models for coating C (CoNiCrAlY-BN-polyester material, normalized data).

Figure 4 - 24 :

 424 Figure 4-24: Comparison of the experimental and computed transverse thermal conductivity for coating D (CoNiCrAlY-BN-polyester material, normalized values).

Figure 4 -

 4 [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] represents the Young's modulus of YSZ-polyester coatings by comparing the calculated results for both modeling methods.

Figure 4 - 25 :

 425 Figure 4-25: Calculated transverse Young's modulus for the different YSZ-polyester coatings using the 2-scale and conventional models (left) -comparison between the transverse and longitudinal moduli (right).

Figure 4 - 26 :

 426 Figure 4-26: Calculated thermal conductivities for the different YSZ-polyester coatings using the 2-scale and the conventional models.

Figure 4 - 27 :

 427 Figure 4-27: Computed thermal conductivity (left) and Young's modulus (right) for the one-scale and two-scale modeling methods. The highest and lowest coating properties have been calculated by considering the grey phase as the matrix (highest limit) and as pores (lowest limit).

  eq ≤10 µm 10 < D eq ≤ 20 µm 20 < D eq ≤ 40 µm 40 ≤D eq The fraction of the surface area of voids of three different coatings is represented on Figure 4-28

  is few coarse voids and no Very coarse voids on contrary with the two other coatings. The less porous coating is constituted of Fine Medium and Coarse voids size but on contrary to the More porous coating, the very coarse voids class is less represented.

Figure 4 - 28 :

 428 Figure 4-28: Surface area fraction of voids versus equivalent diameter.

  eff represents the effective Young's modulus of the grey phase, E 1 and x are the Young's modulus and volume fraction of the continuum media (YSZ), and E 2 and y corresponds to the Young's modulus and volume fraction of dispersed spherical phase (nano-voids here).The thermal conductivity of the grey phase areas were estimated from the Maxwell equation which considers the effective thermal conductivity of a heterogeneous medium composed of dispersed spheres in a continuous medium The effective thermal conductivity of the grey phase areas (κ eff ) can thus be estimated from equation (IV-3):

Figure 4

 4 Figure 4-30.

Figure 4 - 30 :

 430 Figure 4-30: Calculation of the coating extrinsic properties by changing the intrinsic properties of the dark grey phase areas, which are a mix between "nano-porosities" and small grains of YSZ.

Table 4 - 28 :

 428 Estimation of the time required by the different tests.

Figure 4 - 31 :

 431 Figure 4-31: Description of the entire procedure to conduct the modeling method, from the sample preparation to the calculation.

Table 4 - 29 :

 429 Example of experimental and numerical properties values of the abradable microstructure.

Table 0 -

 0 1: Information about abradable materials studied. .................................................................................

Table 1 -

 1 1: Experimental extrinsic properties of several abradable materials. ......................................................

Table 1 -

 1 2: Hardness of two different coating materials having the same chemical composition but different microstructures. ....................................................................................................................................................

Table 1 -

 1 3: Equivalent thermal conductivity for porous media with isolated pores. ..............................................

Table 1 -

 1 4: list of experimental tests applied for the different abradable coatings. ..............................................

Table 2 -

 2 1: Powder information -data that are more complete are provided in annexes .....................................

Table 2 -

 2 2: Coating manufacturing information [174-176] ....................................................................................

Table 2 -

 2 3: Matrix of tests for YSZ-polyester coatings ............................................................................................

Table 2 -

 2

4: Different test temperature for the abradables coatings ......................................................................

Table 2 -

 2

5: Parameter of the different tests. .......................................................................................................... Table 2-6: sample preparation steps .....................................................................................................................

Table 2 -

 2

7: sample preparation protocol for thermal sprayed NiCrAl-bentonite, CoNiCrAlY-BN-polyester and YSZpolyester samples .................................................................................................................................................. Table 2-8: Void content of each size class for all coatings. ................................................................................. Table 2-9: list of the different data ..................................................................................................................... Table 2-10: Bulk properties of each phase according to the literature ...............................................................

Table 2 -

 2

11: Range of variation of the properties according to the literature .....................................................

Table 2 -

 2

12: Young's modulus measured by nano-indentation (directly in coating samples). .............................

Table 3 -

 3

1: Void content of each size-class for the reference coating ..................................................................

Table 3 -

 3

2: Description of features observed in function magnification level ......................................................

Table 3 -

 3

3: Variability study of porosity rate to determine appropriate magnification level. ..............................

Table 3 -

 3

4: Minimum number of SEM images necessary to get representative properties of coatings. ..............

Table 3 -

 3

5: Summarize of the different magnification level REV for each coating and magnification level. .......

Table 3 -

 3

6: Presentation of the two different microstructures studied for NiCrAl-Bentonite coating. .................

Table 3 - 7 :

 37 Presentation of the two different microstructures studied for CoNiCrAlY-BN-Polyester coating. ......

Table 3 -

 3

8: Presentation of four different microstructures studied for YSZ-Polyester coating. ............................

Table 3 -

 3

9: Bulk properties of various phases published in literature. .................................................................

Table 3 -

 3 10: Nanoindentation Young's modulus values of CoNiCrAlY and NiCrAl. ...............................................

Table 3 -

 3 11: Input data used for the model ..........................................................................................................

Table 3 -

 3

12: Nanoindentation Young's modulus values of Bentonite and grey phases of CoNiCrAlY-BN-Polyester coating ................................................................................................................................................................ Table 3-13: Influence of the variation in the selected intrinsic thermal conductivity of the Al2O3-BN compound on the calculated coating apparent thermal conductivity.. ................................................................................

Table 3-14: Input data used for the model. .........................................................................................................

Table 4 -

 4 1: List of experimental tests applied depending on the abradable coatings. .........................................

Table 4 -

 4 2: Young's modulus of NiCrAl-bentonite coatings at room temperature (normalized results). .............

Table 4 -

 4 3: Experimental mechanical properties of CoNiCrAlY-BN-polyester coatings at room temperature (normalized results). ............................................................................................................................................

Table 4 -

 4 4: HR15Y hardness and erosion resistances of YSZ-Polyester coatings. .................................................

Table 4 -

 4 5: Calculated Young's modulus of NiCrAl-bentonite coatings at room temperature (normalized results). ............................................................................................................................................................................

Table 4 -

 4 6: Calculated mechanical properties of CoNiCrAlY-BN-Polyester coatings at room temperature (normalized results). ............................................................................................................................................

Table 4 -

 4 7: Normalized calculated Young's modulus of YSZ-polyester coatings. .................................................

Table 4 -

 4 8: Experimental thermal conductivity of NiCrAl-bentonite coatings at room temperature (normalized results). ................................................................................................................................................................

Table 4 -

 4

9: Experimental thermal conductivity of CoNiCrAlY-BN-polyester coatings at room temperature (normalized results). ............................................................................................................................................

Table 4 -

 4

10: Calculated through-thickness thermal conductivity of NiCrAl-bentonite coatings at room temperature (normalized results based on the experimental data measured for coating B). ............................ Table 4-11: Calculated through-thickness thermal conductivity for CoNiCrAlY-BN-polyester coatings at room temperature (normalized results based on the experimental measure for case D). ........................................... Table 4-12: Calculated through-thickness thermal conductivity of YSZ-polyester coatings (normalized from data computed for the reference coating). .................................................................................................................

Table 4 -

 4

13: Description of features observed in function the magnification level. .............................................

Table 4 -

 4

14: Input properties implemented for the 2-scale modeling method.....................................................

Table 4 -

 4

15: Representative elementary volume of the 2-scale modeling method. .............................................

Table 4 -

 4

16: Input data used for the model. .........................................................................................................

Table 4 -

 4

17: Microstructures of Coating C and D at the twoscales used for modelling. ......................................

Table 4 -

 4

18: Experimental and calculated Young's moduli of CoNiCrAlY-BN-polyester coatings at room temperature (normalized results from RFDA measure for coating D). ...............................................................

Table 4 -

 4

19: Relative difference between experimental and calculated Young's modulus (in-plane direction) of CoNiCrAlY-BN-polyester coatings. .......................................................................................................................

Table 4 -

 4

20: Experimental and calculated thermal conductivities of CoNiCrAlY-BN-polyester coatings at room temperature (normalized values). .......................................................................................................................

Table 4 -

 4 21: Relative difference between the computed and experimental thermal conductivity of CoNiCrAlY-BN-

polyester coatings. ..............................................................................................................................................

Table 4 -

 4

22: Normalized Young's modulus of YSZ-polyester coatings, calculated with the 2-scale modeling method. ............................................................................................................................................................... Table 4-23: Normalized thermal conductivities of YSZ-polyester coatings, calculated with the 2-scale modeling method. ............................................................................................................................................................... Table 4-24: Experimental thermal conductivity taken from the literature for similar coatings. .........................

Table 4 -

 4 25: Different classes of the equivalent diameter of voids. .....................................................................

Table 4 -

 4 26: Area fraction of each voids class in coatings. ...................................................................................

Table 4 -

 4 27: Modeling Input data for the dark grey phase. .................................................................................

Table 4 -

 4 28: Estimation of the time required by the different tests. ....................................................................

Table 4 -

 4 29: Example of experimental and numerical properties values of the abradable microstructure. (Normalized values) ............................................................................................................................................

Table 44 -

 44 30: Estimation of the time preparation mandatory for the different tests. .........................................

Table 4 -

 4 31: SEM microstructure images. Microstructures elaborated with F4 torch. ........................................

Table 4 -

 4 32: SEM microstructures images. Microstructures elaborated with 9MB torch; ...................................

Table 4 -

 4 33: CoNiCrAlY-BN-Polyester coatings thermal expansion ......................................................................

Table 4 -

 4 34: NiCrAl-Bentonite thermal expansion ................................................................................................

Table 4 -

 4 35: properties calculated at the high level of the 2-scales modelling for CoNiCrAlY-BN-Polyester (less

porous coating) ...................................................................................................................................................

  Figure 1-9: HR15Y test schematization (modified from Peyraut et al. [67]). ........................................................Figure 1-10: Examples of erosion test devices. ......................................................................................................Figure 1-11: Thermal cycling test schematization [102]. ......................................................................................Figure 1-12: Thermal Spray techniques in function of the energy sources. ........................................................... In series and in parallel representation of a two-phase model [103, 161, 162]. ...............................Figure 1-18: Shape factor F as a function of the axial ratio a/c of the spheroid [156] ..........................................Figure 2-1: Summary of chapter 2. ........................................................................................................................

	Figure 2-2: Surface of the NiCrAl-Bentonite powder (left) and powder cross section (right) observed by SEM.

-7: Several lamellae shapes after droplets crushing onto the substrate

[START_REF] Fauchais | Knowledge concerning splat formation: An invited review[END_REF]

. ..........................................

Figure 0-8: Microstructures of YSZ-Polyester coatings. .........................................................................................

Figure 0-9: Integration of the thesis work into the process-microstructure-properties triptych link. ...................

Figure 0-10: Schematic of the thesis plan organization. .......................................................................................

Figure 1-1: Summary of chapter 1. ........................................................................................................................

Figure 1-2: Abradable application into turbo engine, general localization (left), zoom localization (right). ........

Figure 1-3: Abradable coatings applied in each engine part, function of exposition temperature. Picture modified from [4]. ................................................................................................................................................................

Figure 1-4: Erosion resistance VS abradability performance function of ultimate tensile strength, from Douglas E. Chupp from [83]. ...............................................................................................................................................

Figure 1-5: Properties of interest concerning abradable coating description. ......................................................

Figure 1-6: High temperature Abradability rig test from Oerlikon-Metco [59, 85]. ..............................................

Figure 1-7: High speed rig-test: HSR-ALSTOM IPEK [86]. ......................................................................................

Figure 1-8: Abradability wear marks observed on coatings -Measurements realized with the high-speed rig-test from HSR-ALSTOM IPEK (Rapperswil, CH) in the frame of the E-BREAK European project. ..................................

Figure 1-13: Schematization of the flame/plasma process (left) and coating formation process (right). ............

Figure 1-14: Example of different splat shapes obtained by Fukumoto [13]. .......................................................

Figure 1-15: Plasma coating microstructure features [12]. ..................................................................................

Figure 1-16: Schematization of a heterogeneous structure, REV and equivalent homogeneous structure. .........

Figure 1-17: Micrographs taken from Oerlikon-metco material product data sheet [174]. .....................................................

Figure 2-3: Surface of CoNiCrAlY-BN-polyester powder (left) and cross-section of the metal phase particles (right) observed by SEM ........................................................................................................................................

  4-6: Representation of the porosity network difference between one image and another from the same

coating. ............................................................................................................................................................... Figure 4-7: Comparison of the experimental and calculated in plane normalized Young's modulus of NiCrAl-Bentonite coatings. .............................................................................................................................................

Figure 4-8: Comparison of the experimental and calculated in plane normalized Young's modulus of CoNiCrAlY-BN-polyester coatings. ........................................................................................................................................

  4-20: Up: a smaller magnification level (x600) is not sufficient to observe only the NiCrAl matrix. Down: Zoom in the NiCrAl matrix to avoid bentonite particles, the issue is that this low magnification level only represents a single lamellae splat. ......................................................................................................................Figure 4-21: Comparison between the experimental and calculated Young's modulus (longitudinal direction) for coating C (CoNiCrAlY-BN-polyester coating, normalized values). ....................................................................... CoNiCrAlY-BN-polyester material, normalized values)................................................................Figure 4-23: Comparison of the measured transverse thermal conductivity with the values computed with the two models for coating C (CoNiCrAlY-BN-polyester material, normalized data). ...............................................Figure 4-24: Comparison of the experimental and computed transverse thermal conductivity for coating D (CoNiCrAlY-BN-polyester material, normalized values).......................................................................................

	Preamble
	Compliance with environmental regulations is an important innovation vector in all technology
	sectors. The aviation world faces new challenges for future generations of aircraft. The Advisory
	Figure 4-22: Comparison of the experimental and calculated Young's modulus (longitudinal direction) for Council for Aeronautics Research in Europe (ACARE) has set during year 2000 a strategic agenda of
	technological innovations breakthrough in this sector for 2020. The objectives are to reduce by 50% coating D (i.e., Figure 4-25: Calculated transverse Young's modulus for the different YSZ-polyester coatings using the 2-scale CO 2 emissions, from 15 to 20% the specific fuel consumption whereas the nitrogen oxide (NOx)

and conventional models (left) -comparison between the transverse and longitudinal moduli (right). ............ Figure 4-26: Calculated thermal conductivities for the different YSZ-polyester coatings using the 2-scale and the conventional models. ..........................................................................................................................................

Figure 4-27: Computed thermal conductivity (left) and Young's modulus (right) for the one-scale and two-scale modeling methods. The highest and lowest coating properties have been calculated by considering the grey phase as the matrix (highest limit) and as pores (lowest limit). .........................................................................

Figure 4-28: Surface area fraction of voids versus equivalent diameter. ............................................................

Figure 4-29: Image of artefacts occurring on the grey phase areas during the sample preparation (cutting, mounting, and polishing). ...................................................................................................................................

Figure 4-30: Calculation of the coating extrinsic properties by changing the intrinsic properties of the dark grey phase areas, which are a mix between "nano-porosities" and small grains of YSZ. ...........................................

Figure 4-31: Description of the entire procedure to conduct the modeling method, from the sample preparation to the calculation. ...............................................................................................................................................

Acknowledgment _________________________________________________________ Index ___________________________________________________________________ List of tables _____________________________________________________________ List of figures ____________________________________________________________ Summary ______________________________________________________________ Chapter 1. : State of Art _________________________________________________ I Abradable composite coatings ______________________________________________

	I.1.	Role of Abradable seals ________________________________________________________
	I.2.	Expected properties of abradables _______________________________________________
	I.3.	Abradable composition ________________________________________________________
	I.4.	Abradable families ____________________________________________________________
	I.5.	Abradable performances indicator _______________________________________________
	I.6.	Elaboration processes of abradable materials ______________________________________

• Database of structural coated materials • Database of properties of coated materials • Preparation of coated samples for the database abradability tests • Preparation of coated samples for validation of abradability tests • Establishment of a micro model for determining the properties of abradable materials Summary II Thermal spraying -Elaboration process _______________________________________ II.1. Process description ____________________________________________________________ II.2. Coating formation process ______________________________________________________ II.3. Microstructure influence on coating properties _____________________________________ II.4. Residual stresses ______________________________________________________________ II.5. Abradable problematic _________________________________________________________

III Modeling of thermal and mechanical properties _____________________________

		III.1. Homogenization theories _______________________________________________________
		III.2. Analytical models _____________________________________________________________
		III.3. Numerical models _____________________________________________________________
	IV	Conclusion -

Thesis strategy _____________________________________________ Chapter 2. : Experimental and numerical protocols and methodologies ___________ I Manufacturing process ____________________________________________________

	II.1. Thermal properties ____________________________________________________________
	II.2. Mechanical properties _________________________________________________________
	III	Image analyses protocols and preparation of the samples. _____________________
	III.1. Preparation protocols of the samples _____________________________________________
	III.2. Observation of the coating microstructures ________________________________________
	III.3. Image analysis _______________________________________________________________
	III.4. Representative Elementary Volume (REV) ________________________________________
	III.5. Void Content and size determination ____________________________________________
	IV	Modeling methodology ________________________________________________
	IV.1. Modeling Steps ______________________________________________________________
	IV.2. Input Data __________________________________________________________________
	IV.3. Routines of the model ________________________________________________________
	Chapter 3.
	I.1.	Presentation of Materials _______________________________________________________
	I.2.	Presentation of the coating elaboration processes ___________________________________

II Experimental protocols to determine coating properties _________________________

: Model input data and computational domain ____________________ I Selection of an appropriate magnification level _______________________________

	alternative solutions ___________________________________________________________________
	I.1.	Conclusion __________________________________________________________________
	II Conclusion _____________________________________________________________

I.1. Microstructure description of the coatings ________________________________________ I.2. Representative Elementary Volume (REV) ________________________________________ II Phase discrimination ____________________________________________________ II.1. Detailed protocol ____________________________________________________________ II.2. Some specific issues encountered in image processing and implemented solution ________ II.3. A few results ________________________________________________________________ III Intrinsic properties of phases ____________________________________________ III.1. Intrinsic properties from databases available in the literature _________________________ III.2. Issues in the identification of intrinsic properties of some features and implemented

Chapter 4. Results and discussions -Improvement of the modeling method ______ I. Mechanical properties ___________________________________________________

	I.1.	Experimental values __________________________________________________________
	I.2.	Calculated Young's modulus by "conventional" 1-scale modeling ______________________
	I.3.	Comparison of experimental and calculated properties ______________________________
	II Thermal properties ______________________________________________________

II.1. Experimental values __________________________________________________________

II.2. Calculated properties by "conventional" 1-scale modeling ___________________________

Table 0 -1: Information about abradable materials studied.

 0

	Coating	NiCrAl-Bentonite	CoNiCrAlY-BN-Polyester	YSZ-Polyester
	Powder name Metco 314 NS	Metco 2043	Metco 2460 NS
	Family	Metallic Matrix with addition of solid lubricant	Metallic Matrix with addition of solid lubricant	Modified Thermal Barrier coating
	Operating temperature	From 300 to 650°C	From 300 to 750°C	From 750 to 1150°C

Table 1 -1: Experimental extrinsic properties of several abradable materials.

 1

Table 1

 1 -2 presents an example showing two microstructures having the same chemical composition but different microstructures leading to different hardness values. In order to improve the coating properties for a

Table 1 -2: Hardness of two different coating materials having the same chemical composition but different microstructures.

 1

	Porosity (%)	50	35
	HR15Y	55	75

Coating

CoNiCrAlY-hBN-polyester-Microstructure 1

CoNiCrAlY-hBN-polyester-Microstructure 2

Table 1 -3: Equivalent thermal conductivity for porous media with isolated pores.

 1

Table 1 -4: list of experimental tests applied for the different abradable coatings.

 1 Regarding the YSZ-Polyester coatings, that suffered from a lack of feedback (unlike the two previous coatings), a series of microstructures were manufactured with three different APS spraying torches. All microstructures obtained were analyzed, but only the most eloquent are considered in this manuscript. This chapter 2 describes all the experimental technics and the images treatment methodology used for this study, see Figure 2-1.

	Coating type	Experimental	mechanical	Experimental thermal property
		property measured		measured
	NiCrAl-Bentonite	Young's modulus by tensile test	Thermal conductivity by Laser flash
		and RFDA		analysis
	CoNiCrAlY-BN-Polyester Young's modulus by tensile test	Thermal conductivity by Laser flash
		and RFDA		analysis
	YSZ-Polyester	Erosion resistance		No experimental properties
		Hardness HR15Y test

Chapter 4 is a discussion concerning results obtained by numerical calculations performed on the three different coatings, in comparison with the corresponding measured experimental properties. Due to the differences of strategy between the two sub-working groups of the E-BREAK European project (explained in the introduction), experimental properties measured are not the same on the three abradable coatings and are listed in Table

1

-4.

In the frame of the EBREAK project other properties have been investigated, i.e, thermal expansion coefficient, Martens Hardness, Poisson's ratio, heat capacity, thermal diffusivity, abradability. However, those properties will not be taken in consideration in the manuscript.

Table 2 -1: Powder information -data that are more complete are provided in annexes

 2

	Reference Name from Oerlikon-Metco and powder nature	Composition (% wt.)	Manufacturing process of the powder	Size distribution range (µm)
	Metco 314NS NiCrAl-Bentonite	Ni balance, Cr: 4, Al: 4, Bentonite: 21	Chemical cladding	-177 +74
	Metco 2043 CoNiCrAlY-hBN-Polyester	Co: 30, Ni: 25, Cr: 16, Organic solids: 3 Al: 6, Y: 0.3, BN: 4, Polyester: 15,	Mechanical cladding and blending	-176 +11
		ZrO 2 : balance, Y 2 O 3 :
	Metco 2460NS YSZ-Polyester	7.5, Polymer: 4, Binder: 4,	Agglomeration	-125 +16
		Impurities (max): 0.9

Table 2 - 2 :

 22

Table 2 -2: Coating manufacturing information [174-176]

 2

	Coating	Manufacturing process	Substrates
	NiCrAl-bentonite	Combustion powder Thermospray TM	Nature: Inconel 718 Dimension: plates of 80x30x2 mm 3 with a coating thickness of 2 mm
	CoNiCrAlY-hBN-polyester	Atmospheric Plasma Spraying (F4-torch).	Nature: Inconel 718 Dimension: plates of 80x30x2 mm 3 with a coating thickness of 2 mm
		Atmospheric Plasma	Nature: Hastelloy X
	YSZ-polyester	Spraying (F4, 9MB, and SINPLEX torches)	Dimension: plates of 50x25x1 mm 3 with a coating thickness of 1 mm
	NiCrAl-bentonite coatings are coated directly on the substrate. Coatings were elaborated by the
	project partner MTU.

CoNiCrAlY-hBN-polyester coatings are coated directly on the substrate. The CoNiCrAlY matrix improves oxidation and corrosion resistances. Boron nitride provides lubrication to improve abradability and avoid wearing of the blades. The polyester phase allows excellent friability against titanium alloy, steel or superalloy components.

YSZ-polyester coatings are made of three different layers. The first one is a bond coat composed of NiCrAlY (AMDRY 962, Oerlikon-metco group, Switzerland), which is known for its excellent resistance against oxidation and hot corrosion. The second layer is made of Yttria-Stabilized Zirconia (8Y 2 O 3 -ZrO 2 ) manufactured with HOSP™ process (Metco 204NS powder, Oerlikon-metco group,

Table 2 -3: Matrix of tests for YSZ-polyester coatings

 2

Table 2 -4: Different test temperature for the abradables coatings

 2

	Abradable type	Test temperature
	CoNiCrAlY-BN-	50°C	450°C	750°C
	polyester
	NiCrAl-bentonite	50°C	300°C	600°C
	II.2. Mechanical properties
	II.2.1. Rockwell superficial hardness test (HR15Y)

Table 2 -5: Parameter of the different tests.

 2

	Test parameters	NiCrAl	Bentonite	CoNiCrAlY	Grey phase (BN+oxides)
	Indenter type	Berkovich	Berkovich	Berkovich	Berkovich
	Approach speed (nm.min -1 )	4000	4000	4000	4000
	Load type	Quasi-static	Quasi-static	Quasi-static	Quasi-static
	Load time (s)	30	30	30	30
	Break period (s)	15	30	15	15
	Unload time (s)	30	30	30	30
	Maximum Load (mN)	25	2.5	25	5

Table 2 -7: sample preparation protocol for thermal sprayed NiCrAl-bentonite, CoNiCrAlY-BN-polyester and YSZ-polyester samples

 2

			NiCrAl-bentonite	CoNiCrAlY-BN-polyester	YSZ-polyester
	Sectioning	Cut-off wheel Transverse speed feed Specimen orientation	Hard material 0.02 mm.s -1 compressive	Hard material 0.02 mm.s -1 compressive	Medium hard materials 0.005 mm.s -1 compressive
		Type	Cold Vacuum Mounting Cold Vacuum Mounting Cold Vacuum Mounting
	Mounting	Resin Curing time	EpoFix, 8 hours	EpoFix fluidization by water bath before mounting 8 hours	EpoFix 8 hours
			Disc/cloth: SiC paper	Disc/cloth: SiC paper	Disc/cloth: SiC paper
		Step 1	Force (N) by sample: 30 Abrasive: 600 grit SiC	Force (N) by sample: 30 Abrasive: 600 grit SiC	Abrasive: 600 grit SiC
			Time : 60 s		Time : 60 s
	Polishing

Table 2 -8: Void content of each size class for all coatings.

 2

Table 2 -9: list of the different data

 2

				Properties
			Intrinsic			Effective
		Nano scale		Micro scale	Micro scale
	Literature	Feedstock measurement	Bulk measurement	Literature	Bulk measures	Literature	Measures on coatings
	E, α, ν, κ, Cp, Hardness	E, ν, Hardness (nano-indentation)	E, ν, Hardness (nano-indentation)	E, α, ν, κ, Cp, Hardness, T melting	X	E, α, ν, κ, Cp, Hardness, T melting	E, α, ν, κ, Cp, Hardness, T

melting

Several data can be identified in the literature and are listed in Table

2-9:

Table 2 -

 2 10: Bulk properties of each phase according to the literature

	Phase	YSZ	voids	CoNiCrAlY	NiCrAl
	E (GPa)	215

Table 2 -11: Range of variation of the properties according to the literature

 2

	Phase	Bentonite [204-206] [207]	Aluminum Oxide [208-214]	hBN [210, 213, 215-220]
	E (GPa)	0.02-50	200-400	11-74
	ν (-)	0.1-0.5	0.22-0.27	0.05
	к (W.m -1 .K -1 )	0.2-1.4	10-35	15-78

Table 2 -

 2 12. The Young's modulus of Al 2 O 3 and bentonite was measured at 190 GPa and 4 GPa respectively.In comparison with data from the literature, the measured Young's modulus of the grey phase in CoNiCrAlY-BN-polyester coatings is lower than data found for Al 2 O 3 and much higher than data concerning hBN. The Young's modulus deduced from nano-indentation in the grey phase of the coatings seems thus intermediate between that of Al 2 O 3 and that of hBN.

Table 2 -12: Young's modulus measured by nano-indentation (directly in coating samples).

 2

	Phase	CoNiCrAlY	NiCrAl	Bentonite	Oxide+BN
	E (GPa)	150	142	4	190

Table 3 -

 3

1 displays the voids content repartition of each size-class into the reference coating. Except for the voids having equivalent diameters smaller than 1.5µm, one can conclude that each voids sizeclass is equally represented in the reference coating. There is no preponderant voids size-class.

Table 3 -1: Void content of each size-class for the reference coating

 3

		D eq ≤1.5 µm 1.5<D eq ≤10 µm	10<D eq ≤20 µm 20< D eq ≤40 µm	40 µm<D eq	Total void
	Reference coating	0.1%	2.4%	1.2%	2.2%	2.0%	7.9%

CoNiCrAlY-BN-Polyester-Low level

CoNiCrAlY-BN-Polyester-High level YSZ-Polyester-Low level YSZ-Polyester-High level

Table 3 -2: Description of features observed in function magnification level

 3

	Coating	1 scale	Index of complexity 2 scales	3 scales
		 Two phases
	NiCrAl-Bentonite	 Globular voids
		 Lamellar cracks
	CoNiCrAlY-BN-polyester	 Coarse globular voids due to the polymer acting as a pore former  Matrix phase (large cracks and voids)	 Voids network intrinsic to the APS process  Lamellar cracks  Aluminum oxides and BN
		 Coarse voids due
		to the polymer
		acting as pore
	YSZ-polyester	former and to voids formed during the spray process	 Lamellar cracks  Voids network intrinsic to the APS process	 Unmolten cores of feedstock powder particles
		 Matrix phase
		(large cracks and
		voids)
	Coating	NiCrAl-Bentonite	CoNiCrAlY-BN-polyester	YSZ-polyester
	Number of scales
	required to depict the	1	2	3
	structure
	Number of phases
	identified in the	3 [note 1]	3 [note 2]	3 [note 3]
	coating structure
	index of complexity	4	5	6
	[note 1] NiCrAl, Bentonite, voids.

[note 2] CoNiCrAlY, BN, voids.

Table 3 -3: Variability study of porosity rate to determine appropriate magnification level.

 3

Table 3 -4: Minimum number of SEM images necessary to get representative properties of coatings.

 3

Table 3 -5: Summarize of the different magnification level REV for each coating and magnification level.

 3

	Coating nature	Modeling method	Magnification level	Number of images
	NiCrAl-Bentonite			15
	CoNiCrAlY-BN-Polyester	"Conventional" 1-scale	×150	10
	YSZ-Polyester			15

Table 3 -6: Presentation of the two different microstructures studied for NiCrAl-Bentonite coating.

 3

Table 3 -7: Presentation of the two different microstructures studied for CoNiCrAlY-BN-Polyester coating.

 3

Table 3 -8: Presentation of five different microstructures studied for YSZ-Polyester coating.

 3

Table 3 -9: Bulk properties of various phases published in literature.

 3

	at room temperature	8YSZ	CoNiCrAlY	NiCrAl
	E (GPa)	200-220 [228, 230,	100-200 [202, 203, 231,	100-178 [202, 203,
		235-238]	232]

Table 3 -10: Nanoindentation Young's modulus values of CoNiCrAlY and NiCrAl.

 3

		average Young's modulus and
	Phases	association standard deviation	Variability (-)
		(GPa)
	CoNiCrAlY	148 ±15	0.20
	NiCrAl	141 ±32	0.45

Table 3 -11: Input data used for the model

 3

	Phases	Void	Assumption
	E (GPa)	1.10 -5 *

 Voids are filled with dry air.

Table 3 -14: Input data used for the model.

 3

	Phases / features	YSZ	Air in voids	CoNiCrAlY	NiCrAl	Bentonite	Al 2 O 3 +BN composite compound
	E (GPa)	215 [192]	1.10 -5 **	150*	142 *	4*	190*
	ν (-)	0.3 [192]	0.33 **	0.3 [197,

Table 4 -4: HR15Y hardness and erosion resistances of YSZ-Polyester coatings.

 4

		normalized HR15Y hardness	resistance normalized erosion	(%) Porosity rate	Grey phase rate (%)
	Reference	1.000 ± 0.042	1.00	8	18
	Similar coarse				19
	microstructure to	1.058 ± 0.033	0.92	8
	the reference
	More porous	1.041 ± 0.024	0.38	12	12
	Less porous	1.028 ± 0.057	1.51	3	18
	Dense, no coarse porosities	1.179 ± 0.060	1.84	3	21

Table 4 -5: Calculated Young's modulus of NiCrAl-bentonite coatings at room temperature (normalized results).

 4

	Young's modulus		Microstructure type
	Numerical method	Grade A (less porous)	Grade B (most porous)
	E in plane	1.4 ± 1.18	1.3 ± 0.88
	Table 4-5 regroups the through-thickness Young's modulus calculated for both types of NiCrAl-
	bentonite coatings. The calculated Young's moduli are pretty low, as expected for abradable
	performances. The average Young's modulus calculated for the most porous coating (i.e.,
	microstructure B

Table 4 -6: Calculated mechanical properties of CoNiCrAlY-BN-Polyester coatings at room temperature (normalized results).

 4

	Microstructure type

Table 4 -7: Normalized calculated Young's modulus of YSZ-polyester coatings.

 4

		normalized through-	Porosity rate	Grey phase
		thickness Young's modulus	(%)	rate (%)
	Reference	1.00 ± 0.14	8	18
	Similar coarse microstructure as ref	1.07 ± 0.14	8	19
	More porous than ref	0.79 ± 0.14	12	12
	Less porous than ref	1.24 ± 0.09	3	18
	Dense, no coarse porosities	0.69 ± 0.23	3	21

Table 4

 4

	-8 and Table 4-9). For

Table 4 -8: Experimental thermal conductivity of NiCrAl-bentonite coatings at room temperature (normalized results).

 4 Similar measurements have also been performed for CoNiCrAlY-BN-polyester coatings. The results are provided in the Table 4-9. In this table, the thermal conductivity is normalized by the experimental value measured by Laser Flash Technique for the more porous coating (namely D).

	Experimental method	Grade A (less porous)	Grade B (most porous)
	Laser flash analysis	1.13± 0.04	1.00 ± 0.02
	NiCrAl-Bentonite coatings seem to show good thermal insulation properties. The lowest thermal
	conductivity measured corresponds to the most porous coating (B). However, the thermal
	conductivities measured for both coatings are quite close one from the other. The coefficient of
	variation represents 3 and 2 % of the absolute value of the thermal conductivity measured by laser
	flash analysis for coatings A and B respectively.
	II.1.2. CoNiCrAlY-BN-polyester coatings

Table 4 -9: Experimental thermal conductivity of CoNiCrAlY-BN-polyester coatings at room temperature (normalized results).

 4

Table 4 -11: Calculated through-thickness thermal conductivity for CoNiCrAlY-BN-polyester coatings at room temperature (normalized results based on the experimental measure for case D).

 4

Table 4 -12: Calculated through-thickness thermal conductivity of YSZ-polyester coatings (normalized from data computed for the reference coating).

 4

		normalized through-	Porosity rate (%)	Grey phase rate
		thickness thermal		(%)
		conductivity
	Reference	1.00 ± 0.07	8	18
	Similar coarse	1.03 ± 0.08	8	19
	microstructure than ref
	More porous	0.88 ± 0.07	12	12
	Less porous	1.14 ± 0.04	3	18
	Dense, no Coarse	0.97 ± 0.14	3	21
	porosities

Table 4 -13: Description of features observed in function the magnification level.

 4 Due to the coarse porosity network provided by the polymer contained in the powder, it becomes difficult to find an adequate magnification level to represent the YSZ polyester and CoNiCrAlY-BN-

			Level of complexity
	Material
		1 scale	2 scale	3 scale
	NiCrAl-Bentonite	 The two phases  Porosity  Thin cracks
			 Porosity due to
		 Coarse porosities due to the	the APS process
	CoNiCrAlY-	polymer removal	 Thin cracks in the
	BN-	 Matrix phase without the	matrix
	polyester	thinnest cracks	 Aluminum oxides
			/ BN phases
	YSZ-
	polyester

 Coarse porosities due to the polymer removal  Matrix phase without the thinnest cracks  Thin cracks  Porosity due to APS process  Details of the grey phases

Table 4 -14: Input properties implemented for the 2-scale modeling method.

 4

		Phase	E (GPa)	к (W.m -1 .K -1 )
		Voids	1.10 -4	0.025
		YSZ bulk	215	3
	1 st magnification level
		CoNiCrAlY	150	25
		BN/Oxide	190	50
		Voids	1.10 -4	0.025
	2 nd magnification level	YSZ matrix	Calculated in the first stage of the model	Calculated in the first stage of the model
		CoNiCrAlY-	Calculated in the first	Calculated in the first
		BN/oxide matrix	stage of the model	stage of the model

Table 4 -15: Representative elementary volume of the 2-scale modeling method.

 4

	Magnification	Image number

Table 4 -16: Input data used for the model.

 4

					CoNiCrAlY	NiCrAl	Bentonite Grey phase
	Phases	YSZ	Void				(Oxide+BN)
	E (GPa)	215 [192]	1.10 -5 **	150*	142 *	4*	190*
	ν (-)	0.3 [192]	0.33 **	239] 0.3 [197,	239] 0.3 [197,	205] 0.3 [204,	0.3
	к (W.m	-1 .K -1 )	3-2.5 [192]	0.025 [193]	25 [199]	201] 15 [200,	206] 1.4 [204,	5-90 [215-217]

Table 4 -18: Experimental and calculated Young's moduli of CoNiCrAlY-BN-polyester coatings at room temperature (normalized results from RFDA measure for coating D).

 4 conventional modeling method. Indeed the normalized Young's modulus of coating C calculated with the 2-scale modeling method is 3.4, whereas the values measured experimentally with the RFDA and Tensile techniques are 3.1 and 2.8 respectively. Those two values are very close and the computed Young's is in better accordance in comparison with the conventional model which provided a Young's modulus twice higher than the experimental value (normalized Young's modulus of 7.1). It appears also that the 2-scale modeling method also provides a decrease of the Coefficient of variation of the calculated Young's modulus. Indeed the coefficient of variation decreases from 25% to 14% by using the 2-scale modeling.The relative difference between the computed and the experimental values of the Young's modulus was calculated with equation (VI-1) and is listed in

		Coating C	Coefficient of variation	Coating D	Coefficient of variation
	RFDA
		3.1 ± 0.2	8%	1.0 ± 0.6	77%
	(Experimental method)
	Tensile test
		2.8 ± 1.1	39%	1.1 ± 0.6	55%
	(Experimental method)
	E longitudinal	7.1 ± 1.8	25%	8.8E -4 ± 1.76E -3	200%
	(Conventional modeling)
	E longitudinal	3.4 ± 0.5	14%	8.8E -4 ± 5.88E -4	67%
	(2-scale modeling)
	E transversal	5.0 ± 0.9	15%	5.88E -4 ± 1.18E -4	20%
	(2-scale modeling)

Considering firstly coating C (the less porous one), the Young's modulus calculated with the 2scale modeling method is in better accordance with the experimental measure than the one calculated with the

Table 4 -19: Relative difference between experimental and calculated Young's modulus (in-plane direction) of CoNiCrAlY-BN-polyester coatings.

 4

	Coating C	Coating D

Table 4 -

 4

19. As represented in

Table 4 -21: Relative difference between the computed and experimental thermal conductivity of CoNiCrAlY-BN- polyester coatings.

 4

	Coating C	Coating D

Table 4 -22: Normalized Young's modulus of YSZ-polyester coatings, calculated with the 2-scale modeling method.

 4

		normalized calculated	normalized calculated
		transversal Young's modulus	longitudinal Young's modulus
	Reference	0.29 ± 0.01	0.46 ± 0.01
	Similar coarse microstructure as ref	0.37 ± 0.01	0.54 ± 0.01
	More porous	0.08 ± 0.01	0.20 ± 0.01
	Less porous	0.62 ± 0.01	0.87 ± 0.01
	Dense, no coarse porosities	0.66 ± 0.01	0.84 ± 0.01

Table 4 -23: Normalized thermal conductivities of YSZ-polyester coatings, calculated with the 2-scale modeling method.

 4

		normalized calculated	normalized calculated
		transverse thermal conductivity	longitudinal thermal conductivity
	Reference	0.71 ± 0.01	0.80 ± 0.01
	Similar coarse microstructure as ref	0.76 ± 0.02	0.82 ± 0.01
	More porous	0.38 ± 0.01	0.46 ± 0.01
	Less porous	0.92 ± 0.01	0.98 ± 0.01
	Dense, no coarse porosities	0.95 ± 0.01	1.09 ± 0.01

Table 4 -24: Experimental thermal conductivity taken from the literature for similar coatings.

 4 Even if it is not possible to compare quantitatively those experimental results with the ones calculated in this manuscript (due to the normalization of the results), it is however possible to say that the thermal conductivities calculated with the 2-scale modeling model for the different coatings presented in this manuscript are in the same range of value in comparison with experimental results reported in

	Author	Elaboration process	Material	Experimental Process	Thermal Conductivity (W.m -1 .K -1 )
	N. Curry [108]	APS	YSZ-polyester (26% porosity)	Laser flash	0.64 (as sprayed) 1.08 (after thermal
					treatment)
	N. Curry [108]	APS	YSZ (16 % porosity)	Laser flash	0.69 (as sprayed) 1.31 (after thermal treatment)
	Tamarin	APS	YSZ	Laser flash	0.84
	[232]

Table 4 -25: Different classes of the equivalent diameter of voids.

 4

Table 4 -26: Area fraction of each voids class in coatings.

 4

	Coating	Very fine D eq ≤1.5	Fine 1.5 < D eq ≤10	Medium 10 < D eq ≤ 𝟐0	Coarse 20 < D eq ≤ 𝟒0	Very coarse 40 ≤D eq
		µm	µm	µm	µm	µm
	More porous	1%	21%	8%	21%	49%
	Less porous	3%	44%	22%	28%	3%
	Dense with no coarse porosities	3%	71%	24%	2%	0%

Table 4 -27: Modeling Input data for the dark grey phase. YSZ and porosity volume fractions in the grey phase areas E grey phase areas (GPa) к grey phase areas

 4

	(W.m -1 .K -1 )

Table 4 -34: NiCrAl-Bentonite thermal expansion

 4

Table 4 -35: properties calculated at the high level of the 2-scales modelling for CoNiCrAlY-BN-Polyester (less porous coating)

 4

		кy	кy	Ey	Ex	νyx	νxy	αy	αx
	High	9±1 10±1	41±8	51±6	1.2E-01±	1.4E-01±	1.3E-05 ±	1.3E-05±
	level					1.4E-02	1.0E-02	3.7E-07	4.5E-07

Figure 1-1: Summary of chapter 1.

Figure 2-1: Summary of chapter 2.

Figure 3-1: Summary of chapter 3.

A complete study of this void network has been published in a contribution entitled "Influence of APS Process Parameters on Morphologies of YSZ-Polyester Abradable Coatings"[START_REF] Aussavy | Influence of APS Process Parameters on Morphologies of YSZ-Polyester Abradable Coatings[END_REF] D. Aussavy, R. Bolot, G. Montavon, F. Peyraut, G. Szyndelman, J. Gurt-Santanach,S. Selezneff, "Influence of APS Process Parameters on Morphologies of YSZ-Polyester Abradable Coatings," in Thermal Spray

2015: Proceedings of the International Thermal Spray Conference, Longbeach,CA,USA, 11-14 th May 2015, pp. 99 -106.

Depending upon the coating structure; i.e., NiCrAlY-Bentonite, CoNiCrAlY-BN or YSZ-polyester, adaptation in terms of threshold values have been introduced.

Figure 4-1: Summary of chapter 4.

Acknowledgment

Chapter 1. : State of Art

The sample preparation is described once below for YSZ-polyester coatings: finding an appropriate preparation protocol of the samples was the most difficult for this type of coatings. The sample preparation of CoNiCrAlY-BN-polyester and NiCrAl-Bentonite coatings are described in Table 2-7.

The equipment chosen for sectioning is an automatic micro abrasive cutting, Accutom-5 precision cut-off machine from Struers Company. An automatic constant feed system supplies a constant cutting pressure which induces uniform sectioning conditions and a uniform quality of the cut surface. This process is not operator dependent. The micro-sectioning quality of the sample can also be influenced by the blade nature, the cut speed and the sample fixing. The cut-off wheel chosen was a saw recommended for cutting soft materials. A saw adapted for hard material was first used, but a lot of pull out damages were observed near from the polyester phase of the coating. Because of the brittle nature of abradable coatings, it was thus chosen to work with the lowest cutting velocity to realize a clean cut and decrease the possibility to provide damages to the coating. Finally, the transverse feed speed was 0.005mm.s -1 . The specimen is orientated in such way that the coating is pushed toward the substrate during the blade penetration. It permits to keep the coating in compression and prevents delamination, Figure 23456789101112131415161718. For the second step, a cold vacuum mounting was chosen, Figure 2345678910111213141516171819. Indeed hot mounting is not appropriate for porous coatings. This method consists in heating epoxy powders up to a temperature of 150°C at a constant pressure, which induces a short time of mounting. Those short cycle time do not permit a deep penetration of epoxy in the coating. But porous structures rely on support of epoxy resin impregnation to limit mechanical damages which could occur during polishing. Adequate mounting for abradable coatings is then a long cure time and requires a high viscosity epoxy to permit a deep penetration of epoxy in the coating. Indeed, vacuum cold mounting is mandatory for those types of structures due to their high void content. Vacuum mounting permits to remove the air from the sample σ and μ are respectively the standard deviation and average value of the calculated properties.

For this reason, image analysis (i.e.: total void content) and modelling of the coating properties (i.e.: E, α, κ) are conducted on several pictures of the same coating. The measured and calculated data may be randomly selected (order) and used to plot the variability versus the number of analyzed images. The shape of the curve, e.g. when the function tends toward an asymptote, determines the required number of images (i.e. the asymptote plateau value give the number of images). [START_REF] Antou | Quantifying Thermal Spray Coating Architecture by Stereological Protocols: Part II. Key Points to be Addressed[END_REF] As an example, the void content has been considered on Figure 2-25 as the discriminative feature for determining the REV of YSZ-polyester coatings. Several steps are required. The first one consists in the analysis of the void content on 20 images randomly located on the sample (all images captured for the same coating) using the Delesse stereological protocol [START_REF] Russ | Practical Stereology[END_REF]. This work was performed at several magnification levels (depending on the material studied). The average content of voids (µ cumul ) is then calculated, as well as the related standard deviation (σ cumul ) for various numbers of randomly selected data issued from the 20 images. From that information it is possible to determine the variability by using II-22, for each random selection (the order of images influences the curve). To finish, the variability may be plotted versus the number of analyzed images and analyzed to deduce the appropriate magnification level and the number of images to consider. The detailed results of the REV study will be given in chapter 3.

III.5. Void Content and size determination

The estimation of the void content was conducted by image analysis. Thus calculation was implemented with Image J Software (http://rsbweb.nih.gov/ij/download.html). The process consists in discriminating all the coating phases, meaning using one or more threshold levels on SEM or OM images (e.g., 2 thresholds for 3 phases) to transform the grey-scale images in binary (2 phases) or ternary images (3 phases). This allows the discrimination of voids from other coating phases. With Image J software, it is possible to measure the surface areas of the different phases. The rate of the different phases can then be calculated with equation below corresponding to the Delesse protocol [START_REF] Russ | Practical Stereology[END_REF]. Therefore, it is assumed that they correspond mostly to aluminum oxides formed during the spray process, either by in-flight oxidation of particles (liquid-gas interaction) or, after particles flattening, on the coating surface (solid-gas interaction). Bore is a light element having a low energy. This makes impossible its identification by this technique. Beside, nitrogen and oxygen excitation energy levels are very close from each other (0.40 and 0.53 keV, respectively). For those two reasons, detection of the BN phase is impossible with this technique.

FOCUS : Microstructure of CoNiCrAlY-BN-Polyester

Features 7 (grey features) identified into the YSZ-Polyester coating, have been studied more precisely to determine their nature. Images at very high magnification (×10 000) have been captured by SEM, in both Secondary (SE) and Backscattered (BSE) modes. The SE mode permits to determine

I.2. Representative Elementary Volume (REV)

Extracting quantitative data from a coating microstructure required, in view of avoiding biases, to collect images representative of the whole structure, that is to say to consider a representative elementary volume (REV). REV is directly related to the magnification at which the structure is studied. For example, Figure 3-4 represents coatings having the same structure at two different magnification levels; i.e. × 150 and ×500, respectively. It is obvious that the nature of information extracted from these two images is different considering one magnification and another. As described before, microstructure features are above a size range inducing to found a magnification level allowing observing as well the global microstructure of such coatings as microstructure details. In the case of plasma spray coatings, an adequate magnification level should i) permit the observation of a fraction of the coating representative of its whole structure: the global structure but also structural details such as phases, voids, cracks, etc. ii) consider a number of images permitting to obtain statistically representative values of coating descriptors. A compromise has to be found. As describe in chapter 2, to determine an appropriate RVE it is also necessary to determine the number of images to study in order to have statistical representative results. The entire study procedure will be depicted for YSZ-Polyester coating which is the microstructure having the highest complexity. For the two other structures the results of the REV procedure will be directly given.

I.2.1. First step, magnification level selection

The determination of the REV was realized in several steps. The first step corresponds to the determination of a relevant magnification level. Voids have been considered here as the discriminative features. The analyses of voids content were conducted on twenty images, randomly located along a cross-section of the same coating, at several magnification levels (for example here; ×150, ×200 and ×300, Figure 345678).

X 150 X200 X300 Then for the data issued from the initial images, data values were randomly sorted to calculate the cumulative void average content (µ cumul ) using the Delesse stereological protocol [START_REF] Russ | Practical Stereology[END_REF], and the related standard deviation (σ cumul ). It was then possible to determine the variability (σ cumul /μ cumul ) for each

I. Mechanical properties

Tensile tests and RFDA measured on NiCrAl-Bentonite and CoNiCrAlY-BN-Polyester coatings provide the Young's modulus along the in plane direction of the coating. To compare coherent properties the Young modulus will be calculated along the in plane direction for those two materials.

The differences of the properties calculated in the in plane or through thickness direction will be discuss later in the study.

I.1. Experimental values

I.1.1. NiCrAl-Bentonite coatings Among the three coatings studied, the less complex one is the NiCrAl-bentonite coating. As described in chapter 3, two different grades of this material were studied (coatings obtained for two different sets of parameters). In practice, the first one called A, is less porous (35%) than the second one called B (40%), see Figure 4-3. Those microstructures show an important network of connected voids, even if the corresponding void rate is lower than 50%. Experimental measurements of the Young's modulus were conducted by IFAM-DD (partner of the project) and are listed in Table 4-2. However, due to confidentiality reasons, the results were normalized by considering the Young's modulus measured by RFDA for the most porous coating (grade B). When comparing the two NiCrAl-Bentonite coatings, the Young's modulus is lower for the most porous one (i.e., microstructure B). This observation is the same for both experimental techniques (i.e., tensile test and RFDA). However, the Young's modulus ratio of the two coatings is a little higher for the RFDA method (about 3) in comparison with the tensile test measure (about 2). Moreover, the order of magnitude of experimental values measured with the two techniques is consistent, especially in the case of sample B (0.6 for the tensile test against 1.0 for the RFDA method, that is to say a factor of about 1.7). However, the discrepancy between the Young's modulus measured with the two techniques is higher for the less porous coating (i.e., microstructure A, factor of 2.7). Indeed, the value calculated by RFDA is higher than the one measured by tensile test. This result can be explained by the brittle behavior of the coating determined by the analysis of the strain-stress curve of the tensile test. Indeed, a detailed analysis of the stress-strain-curves revealed that there is no distinct elastic limit.

A B

The material fails shortly after reaching ultimate tensile strength or shows a large drop of stress till rupture. RFDA technique is lower. On the contrary, when the void rate is higher (i.e., coating D), the standard deviation of the Young's modulus is lower for the tensile test method.

I.1.3. YSZ-polyester coatings

Due to the differences of strategies described in the preface of the manuscript between one work package (abradables for the high-temperature compressor) and the other (abradables for the highpressure turbine), the mechanical properties measured for the YSZ-Polyester coatings are different and consist in their hardness and erosion resistance.

Table 4-4 lists the HR15Y hardness and erosion resistance values for the five different YSZpolyester coatings presented in chapter 3. For confidentiality reasons, the measured data were normalized once again. The erosion resistances and hardness values were normalized in function of the erosion resistance and hardness measured for the reference coating.

[199]. The CoNiCrAlY-BN-polyester coatings also show good thermal insulation properties. The lowest thermal conductivity measured corresponds to the most porous coating (coating D), but thermal conductivities of both coatings are quite close in view of their strong differences of microstructure.

The coefficient of variation represents 10 and 2 % of the thermal conductivity measured by laser flash analysis for coatings C and D, respectively.

Unfortunately thermal conductivity measurements were not performed for YSZ-polyester coatings.

II.2. Calculated properties by "conventional" 1-scale modeling

The thermal conductivities have been estimated using the modeling method described in chapter 2.

Only the thermal conductivity calculated along the through-thickness direction of the coating will be compared to the experimental results. Indeed the laser flash technique allows measuring the thermal conductivity in the through-thickness direction of the coating only. The results obtained are listed in the following 3 tables: Table 4-10, Table 4-11 and Table 4-12.

II.2.1. NiCrAl-Bentonite coatings

The thermal conductivities calculated for NiCrAl-Bentonite coatings are pretty low, thus confirming experimental data. The calculated thermal conductivities and the corresponding standard deviations are similar for both coatings A and B. The differences in the microstructure (low according the micrographs) do not induce a difference in the calculated properties. The standard deviations are quite high: about 32% of the calculated thermal conductivities for both coatings A and B. This can be explained by the porous network which is widely connected: with 2D images of the coating microstructure, the porosity network between one image and another may be quite different, thus providing this high variability. × 900, × 1000) provide a quite similar variability of the porosity. For CoNiCrAlY-BN-polyester, at the magnification level of × 1200, the variability is much higher and the curve is not really stable when considering fifteen pictures only. At the bottom of the figure, the porosity variability of CoNiCrAlY-BN-polyester and YSZ-polyester coatings is represented for the low magnification levels.

CoNiCrAlY-BN-polyester YSZ-polyester

For CoNiCrAlY-BN-polyester, the considered acquisition method was optical microscopy (OM), which allows observing the coatings in their overall with very low magnification levels. Unfortunately the choice of the magnification level is limited to × 25 and × 50. From the previous curves it appears that only few images are mandatory to reach a representative volume. For YSZ-polyester coatings, this time OM does not allow observing sharp images of the coating. For this reason it was chosen to work

with SEM images at a very low magnification level (but not so low in comparison with OM).

However, at low magnification level, another issue has to be considered: the edges of the detector are present on the pictures, so that the images have to be reworked with the help of image J software. The use of a magnification level of × 80 induces a similar variability in comparison with the × 70 and ×

V Calculated properties by applying the 2-scale modeling methodology

Effective properties calculated by applying the 2-scale modeling method will be presented for the CoNiCrAlY-BN-polyester material first, and for the YSZ-polyester coatings then. The results computed for the CoNiCrAlY-BN-polyester coatings will be compared with the properties measured experimentally but also with the properties calculated by applying the conventional one-scale model, in order to show clearly if the 2-scale modeling process is really an improvement of the conventional modeling methodology. For the YSZ-polyester coatings, no experimental data are available. However, the variation of the computed properties by considering conventional or two scale modeling will be studied, as well as the influence of the coating microstructure on the calculated properties.

V.1. CoNiCrAlY-BN-polyester coatings

Representations of the image used at each scale are given in the following Table 4-17. The thermal conductivity was calculated using the 2-scale modeling method for both coatings C and D (CoNiCrAlY-BN-polyester material). The thermal conductivity was evaluated for both directions (i.e., longitudinally and transversally), in order to investigate the anisotropy. The results are listed in Table 4-20 and represented on Figure 4-23 and Figure 4-24.

Only the thermal conductivity calculated in the through-thickness direction (transverse direction) will be compared to the experimental data. Indeed the laser flash method allow measuring the thermal conductivity in the transverse direction of the coating only. Considering firstly coating C, the thermal conductivity calculated for the 2-scale modeling method is lower than the one calculated with the conventional modeling method. However it remains very high in comparison with the measured thermal conductivity. Indeed the normalized thermal conductivity computed for coating C is 10.0 for the conventional modeling method and 6.7 for the 2scale modeling method, whereas the measured value is 1.35. Those two values are very different (factor 5 between the value predicted with the 2-scale model and the measure). However, the coefficient of variation of the effective thermal conductivity computed with the 2-scale model is well decreased in comparison with that previously calculated with the conventional model. Indeed the coefficient of variation decreases from 50% to 15% with the 2-scale modeling. This tendency was also previously noticed for the Young's modulus.

modeling is the decrease of the standard deviation of the calculated results. Using the 2-scale modeling allows decreasing the differences of the calculated properties between on image of the coating and another. The computed result may thus be considered as more reliable. In this context, a way to improve the estimation of the Young's modulus for such a coating (highly porous with a highconnectivity network) could be to combine the two-scale modeling method with 3-dimension modeling. The first step of the methodology would be to estimate the extrinsic properties of the matrix (as described in this manuscript) on the basis of calculations performed on 2D images, and then to consider the resulting data as input properties for the matrix to perform calculations on 3-dimension images of the global microstructure of the coating. This comment suggests that the first step of calculation performed here is good, but that the discrepancies come from the second step of calculation (i.e., representation of the global structure not possible using 2D images).

The analysis of the thermal conductivity results is more problematic due to the incertitude concerning the input properties to consider for the grey phases embedded in the CoNiCrAlY-BNpolyester coatings. Indeed, the calculated thermal conductivities are far from the measured data for those coatings. Concerning coating D, the discrepancies could be due to the same explanation as that suggested for the Young's modulus (i.e. misrepresentation of the connectivity of the CoNiCrAlY phase when using 2D images). However this cannot explain the discrepancies observed for coating C for which the computed results are also far from the corresponding experimental data. On contrary with the intrinsic Young's modulus of the unknown grey phase (which was estimated from nanoindentation tests), the thermal conductivity of these areas was estimated to 50 W.m -1 .K -1 by the combination of literature researches and assumptions. The grey phase has been assumed to be a mixing of BN phase already existing in the powder and Aluminum oxide created during the plasma spraying elaboration process. In the literature [START_REF] Lipp | Hexagonal boron nitride: Fabrication, properties and applications[END_REF][218][219], the properties of those two phases were found to vary in between 10 to 90 W.m -1 .K -1 in function of their real composition (impurities or not) and in function of the elaboration process of the powder itself. In chapters 3 and 4, the influence of the variation of this input thermal conductivity on the calculated data was studied and discussed, see Figure 4-19. This influence was qualified as low if the thermal conductivity of this phase remains in between 20 W.m -1 .K -1 to 90 W.m -1 .K -1 . However, for lower values (lower than 20 W.m -1 .K -1 ) of the input thermal conductivity of the grey phase regions, the influence on the coating calculated properties was found to become stronger. The coming questions are thus the following ones:

-does the difference between the experimental and computed thermal conductivity can be explained by the assumptions made for the input properties of the grey phase of the CoNiCrAlY-BN-polyester coatings?

-does coating D represent a limitation to the use of 2-D images for the modeling method?

Two other observations can be made from those results:

 When applying the 2-scale model, the standard deviation is decreased whatever the calculated property (Young's modulus and thermal conductivity).

 The Young's modulus seems to be anisotropic for such microstructures. However the thermal conductivity of the coating is pretty similar whatever the considered direction.

VI Solution to better estimate the properties of the YSZ grey phase

VI.1. Why using a 3-scale modeling is not possible?

The use of calculations performed on micrographs captured at two different magnification levels permits to take into account the influence of inter and intra lamellar voids, as well as that of the coarsest voids, on extrinsic properties of the coatings. Nevertheless, one question appeared regarding the intrinsic properties to consider for the dark grey areas of the YSZ-polyester coating (of unknown exact nature or unknown properties).

Indeed, as explained in previous sections describing those grey phases, they are composed of both YSZ small grains and "nano-porosities". Since this phase has an influence on the calculated extrinsic properties, it is thus mandatory to determine precisely their intrinsic properties. One solution could be the use of a third magnification level into the model to calculate the extrinsic properties of these dark grey phases and to implement them into the calculations performed at a lower scale. However, for the following reason this solution is not applicable; one of the most important reasons concerns the artefacts occurring during the metallographic preparation for those phases. Indeed, those artefacts increase the difficulty to separate the features which constitute those grey phases, see Figure 4-29, so that it does not permit to have representative images of their real microstructure suitable for the modeling works. Moreover, the addition of a third scale would be time consuming with no guarantee of the good reliability of the properties calculated due to the misrepresentation of the microstrucre.

VI.2. Estimation of the grey phase properties

The solution applied here will permit to study more precisely the influence of the input properties of these grey-phase areas on the final coating effective properties. The considered approach stands in the choice of intrinsic properties in order to calculate the upper and lower limits of the coating For YSZ-polyester coatings, both modeling methods were discriminant (between the different coating microstructures) for both computed Young's moduli and thermal conductivities. Even by considering the absence of experimental data to discuss the relevancy of the 2-scale modeling method, the calculated results were found in the same range as experimental data obtained in the literature for similar coatings, compared to the properties derived from the conventional modeling method. The Young's moduli and thermal conductivities calculated with the 2-scale modeling method were lower than those calculated with the conventional model, which shows that the smallest microstructural details were taken into account in a better way. However, there is one exception to this conclusion: the properties calculated for the coating with the densest microstructure (i.e., with no coarse porosities) are similar for both modeling methods. This allows concluding that applying the 2-scale modeling does not represent an interest for coatings with no coarse porosities (i.e., single size range of the pores).

From all the computed results, it appears that numerical modeling is a good way to discriminate different coatings of the same nature. This method could permit to avoid performing a long and fastidious series of tests to select only the best candidates for abradable or erosion resistance application. In most cases, using the 2-scale modeling allows obtaining calculated properties more in accordance with experimental data. Nevertheless, some limitations to the use of 2-scale modeling were pointed out during this study. From this conclusion, using a 2-scale modeling is not a solution to apply constantly as a reference modeling method. Some criteria must be fulfilled and will be listed in the following paragraph. Another interest of the modeling method for abradable coatings is the possibility to study the anisotropy of the coating (i.e., amplitude of the property differences for both the in-plane and through-thickness directions). Very often, only one direction can be considered for experimental tests. Combining experimental tests and numerical modeling allows obtaining information for both coating directions, and allows verifying if the calculated results for at least one direction are relevant with experimental properties.

Most of the time, modeling is timeless consuming than experimental measurements, and the preparation of the samples is less complicated. Indeed due to the brittle behavior of such coatings, the preparation of samples for experimental measurements often leads to a damaging of these samples.

Table 4-28 lists an estimation of time consumption for the modeling method and experimental tests.

When and why do not use the 2-scale modeling methodology?

Using the 2-scale modeling method must not be an automatic choice. The material must show a complex microstructure, otherwise conventional 1-scale modeling may be as useful. The microstructural features of the coating must have an influence on the coating properties. for example, different phases of a single element can have very similar properties, which should not influence a lot the coating properties (2-scale modeling method is there useless)..

If the aim is to differentiate different coatings, with a porosity level higher than 40%, the relevancy of 2-scale modeling is limited, especially if the pore network is continuous from one edge of the image to the opposite one. In that case, the phase connectivity is misrepresented on two-dimension images of the microstructure. This leads to an overestimation of the influence of the porous phase on the coating properties, especially on the mechanical properties. The modeling methodology is not sensitive to the variation of the microstructure when the porosity level becomes as high as 40%.

If the intrinsic properties of the some phases embedded in the coating are unknown, the calculated results will not correspond to the material performance. If, the aim is to obtain real precise values of the coating properties, unknown input data of the modeling become a limitation. Some solutions must be found to determine the required properties. Using nano-indentation to determine the Young's modulus of unknown phases is a good example. However, the case of thermal conductivity is more problematic.

To conclude, the limitations of the 2-scale modeling methodology are:

 If the porosity level is higher than 40% and the pore network is highly connected, it becomes difficult to differentiate the coatings even if the results are close to the experimental ones  If the porosity level is 50%, the results become far from experimental ones whatever the modeling method applied, especially if the pore network contains continuous cracks connecting opposite edges of the considered image.

 The availability of appropriate input properties is a capital condition to have the most reliable results as possible and to compare them with experimental data The Young's modulus and thermal conductivity corresponding to different

Conclusions and perspectives

The objective of this work was to estimate apparent thermomechanical properties of abradable coating. Abradable materials are strongly heterogeneous structures and the aim was to consider their detailed microstructure. The interest of this work is to overstep one difficulty when manufacturing abradable coating which is their dimensioning. For this, coating properties are mandatory to evaluate, calculate and study strength of materials.

Three abradable coatings having different nature and different microstructures have been studied.

Their microstructure complexity was different from each other. Due to the difficulty to provide sample for experimental testing the solution explored here was to determine their properties by a 2D modeling method. To validate the modeling method, the results calculated were coupled and compared to experimental ones. The modeling method applied was directly pixel by pixel to take account of all the microstructural details.

In the homogenization theory the mean values of the stress and strain fields are integrated over the RVE and linked together by the equivalent elastic properties. Here, the general equivalent behavior laws were not determined because the calculations were only performed in simple test case.

Nevertheless, the numerical part that has been developed can be considered as a first insight on the application of a simplified homogenization approach to abradable coatings. Two modeling methods have been applied:

 A conventional one, with image representing coating microstructural details as well as the global coating microstructural distribution.

 A 2-scale modeling method, with one scale representing the microstructural details of the matrix and a second one representing the global coating coarse pores distribution.

The experimental tests conducted in parallel to validate modeling results were; Laser Flash Analysis, Tensile test, Erosion Resistance, Resistance Frequencies Damping Analyses. This manuscript details all the protocols mandatory to conduct this modeling method, from the coating surface preparation to the properties calculated.

The Figure 4-31 schematizes the methodology process to conduct in order to determine coating properties by modeling.

To observe the coating microstructure, the main image acquisition tools used are the SEM and OM.

Several tool combinations are possible to observe the more precisely the coating microstructure. Three of them have been used in this manuscript:

 SEM + OM: To observe as well the detail of the microstructures (SEM) as the global distribution and shape of the coarse pores (OM).

 SEM + SEM: To observe as well the detail of the microstructures (SEM, high magnification level) as the global distribution and shape of the coarse pores (SEM: low magnification level). This solution have been applied when Optical microscope does not allow obtaining image of the microstructure due to low reflective material nature.

 SEM: To observe the coating at a magnification level representing as well coating details as global microstructure.

Then, all the phase are discriminate by images analysis and analyzed by stereological protocols.

The choice of threshold to apply is important to discriminate all the phase of the study. For bi-phase microstructure a lot of automatic mathematical tool are available and one of them must be selected.

However, for three or more phases microstructures other threshold methodology must be applied. In order to compare microstructure from each other, the same image analyses method must be applied.

The one applied in this manuscript induces a study of the pixels grey level histogram.

After phases discrimination, stereological protocols are used to quantify the information obtained on each phase. The conclusions of the chapter 4 have highlighted the necessity to used different modeling methodology in function of the coating microstructure. From the studies made on all the different coating, it appears that coating porosity rate and pore equivalent surface diameter are the main tool indicating which methodology to conduct. Indeed, 2D-modeling can give relevant results only if the porosity rate is lower than 50%. For the coating having a porosity rate lower than 50%, a study on the equivalent surface diameter of their voids will be conducted. This will allows determining which 2D-modeling methodology should be applied. For each coating, the surface equivalent diameter of the biggest pores and the smallest pores must be compared. If the surface equivalent diameter of the biggest pores is forty times higher than the surface equivalent diameter of the smallest pores, the modeling methodology to apply is the 2scale modeling methodology otherwise the conventional modeling method should be apply.

When the choice of the modelling method is made, the determination of the Representative Elementary Volume is the next step. The protocols to conduct this study have been describe in Chapter 2 and applied in Chapter 3. For the conventional modeling method the REV must be determine at one magnification level which is a compromise between representing as well the matrix details and the global coating microstructure. For the 2-scale modeling method the REV must be determine at two guarantee of success of the sample preparation on contrary with those mandatory for the experimental measures.

Tensile test

No data available because the test was performed by IFAM, partners of E-Break project. But for tensile test, removing the coating substrate is mandatory which do not guarantee the success of the sample preparation. RFDA This work helps to provide a tool for digitally select the most promising abradable layers with the effect of reducing the number of experimental tests, which are longer and more expensive to implement. It is a tool for decision support in the abradable coating development.

The results were valued in one Journal publications 3 , two proceeding of international conference 4 , and three other oral communications 5 . One other article publication is in process. However, even if the calculated properties of some abradable coating microstructures correspond to the measured experimental property, we are aware of the limitations of this work. Indeed, three major limitations may be listed:  Due the use of 2D image of the microstructure, the tridimensional effects are ignored during the modeling process. These effects, however, are considered in experimental properties measurements on these deposits. This can explain some of the differences between the properties values obtained by measure and by calculation. The results concerning the thermal conductivity of CoNiCrAlY-BN-Polyester may be an example of this limitation.

 The domain of applicability of this method is not infinite, two important conditions must be fulfilled. The resolution of coating image should be greater than the size of its microstructural features, in order to have a real representation. In addition, the porosity of the coating levels should be less than 50%. Indeed, due to the 2 dimension of images, a porosity rate higher than 50% induces real misrepresentation of the connectivity of the  An important question regarding these complex structures is the management of interfaces, which are considered perfectly cohesive in this manuscript. However, this is not the case in actual microstructures and this can lead to variations between the measured properties and calculated. For example it can there be diffusion or degradation of the surface of the particles and therefore the interphase. This degradation can be over a short distance and therefore not visible with the used imaging tools acquisition because resolutions are not high enough.

From those conclusions and limitations, on a short term basis, two perspectives can be considered:

 A more detailed description of the thermomechanical properties of the abradable coating, such as computing their Poisson's ratio and coefficient of thermal expansion. These Other parameters have been applied to manufacture YSZ-Polyester coatings with the F4-Torch.

However the surface preparation of the samples induces damages certainly due to a very low cohesion of particles lamellae in the coating. Those coatings cannot be presented.

Annex n°3-Thermal expansion results

Table 4-33: CoNiCrAlY-BN-Polyester coatings thermal expansion

Whatever the porosity rate or the microstructure or CoNiCrAlY-BN-Polyester coatings, their expansion behaviour and coefficient of expansion are similar under temperature changes. At higher temperature there is a small difference of the displacement behaviour of those two coatings. The displacement of the most porous coating is a little lower.

The objective of this work was to estimate apparent thermomechanical properties of abradable coating. Abradable materials are strongly heterogeneous structures and the aim was to consider their detailed microstructure. The interest of this work is to overstep one difficulty when manufacturing abradable coating which is their dimensioning. Three abradable coatings having different nature and different microstructures have been studied. Their microstructure complexity was different one from each other. Their properties were determined by a 2D modeling method which was applied directly pixel by pixel to take account of all the microstructural details. To validate the modeling method, the results calculated were

Abstract Résumé