Merci À Jean

Claude Grandidier

Frédéric Jacquemin

Lénaïk Belec

Thomas Bonnemains

Bruno

Mickael Merci À Christophe

Alexis Luc

Pierre-Yves Et

Romain Alban

Sabrina Robin

Je n'aurai jamais réussi ce travail tout seul, et je tiens à remercier sincèrement toutes les personnes qui ont pu aider ce projet de près ou de loin. Bon, je ne suis pas très doué pour les discours donc ça sera concis, mais mieux valent quelques mots pleins de sentiments que mille mots de décoration. Mes remerciements vont d'abord à Daniel, pour avoir accepté d'être mon directeur de thèse. Sans toi cette thèse n'aurait pas pu avoir lieu. Merci également à mes encadrants Maelenn, Pierre-Yves, Denis, pour m'avoir guidé, encouragé et fait confiance durant ces trois ans. Vous m'avez fait énormément progresser et redonné confiance, et cette thèse ne serait pas ce qu'elle est sans vous. J'espère qu'on se reverra bientôt (et même souvent) pour rire ensemble de ma gestion des délais autour d'une bière.

List of Figures

(2), 0,2 (3) 0,3 (4), 0,4 (5), 0,5 (6) et 0,6 [START_REF] Saunders | Conversion of pressure to depth in the ocean[END_REF] [START_REF] Kochetkov | Water absorption and swelling of glass/epoxy syntactic foams[END_REF] Table 1.1 Homogenization methods considering the effect of fibers on the water diffusion in a composite [START_REF] Dana | Identification of moisture diffusion parameters in organic matrix composites[END_REF]

List of Tables

General introduction

The oceans represent 70% of our planet surface, however it is the largest unexplored place on Earth:

less than 5% of it has been explored [START_REF] Schoedinger | Ocean literacy through science standards[END_REF]. The oceans are a complex and aggressive environment, limiting their exploration and exploitation. Marine exploration systems must resist sea water, withstand the pressure induced by the water column, support storms and winds while not perturbing the marine environment with little maintenance. This leads to major technological challenges. One of them is to reduce the weight of the structures or vehicles used at high depths in the oceans, using buoyancy solutions, to prevent them to sink or allow them to be brought back at the surface. This is one of the issues that the oil companies are facing. They extract the petroleum from the reserves below the sea floor. Those oil deposits are located at depths down to 3000 m, requiring the use of floating platforms and pipelines of several kilometer length (Fig. 0.1) [START_REF]Pazflor, one of the world's largest ever deepwater development[END_REF]. The oil flows from up to the floating platform through pipelines. Because of their length, these pipelines are heavy, and are subjected to high fatigue stress because of their own weight, that can lead to failure of the structure and disastrous events, such as oil spill in the ocean. So, for safety reasons, it is necessary to add buoyancy to them to reduce their apparent weight.

To do that, buoyancy modules are attached (Fig. 0.2). Those modules are designed either for temporary structures, such as drilling risers, with a service life of around 1 year, or for permanent oil production structures, such as jumpers, flexibles and umbilical riser towers, with a service life up to more than 20 years. They are exposed to cold water (around 4°C in deep water), and to high hydrostatic pressure of 30 MPa at 3000 m depth (pressure increases by around 0.1 MPa for each 10 m depth increase). Design of these buoyancy modules is based on short term testing but also on feedback from use in the field. The materials developed in response to this need for buoyancy are syntactic foams [START_REF] Fonblanc | Mousses syndactiques: matériaux composites pour grandes profondeurs[END_REF]. These are composite materials made of hollow spheres of different sizes (from micrometers to centimeters), surrounded by a polymer resin. These materials show a low density and a hydrostatic pressure resistance up to more than 100 MPa depending on their composition, and the use of an epoxy resin matrix makes them stable in marine environment.

One of the challenges of the oil industry is to access oil reserves located at greater depths (up to 4000 meters). But for that, it is necessary to ensure that the buoyancy modules made of this complex material can be used at such depths, in an even harsher environment than current applications with higher loads.

Another important point is to ensure that the methods to qualify these materials are appropriate to estimate their long-term behavior. This is of great importance as the suppliers recently received request of buoyancy modules with service life up to 40-50 years. This thesis, carried out at the Marine Structures Laboratory (LCSM) at IFREMER, in collaboration with the Exploration and Production branch of the TotalEnergies company, aims to address these needs.

The first chapter of this thesis is focused on today's oil company practice, and on the current scientific knowledge, with respect to syntactic foams used as buoyancy modules. By presenting the state of the art, the scientific issues, that need to be investigated, will be highlighted, which will define the scientific approach used in this study.

CHAPTER 1 Bibliography

Introduction

This work addresses the behavior of syntactic foams used as buoyancy materials in a deep-sea environment. The knowledge of the behavior of syntactic foam in the harsh deep ocean environment is of primary interest, especially their resistance to hydrostatic pressure and their long-term behavior, to ensure that they will keep their integrity over their lifetime. This chapter focuses on the current practices of industrial operators and the state of the art on syntactic foam; it highlights that the knowledge of the hydrostatic strength and durability of syntactic foams is not complete and should be improved. But first, a reminder of what is buoyancy is.

Buoyancy: definition

In this context, syntactic foams are used for buoyancy applications, which aim to reduce the apparent weight of structures immersed in seawater. The apparent weight of these structures placed in water follows Archimedes' principle (Eq. 1.1). In this work, the term buoyancy refers to this apparent weight.

𝐹(𝑃, 𝑇, 𝑡) = (𝜌 𝑤𝑎𝑡𝑒𝑟 (𝑃, 𝑇). 𝑉(𝑃, 𝑇, 𝑡) -𝑚(𝑡)). 𝑔

Eq. 1.1

With F (N) the apparent weight or buoyancy of the immersed object, V (m 3 ) its volume (or displaced volume of water), m its mass, ρwater (kg.m 3 ) the water density, P (MPa) the environmental pressure, T (°C) the environmental temperature, t (s) the time spent in water and g (m/s 2 ) the gravitational constant.

For a non-buoyant structure, F is negative and the structure sinks, while F is positive for buoyant structures that float. The positive effect of buoyancy modules is used to adjust the negative buoyancy of pipelines to achieve the desired apparent weight.

The hydrostatic pressure and temperature need to be considered because they affect the physical properties of water, and hence its density [START_REF] Fine | Compressibility of water as a function of temperature and pressure[END_REF]. The density of water is related to its compressibility defined by Eq. 1.2:

𝛽 = 1 𝐾 = - 1 𝑉 𝜕𝑉 𝜕𝑃 ≡ 1 𝜌 𝜕𝜌 𝜕𝑃 Eq. 1.2
With β (MPa -1 ) the compressibility, K (MPa) the bulk modulus, V (m 3 ) the volume, ρ (kg.m 3 ) the density and P (MPa) the pressure.

The properties of sea water are generally defined from the state equation of seawater provided by Unesco under the reference TEOS-10, the International Thermodynamic Equation of Seawater, which allows all the thermodynamic properties, and especially the density, of pure water, ice, seawater, and moist air to be evaluated depending on the water salinity (null for pure water), temperature and pressure [5].

The compressibility of buoyancy modules must also be considered because hydrostatic pressure will change their volume, and therefore their buoyancy. For materials with a compressibility higher than that of water, the module will lose buoyancy when it dives in the oceans since it compresses itself faster than water. On the contrary, it will gain buoyancy if its compressibility is lower than water [START_REF] Watkins | Buoyancy Materials for Offshore Riser Pipe[END_REF].

As noted previously, the hydrostatic pressure depends on the immersion depth. The usual approximation made is that the pressure increases by 0.1 MPa for every 10 meters depth [START_REF] Saunders | Conversion of pressure to depth in the ocean[END_REF]. To be more accurate, the water compressibility, and dependence of the Earth's gravity on latitude and depth can be considered to have an exact relationship between depth and pressure [START_REF] Copin-Montegut | Propriétés physiques de l'eau de mer[END_REF]. In this work the first approximation is used.

The need for buoyancy modules in the oil industry requires adapted materials with a lower density than water, to make buoyancy modules float. They also need resistance to hydrostatic pressure, and longterm durability (up to 20 years) in a cold (4°C) seawater environment. The materials usually chosen for this application are syntactic foams [START_REF] Fonblanc | Mousses syndactiques: matériaux composites pour grandes profondeurs[END_REF], which are presented in the next section.

Syntactic foams 1.2.1. Generalities on syntactic foams

Syntactic foams are composite materials, designed in the 1960's, in response for the need for materials with low densities [START_REF] Fonblanc | Mousses syndactiques: matériaux composites pour grandes profondeurs[END_REF][START_REF] John | Update on syntactic foams[END_REF]. Their name come from the Greek term "suntaktikos" (taktikos = disposal and sun = with), syntactic means global arrangement [START_REF] John | Update on syntactic foams[END_REF], and the term "foams" is used because of the cellular nature of the material. They are made of hollow spherical filler incorporated in a matrix (Fig. 1.1). Filler sizes vary from several micrometers (called hollow microspheres) to several centimeters (called hollow macrospheres). In this work, the syntactic foams containing only microspheres are called "pure syntactic", while syntactic foam combining the use of microspheres and macrospheres are called "composite foam" (Fig. 1.1). In this chapter, the term "syntactic foam", used by the authors in the literature, refers to "pure syntactic" materials. Such materials differ from other types of foams in that they be classed as physical rather than chemically blown [START_REF] Hobaica | The characteristics of syntactic foams used for buoyancy[END_REF]. Since their voids are formed by hollow inclusions, they are truly unicellular foams.

Fig. 1.1 Composition of a syntactic foam

Fillers and matrix can be metallic, polymeric, or ceramic depending on the application [START_REF] John | Update on syntactic foams[END_REF]. The spherical hollow fillers give the material its low density (between 0.4 and 0.8 g/cm 3 ). The nature of the filler and the matrix depends on the applications and functional properties needed. John et al. provides an exhaustive review of the possible applications in his work [START_REF] John | Update on syntactic foams[END_REF], with description of the matrix and spherical fillers used, general methods for manufacturing, and material properties. Those low-density materials are interesting for aeronautical, aerospace and automotive applications, and the presence of cavities in the material make them suitable for insulating applications, such as thermal insulation in the oil industry, and acoustic insulation for radomes or microwave applications. Their energy absorption properties also make them suitable for applications involving shock absorption such as protective materials for military vehicles. Recent advances in syntactic foam, such as syntactic foams containing carbon nanofibers [START_REF] Poveda | Carbon nanofiber reinforced syntactic foams: Degradation mechanism for long term moisture exposure and residual compressive properties[END_REF] or functionally graded syntactic foams [START_REF] Gupta | A functionally graded syntactic foam material for high energy absorption under compression[END_REF], have also been described.

It should be mentioned that the primary application for these materials is buoyancy in marine applications, for instance buoyancy for AUV and submersible vehicles for deep sea exploration [START_REF] Gupta | Reinforced Polymer Matrix Syntactic Foams[END_REF][START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF].

The early knowledge of syntactic foams for those applications was described by Shutov with a review of the materials chosen for the hollow inclusions and the matrix [START_REF] Shutov | Syntactic polymer foams[END_REF].

For buoyancy applications, the fillers used are hollow glass microspheres and the matrix is a thermoset epoxy resin [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]. Nowadays, syntactic foam producers add glass fiber or carbon fiber macrospheres in epoxy syntactic foams to further reduce their densities. A detailed description of existing knowledge in literature about syntactic foam behavior used for marine applications is proposed later.

Syntactic foams for buoyancy applications: testing and requirements

The present study concerns the use of syntactic foams for buoyancy application in deep offshore oil exploration and production. For those applications, buoyancy modules are exposed to high hydrostatic pressure (up to 40 MPa at 4000 m depth) and cold seawater for several years (1 to 27 years). Syntactic foams must therefore retain their buoyancy properties during their installation in deep sea and over their service life to prevent catastrophic failure of the global structure. Appropriate testing of syntactic foam is essential to ensure their reliability.

The testing of syntactic foam for buoyancy application is mainly based on the work of Watkins who is one of the first to present characterization of the buoyancy of syntactic foams (in his work the material is a pure syntactic with a thermoset polymer matrix containing hollow glass microspheres) [START_REF] Watkins | Buoyancy Materials for Offshore Riser Pipe[END_REF][START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF][START_REF] Watkins | Syntactic Foam Buoyancy for Ultradeep Marine Riser[END_REF][START_REF] Watkins | Syntactic foam thermal insulation for ultra-deepwater oil and gas pipelines[END_REF].

Instrumented Buoyancy Loss (IBL) tests were used to measure in situ the buoyancy loss of syntactic foam under pressure. Those tests consisted of pressurizing the sample in a pressure vessel in water up to their service pressure, maintaining the applied pressure for a given period. Watkins measured the buoyancy change of the sample during the test using remote-reading gauges. A typical buoyancy loss curve is shown Fig. 1.2. The first part corresponds to the increasing of pressure in the vessel, leading to buoyancy loss due to elastic deformation of the material [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF]. The volume change due to elastic deformation is related to the bulk modulus of the material by Eq. 1.3, analogous to Eq.

1.2.

During the second part of the test, Fig. 1.2, where the pressure is kept constant, a time-dependent buoyancy loss is observed. Finally, the sample is brought back to atmospheric pressure and its residual buoyancy loss is measured. Watkins neglected the viscoelastic deformations because the final buoyancy loss recorded corresponded to the weight gain of the sample, and stated that buoyancy loss is due to water absorption of the material [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF]. The buoyancy loss is usually described by Eq. 1.4.

∆𝑉 𝑉 0 = 𝐾 * ∆𝑃

Eq. 1.3

With ΔV (m 3 ) the volume change of the material, V0 (m 3 ) its initial volume, K (MPa) its bulk modulus, and ΔP (MPa) the pressure change.

𝐹 = 𝑥 * log(𝑡) + 𝑦

Eq. 1.4

With F (N) the buoyancy loss of the material, x (N) and y (N) empirical constant, and t (s) the time at pressure.

Using those results, Watkins proposes a way to predict the long-term buoyancy loss of syntactic foams by determining the x and y constant experimentally, and extrapolating Eq. 1.4 to the material service life. At that time, it was used for pure syntactics but currently it is also used for composite foam.

It should be noted that while buoyancy loss is described by a logarithmic law and is assumed related to water absorption, this is not in agreement with usual description of water absorption in polymer that are usually proportional with the square root of time [START_REF] Fick | On liquid diffusion[END_REF][START_REF] Crank | The mathematics of diffusion[END_REF]. This point is discussed latter in this document.

Watkins also proposed ways to accelerate buoyancy loss to cut down the test duration and accelerate the appearance of possible phenomena that would not be seen during short test. He states, without much evidence, that the factors that accelerate buoyancy loss are the following [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF]:

• Increasing temperature. But no formula on the impact of temperature on buoyancy loss exists.

In the polymer's field, Arrhenius' Law is often used to accelerate aging so it may be an appropriate method [START_REF] Porfyris | Accelerated ageing and hydrolytic stabilization of poly (lactic acid)(PLA) under humidity and temperature conditioning[END_REF][START_REF] Lei | Kinetic studies on the rate of hydrolysis of N-ethyl-N'-(dimethylaminopropyl) carbodiimide in aqueous solutions using mass spectrometry and capillary electrophoresis[END_REF][START_REF] Bardin | Durabilité des élastomères thermoplastiques pour applications marines[END_REF][START_REF] Arhant | A study of pure hydrolysis of carbon fibre reinforced polyamide 6 composites tested under mode I loading[END_REF], but no examples have been found in the literature for syntactic foam materials.

• Cyclic pressurization. In his publication, Watkins proposed a formula (Eq. 1.5) to accelerate buoyancy loss using cyclic tests [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF], but the validity of this expression is based on the testing of one sample only and is therefore questionable.

𝐹 = 𝑥 * log(𝑛 2 * 𝑡) + 𝑦 Eq. 1.5
With F (N) the buoyancy loss, x (N) and y (N) empirical constants, n (no unit) the number of cycles and t (s) the cycle period.

• Increasing pressure. It accelerates water uptake of syntactic foams [START_REF] Hobaica | The characteristics of syntactic foams used for buoyancy[END_REF], therefore buoyancy loss. Watkins proposes a formula (Eq. 1.6) to describe the impact of pressure on buoyancy loss of syntactic foams [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF], but there is no experimental validation of the method. The validity of such a relationship will be evaluated in this study.

𝑅 = 1

1 -𝑃 𝑃 𝑐

Eq. 1.6

With R the buoyancy loss rate, P (MPa) the test pressure, Pc (MPa) the foam crush pressure.

• Decreasing the sample volume (Eq. 1.7). Watkins states that according to Fick's Law of water diffusion in polymers, reducing the sample volume should accelerate water uptake, and so buoyancy loss. This acceleration method is common knowledge for polymeric material, as the study of water absorption kinetics in polymer and composite materials is made on small size samples to accelerate the diffusion process [START_REF] Bardin | Durabilité des élastomères thermoplastiques pour applications marines[END_REF][START_REF] Arhant | A study of pure hydrolysis of carbon fibre reinforced polyamide 6 composites tested under mode I loading[END_REF][START_REF] Gac | Yield stress changes induced by water in polyamide 6: characterization and modeling[END_REF]. But here the presence of microspheres could have an impact on the water absorption kinetics of such materials, and therefore limits the use of such a method. At least, experimental validation is needed.

𝑊2 𝑊1 = 𝑆2 𝑉2 ⁄ 𝑆1 𝑉1 ⁄ Eq. 1.7
With W1, W2 the water uptake (%) of small and large sample immersed during the same period, S1 and S2 (m²) their surface exposed to water, and V1 and V2 (m 3 ) their volume.

Based on this work, oil companies provide specific qualifications for buoyancy materials to ensure their usability. The specifications used by the companies are API-16F [START_REF]API Spec 16F Cementing Float Equipment[END_REF] for marine drilling risers and API-17L [START_REF]API Spec 17L Specification for Flexible Pipe Ancillary Equipment[END_REF] for flexible pipe ancillary equipment. The main qualifications required are:

• Assurance that buoyant materials do not absorb water at an excessive rate (without giving any definition of what an excessive rate is),

• Crush pressure (hydrostatic strength) of at least 1.25 (API-16F) or 1.5 (API-17L) times the hydrostatic pressure at service depth. These safety factors are needed to ensure that the material will keep its integrity at the desired service depth. The value for API-17L is chosen for consistency with the safety factors applied to the pipeline. The origin of the value for API-16F is unclear. The crush pressure of syntactic foam is usually named Hydrostatic Crush Pressure (HCP),

• Compression of material at service depth shall not cause more than 1.5% loss of net volume,

• API-16F hydrostatic testing: The total buoyancy loss of a module shall not exceed 5% after 1 year of immersion. Extrapolation is made based on the results of the 96-hour hydrostatic tests at service pressure, with the last 72 hours of data extrapolated over a 12-month period,

• API-17L hydrostatic testing: the module shall be tested in an IBL test at 110% of the module service pressure, for a period specified by the purchaser of the module. Then the buoyancy loss shall be extrapolated to the module service life, and the final buoyancy value must be in the range of the purchaser specification.

There is a new standard that is currently being written by the offshore industry to improve the specification. Today, the existing specifications are satisfactory, being based on the feedback from service after years of use of buoyancy modules, but that feedback concerns materials used at 3000 m depths, and nothing is certain concerning the use of such materials at 4000 m depth and more.

Those requirements highlight the need to characterize and study the following mechanical and longterm properties of syntactic foam used for buoyancy module:

• Their mechanical behavior under hydrostatic pressure: how does the material volume change under increasing hydrostatic pressure? What is the material crush pressure?

• What is their water absorption rate that leads to direct buoyancy loss? Is it really the only factor leading to buoyancy loss over time, or can viscoelastic strains also be an important contributor?

There is also a need to establish adequate testing method (equipment, measurement methods, etc.) to characterize the behavior of such materials based on the physical mechanisms involved in the process that must be characterized. Furthermore, currently, testing method are based on tests of duration of usually 96 hours, up to 2 months in some cases. This is short compared to the materials service life of

The state of the art on those point of interest is developed in the following part of this chapter, with a focus on epoxy syntactic foams containing hollow glass microspheres, being the type of syntactic foams used for most deep-sea buoyancy applications [START_REF] Shutov | Syntactic polymer foams[END_REF].

Mechanical behavior of syntactic foams

In this section, the knowledge of the mechanical behavior of syntactic foams is presented, as well as the testing methods used to characterize it. It mostly concerns the behavior of pure syntactics since they are mostly studied by the scientific community (the literature specific to composite foam will be investigated later). The hollow inclusions are the basis of pure syntactics, so their mechanical behavior is investigated first.

Mechanical behavior of hollow microspheres

Hollow glass microspheres, also called glass bubbles (GB), are the hollow inclusions usually used in pure syntactics for marine applications [START_REF] Watkins | Buoyancy Materials for Offshore Riser Pipe[END_REF][START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]. They were developed by 3M in the 1960's. Nowadays they are used in a wide range of applications such as in aerospace and military applications, offshore applications, paints and coatings, etc. [START_REF] John | Update on syntactic foams[END_REF][START_REF] Trofimov | Hollow glass microspheres for high strength composite cores[END_REF]. Microspheres are described by Ruckebush [START_REF] Ruckebusch | Microsphères creuses de verre pour mousses syntactiques[END_REF], as well as their manufacturing process. Let us here recall some general features. The glass bubbles generally used are made of sodium and borosilicate glasses [START_REF] Ruckebusch | Microsphères creuses de verre pour mousses syntactiques[END_REF]. They have a good sphericity (Fig. 1.3) and a low density. Due to their manufacturing process, it is impossible to have a unique size of microsphere: a batch of microspheres with a size distribution is obtained (Fig. 1.3). The microspheres size range can be adjusted by selecting them using adapted filters.

By adjusting their size, it is possible to optimize their nominal density (i.e., the density of a batch of microspheres) [START_REF] Ruckebusch | Microsphères creuses de verre pour mousses syntactiques[END_REF].

𝜌 𝑝𝑢𝑟𝑒-𝑠𝑦𝑛𝑡𝑎𝑐𝑡𝑖𝑐 = 𝜌 𝑚𝑖𝑐𝑟𝑜 * 𝑓 + 𝜌 𝑚𝑎𝑡𝑟𝑖𝑥 (1 -𝑓)

Eq. 1.8

With ρpure-syntactic (kg/m 3 ) the foam density, ρmicro (kg/m 3 ) the microspheres density, ρmatrix (kg/m 3 ) the matrix density and f (no unit) the microspheres volume fraction in the pure syntactic.

The law of mixtures (Eq. 1.8) describes the pure syntactic density (considering no voids in the material due to process) [START_REF] Ruckebusch | Microsphères creuses de verre pour mousses syntactiques[END_REF]. The lower the microspheres density, the lower the foam density will be, the higher the buoyancy it will bring. It can also be adjusted by choosing the appropriate volume fraction of microspheres introduced in the matrix.

Microspheres are usually described by their nominal density, and their size distribution, but are also chosen according to their hydrostatic strength. Fig. 1.4 presents a non-exhaustive list of microspheres grading by 3M TM and their given properties. This table presents the 3 main properties of glass bubbles, i.e., mechanical resistance, density, and particle size distribution. But here, it is necessary to mention that an increase in wall thickness of the glass bubble leads to an increase in hydrostatic strength, but also in density. In other words, the choice of microsphere type will always be a compromise between density and mechanical strength. Scanning Electron Microscopy (SEM) is an effective tool to measure wall thickness and diameter of microspheres to calculate their aspect ratio [START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF]. But using this method, wall thickness can be measured only on collapsed microspheres, and the observation angle, as well as the crack angle with the microsphere surface tangent, must be considered with care [START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF]. Laser Granulometry used at IFREMER (presented in Chapter 2), is also an efficient tool to characterize microspheres diameter distribution, but it does not provide thickness measurements. In their work, Adrien et al have measured the thickness of microspheres using X-ray microtomography on pure syntactics [START_REF] Adrien | Experimental study of the compression behaviour of syntactic foams by in situ X-ray tomography[END_REF]. This method requires the use of advanced image processing methods, and the precision is limited to the imaging technique resolution.

In the work of Adrien, the resolution of 0.7 µm 3 voxel (resolution of 3D imaging) is close to the microspheres thickness of around 1.5 µm, making it difficult to precisely measure the thickness distribution of the microspheres, but a range of microspheres thickness can be estimated; in Adrien's work this was 1-2 µm.

An interesting point is that increasing the microspheres density increases their strength (Fig. 1.4) and allows them to be used at higher depth, but at the cost of lower buoyancy. Furthermore, for the same nominal density, their strength might be improved by selecting the right choice of microspheres size distribution.

Several methods have been used to estimate the hydrostatic strength of microspheres in the past [START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF].

Here, a focus is made on the method used by 3M TM , which is based on the testing of microspheres by batch. At one time, the procedure was an ASTM Standard, designated D3102-78, but was withdrawn for unspecified reasons. However, the method is still used by the manufacturer. The basic premise of the test is to expose a batch of microspheres to a high isostatic pressure, during which some of the microspheres will collapse [START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF]. The percent of microspheres that collapse (or survive) is measured, and the microspheres are classified according to the isostatic pressure at which a given percent of the tested microspheres collapse (or survive, as the classification used on Fig. 1.4). During the test, microspheres are mixed with either glycerin or isopropyl alcohol, probably to prevent contact between microspheres, and the mixture is placed in a rubber container inside a pressure chamber. Once the microspheres sample is in the pressure chamber, it is pressurized to a predetermined pressure while recording both pressure and volume. This process is repeated to get a second set of pressure-volume data, and both data sets are plotted as pressure vs. volume curves. The initial pressurization is known as the collapse curve, and the second pressurization provides the system compression curve. To overcome the limitation of the method presented above, another method has been developed where microspheres are tested individually. For that, a microsphere is placed into a special holder in a pressure chamber with a quartz window. The window allows to see when the microsphere collapse, and the pressure of the chamber is then recorded [START_REF] Bratt | Mechanical testing of glass hollow microspheres[END_REF]. Buckling is the commonly accepted failure mechanisms of thin-walled hollow sphere under hydrostatic loading [START_REF] Zoelly | Ueber ein Knickungsproblem an der Kugelschale[END_REF][START_REF] Gere | Theory of elastic stability[END_REF][START_REF] Wunderlich | Buckling behaviour of imperfect spherical shells[END_REF]. The equation (Eq. 1.9) used to describe shell wall buckling was first found by Zoelly [START_REF] Zoelly | Ueber ein Knickungsproblem an der Kugelschale[END_REF] and is presented in the work of Timoshenko [START_REF] Gere | Theory of elastic stability[END_REF]. The experimental data of microspheres collapse pressure obtained by Bratt et al [START_REF] Bratt | Mechanical testing of glass hollow microspheres[END_REF] have been compared with the buckling theory in Fig. 1.6, where the line represents the estimated buckling pressure of a microsphere (calculated with a Young's modulus of 56.5 GPa and a Poisson's ratio of 0.21 [START_REF] Bratt | Mechanical testing of glass hollow microspheres[END_REF]). A good agreement was found between theory and experiment. But it should be mentioned that this value of glass Young's modulus is low compared with usual values for borosilicate glass of at least 70 GPa [START_REF] Deriano | Physical and mechanical properties of a new borosilicate glass[END_REF][START_REF] Paufler | Mechanical properties and structure of a nanoporous sodium borosilicate glass[END_REF][START_REF] Chen | Radiation effects on structure and mechanical properties of borosilicate glasses[END_REF].

𝑃 𝑏,𝑠𝑝ℎ𝑒𝑟𝑒 = 8𝐸 √3(1 -𝑣 2 ) * ( 𝑒 𝑑 ) 2 = 2𝐸 √3 * (1 -𝑣 2 ) * (1 -𝜂) 2
Eq. 1.9

With E (MPa) the glass Young's modulus, ν (no unit) the glass Poisson's ratio, e (mm) the microsphere wall thickness, d (mm) the microsphere outer diameter, η (no unit) the microsphere radius ratio, Pb,sphere (MPa) the buckling pressure of the sphere.

The radius ratio η used in Eq. 1.9 is another way to describe the microsphere geometry [START_REF] Gupta | Tensile properties of glass microballoon-epoxy resin syntactic foams[END_REF]. The radius ratio is defined by the following Eq. 1.10.

𝜂 = 𝑅 𝑖 𝑅 𝑜

Eq. 1.10

With Ri (mm) the microsphere intern radius, Ro (mm) its outer radius, η (no unit) the radius ratio of the microsphere By considering that all the microspheres have the same internal and external radius Ri and Ro, the radius ratio can be expressed as a function of the average density of the microspheres [START_REF] Gupta | Tensile properties of glass microballoon-epoxy resin syntactic foams[END_REF]:

𝜂 = (1 - 𝜌 𝑚𝑖𝑐𝑟𝑜 𝜌 𝑔 ) 1 3 ⁄

Eq. 1.11

With η (no unit) the radius ratio of the microspheres, ρmicro (kg/m 3 ) the microspheres average density, ρg (kg/m 3 ) the glass density

Other authors have studied the failure of microspheres, and it is usually accepted that Eq. 1.9 will always overestimate the strength of hollow spheres by a large amount, because this assumes that the sphere geometry is perfect, which it never is [START_REF] Wunderlich | Buckling behaviour of imperfect spherical shells[END_REF].

The other possible failure mechanism of a microsphere is the failure of the microsphere shell material because of the glass compressive strength being reached [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]. Knowing the microsphere dimensions, the collapse pressure of microsphere under this failure is described by Eq. 1.12.

𝑃 𝑐,𝑠𝑝ℎ𝑒𝑟𝑒 = 𝜎 𝑚 2 3

𝑑 3 -(𝑑 -2e) 3 𝑑 3 = 𝜎 𝑚 2 3 (1 -𝜂 3 )
Eq. 1.12

With σm (MPa) the maximal compressive stress of the glass, d (mm) the microsphere outer diameter, e (mm) the microsphere thickness, Pc,sphere (MPa) the collapse pressure of the sphere, η (no unit) the sphere radius ratio

This equation comes from the expression of the circumferential stress in a hollow sphere subjected to hydrostatic loading [START_REF] Young | Roark's formulas for stress and strain[END_REF], coupled with the failure criterion of maximal admissible stress in the material.

The exact determination of the failure mode of microspheres is difficult. But all theories agree with the experimental observations that the microspheres strength increases when their thickness increases and their radius decreases (i.e., when their density increases). This strength improvement of inclusions is made at the cost of increasing the density of the pure syntactic, lowering its buoyancy.

Mechanical characterization of syntactic foams

The characterization of the mechanical behavior of pure syntactics has been studied by several authors [START_REF] Gupta | A functionally graded syntactic foam material for high energy absorption under compression[END_REF][START_REF] Gupta | Reinforced Polymer Matrix Syntactic Foams[END_REF][START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF][START_REF] Jayavardhan | Quasi-static compressive response of compression molded glass microballoon/HDPE syntactic foam[END_REF][START_REF] Gupta | Compression properties of syntactic foams: effect of cenosphere radius ratio and specimen aspect ratio[END_REF][START_REF] Gupta | Comparison of compressive properties of layered syntactic foams having gradient in microballoon volume fraction and wall thickness[END_REF][START_REF] Gupta | Comparison of tensile and compressive characteristics of vinyl ester/glass microballoon syntactic foams[END_REF][START_REF] Gupta | Hygrothermal studies on syntactic foams and compressive strength determination[END_REF][START_REF] Gupta | Characterization of mechanical and electrical properties of epoxy-glass microballoon syntactic composites[END_REF][START_REF] Song | Dynamic compressive response and failure behavior of an epoxy syntactic foam[END_REF][START_REF] Huang | Elastic behaviour and failure mechanism in epoxy syntactic foams: The effect of glass microballoon volume fractions[END_REF][START_REF] Ale | Dynamic attenuation and compressive characterization of syntactic foams[END_REF][START_REF] Huang | X-ray microtomography and finite element modelling of compressive failure mechanism in cenosphere epoxy syntactic foams[END_REF][START_REF] Gall | Novel mechanical characterization method for deep sea buoyancy material under hydrostatic pressure[END_REF][START_REF] Liu | Novel test method for the hydrostatic properties of syntactic foams[END_REF]. First, we consider the mechanical behavior of syntactic material using a simple test, the uniaxial compression test. In fact, in the literature, most of the work on syntactic foams refers to uniaxial compressive behavior [START_REF] Jayavardhan | Quasi-static compressive response of compression molded glass microballoon/HDPE syntactic foam[END_REF][START_REF] Huang | Elastic behaviour and failure mechanism in epoxy syntactic foams: The effect of glass microballoon volume fractions[END_REF]. Under this type of loading, epoxy pure syntactics show an elastic response before exhibiting densification due to the material collapse. A typical curve obtained during uniaxial compression is presented in Fig. 1.7. At first, the material presents an elastic behavior. Then, it reaches a densification plateau corresponding to the collapse of the material [START_REF] Gupta | Compression properties of syntactic foams: effect of cenosphere radius ratio and specimen aspect ratio[END_REF]. Gupta et al have revealed that decreasing the microspheres radius ratio η in the foam increases its compressive strength [START_REF] Gupta | Compression properties of syntactic foams: effect of cenosphere radius ratio and specimen aspect ratio[END_REF]. This was attributed to the microspheres being the load bearer in the foam: when η decreases they become stronger as shown previously, and so will be the pure syntactic. Gupta and other authors also revealed that increasing the microspheres volume fraction decreases the material strength [START_REF] Jayavardhan | Quasi-static compressive response of compression molded glass microballoon/HDPE syntactic foam[END_REF][START_REF] Ale | Dynamic attenuation and compressive characterization of syntactic foams[END_REF]. This was attributed to the matrix enhancing the microspheres strength: the more matrix in the foam, the more it enhances the microspheres. But no explanation of the origin of this enhancement is given. Song has shown that epoxy foam strength increases with the test strain rate [START_REF] Song | Dynamic compressive response and failure behavior of an epoxy syntactic foam[END_REF]. Adrien et al also used in-situ confined-compression test under X-ray microtomography [START_REF] Adrien | Experimental study of the compression behaviour of syntactic foams by in situ X-ray tomography[END_REF] to study damage mechanisms in epoxy, polypropylene, and polyurethane pure syntactics, revealing that X-ray tomography seems to be an interesting way to study this type of material. Those results revealed that the damage mechanisms of pure syntactics (collapse of microspheres, and matrix cracking/plastic deformations) depend on the type of matrix used.

However, Choqueuse [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF] have shown that uniaxial compressive tests using different sample geometries lead to different results (Fig. 1.8). The linear region is not always observed, nor the densification plateau.

Results can easily be polluted by damage at the edges of the sample. It seems therefore difficult to measure syntactic foam properties precisely using uniaxial compression tests. Furthermore, most of those previous studies were concerned with sandwich applications for marine or aeronautical applications, justifying studying the material under uniaxial compression, while for buoyancy applications, the loading is hydrostatic loading. We will therefore now focus on the mechanical behavior of syntactic material under hydrostatic loading.

The characterization of syntactic foams under hydrostatic loading is not commonly used since experimental methods are not easy to implement. Choqueuse [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF] has presented most of the usual techniques to pressurize samples. The simplest approach is to place a sample of syntactic foam in a pressure vessel and increase the pressure by injecting fluid (usually water).

During the pressurization, the volume change of the sample can be estimated by measuring the buoyancy of the material using an immersed weighing device [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF][START_REF] Gall | Novel mechanical characterization method for deep sea buoyancy material under hydrostatic pressure[END_REF], or by estimating the amount of water introduced in the vessel [START_REF] Liu | Novel test method for the hydrostatic properties of syntactic foams[END_REF]. Another way to pressurize a sample is to place it in a pressure vessel, and then increase the pressure in the vessel using piston displacement, with the help of a tensile machine to compress the piston in the chamber to reduce the chamber volume [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF][START_REF] Gall | Novel mechanical characterization method for deep sea buoyancy material under hydrostatic pressure[END_REF][START_REF] Viot | Hydrostatic compression on polypropylene foam[END_REF][START_REF] Wang | Viscoelastic properties of foam under hydrostatic pressure and uniaxial compression[END_REF][START_REF] Lachambre | In situ observation of syntactic foams under hydrostatic pressure using X-ray tomography[END_REF].

Another way to measure the sample volume is to make strain gauge measurements during the test, but this requires the use of specially designed electrical connectors and implementing ways to waterproof the gauges, and no such test, to the author's knowledge, is presented in the literature. A detailed description of these test methods is proposed in Chapter 2.

The behavior of pure syntactics under hydrostatic loading is shown in Fig. 1.9. First, a linear behavior is observed, before a densification plateau appears [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]. The pure syntactic strength is often defined as the pressure at which the densification plateau appears. When looking at the results from Choqueuse, it appears that the strength of pure syntactics under hydrostatic loading differs from what is measured under uniaxial compression. This emphasizes the need to use hydrostatic testing when estimating syntactic foams crush pressure. Through Choqueuse's characterization of two different pure syntactics made with the same type of microspheres (Fig. 1.9), we understand that the choice of matrix influences the syntactic foam behavior, as epoxy syntactic foams present an elastic behavior until failure, while polyurethane syntactic foams present elasto-plastic behavior [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]. The choice of resin also influences the foam hydrostatic strength.

Liu showed in his work, as expected, that the microspheres grade also influences the material hydrostatic strength [START_REF] Liu | Novel test method for the hydrostatic properties of syntactic foams[END_REF]. Furthermore, Le Gall et Al. showed that increasing the test temperature decreases the foam hydrostatic strength [START_REF] Gall | Novel mechanical characterization method for deep sea buoyancy material under hydrostatic pressure[END_REF].

Choqueuse also used in situ X-ray microtomography to study the failure mechanisms of pure syntactics under hydrostatic loading, revealing that when using epoxy, the matrix skeleton of the material does not present plastic deformations when microspheres collapse in the material (Fig. 1.10). On the contrary, when a polyurethane matrix was used it acted as a fluid that filled the space created by the collapse of a microsphere [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF] (Fig. 1.11). In his work with Lachambre [START_REF] Lachambre | In situ observation of syntactic foams under hydrostatic pressure using X-ray tomography[END_REF], a behavior similar to polyurethane syntactic foam was observed on polypropylene pure syntactics. The use of acoustic emission, often employed to study the damage mechanisms of composite materials [START_REF] Lainé | Acoustic emission description from a damage and failure scenario of rotomoulded polyolefin sandwich structure subjected to internal pressure for storage applications[END_REF][START_REF] Ramirez | Assessment of a damage model for wound composite structures by acoustic emission[END_REF], was also used to estimate initiation of damage in epoxy and polyurethane pure syntactics [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF][START_REF] Grosjean | Acoustic emission analyses of the hygrothermal ageing of glass syntactic foams[END_REF]. This revealed that early damage in the material appears as soon as pressure is applied. But this method cannot be used to quantify this damage. An approach using calcination of samples and density measurements of microspheres was used to estimate damage levels in polyurethane foams, but this method could not be used for epoxy pure syntactics due to the difficulty of obtaining only powder of microspheres after burning off the matrix [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF].

To conclude, it clearly appears that the hydrostatic resistance of a pure syntactic depends on the type of microspheres used in the foam, as well as on the quantity of microspheres, and on the matrix mechanical properties. However, based on existing data there is no quantitative relationship between the crush pressure of a syntactic foam and its composition. This point will be considered in detail in this study.

Moreover, when looking at all the studies of syntactic foams under hydrostatic loading, epoxy pure syntactics always exhibit strength higher than the microspheres' hydrostatic strength [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF][START_REF] Gall | Novel mechanical characterization method for deep sea buoyancy material under hydrostatic pressure[END_REF][START_REF] Liu | Novel test method for the hydrostatic properties of syntactic foams[END_REF].

Surrounding the microspheres by a resin reinforces them, but the origin of this improvement is not explained by any author. An explanation of this behavior, usually called the amplification factor, is proposed in Chapter.4, based on analytical modelling of syntactic foams. Let us present the existing work in this field in the next section.

Modelling of syntactic foams

A certain number of studies have been performed to implement models to predict the thermomechanical behavior of syntactic foams. Those studies are focused only on the elastic behavior of those materials using the homogenization method. The models are mainly based on a three-phase method (Fig. 1.12), initiated by Hashin [START_REF] Hashin | The elastic moduli of heterogeneous materials[END_REF] and Christensen [START_REF] Christensen | Solutions for effective shear properties in three phase sphere and cylinder models[END_REF], and extended by Hervé and Pellegrini [START_REF] Hervé | The elastic constants of a material containing spherical coated holes[END_REF]. Recently, this approach has been completed by several authors in different ways [START_REF] Marur | Effective elastic moduli of syntactic foams[END_REF][START_REF] Bardella | On the elastic behavior of syntactic foams[END_REF] to include the variability of the size particle in the model, or the presence of voids in the foam. This approach, that will be presented in more detail in Chapter 4, allows the elastic behavior of syntactic foams to be described, knowing the mechanical properties of both matrix and microspheres as well as the microsphere volume fraction in the foam. The authors always focus on pure syntactic materials, but the approach can be adapted to composite foams by using the pure syntactic as the matrix, and the macrospheres as the inclusion.

The reliability of this approach has been recently considered by Bardella [START_REF] Bardella | On the elastic behavior of syntactic foams[END_REF] with a comparison of an analytical model prediction with experimental and numerical data. This revealed that the microspheres size distribution has little influence on the elastic properties of the pure syntactics. Some authors, such as Zouari [START_REF] Zouari | A micromechanical iterative approach for the behavior of polydispersed composites[END_REF][START_REF] Zouari-Trabelsi | Approche itérative d'homogénéisation pour le comportement des composites polydisperses: applications à l'endommagement des mousses syntactiques immergées[END_REF], have applied this model to introduce damage parameters to describe the mechanical behavior of syntactic foams. Choqueuse has also introduced a hydrostatic damage parameter to describe the behavior of polypropylene syntactic foams [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]. It is efficient to describe the volume variation of polypropylene pure syntactics, but the damage parameter is not based on damage measurements, and the damage rate is chosen arbitrarily to fit the model to the experimental data, without considering any physical mechanism.

Several authors have also estimated syntactic foam elastic properties through numerical approaches using finite element modelling [START_REF] Antunes | Numerical modelling of the Young's modulus of syntactic foams[END_REF]. One method is to use homogenization method by meshing a representative cell of the material. Marur presents in his work [START_REF] Marur | Numerical estimation of effective elastic moduli of syntactic foams[END_REF] the different possible representative cells that can be used for modelling (Fig. 1.13), with the appropriate meshing, and also obtained good agreement between analytical and numerical results. Another method is to mesh a cell of syntactic foam containing several microspheres (Fig. 1.14). The sphere position can be set randomly using an algorithm [START_REF] Carolan | Micromechanical modelling of syntactic foam[END_REF], or by using X-ray tomography to retrieve the true position of the inclusions in a representative volume to mesh the cell [START_REF] Charière | Use of composite voxels in FFT based elastic simulations of hollow glass microspheres/polypropylene composites[END_REF]. However, those studies are focused on uniaxial compressive or tensile loading and have not studied the hydrostatic behavior of syntactic materials. A novel numerical modelling method, based on FFT modelling, has been recently presented by Charière [START_REF] Gimenez-Fouque | Vieillissement hydrolytique de mousses syntactiques époxydeamine/verre pour l'isolation thermique sous hautes pression: mécanismes de dégradation et simulation de la prise en eau[END_REF] as an effective way to implement numerical modelling of pure syntactics to estimate their elastic properties.

Modelling of syntactic foams can be a useful tool to better understand the complex behavior of syntactic foams. Most of the existing studies focus on the elastic behavior of such materials, based on homogenization methods. In the literature, for the time being, there is no model to describe the collapse of syntactic foams under hydrostatic loading.

In service, syntactic foams are under pressure for long durations (up to 20 years), let us now focus on existing knowledge about the long-term behavior under pressure of these materials.

Long term behavior of syntactic foams

When used for buoyancy applications, the long-term behavior of syntactic foams is of primary interest to ensure that they can maintain their function over their service life.

The two phenomena that can lead to buoyancy loss in syntactic foams are weight gain or volume loss.

It is assumed that buoyancy loss in epoxy syntactic foam comes from water absorption [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF], and so water absorption will be investigated first.

Water absorption of syntactic foams

Water absorption is known to occur in all polymers or composite materials placed in a humid environment, to a lesser or greater extent. It has been extensively studied in polymers because it can dramatically affect their material properties. Here we obviously consider data generated only when samples are placed at pressures below the hydrostatic crush pressure.

When immersed in water, polymers absorb water through their surfaces following two mechanisms:

water molecule vaporization at the sample surface, followed by diffusion of the molecule in the polymer, until the water concentration in the polymer reaches an equilibrium. Numerous studies focus on the water diffusion behavior in polymers, and Fick's second Law is usually considered to describe water uptake in these materials (Eq. 1.13) [START_REF] Fick | On liquid diffusion[END_REF]:

𝜕𝐶 𝜕𝑡 = 𝑑𝑖𝑣(𝐷𝑔𝑟𝑎𝑑 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (𝐶))
Eq. 1. [START_REF] Gupta | Reinforced Polymer Matrix Syntactic Foams[END_REF] With C (no unit or % or mol.L -1 ) the water concentration in the polymer, D (mm²/s) the diffusion coefficient.

Following that, by using the appropriate geometry of the studied sample (usually plate or cubic sample) and assuming that D is homogeneous and equal in every direction, a mathematical solution for this equation exists. For plate samples (i.e., diffusion in 1D direction), this solution results in Eq. 1.14 [START_REF] Crank | The mathematics of diffusion[END_REF][START_REF] Humeau | Contribution to the study of coupling between moisture diffusion and mechanical stress, in high performance marine materials[END_REF].

𝐶(𝑡) 𝐶 𝑠𝑎𝑡 = 1 - 8 𝜋 2 ∑ 1 2𝑖 + 1 exp (-𝐷(2𝑖 + 1) 2 𝑖 𝜋 2 𝑡 ℎ 2 )
Eq. 1.14

With C (no unit or % or mol.L -1 ) the water concentration, Csat (no unit or % or mol.L -1 ) the water concentration at equilibrium, D (mm²/s) the diffusion coefficient, t (s) the immersion time, and h (mm) the plate sample thickness.

This solution is usually represented as a function of the square root. This representation is the black line (0) on Fig. 1.15. Initially, the water concentration is linear with the square root of time, before reaching the equilibrium value. Some polymer behavior differs from this theory, and those divergent behaviors can be described by alternative expressions such as Dual Fick (red line (2) on Fig. 1.15) and Langmuir (pink line (1)) [START_REF] Weitsman | Moisture in composites: sorption and damage[END_REF].

Some polymers show acceleration of the diffusion kinetics (green curve (3)), while others highlight a weight reduction (blue curve (4)). For those examples, specific theories must be established [START_REF] Gac | Water clustering in polychloroprene[END_REF].

In the case of Fickian behavior, the water absorption in polymers can be accelerated by using samples of small geometries to determine the diffusion coefficient and water absorption at equilibrium, and then applying the Eq. 1.14 on samples of larger size to predict their water absorption over time.

Water absorption in composite materials is different due to the presence of inclusions which change the water absorption kinetics. If we consider a unidirectional fiber composite with hydrophobic fibers perfectly aligned along one axis (Fig. 1.16), the fibers will act as an obstacle to water absorption in the transverse direction. Alongside their orientation, the fibers will not be an obstacle and the diffusion will be equivalent to the water diffusion in the resin. Furthermore, the interphase between the fiber and the resin will also play a role in the diffusion kinetics [START_REF] Joliff | Experimental, analytical and numerical study of water diffusion in unidirectional composite materials-Interphase impact[END_REF][START_REF] Joliff | Modified water diffusion kinetics in an unidirectional glass/fibre composite due to the interphase area: Experimental, analytical and numerical approach[END_REF]. Many studies have tried to quantify the transverse kinetics of diffusion through homogenization methods [START_REF] Dana | Identification of moisture diffusion parameters in organic matrix composites[END_REF]. The principal theories that estimate the transverse diffusion coefficient (DT = Dy = Dz) are summarized in Table 1.1, and consider the volume fraction of fiber vf in the composite.

Method Effective diffusion coefficient

Hahn-Tsai [START_REF] Tsai | Introduction to composite materials[END_REF][START_REF] Gueribiz | Homogenization of moisture diffusing behavior of composite materials with impermeable or permeable fibers-application to porous composite materials[END_REF] 𝐷 𝑇 = 𝐷 𝑟 (1 -2𝑣 𝑓 )

Shen-Springer [START_REF] Shen | Moisture absorption and desorption of composite materials[END_REF] 𝐷 𝑇 = 𝐷 𝑟 (1 -2√ 𝑣 𝑓 𝜋 )

Springer-Tsai [START_REF] Springer | Thermal conductivities of unidirectional materials[END_REF] 𝐷 𝑇 = 𝐷 𝑟 (1 -2√ 𝑣 𝑓 𝜋 )

1 -𝑣 𝑓 Shirrel-Halpin [START_REF] Shirrell | Moisture absorption and desorption in epoxy composite laminates[END_REF] 𝐷 𝑇 = 𝐷 𝑟 1 + 𝑣 𝑓 Rayleigh [START_REF] Rayleigh | XIX. On the instability of cylindrical fluid surfaces[END_REF] 𝐷 𝑇 = 𝐷 𝑟 (1 -𝑣 𝑓 -0.3058𝑣 𝑓 4 )

(1 -𝑣 𝑓 )(1 + 𝑣 𝑓 -0.3058𝑣 𝑓 4 )

Table 1.1 Homogenization methods considering the effect of fibers on the water diffusion in a composite [START_REF] Dana | Identification of moisture diffusion parameters in organic matrix composites[END_REF] For syntactic foam materials, being composite materials, the presence of filler also changes the water absorption behavior of the polymer. Kochetkov presents the longest available experimental data on epoxy pure syntactics [START_REF] Kochetkov | Water absorption and swelling of glass/epoxy syntactic foams[END_REF], with experimental results over 10 years of immersion at atmospheric pressure. He highlights that adding inclusions in an epoxy resin decreases the water absorption rate (Fig. 1.17), but after some time the water content in pure syntactics surpasses the amount in the neat resin.

He has studied the swelling of those materials and has revealed that the water uptake for syntactic foams with high microspheres content happens without additional swelling compared to the neat resin, indicating that water fills the microspheres and porosity at interfaces. This was also observed by Fontblanc, who observed that for two syntactic foams made with two types of epoxy resins (one epoxy amine and one epoxy anhydride), after some time at 20°C and atmospheric pressure, the water absorption in the syntactic foam becomes higher than the water uptake in the neat resin [START_REF] Fonblanc | Mousses syndactiques: matériaux composites pour grandes profondeurs[END_REF]. This has been attributed to either hydrolysis of the microspheres and resin, or by the presence of unwanted porosities in the material. Fontblanc has also revealed the difficulty to use a time-temperature equivalence for this type of material to predict their lifetime [START_REF] Fonblanc | Mousses syndactiques: matériaux composites pour grandes profondeurs[END_REF]. Avena performed water absorption tests under hydrostatic pressure [START_REF] Avena-Barthelemy | Comportement a long terme de materiaux composites immerges a grande profondeur[END_REF]. She revealed that applying pressure increases the water absorption. After 3500 s under 0.1 MPa, the studied syntactic foam presents a water uptake of 0.52%, and of 2% after 3500 s under 20 MPa hydrostatic pressure. This has been attributed to collapse of microspheres. Hobaica, using water absorption tests on epoxy syntactic foams up to 7 days under hydrostatic pressure, revealed that the higher the pressure, the higher the water absorption (Fig. 1.18) [START_REF] Hobaica | The characteristics of syntactic foams used for buoyancy[END_REF]. He attributed the high water absorption under high pressure to structural failure of the material. Gimenez gives a detailed study on the hot wet ageing of epoxy syntactic foams, up to a temperature of 120 °C [START_REF] Gimenez-Fouque | Vieillissement hydrolytique de mousses syntactiques époxydeamine/verre pour l'isolation thermique sous hautes pression: mécanismes de dégradation et simulation de la prise en eau[END_REF]. A phenomenological model of water absorption was proposed including diffusion through the resin, hydration of the glass surface/interface and filling of the microspheres. This model is based on the hydrolysis of the microsphere when the hot water reaches it. This chemical reaction at high temperature induces a degradation of the microsphere wall thickness with time, and so a decrease in hydrostatic resistance. This model has been used by Lefebvre and Sauvant-Moynot to describe the water absorption of pure syntactics under high-temperature and high-pressure conditions [START_REF] Lefebvre | Durability of syntactic foams for deep offshore insulation: modelling of water uptake under representative ageing conditions in order to predict the evolution of buoyancy and thermal conductivity[END_REF] (Fig. 1.19). But hydration of glass being extremely slow at low temperature, this does not seem enough to explain how the water fills the microspheres when pure syntactics are immersed in cold water. Water absorption mechanisms in syntactic foams under those conditions have yet to be found. This is one aim of the present study, that is considered in Chapter 5.

Creep of syntactic foams

Concerning the long-term behavior of syntactic foam, volume loss due to creep deformations can be a factor generating buoyancy loss.

Creep is a slow continuous deformation of a material under constant stress. If the load is removed, a reverse elastic strain followed by recovery of a portion of the creep strain will appear (Fig. 1.21). The time-dependent recovery strain is related to viscoelastic deformations, while the non-reversible strain is related to plastic and damage deformations [START_REF] Findley | Creep and relaxation of nonlinear viscoelastic materials[END_REF].

Fig. 1.21 Creep and recovery of materials [87]

Many studies focus on the creep behavior of polymer and fiber composite material, and use empirical equations to represent the creep curves of such materials under different types of loads, as described by

Findley [START_REF] Findley | Creep and relaxation of nonlinear viscoelastic materials[END_REF]. The power law (Eq. 1.15) has been widely used by various authors to describe creep with good accuracy over a wide span of time [START_REF] Humeau | Tension fatigue failure prediction for HMPE fibre ropes[END_REF][START_REF] Lainé | Methodology of Accelerated Characterization for long-term creep prediction of polymer structures to ensure their service life[END_REF].

𝜀 𝑐 = 𝜀 0 + 𝐴𝑡 𝑛 Eq. 1.15
With ε c (no unit or percent) the creep strain, ε 0 (no unit) the elastic strain, A (s -n ) a coefficient that depend on the stress and temperature, and n a material constant independent of the stress.

Another empirical method to describe creep, which is very practical, is the use of a logarithmic law Eq.

1.16, which provides a good description of creep behavior in which the creep rate approaches zero at long times [START_REF] Findley | Creep and relaxation of nonlinear viscoelastic materials[END_REF]. This representation is often used in industry to describe creep of material and make prediction of creep strain over long period.

𝜀 𝑐 = 𝜀 0 + 𝐴𝑙𝑜𝑔(𝑡)

Eq. 1.16

With ε c (no unit) the creep strain, ε 0 (no unit) the elastic strain, and A (no unit) a material constant that depend on the stress and temperature.

To describe creep of materials, various rheological viscoelastic models are also used. Those are models based on the association of spring and dashpot elements that represent instantaneous elasticity or delayed strain. The most basic models are the Kelvin-Voigt and Maxwell models, presented in Fig. 1.22. Base on those two models or on more complex rheological analyses, mathematical descriptions of creep can be proposed. Those are used to describe creep in epoxy polymers [START_REF] Majda | A modified creep model of epoxy adhesive at ambient temperature[END_REF], but require finding an appropriate relaxation spectrum to represent the material behavior, that can require extensive experimental characterization of the material.

While creep has been studied in polymer and fiber composite, this aspect is rarely studied in epoxy syntactic foams, much less concerning creep under hydrostatic loading. The only study we found on volume creep of composite foam was made by Phan [START_REF] Phan | Experimental analysis and modelling of the long term thermo-mechanical behaviour of glass/polypropylene syntactic used on thermally insulated offshore pipeline[END_REF]. In his work, he showed that polypropylene pure syntactics present volume creep under hydrostatic loading (Fig. 1.23) and he used a rheological model to describe it. As far as we know, there is no study of epoxy based syntactic foam creep under hydrostatic pressure. This point is considered and presented in Chapter 5 and 6. To conclude, it appears that when syntactic foams are immersed in water, the water absorption in the material is higher than the amount absorbed by the resin, suggesting damage and filling of microspheres with time. This water absorption is higher when hydrostatic pressure is applied. The microspheres failure under hot wet aging is attributed to reduction of their strength due to hydrolysis, but at low temperatures nothing is certain. Furthermore, none of those studies related water absorption with buoyancy, and no studies of creep of epoxy syntactic foam exist. It is not clear which phenomena would lead to more buoyancy loss in epoxy syntactic materials. This will be investigated in Chapter 5 and Chapter 6.

Syntactic foams containing macrospheres: composite foams

Most of the studies presented above focus only on the behavior of pure syntactic (i.e., syntactic foams containing only microspheres). But the possibility to use hollow macrospheres (i.e., hollow spheres of diameter of several millimeters or centimeters) has been highlighted by Shutov [START_REF] Shutov | Syntactic polymer foams[END_REF] and Watkins [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF],

and nowadays such materials are used by the oil industry for buoyancy applications.

The early knowledge on macrospheres concerned the process of making such objects [START_REF] Shutov | Syntactic polymer foams[END_REF], which are usually made from short fiber composite. The fabrication process of macrospheres often involves the application of a layer of composite material around a polystyrene sphere. The full processing of macrospheres is complicated, and often kept confidential by companies.

New studies on macrospheres and composite foam (i.e., syntactic foam with both microspheres and macrospheres) were found, but again they mainly focus on the manufacturing process. Several types of macrospheres are proposed, such as ceramic [START_REF] Jiang | Ceramic spheres-A novel solution to deep sea buoyancy modules[END_REF], polymeric, metallic [START_REF] Huang | Preparation, compression behavior and 3D damage evolution of epoxy syntactic foam with nickel hollow spheres[END_REF] or fiber composite [START_REF] Wu | Preparation and characterization of three phase epoxy syntactic foam filled with carbon fiber reinforced hollow epoxy macrospheres and hollow glass microspheres[END_REF] macrospheres. Concerning composite foam, Wu et Al [START_REF] Wu | Preparation and characterization of three phase epoxy syntactic foam filled with carbon fiber reinforced hollow epoxy macrospheres and hollow glass microspheres[END_REF] present the manufacturing process of a composite foam made of glass microspheres and carbon fiber macrospheres (Fig. 1.24) and characterized it under uniaxial compression loading. But no studies on the behavior of macrospheres or composite foam under hydrostatic loading could be found. Several technologies that use macrospheres are available, and those are used in the oil industry, but there are no published studies, to the author's knowledge, on the prediction and characterization of the mechanical and long-term behavior of such objects under hydrostatic loading. This is a new area for the scientific community.

Conclusions

The functional properties of syntactic foams for buoyancy applications are their strength and their longterm behavior in a water environment under hydrostatic pressure. Epoxy syntactic foams are the main materials used for buoyancy applications.

Study overview

Based on this overview of existing knowledge a scientific approach is proposed to answer some of the remaining questions related to the buoyancy loss in syntactic foams used in deep sea. Our approach is summarized by the schematic diagram given in Fig. 1.25 below.

As syntactic foam is a composite material, we have chosen to study the behavior under hydrostatic pressure of each component, in order to understand its role in the overall behavior through a step-bystep approach. In this study, microspheres, polymer resin, macrospheres, pure syntactic materials and finally composite foams were all studied. All the materials used here are described in Chapter 2, together with the experimental methods.

Chapter 3 will focus on the hydrostatic strength (i.e., the short-term behavior in Fig. 1.25) of each component of syntactic foams through an experimental study using more than 100 samples of different compositions. Through those results, a better understanding of the hydrostatic response will be given.

They will also emphasize the parameters influencing syntactic foam crush pressure, and an empirical model to describe it will be proposed. Chapter 4 also focuses on the short-term behavior of syntactic foams. This time, analytical and numerical models will be used to understand the mechanical mechanisms involved in the hydrostatic strength of buoyancy materials, and those results are compared with the experimental observations made previously.

The two next chapters focus on the long-term behavior of buoyancy materials under hydrostatic pressure. In Chapter 5, a focus is made on the behavior of pure syntactic (i.e., syntactic foam without macrospheres) considering the origin of the buoyancy loss in this material: water absorption.

Experimental results are presented to investigate the mechanisms involved in the water absorption as well as influencing parameters. Finally, Chapter 6, is dedicated to syntactic foam in presence of macrospheres (i.e., composite foam), to understand the origin of buoyancy loss in these materials and propose ways to make predictions over the material service life (from 1 year up to 20 years).

Introduction

This chapter describes all the materials and experimental test methods that are used in this study. The materials are a set of model materials, specially designed for this study. With a few exceptions, all the experimentations have been improved, developed, and performed by the author.

Materials

All the materials studied here were provided by Balmoral Comtec Ltd. The material set comprises several pure syntactics and composite foams, as well as the pure resins, microspheres, and macrospheres constituting the foams.

Resins

This study was fortunate to have not only pure syntactic materials, but also the resins constituting them.

This allowed us to study the impact of the type of resin on the behavior of syntactic foams. Four resins are available, 3 epoxy resins, and one paraffin resin. In the industrial context, the choice of resin will depend on the application. The epoxies made with anhydride hardener, here named E1 and E3, are usually chosen for temporary applications because of their high glass transition temperature (Tg) that allows the materials to support temperature variations. The epoxy made with amine hardener (E2), has a lower Tg and is used for long term oil production where the environmental temperature is stable around 4°C. The wax is used for buoyancy applications at lower depths (around several hundred meters) because the materials designed with it have lower hydrostatic strength than epoxy syntactic foams.

The resin densities and glass transition temperatures are given in Table 1.1. In the range of -50/100°C, no glass transition could be seen for the paraffin resin, but a melting temperature of 59°C was measured.

Microspheres

The three types of hollow glass microspheres considered in this study, as well as their nominal density, are given in Table 2.2. They are used to study the impact of microsphere size and density on syntactic foam properties. The notation GB in their names refers to "Glass Bubble", and the following number refers to their density. They are made of borosilicate-soda-lime glass and have a white powder aspect.

MEB observations show that microspheres have a round shape but are not perfectly spherical (Fig. 2

.1).

It is also apparent on the picture in Fig. 2.1 that during the manufacturing process, some microspheres are already broken, and some of them are attached to one another (Fig. 2.1). The glass density of the microspheres shell is 2.54 g/cm 3 , value given by the manufacturer.

Microspheres

Density given by manufacturer (g/cm mm (MC2). The macrospheres are described in Table 2.3. The manufacturing process involves surrounding a polystyrene sphere by the shell material, the polystyrene sphere can be seen after cutting the sphere in half (Fig. 2.2).

Pure syntactic

The pure syntactics are model materials made with the resins and microspheres presented before, with ten types of pure syntactic made with different amounts of microspheres in the material. This allowed the influence of the composition of the material on its mechanical and long-term properties to be studied.

Their compositions are presented in Table 2.4. The pure syntactics were provided in large blocks of several liters or in small blocks of around 1L depending on the quantity needed for the study.

Material

Composite foams

The composite foams, being the final material for the application, are made of the pure syntactics and macrospheres presented previously. Their composition and density are presented in Table 2.5. The term CF in their names refers to the appellation "Composite Foam" of the material. The samples are all available in the same geometry: a right cylinder of 1-meter height per 0.3-meter diameter (Fig. 2.3). The macrospheres located at the sample surface are visible (black spots on the sample surface, Fig. 2.3).

Name

Pure syntactic (Table 2.4)

Macrospheres

Methods

The experimental procedures used in this project are presented in this section.

Sample preparation

Microspheres samples for hydrostatic compression tests.

To study the behavior of microspheres under hydrostatic loading, microspheres are mixed with glycerol, with an amount of 38% of microspheres in the mixture, by volume. This amount is chosen according to the withdrawn standard ASTM D3102-78, to prevent microspheres to be in contact with each other during the test. Then, around 1L of the mixture is placed in a hermetic plastic bag. Finally, the plastic bag is placed under vacuum to remove the air entrapped in it, before being sealed (Fig. 2.6). Mass measurements are made at every step of the preparation to control the proportions and measure the exact volume of microspheres and glycerol in the mixture, as well as the exact volume of mixture in the bag.

Macrospheres sample preparation

The macrospheres received are not fully cured, whereas in the composite foam they are. So, to test macrospheres in the same state as they would be when placed in composite foam, a post curing of 3 hours at 120°C was applied before each testing, to ensure they are fully cured.

Composite foam sample preparation

Before being tested, the composite foam samples (Fig. 2.3), have their opposite sides polished using sandpaper (grain P60) to have smooth surfaces. This allows the sample to be trapped between two steel plates, connected by two threaded rods, that form the support of the sample during the test.

Initial characterization 2.2.2.1. Density measurements

Microspheres and resin density measurements: gas pycnometer

When measuring density of pure resins or microspheres, a gas pycnometer is used. This technique is based on gas displacement [START_REF] Tamari | Optimum design of the constant-volume gas pycnometer for determining the volume of solid particles[END_REF]. The equipment used is the AccuPyc II from micromeritics®. Around 1 cm 3 of microspheres or resin is used when the analysis is made. The sample mass is measured using a Sartorius balance (0.0001 g precision) before being placed in the pycnometer chamber. The analysis consists of 10 cycles of purge to remove the air in the chamber, then the density is measured from the mean value of 10 analysis cycles. In each cycle, the pressure rate level for considering the equilibrium is 0.015 psig/min.

Pure syntactic and macrospheres density measurements: Archimedes' Principle

To measure pure syntactic sample volume and density, the samples weight in air (mass) and in water (buoyancy) are measured using a Sartorius balance (0.1 mg precision), as seen in Fig. 2.8. This technique is used because the samples presented in 2.2.1.3. are too large to use the pycnometer. The sample volume is calculated using Eq. 2.1. The same method is applied for measuring macrospheres volume and density.

𝑉 = 𝑚 𝑤𝑎𝑡𝑒𝑟 + m 𝜌 𝑤𝑎𝑡𝑒𝑟

Eq. 2.1

With mwater (g) the sample weight in water, m (g) its mass, V (cm 3 ) its volume, ρwater the water density (g/cm 3 ).

Then the material density is the quotient mair/V. It is possible to calculate the volume fraction of microspheres in pure syntactics using the law of mixtures Eq. 1.8, considering that there are no voids within the material. The expression of the microsphere volume fraction is given in Eq. 2.2:

𝑓 = 𝜌 𝑝𝑢𝑟𝑒-𝑠𝑦𝑛𝑡𝑎𝑐𝑡𝑖𝑐 -𝜌 𝑚𝑎𝑡𝑟𝑖𝑥 𝜌 𝑚𝑖𝑐𝑟𝑜 -𝜌 𝑚𝑎𝑡𝑟𝑖𝑥 Eq. 2.2
With ρpure-syntactic (g/cm 3 ) the pure syntactic density, ρmicro (g/cm 3 ) the microspheres density, ρmatrix (g/cm 3 ) the matrix density and f (no unit) the microspheres volume fraction.

Fig. 2.8 Set up for weight measurements in air and in water

Composite foam density measurements: Archimedes' Principle

For composite foams, the same method as for pure syntactics is applied but the mass sample is measured using a balance of 1g precision. After that, the sample buoyancy is measured by ballasting the sample to make it sink, and then placed in a water tank (Fig. 2.9). The sample plus lest buoyancy is measured using a 1000 N load cell type TS from AEP transducer®, 1 N precision. The ballast weight in water is subtracted from the measurement to retrieve the sample buoyancy. The volume of the sample is calculated using Eq. 2.3. Then the material density is the quotient mair/V. the macrosphere volume fraction can be estimated using Eq. 2.2, by substituting the microspheres density and the resin density by the macrospheres density and pure syntactics density respectively.

𝑉 = 𝐹 0 /𝑔 + 𝑚 𝜌 𝑤𝑎𝑡𝑒𝑟

Eq. 2.3

With F0 (N) the sample initial buoyancy, m (kg) its mass, V (m 3 ) its volume, ρwater (kg/m 3 ) the water density, and g (m/s²) the gravitational constant.

Glass transition temperature measurements

The glass transition temperature (Tg) of the materials is measured through differential scanning calorimetry (DSC) methods. This technique is based on the analysis of the heat capacity variation with the temperature. Tg is then defined as the inflexion point of the curve, which gives a value of Tg. In the present study, the Tg was measured on Q200 DSC equipment from TA Instruments® at a heating rate of 10°C/min from the ambient temperature to 150°C.

Uniaxial compressive tests on neat resin

Compression tests followed the standard ASTM D695M [START_REF]ASTM D695M Standard Test Method for Compressive Properties of Rigid Plastics[END_REF]. Samples sizes are right circular cylinders of 12.5 mm Ø * 25 mm (Fig. 2.5). Tests are conducted using an Instron machine by compressing the sample between parallel plates (Fig. 2.10). The machine is equipped with an oven for temperature control from 21°C to 100°C.

Samples are heated for at least 30 minutes at test temperature before being tested, to ensure the temperature is homogeneous in the sample. The test speed rate is 1 mm/min. The load during the test is measured using a 10 kN load cell. Sample strain is calculated from the compression plateau displacement divided by the initial sample length.

Microspheres size distribution measurements: laser diffraction analysis

The microspheres size distribution of each microsphere type must be known and is measured using laser diffraction. The technique allows the size of microparticles to be measured, and was available at IFREMER. It consists of mixing the particles in water, then flowing the mixture water/particle through an infrared laser beam [START_REF] Mccave | Evaluation of a laserdiffraction-size analyzer for use with natural sediments[END_REF]. The laser beam is made parallel by using a lens, and its image is recovered through a second lens (Fig. 2.11). When the particle goes through the laser, it leads to diffraction of the beam, that is observed on the laser image after the second lens, as represented on Fig. 2.11. Then, the particle size is estimated from the diffraction angle.

Hydrostatic Crush Pressure (HCP) characterization

The behavior of the materials under hydrostatic loading is studied through the methods that are presented in the following section. The method is chosen considering sample size, maximal pressure, testing temperature, and type of measurement needed during the test. In all the tests, all the tested samples present negative volume change due to the applied hydrostatic compression. However, the results will always be presented using the absolute value, for consistency with the usual presentation in literature.

Microspheres testing

The hydrostatic strength of microspheres (i.e., collapse pressure) is characterized in a hyperbaric tank with fluid injection. The principle (Fig. 2.12) of the test is to apply a hydrostatic pressure to a specimen through a loading fluid (in our case tape water) and to follow its buoyancy throughout the duration of the test. The sample is placed in a hyperbaric tank which is temperature and pressure controlled. The sample is kept 30 min in the tank before being pressurized to ensure the temperature is homogeneous.

The test temperature is 21°C, and the loading speed is 1.5 MPa/min. The buoyancy is directly measured by means of an immersed load cell placed at the bottom of the vessel, on which the sample is attached.

The load cell is a 5N load cell from SIXAXES, with accuracy of 0.01 N. The equipment for testing microspheres is a hyperbaric vessel of 60 MPa maximal pressure (Fig. 2.12), with temperature control from 20 to 120°C. The chamber dimensions are 300 mm diameter and 600 mm length. The minimal pressure of this hyperbaric vessel is 0.4 MPa, the pressure of the water network. With V (cm 3 ) the microspheres volume, m (kg) the sample mass, F (N) the recorded buoyancy, g (m/s²) the gravitational constant, and ρ water (P, T) (kg/cm 3 ) the water density at pressure P (MPa) and temperature T (°C), β (MPa -1 ) the glycerol compressibility, f (no unit) the volume fraction of microspheres in the glycerol (in our case, f = 0.38), and V0 (cm 3 ) the initial volume of the mixture microspheres/glycerol.

Pure syntactic testing

Two methods are used to characterize the hydrostatic strength (i.e., crush pressure) of pure syntactic.

Hydrostatic compression through fluid injection with volume measurements up to 60 MPa

The first one uses the same method and device as for the microspheres (Fig. 2.12). This procedure allows us to make hydrostatic compression tests at different temperatures (between 20 and 120°C), changing the temperature being a way to change the matrix mechanical properties. The chamber volume allows us to test cuboid samples of 50 or 100 mm 3 . The sample must be heated at least 1 hour in an oven before being tested, and the water in the tank is directly heated using heating resistances. Then, the sample is placed in the water and kept in it for at least 30 min to homogenize the temperature. The sample is finally pressurized at a speed of 1.5 MPa/min, with a maximal pressure of 60 MPa.

Thanks to the weighing device, the buoyancy of the sample is directly measured. Those tests are short, less than 2 hours, and water absorption in the samples during the test is too low to be measured. Hence, the sample mass can be considered constant, and with this hypothesis the sample volume is calculated using Eq. 2.5. The sample mass is measured before the test using a Sartorius balance (0.1 mg precision).

𝑉 = 𝐹 𝑔 + 𝑚 𝜌 𝑤𝑎𝑡𝑒𝑟 (𝑃, 𝑇)

Eq. 2.5

With V (cm 3 ) the pure syntactic volume, m (kg) the sample mass, F (N) the recorded buoyancy, g (m/s²) the gravitational constant, and ρ water (P, T) (kg/cm 3 ) the water density at pressure P (MPa) and temperature T (°C)

Hydrostatic compression through piston displacement up to 200 MPa

The second method is based on piston displacement. The principle is to generate pressure in a hyperbaric tank by piston displacement as shown on Fig. A pressure vessel was specially designed for this test during this thesis, to perform tests up to 200 MPa (Fig. 2.13). The tank has an internal chamber of 80 mm diameter per 80 mm length, allowing small samples such as plate samples of 5 mm thickness to be tested. The temperature for those tests is limited to 21°C (controlled room temperature), and samples must be placed in the room at least 1 hour before being tested to be at the right temperature. The piston displacement speed is 0.51 mm/min, chosen to be equivalent to a load speed of 1.5 MPa/min.

The volume change of samples is difficult to measure due to the tank dilatation and will not be presented in this study. But once the pressure vessel is placed in the tensile machine, those tests are easier to perform than the other methods.

Macrospheres testing

Macrospheres collapse pressure was studied in a hyperbaric tank designed for this study (Fig. 2.

14).

This tank allows hydrostatic pressure tests up to 100 MPa. The principle of this test is to pressurize the macrospheres using fluid injection up to failure. The circumferential strain of the macrospheres is measured using 3 strain gauges; this is possible using special waterproof connectors, and by protecting the gauges from water using waterproof resin. The gauges are KYOWA gauges of gage factor 2.12, resistance 120Ω, and length 30 mm (reference KFGS-30-120-C1-11). Those measurements are preferred because when the macrospheres implode, the shock can damage the load cell.

The vessel chamber dimensions are 80 mm diameter by 100 mm height. The temperature is controlled by placing the vessel in a tray filled with water at temperature controlled between 10 and 25°C. The loading speed is 2.5 MPa/min, and the maximal pressure is 100 MPa. The gauges measurement gives the circumferential strain of the macrospheres, from which volume variations can be estimated. By considering the strain homogeneous in the shell, the volume change of the macrospheres is related to the circumferential strain by Eq. 2.6.

This test procedure can also be used on pure syntactic to measure its bulk modulus. Cubic samples of 50 mm 3 are used, and 3 gauges are placed parallel to its sides, to measure strain in the Cartesian coordinates. By assuming the strain to be homogeneous in the material, the sample volume change can be calculated using Eq. 2.6, and replacing the mean circumferential strain by the mean of the Cartesian strain.

∆𝑉 𝑉 0 = (1 + 𝜀) 3 -1

Eq. 2.6

With ∆𝑽 𝑽 𝟎 (no unit) the volume strain of the macrosphere, ε (no unit) the mean circumferential strain of macrospheres, or mean of the Cartesian strain for pure syntactics.

It should be mentioned that the characterization of mechanical strength of glass fiber macrospheres was made prior to the beginning of this thesis.

Composite foam testing

When characterizing the hydrostatic strength (i.e., crush pressure) of composite foam, the same principle as presented in Fig. 2.12 is used, but in a larger hyperbaric vessel presented in Fig. 2.15, of diameter 1 m and length 2.5 m with a maximal pressure of 100 MPa, and temperature control from 2 to 25 °C. The collapse of composite foams leads to the formation of debris that can obstruct the pipes of the pressure vessel. To prevent that, the sample must be placed in a bag when being tested (Fig. 2.16). The bag will retain the debris but let the water flow through it since it is permeable. A metallic structure is needed (Fig. 2.16) to prevent the bag from being sucked in the sample when it implodes, otherwise the bag tears itself when the sample crushes and can no longer retain the debris.

The sample mass is measured prior to testing using the 1 g precision balance. The buoyancy of the sample is recorded during the test with a 500 N load cell from GAROS, with 1 N precision. The volume change of the sample is calculated using Eq. 2.5.

Long term behavior characterization

This section presents the methods to characterize the long-term behavior of syntactic foams. The experimental procedure focuses on the water absorption, volume loss and buoyancy loss of the studied materials. The method is chosen depending on the type of material studied and on the needed measurements. In all the test, all the tested samples present negative volume change due to the applied hydrostatic compression. However, the results will always be presented using the absolute value (positive value), for consistency with the usual presentation seen in literature.

Water absorption of neat resin and pure syntactic

To characterize water absorption of resin and pure syntactics, plate samples of 50*50*e mm 3 (e = 1, 4, 5, 6 mm) or cubic samples of 25 or 50 mm 3 are placed in water and weighed regularly to measure their weight gain. The weight gain is defined by Eq. 2.7.

𝑊 = 𝑚(𝑡) -𝑚 0 𝑚 0

Eq. 2.7

With W (no unit) the weight gain, m (g) the sample mass, m0 (g) the initial mass of the dried sample, and t (s) the immersion time.

In this study, all water absorption tests were made at 15 °C. For water absorption without additional pressure, samples are placed in water tanks in either tap water or sea water. Before doing that, samples were dried at 40°C in a desiccator at 0% relative humidity (RH) until the mass reached a stable value.

When immersed in water, samples of pure syntactics present an instantaneous initial water absorption due to the filling of the broken microspheres at the sample surface (due to manufacturing of the sample).

This initial step is subtracted from the weight gain measurements to consider only the water absorption inside the material, and not the water at the sample surface filling the already open cells.

For water absorption with additional pressure, samples are placed in pressure vessels (Fig. 2.17) with maximal admissible pressure of 100 MPa, and internal cell of 80 mm height per 80mm diameter.

Pressure is increased using a manual pump at a speed around 40 MPa/min. The samples are immersed in tap water because the pressure vessels do not support sea water. However, one pressure vessel was designed for sea water and was used for some immersions. Temperature control is made by placing the vessels in a water tank with a water cooler. (Fig. 2.17). For water absorption at pressures higher than 100 MPa, the 200 MPa piston vessel presented in 532.2.3.2.

was used by replacing the piston by a special plug.

It should be mentioned that the water absorption of the resins E1 and E2, in sea water without additional pressure, was characterized before the start of the present study.

Creep tests on pure syntactics and macrospheres

Creep tests on pure syntactics and macrospheres follow the same procedure as for the HCP tests with gauge measurements (presented in 2.2.3.3. ), but instead of increasing the pressure up to failure, it is maintained at the desired pressure for the desired period (usually until collapse of the macrosphere), and the volume strain is measured throughout the test.

Long term buoyancy loss of composite foam

To measure buoyancy loss of composite foam sample with accuracy, a special test procedure, named Instrumented Buoyancy Loss (IBL) test, was implemented, and significantly improved during this thesis. The principle of the test is presented in Fig. 2.18. A sample of composite foam is ballasted and attached to the pressure vessel plug using 3 springs of constant k. As the sample buoyancy changes, the spring length will change as well (decreasing/increasing when the sample becomes more/less buoyant).

Then, the sample buoyancy change is measured by measuring the spring length variation, using a LVDT sensor. Fig. 2.18 presents the set up used for those tests.

The measurement set up is calibrated before each test and has a 0.01N precision. Since the sample is ballasted, the initial sample buoyancy/volume must be measured externally using the procedure presented in 2.2.2.1. Sample mass is measured before/after the test using a 1g precise balance to measure the weight gain of the sample during the test.

Water absorption of composite foam

Composite foam water absorption with additional pressure is measured during the IBL test by measuring the sample mass before and after the test, or between each pressurizing cycle, with or without buoyancy loss measurements.

The water absorption can also be characterized without additional pressure, but the size of the sample requires it to be placed in the hyperbaric vessels used for the IBL tests (Fig. 2.19).

Post-testing characterization 2.2.5.1. Pure syntactics calcination/acid digestion

To study the damaging of pure syntactic with a quantitative approach, the microspheres inside pure syntactics after hydrostatic pressure are extracted from the material to perform density measurements to evaluate the number of collapsed microspheres.

To recover those microspheres, burning off the epoxy is not sufficient. Even after long calcination times aggregates of microspheres are still present. Those aggregates pollute the density measurements, preventing accurate estimation of damage inside the material.

To overcome this problem, a washing method was implemented during the project. The microspheres recovered after calcination are washed with acid at 100°C for at least 1 hour, to remove the resin still present. Then, the solution containing acid and microspheres is filtered using a 1.2 µm filter to retain only the microspheres. After that, the microspheres are washed again using water and acetone, and dried in an oven for 1 hour at 120°C.

The procedure for washing the microspheres follows the standard ASTM D3553 [START_REF]ASTM D3553 Standard Test Method for Fiber Content by Digestion of Reinforced Metal Matrix Composites[END_REF]. This standard recommends the use of nitric acid for washing amine-based epoxy, and sulfuric acid for anhydride-based epoxy. For safety reasons, only nitric acid is used in the laboratory, so only pure syntactic made with epoxy amine was studied with this method.

Imaging: X-ray tomography

X-ray tomography is a nondestructive method to provide observations of syntactic foams microstructure.

It allows the different phases of the material to be observed: spherical inclusions, matrix, and voids [START_REF] Adrien | Experimental study of the compression behaviour of syntactic foams by in situ X-ray tomography[END_REF].

For pure syntactic material, X-ray microtomography is used. Those analyses were performed by Jerome For macrospheres and composite foams, X ray tomography was used. Those analyses were performed by the technical resource center CRT Morlaix. They provide radius and thickness measurements of macrospheres, locate the positions of macrospheres inside composite foam samples, and allow observation of the state of the macrospheres inside the structure.

Modelling

Abaqus® software was used to model composite foam elastic behavior under hydrostatic loading. The localization of macrospheres in the modelled sample were retrieved using X-ray tomography, and the sample geometry was reconstructed by placing hollow spheres of equal size and thickness in a right circular cylinder. The hollow spheres were meshed using 8-noded hexahedrons elements, and the matrix material was meshed using 4-noded tetrahedrons elements. A tie constraint was imposed between the cylinder and sphere surface.

Conclusions

For this study, model materials of different composition were obtained. These will allow us to study the influence of each component on the material properties. The composite foam available will allow us to investigate the impact of adding macrospheres in the material, which is rarely studied. The experimental method for testing composite foam, that existed at the laboratory, has been improved to provide better results.

Now that the materials and method are presented, they will be used to study the behavior of syntactic foams. The next chapter will focus on the hydrostatic strength of syntactic foams.

CHAPTER 3 Hydrostatic strength of syntactic foams

Introduction

As presented in Chapter 1, the hydrostatic strength of pure syntactic is of great importance to define the depth at which the material can be used. This section presents the experimental study of the hydrostatic strength of syntactic foams made in this thesis. It is organized in 3 parts. First an experimental study of the collapse of hollow spherical inclusions was performed, these being glass microspheres and fiber composite macrospheres. Second, the hydrostatic strength of pure syntactic was studied experimentally, and the strengthening effect of the resin on the microspheres highlighted. An empirical model to predict crush pressure of pure syntactic depending on material composition is proposed and compared with experimental results. The damage of epoxy pure syntactic originating from hydrostatic loading is quantified, for the first time at the best of the author's knowledge, to investigate the crush mechanisms of those materials. Finally, the strength of composite foam is investigated.

Hydrostatic strength of hollow spheres

In this section, the hydrostatic strength of the microspheres and macrospheres used in this study is characterized, to be able to later compare their strength with the resistance of syntactic foams.

Hydrostatic strength of hollow glass microspheres

First, the behaviour of glass microspheres without matrix under hydrostatic loading is studied. The HCP test, described in 2.2.3.1. was used to measure GB15 microspheres volume change during the test (Fig. Between 0.4 MPa (which is the minimal water pressure in the hyperbaric vessel) and 0.7 MPa (Fig. 3.1), a linear behaviour is observed that is related to the elastic deformation of the microspheres [START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF]. Then the volume loss increases much faster when higher hydrostatic pressure is applied; this is due to the collapse of most of the microspheres induced by the mechanical loading [START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF]. Based on this in situ measurement of volume variation under hydrostatic pressure it is possible to define a collapse pressure using the tangent method, as shown in Fig. 3.1. This method is different from what the microsphere manufacturer does, as they define the microspheres strength by the pressure at which 10% of the microspheres collapse. In this work, the tangent method is chosen to be consistent with the measurement of crush pressure of pure syntactic materials presented later in this chapter. The collapse pressure of the different microspheres available in this study is reported in Table 3.1. The collapse pressure was measured using 3 samples for GB15 microspheres, and 1 sample for GB25 and GB38 microspheres. The strength of microspheres increases as their average density increases. These collapse values differ from the ones given by the manufacturer due to the difference in the definition of collapse.

To understand where the difference of strength between the different type of microspheres comes from, the microspheres dimensions are measured. From laser granulometry, the diameter distribution of the studied microspheres was obtained (Fig. 3.2). Microspheres show a large variation of diameter. The main difference between the types of microspheres comes from the largest microspheres: they all present small microspheres of minimal size of around 5 μm, but the maximal microspheres diameter diminish as the microspheres apparent density increases, with the maximal diameter for GB15 microsphere being 160 μm, 1.6 times higher than the largest GB38 microspheres of 100 μm.

Fig. 3.2 Microspheres size distribution by number

The thickness of the microspheres GB15 and GB38 was subsequently measured from X ray microtomography imaging on one sample of 5*1.725*1.725 mm 3 of E1_GB15 and E3_GB38 pure syntactic, with a resolution of 0.3 µm (Fig. 3.3). The thickness distribution of those microspheres is presented in Fig. 3.4. The thickness is centered around a value of 1.2 µm for GB15 microspheres, and 1.5 µm for GB38 microspheres. Those values are negligible compared to their diameter, and therefore it can be considered constant for each type of microspheres. The denser the microsphere type is, the thicker the microspheres are, with GB15 being 0.8 times smaller than GB38. From those measurements, the hypothesis being made is that the microspheres collapse is related to the collapse of the largest and thinnest ones, as it is usually assumed in the literature [START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF] (as presented in 1.3.1. ). Since microspheres of the same grading can be considered to have constant thickness, it means that the largest microspheres will collapse first.

To verify this hypothesis, using laser granulometry, the microspheres size distribution was measured before and after pressurization. Fig. 3.5. presents the evolution of size distribution (by number) for microspheres GB25 subjected to hydrostatic loading. After pressurization, the size distribution translates to lower radius. This confirms that the larger microspheres collapsed first. To conclude on this section, the hydrostatic strength of microspheres has been measured, revealing that the denser microspheres collapse at higher pressure. From size measurements, it appears that microsphere hydrostatic strength is correlated with the hydrostatic strength of the largest ones.

Furthermore, the higher the microspheres average density, the smaller and thicker the microspheres are.

The large improvement of strength seen for GB38 microspheres, compared to the others, comes from their higher thickness, and from their much lower diameter.

Hydrostatic strength of macrospheres

The other type of hollow inclusions used in syntactic foam is macrospheres, and their strength is investigated here. The strength of a hollow composite sphere was measured using the method described in 2.2.3.3. During those hydrostatic compression tests, when a maximal pressure is reached, the sphere inside the vessel implodes into numerous small pieces, while emitting a large implosion sound. The strength of the sphere is defined as the pressure at which the sphere imploded. The collapse pressure of all the different spheres studied is reported on Table 3.2. The carbon macrospheres are stronger that the glass fiber macrospheres, and the thickest carbon macrospheres (MC2) are stronger than the smaller ones (MC1). Using carbon macrospheres allows an increase in strength, but they are denser (see Table 3.2), and so increasing the macrospheres strength reduces the buoyancy. To study the collapse mechanism of macrospheres, their dimensions need to be measured. Prior to testing, the glass macrospheres are analyzed using X ray tomography to reconstruct their geometry. Fig. 3.7 presents one 3D reconstruction of a glass macrosphere using X ray tomography. Those analyses give us the radius and thickness distribution of each sphere. Using these measurements, a mean internal and external radius was measured for each macrosphere. The disparity of radius measurement being small, and the thickness of the macrosphere being 10 to 20 times lower than their radius, the macrosphere geometry can be described using mean values of radius and thickness and considering the macrosphere geometry as perfect.

Then, using the formula of stress for a perfect hollow sphere subjected to external pressure (Eq. 1.12), the circumferential stress at break in the macrosphere shell is estimated. The evolution of stress at break is presented in Fig. 3.8. The measured circumferential stress at break does not depend on the macrosphere geometry (Fig. 3.8).

The collapse mechanism of macrosphere is the failure of the shell material due to the compressive strength being reached. The maximal compressive stress at break for the glass fiber composite constituting the macrosphere is evaluated at 245.5 ± 10.0 MPa.

Conclusion

In this section, the hydrostatic strength of microspheres has been characterized. While their exact collapse mechanism is not clear, their collapse is related to the failure of the largest ones, that collapse first. The microsphere thickness also plays a role in the strength of such objects, but each microsphere grade can be considered to have a constant thickness.

The hydrostatic strength of macrospheres was measured experimentally and it is revealed that those hollow spherical elements collapse due to the material shell failure in compression (and not because of buckling). The use of hydrostatic compression tests up to failure on macrospheres allowed us to determine the compressive strength of the glass fiber composite material. The compressive failure of carbon has not been investigated yet, and this should be done to confirm that those elements also collapse because of compressive failure of their shell material. For now, this hypothesis will be assumed.

Experimental study of the hydrostatic strength of pure syntactic

Now that the behavior of hollow spheres under hydrostatic compression has been studied, this section focuses on the crush pressure (i.e., hydrostatic strength) of pure syntactic. Here, the strengthening of the microspheres by the resin is first highlighted. Then, the influence of the type of matrix used, and microspheres volume fraction are investigated. From those results a mathematical description of pure syntactic crush pressure is proposed as a tool to predict the strength of pure syntactic. Then, the influence of the microspheres' density on pure syntactic crush pressure is studied, and the model parameters are estimated for other types of microspheres. Finally, the crush mechanisms of pure syntactic are investigated through the study of damage inside the materials during pressurization.

Pure syntactic crush pressure

The behavior of a pure syntactic E1_GB15 (microspheres volume fraction of 0.56) was studied at 21°C using the hydrostatic compression test with volume measurements (presented in 2.2.3.2. ) on one cubic sample of 100 mm 3 and three cubic samples of 50 mm 3 . Fig. 3.9 presents the pressure/volume strain curve of the material. In Fig. 3.9, a linear behavior of the pure syntactic is observed before a densification plateau appears, in agreement with literature. Several ways exist to measure the crush pressure. In this work it has been defined using a tangent method between the elastic part and densification part of the curve, as shown in red on Fig. 3.9. In this case, the crush pressure of E1_GB15, at 21°C, is 38.0 ± 1.0 MPa for cubic sample of 50 mm 3 , and 37.7 MPa for the 100 mm 3 cubic sample. There is no significant impact of the sample size on the material crush pressure.

Two types of hydrostatic compression tests were used in this work, one based on fluid injection and the other on piston displacement (see 2.2.3.2. ). The one used depended on whether the test temperature or the maximal test pressure was to be increased. Both techniques must be compared first to ensure that they give the same results. Furthermore, they use different sample sizes. So, it must be ensured that the sample size has no impact on the material strength. The previous tests revealed that there are no differences in crush pressure of cubic samples of 100 mm 3 and 50 mm 3 . But in the piston vessel, usually smaller plate samples are used.

To compare both techniques, and further ensure that the sample size has no influence on the material strength, the same pure syntactic E1_GB15, with the same microspheres volume fraction, was pressurized at 21°C in the piston vessel. Three plate samples of 5 mm thickness and one cubic sample 50 mm 3 have been used. Fig. 3.10 presents the results by plotting a typical pressure/piston displacement curve. During HCP testing with piston displacement, at first the pressure inside the vessel increases linearly while the piston moves in the tank (Fig. 3.10). Then, when a maximal pressure is reached, a pressure drop appears in the tank. This pressure drop indicates that the material inside the vessel has suddenly lost volume, and reached its densification plateau, giving more space to water that depressurizes. The crush pressure of the pure syntactic is then measured by recording the maximal pressure just before pressure drop occurs. The crush pressure dependence on the test method used and on the sample size, are shown on Table 3.3. The deviation between each test method being less than the standard deviation of the measurements, this indicates that the testing method and the sample size have no impact on the material crush pressure. The most interesting point is that the microspheres resistance is greatly improved by using epoxy as a matrix. Indeed, as shown in 3.1.1. , the collapse pressure of the GB15 microspheres alone is 1 MPa (Table 3.1), whereas when the same microspheres are embedded in an epoxy resin the crush pressure of the syntactic foam is 38 ± 1 MPa (Table 3.3). This point is very important for the application, as the collapse pressure of the microspheres used can be significantly below the operating pressure as the microspheres' strength is enhanced due to the resin. As we saw, increasing the microspheres strength is obtained through an increase of density, and so a loss of buoyancy. Here, it is revealed that it is possible to use microspheres that can have strength weaker than the operating pressure, allowing a reduction in the microspheres' density and a gain in buoyancy.

The following part of this section will focus on the study of the key parameters than influence the crush pressure of a pure syntactic (resin and microspheres volume fraction) and propose a mathematical description of the results.

Impact of resin mechanical properties on pure syntactic hydrostatic strength

The first parameter we suppose has an influence on the crush pressure of pure syntactic is the resin, based on literature. A focus on the mechanical properties is made. To characterize the impact of the resin mechanical properties on the failure of syntactic foam, two approaches are used here. The first one is to change the resin in the material from E1 to WAX, with the same GB15 microspheres and the same microsphere volume fraction. The second is to change the test temperature to modify mechanical properties of the epoxy E1 over a wide range. But first, the mechanical properties of the 2 resins must be characterized.

Determination of the mechanical properties of the resin

The resins properties are determined using uniaxial compressive tests (see 2.2.2.3. ), and results are presented in Fig. 3.11. The graph on Fig. 3.11.a plots results obtained at 21°C with the two matrix materials considered here (i.e., E1 and WAX). A large difference in terms of behavior was observed.

For the epoxy matrix a linear behavior was observed up to 60 MPa followed by a yield, with a maximal compressive stress of 71 ± 1 MPa. For the WAX, a linear behavior was also observed but only up to 2 ± 0.1 MPa, without a clear maximal compressive stress. These results are in accordance with existing knowledge in published literature [START_REF] Heinz | Uniaxial compression analysis of glassy polymer networks using digital image correlation[END_REF][START_REF] Chen | Tension and compression tests of two polymers under quasistatic and dynamic loading[END_REF][START_REF] Veale | The structural properties of paraffin wax based hybrid rocket fuels with aluminium particles[END_REF].

As the final objective of this section is to study the impact of the resin mechanical properties on the pure syntactic crush pressure, more data points are needed. Only two types of resins are available, paraffin and epoxy. To get around that lack of different resin, we propose another method to change the matrix mechanical properties : to increase the temperature of the tests, to change the epoxy resin properties.

To consider a wide range of resin properties, the epoxy resin E1 is tested under uniaxial compression at several temperatures, from 21°C to 100°C. Results are presented in Fig. 3.11.b. When the test temperature increases, a decrease in both modulus and yield stress is observed. Measurements of the compressive modulus, made using linear regression on the curve between strain values of 0.01 and 0.015, are recorded in Table 3.4 (3 samples per measurements).

Determination of crush pressure of syntactic foam depending on resin properties

Both pure syntactic materials E1_GB15 and WAX_GB15 are characterized under hydrostatic compression tests with volume measurements presented in 2.2.3.2. Then, the pure syntactic E1_K5 is tested under hydrostatic compression at different temperature (without changing any other parameters).

Curves of pressure versus volume strain are shown in Fig. 3.12.

For all results presented here, the same behaviour is observed: first the foams present an elastic behaviour and then, when microspheres collapse, a densification plateau (i.e., the crush pressure appears). However, large differences in crush pressure are observed. In other words, mechanical properties of the matrix strongly affect the crush pressure of the syntactic foam.

A comparison of crush pressure determined in Fig. 3.12 with uniaxial modulus of the matrix determined in Fig. 3.11 is proposed in Fig. 3.13. It clearly appears that the stiffer the matrix, the higher the crush pressure of the pure syntactic. The dotted black line represents a model of this behaviour that will be described in 3.2.4. . To conclude on those results, there is obviously a relationship between the crush pressure of a syntactic foam and the mechanical behaviour of the matrix used. The stiffer the resin, the higher the crush pressure of the syntactic foam. This should be considered when designing pure syntactic, as it can greatly increase the maximal depth at which the material can be used. Increasing the resin stiffness also increases the reinforcing effect on the microspheres, allowing the use of microspheres of low density at higher depths.

In the next section, the impact of volume fraction of microspheres on the syntactic foam crush pressure will be studied, as it has been proposed as another key parameter influencing the crush pressure.

Impact of microspheres volume fraction on pure syntactic crush pressure

Pure syntactic manufacturers try to increase the microspheres content in a syntactic foam to decrease the material density and to provide maximum buoyancy. Here, the impact of the microspheres volume fraction on pure syntactic crush pressure is studied for the first time, at the best of the author's knowledge. For this study, more than 100 samples have been tested.

The crush pressures of pure syntactics E1_GB15 and WAX_GB15 with different microspheres volume fraction were measured using the piston displacement method (see 2.2.3.2. ). Results are plotted in Fig. 3.14.a and Fig. 3.14.b. For both matrix materials, a decrease in microsphere volume fraction leads to an increase in crush pressure of the syntactic foam. For example, for the epoxy matrix, when the volume fraction is decreased from 0.6 to 0.3, the crush pressure increases from 37 MPa to 57 MPa. Attempting to increase the microspheres volume fraction, and so the material buoyancy, will result in a loss of strength, and so a reduction in the depth at which the material can be used.

With those experimental results, a hypothesis can be made on the origin of the strengthening effect of the microspheres: the resin will reduce the stress applied on them, according to their rigidity, and when the volume fraction of microspheres increases, the amount of resin diminishes, reducing the stress reduction. This hypothesis will be investigated through a model of stresses inside the pure syntactic in Chapter 4.

The next step is devoted to setting up and validating a mathematical model that can describe the influence of both the matrix stiffness and microspheres volume fraction on the crush pressure of a pure syntactic made with GB15 microspheres.

Modelling hydrostatic strength of pure syntactic with GB15 microspheres 3.2.4.1. Model description and identification

From the experimental work, it has been shown that the resin properties and the volume fraction of microspheres influence the crush pressure of pure syntactic, which is a key parameter for using them in deep sea applications.

Considering this, a model is proposed to describe changes in crush pressure considering both matrix stiffness and microspheres volume fraction. As the exact physical mechanisms involved here are not fully understood yet (but will be investigated in Chapter 4), a phenomenological model is preferred here.

The proposed model is based on the results observed on material with GB15 microspheres first, because more data are available. However, it was shown in 1.3.1. that changing the microspheres type changes the material strength [START_REF] Gupta | Compression properties of syntactic foams: effect of cenosphere radius ratio and specimen aspect ratio[END_REF], this point will be studied in 3.2.5.

For describing pure syntactic crush pressure in a simple way, the following equation is proposed:

𝑃 𝑐𝑟𝑢𝑠ℎ = 𝛼𝐸 𝑐𝑜𝑚𝑝 (1 -𝑓) + 𝛿 𝑓 Eq. 3.2
With Pcrush (MPa) the crush pressure of the foam, Ecomp (MPa) the compressive modulus of the resin, f (no unit) the microsphere volume fraction, and α (no unit) and δ (MPa) two empirical constants.

The first term of Eq. 3.2 describes the correlation between the compressive modulus, volume fraction of microspheres, and pure syntactic crush pressure, and the second terms describes the fact that there is no crush pressure for a pure resin (volume fraction of zero).

The parameters α and δ can be determined by solving the following matrix equation:

( 𝐸 1 (1 -𝜈 0 ) 1/𝜈 0 ⋮ ⋮ 𝐸 𝑛 (1 -𝜈 𝑛 ) 1/𝜈 𝑛 ) * ( 𝛼 𝛿 ) -( 𝑃 𝐶𝑟𝑢𝑠ℎ 1 ⋮ 𝑃 𝐶𝑟𝑢𝑠ℎ 𝑛 ) = 0 Eq. 3.3
With Ei, (MPa) the resin compressive modulus, vi (no unit) the microspheres volume fraction, and Pcrush i (MPa) the crush pressure of the tested sample indexed by i, with i between 1 and the number of tested samples n (in our case, n=96).

The Eq. 3.3 was solved using the function linalg.lstsq implemented in python, that computes a leastsquares solution of the matrix equation [102]. Results give α = 0.0519 and δ = 2.79 MPa (coefficient of determination R² = 0.99).

As observed in Fig. 3.13 and Fig. 3.14, the proposed model (dashed line) is able to describe both the impact of matrix stiffness and microspheres volume fraction using a simple model within the range considered here: compressive modulus between 5 Pa and 1600 MPa and volume fraction of microspheres between 0.3 and 0.6.

To check the relevance of this unique model as a predictive tool for the pure syntactic collapse pressure, another pure syntactic, E2_GB15, was studied. The results will be discussed in the following section.

Relevance of the proposed model

To investigate the pertinence of the model, the pure syntactic E2_GB15 was studied (microspheres volume fraction of 0.59). The epoxy used here (E2) has different mechanical properties than the one used for identification (E1).

The resin compressive modulus, and pure syntactic crush pressure, were determined using the same method as described in 3. To illustrate the good description of all our experimental results with the model, the model prediction of crush pressure was plotted versus the experimental data measured on Fig. 3.16. Within this section, a mathematical model has been proposed that is able to describe the crush pressure of pure syntactic containing GB15 microspheres, depending on the microspheres amount and resin compressive modulus. But it is known that the type of microspheres influences the material hydrostatic strength. The influence of the type of microspheres will be studied in the next section, and the model compared with the experimental data.

Influence of the type of microspheres

Until now, only pure syntactics made with GB15 microspheres were studied. The last parameter to be studied here is the impact of the type of microspheres used in the pure syntactic. The crush pressure of pure syntactics made with E1 and WAX resin with different kinds of microspheres is measured (three samples per materials). The volume fraction of microspheres in the pure syntactic is 0. Changing the microspheres changes the strength of the pure syntactic: the stronger and denser the microspheres (i.e., the smaller and thicker), the higher the crush pressure is. Once again, a compromise must be made between material density and hydrostatic resistance.

To verify the relevance of our model presented in 3.2.4.1. it was used to describe crush pressure of pure syntactic made with GB25 and GB38 microspheres.

For pure syntactic with GB25 microspheres, the determination of α and δ was made using results from HCP tests at 21°C. The materials used are the ones described in Table 2.4 containing GB25 microspheres: E1_GB25, E2_GB25 and WAX_GB25. The pure syntactic E1_GB25 was also tested at 40°C and 60°C, and the pure syntactic E2_GB25 was tested at temperature of 21, 25, 30, 35 and 50°C.

For pure syntactic containing GB38 microspheres, the HCP tests were conducted on the pure syntactic WAX_GB38, E1_GB38, E2_GB38 and E3_GB38. Only one volume fraction of microspheres is available for each material (but it differs between each material). The tests were conducted only at 21°C. The values of α and δ in the model were determined for each type of microspheres, using the least square method to fit the model to the experimental data. The values of the model constants are given Table 3 To illustrate the pertinence of our model for pure syntactic with GB25 and GB38 microspheres, the model prediction was plotted versus the experimental measurements on Fig. 3.17 and compared with the identity function. For pure syntactic with GB25 microspheres, the model can be used to describe our experimental results. For pure syntactic containing GB38 microspheres, there is not enough data to conclude, but the trend is observed. To conclude, the parameters influencing the crush pressure of pure syntactic are identified, revealing that increasing the resin rigidity increases the reinforcing effect on the microspheres. In the same time, increasing the pure syntactic density (i.e., decreasing the volume fraction and increasing the microspheres density), increase the material crush pressure, but at a cost of buoyancy.

A new empirical model was implemented, which can predict pure syntactic crush pressure depending on the resin rigidity and microspheres volume fraction for the first time, at the best of the author's knowledge. The model constants were determined using experimental data for 3 types of microspheres.

To further investigate the origin of the strengthening effect of the resin on the microspheres, the damage behavior and failure mechanisms of pure syntactic are studied in the next section.

Damage behavior of pure syntactic

Observation of damage localization in pure syntactic

To understand the failure mechanisms of pure syntactic, X ray microtomography was used to observe the microstructure of a pure syntactic E2_GB15 (crush pressure of 38 MPa, volume fraction of 0.59) in its initial state (Fig. 3.18.a) and after 50 MPa pressurization (Fig. 3.18.b). Let us recall here that we will focus mainly on amine-based epoxy (E2) because of the possibility to burn and use acid digestion on samples to characterize microspheres after testing (see 2.2.5.1. )

In Fig. 3.18.a, the undamaged microspheres inside the pure syntactic can be seen. In Fig. 3.18.b, after failure of the material, the microspheres are heavily damaged, but some remain intact. Concerning the resin, it seems undamaged, as no microcracks can be seen (Fig. 3.18.b). This confirms that the material collapse comes from the collapse of the microspheres. The damage quantification cannot be done through X ray tomographic measurements. The method presented in 2.2.5.1. , implemented to overcome those issues, is used in the next section to quantify the evolution of damage in epoxy pure syntactic with the increase of hydrostatic load.

Quantification of damage in pure syntactic

To better investigate the crush mechanisms of pure syntactic, a focus is made on its damage behavior.

As studies already highlighted the presence of early damage in pure syntactic when pressurized [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF], here a quantification of this damage is performed.

For that, two pure syntactics, E2_GB15 and E2_GB25, were pressurized at different pressure levels.

The pressure was applied on plate sample of 5 mm thickness through HCP tests with piston displacement, but instead of stopping the test after the collapse of the sample, the test was stopped at a set pressure. Then microspheres inside the pure syntactic sample were retrieved, and their density was measured. Fig. 3.19 presents the evolution of the average density of the microspheres with the applied pressure on the sample. Let us focus first on results with GB15 microspheres (red dots in Fig. 3.19), the initial density is equal to 0.155, this value is slightly higher than the microspheres density of 0.15 due to some collapse during the processing. Then for hydrostatic pressure applied between 0 and 30 MPa, no significant change in microspheres density is observed meaning that there is no mechanical damage at this pressure. For higher pressure (i.e., above the crush pressure), a large increase in microspheres density is observed. This increase is due to the collapse of microspheres that has been observed with X-Ray tomography. The same behavior is observed for GB25 (blue dots in Fig. 3.19): before HCP (of 50.5 MPa) no increase of microspheres density is observed and then when the applied pressure is equal to the crush pressure, a large increase occurs. It should be noted that the glass density in the microspheres being 2.54 g/cm 3 , even after pressurizing the material at higher pressure than the crush pressure, the density of the recovered microspheres does not exceed 0.5g/cm 3 , meaning not all the microspheres collapsed. Since the aim of this section is to quantify damage in the material, a damage parameter must be introduced. For that, the microspheres are considered to have the same size, with a mean external radios Ro and a mean thickness e. Their mean internal radius is Ri=Ro-e. The number of microspheres in a pure syntactic sample is noted N, and the number of broken microspheres is noted Nc. The microspheres density can be expressed using Eq. 3.4.

𝜌 𝑚𝑖𝑐𝑟𝑜 = 𝜌 𝑔 𝑉 𝑔 𝑉 𝑚𝑖𝑐𝑟𝑜 = 𝜌 𝑔 4𝜋 3 𝑁(𝑅 𝑜 3 -𝑅 𝑖 3 ) 4𝜋 3 (𝑁𝑅 𝑜 3 -𝑁 𝑐 𝑅 𝑖 3 ) = 𝜌 𝑔 (1 -( 𝑅 𝑖 𝑅 𝑜 ) 3 ) (1 - 𝑁 𝑐 𝑁 ( 𝑅 𝑖 𝑅 𝑜
) 3 )

Eq. 3.4
With ρ micro (g/cm 3 ) the microspheres density, ρ g (g/cm 3 ) the glass density, N (no unit) the number of microspheres, Nc (no unit) the number of broken microspheres, Ro (µm) and Ri (µm) the mean external and internal radius of the microspheres, Vg (µm 3 ) the glass volume and Vmicro (µm 3 ) the microspheres apparent volume.

The damage parameter used here is

D 𝑝 = N c
N the fraction, by number, of collapse microspheres. Then, by using the parameter Dp and the radius ratio η presented in Eq. 1.11 (see 1.3.1. ), Eq. 3.4 becomes Eq.

3.5:

𝜌 𝑚𝑖𝑐𝑟𝑜 = 𝜌 𝑔 (1 -𝜂 3 ) (1 -𝐷 𝑝 𝜂 3 )
Eq. 3.5

With ρ micro (g/cm 3 ) the microspheres density, ρ g (g/cm 3 ) the glass density, Dp (no unit) the number fraction of collapsed microspheres, and η (no unit) the radius ratio of the unbroken microspheres, estimated from Equation10.

Using density measurements (Fig. 3.19) and Eq. 3.5 to calculate the damage level D in our samples, the evolution of the damage level in pure syntactic with the load level P/Pc is plotted on Fig. 3.20. The same behavior is observed for the two types of GB considered here: damage level is almost null when a pressure less than 0.8 times the crush pressure is applied on the pure syntactic (Fig. 3.20). A pressure of 0.95 times the crush pressure needs to be applied, to both materials, to start measuring a damage level higher than 10%.

At crush pressure level, the damage becomes much more significant with more than 20% of the microspheres collapse. After the crush (after the densification plateau of the material), most of the microspheres collapsed (more than 50%). The crush of pure syntactic happens at the same pressure as the collapse of the microspheres in the material.

Those results show that the early damage observed in the literature is insignificant, and that the microspheres, in pure syntactic, collapse at pressure close to the collapse pressure. The microspheres that collapse at low pressure without a resin, are all reinforced, and they all collapse at the same time.

The resin seems to play a role in the decrease of the stress the microspheres are subjected to. This last hypothesis needs to be investigated through the study of stress levels in pure syntactic Since this cannot be done experimentally, it will be done through analytical and numerical modelling in Chapter 4.

Conclusions on the hydrostatic strength of pure syntactic

In this section the hydrostatic strength of pure syntactic was studied via experimental procedures considering more than 100 different materials. It has been shown that the strength improvement brought by the resin increases as the rigidity of the matrix increases. Furthermore, increasing the microspheres volume fraction in the material decreases its strength, and decreasing the microspheres density decreases its strength as well. This means that the hydrostatic strength of pure syntactic increases with their density, at the cost of losing buoyancy.

The crush pressure of pure syntactic can be described using the empirical model described in this section, the first of its kind. It works for GB15 and GB25 microspheres, but more data are needed to validate it for pure syntactic containing GB38 microspheres. The only bias of this model is that the constant needs to be measured each time a new type of microsphere is used. But this allows the pure syntactic collapse pressure to be predicted as a function of its constitution, and can be a useful tool to choose the composition of pure syntactic for buoyancy application, depending on the required buoyancy and service depth.

The damage behavior of epoxy pure syntactic was studied for the first time, as far as the author knows.

It appears that early damage in the material is insignificant, and that all microspheres collapse altogether, at the same time as the pure syntactic crushed. The pure syntactic failure is directly related to the failure of the microspheres in the material.

To understand where the reinforcing effect comes from, and to study the hypothesis that the resin decreases the load on the microspheres, an analytical and numerical modelling approach will be proposed in Chapter 4. But before that, the crush pressure of composite foam will be studied in the next section.

Hydrostatic strength of composite foam

This part focuses on the crush pressure of composite foam (i.e., foams containing macrospheres), to investigate the difference or similarity in behavior between composite foam and pure syntactic.

The strength of composite foam was measured via the HCP tests described in 2.2.3.4. . The load speed for those tests was 0.3 MPa, to be consistent with the manufacturer's testing and the IBL test loading speed. Fig. 3.21 presents a typical pressure/volume strain curve of composite foam during the HCP test. The composite pressure/strain curve is very similar to that of pure syntactic. First, a linear behavior is observed, before a densification plateau occurs. The densification plateau appeared at the same time as implosion sounds could be heard. Those sounds correspond to the collapse of macrospheres in the material. From the retrieved sample, collapsed macrosphere at the sample surface could be seen (Fig.

3.22.a)

. Contrary to pure syntactic, the densification plateau appears more abruptly, immediately when one macrosphere collapses. But the similarity between both materials is that the crush appears when the hollow sphere starts collapsing. The crush pressure of composite foam is defined as the pressure at which the first macrosphere collapses. The crush pressure of the composite foams studied in this work are reported Table 3.7.

Macrosphere

Table 3.7 Crush pressure values of composite foam depending on their composition

From those results, several observations can be made. The hydrostatic strength of composite foam material is in the range of the hydrostatic strength of macrospheres (Table 3.7). The reinforcing effect that could be seen in pure syntactic material does not appear in composite foam.

The composite foam strength increases with the macrospheres strength Table 3.7. But it was shown in 3.1.2. that the denser macrospheres are the carbon ones, which are stronger. Increasing the composite foam strength can be made by using stronger macrospheres, but at a cost of buoyancy.

Since the sample failure came from the collapse of macrospheres located at the sample surface, there is a possibility that the location of macrospheres at the sample surface led to stress concentration and early collapse of the macrospheres, therefore preventing the presence of the strengthening effect.

It was decided to test sample of composite foam made in such a way that a minimal thickness of pure syntactic surround all the macrospheres. Put another way, there is a minimal distance between the macrospheres and the sample surface. This led to an internal "skin" of pure syntactic at the sample From this, we can conclude that macrospheres located at the sample surface will initiate early damage and failure of the sample. But even when preventing this, the sample failure, of 40 MPa, is still close to the macrospheres collapse pressure of 34.6 ± 3.7. The reinforcing effect that was seen in pure syntactic is nonexistent, or insignificant, in the composite foam.

Conclusions on the hydrostatic strength of syntactic foams

Hydrostatic strength of syntactic foam, and their spherical inclusions, was studied experimentally. The collapse of syntactic foams is governed by the collapse of the spherical inclusions in the material, with the weakest sphere (i.e., the largest and thinnest) collapsing first.

For pure syntactic foams, the strength of the spherical inclusions is enhanced by the surrounding matrix, while the strength of the sphere in composite foam is not improved. This difference could not be explained through experiment alone, and a modelling approach will be used in the next section to study this reinforcing effect.

The crush pressure of the studied pure syntactic can be estimated using the empirical model that was established, that describe the influence of resin rigidity, microspheres volume fraction, and microspheres grade on the hydrostatic strength of pure syntactic. This model is unique and should further be improved to better consider the microspheres density in the description. It should be noted that this crush pressure does not depend on the sample geometry.

The crush pressure of composite foam can be estimated by knowing the collapse pressure of macrospheres, that are not reinforced in the material, unlike microspheres in pure syntactic. The macrospheres collapse due to their shell material failure, and their collapse pressure can be estimated knowing the spheres dimensions and the maximal compressive stress of the shell material.

CHAPTER 4 Modelling of stress fields in syntactic foams

Introduction

In the previous chapters, it was revealed that in pure syntactics, the microspheres are strengthened by the surrounding medium, while in composite foams the macrospheres are not. To understand where this reinforcing effect comes from, the approach used in this chapter is to study the stress fields in syntactic foams, especially the maximal stresses inside the hollow inclusions of the material, and compare them with the stresses in hollow spheres when they are not surrounded by a medium. Since no experimental method is available, here the stress fields in syntactic foam are investigated through analytical and numerical modelling.

Estimation of syntactic foam elastic properties

This section will focus on the estimations of elastic properties of syntactic foams. From our experimental studies, some data are available. Most of the experimental studies were made using the hydrostatic compression test with piston displacement, which gives no indication on volume variations of the material, and so on its bulk modulus. Hence, the elastic properties of syntactic foams, depending on their composition, must be evaluated differently. An analytical model available in the literature is used, and briefly presented here.

From the available models, it was decided to use the model proposed by Marur [START_REF] Marur | Effective elastic moduli of syntactic foams[END_REF], based on the results of Christensen and Lo [START_REF] Christensen | Solutions for effective shear properties in three phase sphere and cylinder models[END_REF] and allowing the results of Hervé and Pellegrini [START_REF] Hervé | The elastic constants of a material containing spherical coated holes[END_REF] and Bardella [START_REF] Bardella | On the elastic behavior of syntactic foams[END_REF] to be recovered. This model has been recently used by Brini [START_REF] Brini | Modélisation multi-échelles du comportement et du vieillissement des mousses syntactiques immergées[END_REF] and Choqueuse [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF], and is efficient to estimate elastic properties of epoxy syntactic foams under uniaxial loading [START_REF] Dana | Identification of moisture diffusion parameters in organic matrix composites[END_REF]. In this model, the syntactic foam is idealized as multi-phase concentric spheres as shown in Fig. 4.1 The model consists of placing a Representative Volume Element (RVE), here a composite sphere of radius c containing a hollow sphere with outer radius b and inner radius a, embedded in the Homogeneous Equivalent Medium (HEM) representing the syntactic foam with unknown properties that are to be determined (Fig. 4.1). This model aims to describe the elastic properties of syntactic foam, and therefore only considers the material to be in the elastic domain.

The bulk and shear modulus of the inclusion and matrix are assumed to be known. From a macroscopic point of view, the material is considered isotropic and homogeneous, and can therefore be characterized by two constants, here chosen to be the bulk modulus and Poisson's ratio.

The homogenized elastic properties of the equivalent medium must be chosen such that, under the same loading conditions, the equivalent medium possesses the same energy as the configuration presented in With these hypotheses, by considering that the material is subjected to hydrostatic loading, it is possible to evaluate the bulk modulus of the HEM. When applying hydrostatic loading of pressure P, and considering spherical symmetries, the displacement field can be expressed with the following equations [START_REF] Zouari-Trabelsi | Approche itérative d'homogénéisation pour le comportement des composites polydisperses: applications à l'endommagement des mousses syntactiques immergées[END_REF]:

𝑢 𝑟 (𝑖) = 𝐴 1 𝑟 + 𝐴 2 𝑟² Eq. 4.2 𝑢 𝑟 (𝑚) = 𝐵 1 𝑟 + 𝐵 2 𝑟² Eq. 4.3 𝑢 𝑟 (𝐻𝐸𝑀) = - 𝑃 3𝐾 𝐻𝐸𝑀 𝑟 + 𝐻 𝑟² Eq. 4.4
With A1, A2, B1, B2 and H model constants that need to be determined, P (MPa) the applied pressure on the HEM, K HEM (MPa) the bulk modulus of the equivalent medium, and ur (mm) the radial displacement, with the indices i, m and HEM respectively for the inclusion, matrix, or in the homogenized syntactic foam.

The energy equivalence for hydrostatic state of stress leads to H=0 [START_REF] Marur | Effective elastic moduli of syntactic foams[END_REF], Then, the model constants A1, A2, B1, B2 are determined considering continuity of displacement and normal stress at the interface, and a null internal stress (no pressure in the void phase), using the elastic properties of the inclusion and matrix.

Finally, solving the equation system, the bulk modulus can be expressed following Eq. 4.5 (here the notations differ from Marur for better uniformity in this document, but the analytical solution is the same):

𝐾 𝐻𝐸𝑀 = 𝐾 (𝑚) 𝛿(1 + 𝑓𝛽 (𝑚) ) + (𝛽 (𝑖) + 𝜂 0 3 )(1 -𝑓)𝛽 (𝑚) 𝛿(1 + 𝑓) + (𝛽 (𝑖) + 𝜂 0 3 )(𝛽 (𝑚) + 𝑓)

Eq. 4.5

Where:

𝛽 (𝑚) = 4𝐺 (𝑚) 3𝐾 (𝑚) 𝛽 (𝑖) = 4𝐺 (𝑖) 3𝐾 (𝑖) 𝛿 = 4𝐺 (𝑖) 3𝐾 (𝑚) [1 -(𝜂 0 ) 3 ]
With K (MPa) and G (MPa) the bulk modulus and shear modulus of the wall of inclusion (i) and matrix (m), and f = ( In the model, f is taken such that the volume fraction of the inclusions in the RVE corresponds to the sphere volume fraction in the syntactic foam, and η0 is the average radius ratio of the inclusions in the syntactic foam.

Bardella showed in his work [START_REF] Bardella | On the elastic behavior of syntactic foams[END_REF] that, if the geometry ratio between microspheres inside the foam does not vary excessively, then considering η0 constant in the material gives a good approximation of the effective bulk modulus of the syntactic foam. Hence, considering η0 constant, it can be estimated using Eq. 4.6:

𝜂 0 = (1 - 𝜌 𝑚𝑖𝑐𝑟𝑜 𝜌 𝑔 ) 1 3 ⁄

Eq. 4.6

With η0 (no unit) the average radius ratio of the microspheres, ρmicro (g/cm 3 ) the microspheres density, ρg (g/cm 3 ) the glass density Table 4.1 presents the evaluated bulk modulus of some of our materials alongside the experimental measurements made using the HCP test procedure with gauges measurements at 21°C, (see 2.2.3.3. ). The Young's modulus and Poisson's ratio of the resins were estimated using gauges during uniaxial compressive tests at 21°C (average on 3 samples). It should be remembered that the Young modulus of the resin is not the compressive uniaxial modulus that was measured in Chapter 3. The volume fraction is estimated from density measurements (see 2.2.2.1. ), and the geometric ratio η0 estimated using the average density of microspheres. All these parameters are given in Table 4.1.

The glass density of 2.54 g/cm 3 is given by the manufacturer. The exact glass elastic properties being unknown, its Young Modulus and Poisson's Ratio were arbitrarily chosen to be 74 GPa and 0.20 respectively, to be consistent with values found in literature [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF][START_REF] Paufler | Mechanical properties and structure of a nanoporous sodium borosilicate glass[END_REF][START_REF] Bardella | On the elastic behavior of syntactic foams[END_REF]. When comparing the experimental measurements of bulk modulus and the prediction (Table 4.1), we observe that the model can estimate the bulk modulus of pure syntactic materials with a relative error of less than 6%, which is sufficient for the purpose of this work. This error can be attributed to difficult experimental measurements of material properties, as well as the unknown glass properties that were chosen arbitrarily.

Then, the model is used to estimate bulk modulus of composite foam. For that, we consider macrospheres instead of microspheres, and pure syntactic instead of neat resin, and the composite foam as the HEM instead of the pure syntactic. The macrosphere's shell elastic properties, (i.e., the elastic properties of the glass fiber composite), were estimated experimentally in a study made prior to this work at IFREMER (results have not been published yet). The macrospheres radius ratio η0 was estimated using the mean radius and thickness of macrospheres, respectively of 21.5 mm and 1.75 mm (Table 2.3).

The macrosphere's volume fraction was estimated using density measurements. All those parameters are given in Table 4 In the same way, the Poisson's coefficient of syntactic foam can be calculated. This time, the system presented in Fig. 4.1 is subjected to uniaxial tension. In his work [START_REF] Marur | Effective elastic moduli of syntactic foams[END_REF], Marur gives the displacement fields of the wall of the inclusion (Eq. 4.7 and Eq. 4.8). With u r (i) (mm) the radial displacement, u θ (i) (mm) the circumferential displacement, F1-F5 the model arbitrary constant to determine, (r,θ) the coordinates, ν (i) the wall of inclusion Poisson's coefficient.

The displacement fields in the matrix are obtained by replacing the index of the inclusion (i) by the index of the matrix (m) and replacing the constants Fn by constants Gn in Eq. 4.7 and Eq. 4.8. For the homogenized medium (HEM), the displacement fields are described by Eq. 4.9 and Eq. 4.10:

𝑢 𝑟 (𝐻𝐸𝑀) = 𝑇𝐾 (𝐻𝐸𝑀) (1 -𝜈 (𝐻𝐸𝑀) )𝑟 6𝐺 (𝐻𝐸𝑀) (1 + 𝜈 (𝐻𝐸𝑀) ) - 𝐻 1 𝑟 2 - 3𝐻 2 𝑟 4 + (5 -4𝜈 (𝐻𝐸𝑀) )𝐻 3 3(1 -2𝜈 (𝐻𝐸𝑀) )𝑟 2 + ( 𝑇 4𝐺 (𝐻𝐸𝑀) - 9𝐻 2 𝑟 4 + (5 -4𝜈 (𝐻𝐸𝑀) )𝐻 3 (1 -2𝜈 (𝐻𝐸𝑀) )𝑟 2 ) cos (2𝜃)
Eq. 4.9

𝑢 𝜃 (𝐻𝐸𝑀) = -( 𝑇 4𝐺 (𝐻𝐸𝑀) + 6𝐻 2 𝑟 4 + 2𝐻 3 𝑟 2 ) sin (2𝜃)

Eq. 4.10

With u r (HEM) (mm) the radial displacement, u θ (HEM) (mm) the circumferential displacement, H1-H3 the model arbitrary constant to determine, (r,θ) the coordinates, ν (HEM) , K (HEM) , G (HEM) respectively the effective Poisson's coefficient, bulk modulus and shear modulus of the syntactic foam, and T the applied tensile load.

After solving the equation system, and using the boundary conditions and energy equivalence hypothesis, Marur obtained a last equation that describes the syntactic foam's Poisson's coefficient:

𝐻 1 = 8(1 + 𝜈 (𝐻𝐸𝑀) ) 3(1 -2𝜈 (𝐻𝐸𝑀) ) 𝐻 3

Eq. 4.11

With H1 and H3 the model arbitrary constant, 𝜈 (𝐻𝐸𝑀) the effective Poisson's coefficient of the syntactic foam.

The resolution of the system results in a large equation and is usually solved using numerical method.

In our case, the Poisson's coefficient of syntactic foam is estimated using this method, implemented on the Python software. From the Poisson's coefficient and bulk modulus estimated above, all our material elastic constants can be estimated. Now that the elastic properties of our material have been estimated, this chapter will focus on the estimation of stress inside glass microspheres and macrospheres in composite foam.

Analytical determination of stress inside hollow

sphere in syntactic foams.

To estimate the maximal stress inside the microspheres in a pure syntactic, being the radial stress at its inner surface, the same configuration as presented Fig. 4.1 is considered, with a hollow sphere of internal radius a, outer radius b, surrounded by a sphere of resin of outer radius c. But this time, the problem is reversed: the aim is to evaluate the stress inside a hollow of radius ratio η=a/b placed inside an equivalent medium HEM of known elastic properties. It should be mentioned that here, the elastic properties of the HEM are calculated using the average radius ratio η0 of the hollow inclusions, since it provides a good estimation of the HEM elastic properties. After that, since we want to study the stress inside the shell of all the hollow spheres in the syntactic foam, the hollow inclusion placed in the equivalent medium has a radius ratio η that can vary depending on the inclusion's diameter and thickness measured experimentally. But this radius ratio η has no impact on the HEM elastic properties, that is determined using η0.

All the materials are considered purely elastic. This hypothesis is supported by the observations made by Choqueuse that showed, from his in-situ observation using X-ray microtomography on epoxy pure syntactic, that when a microsphere collapses in the material, there is no volume change of the material, and the matrix skeleton remains intact [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]. This was further confirmed by our investigation of damage in 3.2.6.2. , which showed that no damage appears before a pressure of 0.95 times the crush pressure is applied on the material.

The elastic properties of the HEM are known and determined using the previous analytical model proposed by Marur (Eq. 4.5). The volume fraction of the inclusion in the RVE is equal to the volume fraction of inclusions inside the HEM.

The loading state being hydrostatic, and considering spherical coordinates, the displacement fields can be expressed using Eq. 4.2, Eq. 4.3 and Eq. 4.4. The arbitrary constants to determine in this case are still the constants A1, A2, B1, B2 and H from those equations. This time, the homogenized elastic properties of our syntactic are known, and the aim is to estimate stress in hollow spheres in syntactic foams depending on their radius ratio. Thus, the hypothesis of energy equivalence is not relevant here, and the constant H in Eq. 4.4 is not considered null. Since the material behavior is considered purely elastic, the stress fields can be expressed using Hook's Law.

𝐸 (𝑘) * 𝜀 𝑟𝑟 (𝑘) = (𝜎 𝑟𝑟 (𝑘) -2𝜈 (𝑘) 𝜎 𝜃𝜃 (𝑘) )

Eq. 4.12 𝐸 (𝑘) * 𝜀 𝜃𝜃 (𝑘) = (𝜎 𝜃𝜃 (𝑘) * (1 -𝜈 (𝑘) ) -𝜈 (𝑘) 𝜎 𝑟𝑟 (𝑘) )

Eq. 4.13

With:

ε rr (k) = du r (k) dr , ε θθ = u r (k) r
from the kinematics equations, E (k) (MPa) the Young's modulus, and ν (k) (no unit) the Poisson's ratio, and k = i, m or HEM when indexing the inclusion, matrix or homogenized medium respectively.

Knowing the analytical form of the displacement fields in the matrix and glass (Eq. 4.2 and Eq. 4.3), and with the expression of the bulk modulus 𝐾 (𝑘) = 𝐸 (𝑘) /(3(1 -2𝜈 (𝑘) )) and shear modulus 𝐺 (𝑘) = 𝐸 (𝑘) /(2(1 + 𝜈 (𝑘) )), using these expressions in Eq. 4.12 and Eq. 4.13, the stress fields become:

𝜎 𝜃𝜃 (i) = 3𝐴 1 𝐾 (i) + 2 𝐴 2 𝐺 (i) 𝑟 3
Eq. 4.14

𝜎 𝑟𝑟 (i) = 3𝐴 1 𝐾 (i) -4 𝐴 2 𝐺 (i) 𝑟 3
Eq. 4. [START_REF] Shutov | Syntactic polymer foams[END_REF] 𝜎 𝜃𝜃 (m) = 3𝐵 1 𝐾 (m) + 2 𝐵 2 𝐺 (m) 𝑟 3

Eq. 4. [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF] 𝜎 𝑟𝑟 (m) = 3𝐵 1 𝐾 (m) -4 𝐵 2 𝐺 (m) 𝑟 3

Eq. 4.17

𝜎 𝑟𝑟 (HEM) = -𝑃 -4𝐺 (𝐻𝐸𝑀) 𝐻 𝑟 3

Eq. 4.18

With A1, A2, B1, B2 and H constants depending on the boundary conditions, σrr and σθθ the stress fields (in MPa), K (MPa) the bulk modulus and G (MPa) the shear modulus, and the indices (i), (m) and (HEM) respectively indicating the glass, matrix and homogenized equivalent medium.

The constants A1, A2, B1, B2 and H are determined using the null internal stress condition (Eq. 4.19) and the continuity of normal stress (Eq. 4.20, Eq. 4. The resolution of the system will not be detailed here, but in Appendix A. The expression of the maximal stress in the hollow inclusion (i.e., circumferential stress in r=a) is given in Eq. 4.24. This estimation of maximal stress inside the hollow inclusion, at its inner surface, defines our analytical model of stress inside hollow spherical elements in syntactic foams:

𝜎 𝜃𝜃 (i) (r = a) = -3𝛷(1+𝛽 (𝑚) )(1+𝛽 (𝐻𝐸𝑀) )𝑃 2(𝛷(1-𝜂 3 )(1+𝛽 (𝑚) 𝑓+𝜃𝛽 (𝐻𝐸𝑀) (1-𝑓))+(1+ 𝜂 3
𝛽 (𝑖) )(𝛽 (𝑚) (1-𝑓)+𝜃𝛽 (𝐻𝐸𝑀) (𝛽 (𝑚) +𝑓)))

Eq. 4.24

Where: It should be recalled that here, η depends on the inclusion geometry, and can be different from η0, which is the mean value of the geometry ratio of the hollow spheres used to estimate the elastic properties of the HEM with Eq. 4.5.

𝛽 (𝑖) = 4𝐺 (𝑖) 3𝐾 (𝑖) 𝛽 (𝑚) = 4𝐺 (𝑚) 3𝐾 (𝑚) 𝜃 = 4𝐺 (𝐻𝐸𝑀)
Using Eq. 4.24, the maximal stress inside a hollow sphere inside a syntactic foam can be estimated knowing the following parameters:

• E (i) and ν (i) the Young's modulus and Poisson's coefficient of the wall of the inclusion, that are constants. In the case of microspheres, being made of glass, they are chosen as 74 GPa and 0.20 respectively, according to literature [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF][START_REF] Deriano | Physical and mechanical properties of a new borosilicate glass[END_REF][START_REF] Paufler | Mechanical properties and structure of a nanoporous sodium borosilicate glass[END_REF][START_REF] Chen | Radiation effects on structure and mechanical properties of borosilicate glasses[END_REF][START_REF] Bardella | On the elastic behavior of syntactic foams[END_REF].

• E (m) and ν (m) the Young's modulus and Poisson's coefficient of the matrix, that can vary depending on the studied case.

• f the volume fraction of hollow inclusions in the pure syntactic.

• η the radius that describe the specific hollow inclusion considered.

• K (HEM) and G (HEM) the bulk and shear modulus of the syntactic foam, estimated using Marur's model with the parameters: η0 the average radius ratio of the inclusions, E (i) and v (i) and f.

• The pressure P applied to the pure syntactic.

In the next section the model will first be used to predict stress in pure syntactic and then, the obtained results will be confronted with experimental results.

Evaluation of stress inside hollow spheres inside syntactic foam: origin of the reinforcing effect of the resin

Now that an estimation of stress inside microspheres in pure syntactic can be made, the model Eq. 4.24

will be used to interpret the reinforcing effect that appears in pure syntactic (i.e., a hydrostatic strength several times higher than the microspheres strength) in comparison with the non-reinforcing of macrospheres in composite foams. But for that, the radius ratio of the microspheres must be known. It is evaluated from the microspheres size measurements presented in 3.1.1. .

Microspheres geometry description

The radius ratio depends on the external radius and wall thickness of the microspheres (Eq. 4.25).

𝜂 = 1 -𝑒 𝑟

Eq. 4.25

With e (mm) and r (mm) the microsphere thickness and radius, and η (no unit) its geometry ratio.

The diameter of our microspheres is known from laser granulometry measurements and presented in Fig. 3.2 in 3.1.1. . The microspheres thickness is also known, and measurements showed that the microspheres thickness can also vary (see Fig. 3.4 in 3.1.1. ). But the measurements of the thickness are less precise, and the thickness values are less dispersed than the diameter values. In this work, for a first approximation, the thickness of the microspheres of a given type is considered constant, whatever is the diameter. Since it is considered constant, there is a need to choose its value properly. Here it is estimated according to the microspheres' average density.

The microspheres thickness is chosen considering the microspheres average density, but also considering their radius distribution. The microsphere average density, considering that the thickness of microspheres is constant, is expressed using the microspheres size distribution and constant thickness with Eq. 4.26:

𝜌 𝑚𝑖𝑐𝑟𝑜 = 𝜌 𝑔𝑙𝑎𝑠𝑠 ∑ 𝑎 𝑖 [𝑟 𝑖 3 -(𝑟 𝑖 -𝑒) 3 ] 𝑖 ∑ 𝑎 𝑖 𝑟 𝑖 3 𝑖

Eq. 4.26

With e (mm) the thickness of the microspheres, ai (no unit) the population by number of microspheres of radius ri (mm), ρ micro (g/cm 3 ) the microspheres average density, and ρ glass (g/cm3) the glass density Since e<<ri, Eq. 4.26 can be rewritten using Taylor's polynomial of first order, and after reduction the microspheres average density becomes:

𝜌 𝑚𝑖𝑐𝑟𝑜 = 𝜌 𝑔𝑙𝑎𝑠𝑠 ∑ 𝑎 𝑖 3(𝑟 𝑖 2 𝑒) 𝑖 ∑ 𝑎 𝑖 𝑟 𝑖 3 𝑖

Eq. 4.27

With e (mm) the thickness of the microspheres, ai (no unit) the population by number of microspheres of radius ri (mm), ρ micro (g/cm 3 ) the microspheres density, and ρ glass (g/cm3) the glass density By reversing Eq. 4.27, the microspheres thickness is estimated using Eq. 4.28:

𝑒 = 𝜌 𝑚𝑖𝑐𝑟𝑜 ∑ 𝑎 𝑖 𝑟 𝑖 3 𝑖 3𝜌 𝑔𝑙𝑎𝑠𝑠 ∑ 𝑎 𝑖 𝑟 𝑖 2 𝑖

Eq. 4.28

With e (mm) the thickness of the microspheres, ai (no unit) the population by number of microspheres of radius ri (mm), ρ micro (g/cm 3 ) the microspheres density, and ρ glass (g/cm3) the glass density

The representative thicknesses of our different types of microspheres are given in Table 4.3. They are in the range of the measurements made from X ray microtomography.

Microspheres η0

Estimated Thickness (µm) Range of thickness from X ray microtomography (µm) GB15 0.980 0.90 0.9 -1.5 GB25 0.966 1.30 -GB38 0.947 1.34 0.9 -2.7 Table 4

.3 Estimated mean radius ratio and thickness for the considered microspherescomparison with thickness measurements

Then, using this thickness and the known diameter distribution of microspheres, the radius ratio distribution by volume, for our different microspheres, can be estimated, and is plotted in Fig. 4.2. For each microspheres type, more than 95% of their volume is constituted of microspheres with a radius ratio greater than 0.9. Furthermore, 60% of their volume is constituted of microspheres of radius ratio greater than the average radius ratio η0 (dotted line on Fig. 4.2, values given in Table 4.3). It was shown in 3.1.1. that when subjected to hydrostatic loading, the largest microspheres collapse first. Since their thickness is constant, it means that the microspheres with the highest radius ratio collapse first. Here, to be more representative of the behavior of the material and to focus on the microspheres that will collapse first and lead to the crush of the material, the microspheres considered are the ones representing the 90% of the largest microspheres (i.e., with the greatest radius ratio, considering the thickness constant). This leads to a radius ratio range of 0.96 -0.99 for GB15 microspheres, 0.94 -0.98 for GB25 microspheres, and 0.9-0.97 for GB38 microspheres. Now that the values of the radius ratio of our microspheres are known, it is possible to use our model to estimate the stress in a microsphere, depending on its geometry (i.e., depending on its radius ratio).

Those stress estimations are made in the following section.

Estimation of stress inside pure syntactic

In a first approach, only GB15 microspheres will be considered. The HEM elastic properties and the microsphere stress determination are considered using the parameters presented in Table 4.4, that represent the material E1_GB15 with a volume fraction of microspheres of 0.55. The applied pressure of 40 MPa was chosen to be slightly higher than E1_GB15 crush pressure of 38 MPa. This is to estimate the stress in microspheres when the pure syntactic is in the range of crushing, and compare it with the compressive strength of the glass. The maximal stress of a stand-alone microsphere (without resin) is calculated using Eq. 4.29 [START_REF] Young | Roark's formulas for stress and strain[END_REF]:

𝜎 𝜃𝜃 𝑚𝑎𝑥 = - 3𝑃𝑟 3 2(𝑟 3 -(𝑟 -𝑒) 3 ) = 3𝑃 2(1 -𝜂 3 )

Eq. 4.29

With σ θθ max (MPa) the maximal circumferential stress in a hollow sphere (without surrounding resin) subjected to hydrostatic loading of pressure P (MPa), and e (mm) and d (mm) the thickness and diameter of the sphere, and η (no unit) its geometry ratio.

The stress inside a microsphere in pure syntactic is calculated using Eq. 4.24 and parameters from Table 4.4. It is plotted on Fig. 4.3 and compared with the stress of a microsphere that is not surrounded by resin (Eq. 4.29). Being loaded in compression, the stress is negative, but on the presented graphs the absolute value is represented. The stress model from Eq. 4.24 predicts that the resin will tend to decrease the stress applied to the microsphere. This reinforcing effect increases as the radius ratio of the microsphere increases (i.e., as their radius increases). For microspheres of radius ratio of 0.99 (maximal radius ratio of GB15), the stress is 1900 MPa when not surrounded by a resin, and 1050 MPa when surrounded by a resin, meaning a decrease of stress of 45%. The glass compressive strength is around 1000 MPa, and largely vary depending on the glass (values found in literature between 700 MPa [START_REF] Dan | Etudes micromécaniques de l'endommagement des mousses syntactiques[END_REF] and 2500 MPa [START_REF] Bompard | Mechanical and fracture behaviour of porous materials[END_REF]). Here, the model shows that when the pressure applied to the material is in the range of its crush pressure, it induces a stress in the microspheres in the range of the glass strength, making it likely to collapse. It is in agreement with experimental observations in Fig. 3.20 which reveals that the material crushes at the same time as the microspheres collapse.

As seen experimentally, crushing of pure syntactic is due to the collapse of microspheres inside the foam. The stress on weaker microspheres being greatly reduced, this will increase the overall strength of the material. This model clearly illustrates that the reinforcing effect that was seen experimentally, is due to the reduction of the stress in the material by the resin. To the best of our knowledge, this is the first time, as far as the author knows, that one of the phenomena at the origin of the amplification factor has been explained.

Evolution of the stress depending on the pure syntactic microstructure parameters.

To further investigate the relevance of our model, the evolution of stress inside a microsphere depending on the volume fraction, matrix rigidity, and microspheres density used in the pure syntactic was estimated. ). The stress inside the microsphere being proportional to the applied pressure (Eq. 4.24), changing the pressure does not change the observed behavior. So, the applied pressure is kept at 40 MPa, to be the same for all the stress estimations made with Eq. 4.24. We know that pure syntactics made with epoxy resin and GB15 microspheres show a collapse pressure lower than 40 MPa when the microspheres volume fraction is higher than 0.5, but here the purpose is to estimate stress fields in the material, and not to predict crush pressure of pure syntactics. The model predicts that increasing the microsphere volume fraction tends to increase the stress inside the material. This increase leads to early failure. This is in accordance with what was observed experimentally in 3.2.3. in Fig. 3.14. The weakest microspheres (i.e., radius ratio of 0.99) show an increase in strength from 920 MPa to 1050 MPa when the microspheres volume fraction is increased from 0.30 to 0.55. On the contrary, increasing the matrix rigidity will decrease the stress applied on microspheres (Fig. 4.4.b), increasing the reinforcing effect. When increasing the Young's modulus from 1.3 to 3.3 GPa, the stress in the weakest microspheres (i.e., radius ratio of 0.99) decreases from 1370 to 960 MPa. Hence, the microspheres will collapse at higher pressure. Because the pure syntactic crushes when the microspheres collapse, it will also crush at higher pressure when the resin rigidity is increased. This is also in agreement with the experimental results presented in 3.2.2.2. in Fig. 3.13.

The last parameter to be studied here is the type of microspheres used in the pure syntactic. For that, stress was estimated using the parameters presented in Table 4.6. The evaluated stress is presented in Fig. 4.5 and shows that increasing the microspheres density (i.e., decreasing its average radius ratio) will decrease the stress they are subjected to. This is due to the increase in bulk modulus of the pure syntactic by using more rigid spherical inclusions. The model predicts that making a pure syntactic with denser microspheres should make it stronger, as was observed experimentally in 3.2.5. in Table 3.5. As was seen in 3.1.1. , the largest microspheres (i.e., the microspheres with the highest radius ratio) are the weakest and collapse first. It was supposed in Chapter 3 that pure syntactic materials collapse due to the failure of those largest microspheres, because they collapse first, and induce a larger damaged volume because of their higher diameter. In this section, the model presented in 4.2. predicts that the resin will reduce the stress applied on the weakest microspheres. This decrease in stress leads to the strengthening of those microspheres by the resin, delaying their collapse, and reinforcing the pure syntactic material.

The model predicts that increasing the microspheres volume fraction would lead to increasing the stress on the microspheres, making them collapse earlier, and the pure syntactic crush at lower pressure. On the contrary, increasing the resin rigidity and microspheres density decreases the stress on the microspheres, making them and the pure syntactic crush at higher load. Those observations agree with the experimental results presented Chapter 3.

Estimation of stress inside macrospheres in composite foam: origin of the lack of reinforcing effect in composite foam.

To understand why no reinforcing effect was seen in composite foam, the same model as presented before will be used, but using macrospheres instead of microspheres, pure syntactic instead of resin, and composite foam as the HEM.

The stress inside a composite foam made of E2_GB25 pure syntactic and glass fiber macrospheres, is calculated using the parameters presented Table 4.7. In the real material, all the macrospheres have a radius ratio close to η0 = 0.919. However, to compare with the result shown in 4.3. the stress prediction are made on macrospheres with radius ratio between 0.9 and 0.99. For consistency, the applied pressure on the material is 40 MPa. 4.7 Parameters used for the estimations of stress in macrospheres in composite foam CF_E2_GB25 using the analytical model resented in this work Fig. 4.6 presents the stress inside the macrospheres depending on their radius ratio, compared with the stress if they are not surrounded by pure syntactic. The model predicts that macrospheres with η close to 0.9 will be subjected to stress slightly higher, as if they were not placed in composite foam.

For our macrospheres that have a mean geometry ratio of η0 = 0.919, the stress they are subjected to when a pressure of 40 MPa is applied, is 270 MPa when not surrounded by pure syntactic, and 230 MPa when surrounded by pure syntactic (both values estimated analytically). The decrease of stress is around 15%, much lower than the decrease of stress observed on GB15 microspheres that can reach 45%. It explains why the macrospheres in the material will not be reinforced by the pure syntactic. It should be noted that 20% is the difference of strength between the macrospheres (34.7 MPa) and the composite foam with additional skin of pure syntactic (40.0 MPa) measured experimentally in Chapter 3.3.3. The absence of a clear amplification factor in composite foam is due to the radius ratio of macrospheres being too low. If macrospheres with higher radius ratio were used, they would be reinforced by the surrounded medium (Fig. 4.6). The reinforcing effect in syntactic foam appears when the spherical inclusions have very small thickness compared to their diameter. For the application, reducing the sphere thickness would reduce its density. Therefore, it would be possible to reduce the density of the material, therefore gaining in buoyancy, without losing strength.

Conclusion on the analytical model of stress fields in syntactic foams

An analytical model estimating the stress inside hollow spherical inclusions in syntactic foam, induced by hydrostatic loading, was implemented. This model was used to show that, in syntactic foams, the resin will homogenize the stress on its spherical inclusions. Furthermore, if those inclusions have a high radius ratio (close to 1), the stress they are subjected to will be reduced when compared to those not surrounded by a matrix. This explains why epoxy resins reinforce microspheres that have high radius ratio, and also, why the macrospheres used in this study, having a lower radius ratio, are not reinforced when surrounded by pure syntactic.

One hypothesis being made, with this analytical approach, is that when a syntactic foam is loaded hydrostatically, the hollow spheres are hydrostatically loaded as well. To confirm this, a numerical approach is used in the next section to investigate the stress fields in composite foams.

Numerical estimation of stress inside syntactic foams

In this section, the stress fields in syntactic foams are investigated numerically, to ensure that when subjected hydrostatically, the hollow spheres are hydrostatically loaded as well, and that the position of the spheres in the foam does not impact the loading on it. A focus is made on composite foam, being the final material for the application.

Numerical modelling of a composite foam

To study the stress fields inside composite foam, it was decided to realize a numerical modelling of the structure of the composite foam CF_E2_GB25 using finite elements. A cylindrical sample of composite foam, made with glass macrospheres and E2_GB25 pure syntactic, of 300mm diameter and 1m height, was scanned using X-ray tomography, to recover the position of the macrospheres inside the sample.

The sample, too long to be scanned, was cut perpendicularly to its height, to obtain a cylindrical sample of height around 280 mm. The X-ray tomography scan of the sample is presented Fig. 4.7.

Fig. 4.7 X-ray tomography scan of a composite foam sample

Then, after retrieving the position of the spheres, a simple modelling of the sample structure was made using ABAQUS using the following steps:

1. Create a solid cylinder. 4.8. Two other hypotheses are made:

• All macrospheres are perfectly surrounded by the medium (there is no contact between the macrospheres).

• The macrospheres are perfectly bonded to the medium (tie condition between macrospheres and medium surface).

The radius of macrospheres is chosen to be 20 mm, a little bit lower than what measured experimentally (2.1.3. ). This was done to prevent contact between macrospheres and prevent the appearance of distorted elements between the spheres.

Results of the numerical model

The bulk modulus of the modelled sample is calculated to be 2.0 GPa. This is slightly higher than what was measured experimentally (1.75 GPa) and predicted using the homogenization method (1.94 GPa) (Table 4.2). This difference may come from the macrospheres that have lower radius in our numerical model, making the material stiffer. However, the modulus calculated here is of the same order of magnitude as the measured value with an error of 12.5%.

To appreciate the heterogeneity of stress in composite foams, the Von Mises stress field of the material, when subjected to 40 MPa hydrostatic pressure, is represented Fig. 4.9. The macrospheres are the load bearer in the material, while the stress in the pure syntactic, being less than 1 MPa, is very low. This confirms our hypothesis that, during HCP tests, up until failure of the material (at 30 MPa for composite foam CF_E2_GB25), the medium behavior is purely elastic. Furthermore, the stress in all macrospheres is of the same order of magnitude, meaning the position of the spheres in the material does not impact its loading. The numerical modelling of composite foam suggests that the load on the macrospheres in composite foam is almost hydrostatic and is equivalent between each macrospheres. This numerical approach also reveals that some stress concentrations exist in the material, as the load on the macrospheres is not perfectly hydrostatic. Those stress concentrations may come from the interaction between macrospheres and can lead to stress concentration in the sphere shell, that could then lead to early failure of the macrospheres.

But those stress concentrations are extremely localized, and their dispersion is of 20% or less than the mean value in the hollow inclusions shell. Therefore, the main loading on the hollow spheres is the hydrostatic loading coming from the pressure applied at the sample surface. Furthermore, the stress value estimated using both analytical (215 MPa) and numerical models (mean value of 225 MPa) agree with each other. So, the numerical model confirms that, the analytical model presented in 4.2. , can estimate the main load on the hollow spheres in syntactic foam. Therefore, it confirms the conclusion made previously, which is that one possible explanation of the origin of the amplification factor is that the matrix reduces the stress on the weakest hollow inclusions.

Conclusions on the modelling of stress inside syntactic foams

For the first time, to the best of the author's knowledge, the stresses inside the hollow sphere wall thickness in syntactic foams, when subjected to hydrostatic loading, were evaluated using an analytical model. This model explains that while microspheres have all different sizes in the material, the resin homogenizes and reduces the stress in them, explaining the origin of the strengthening effect observed experimentally in Chapter 3. This model predicts that, increasing the microspheres volume fraction in the material reduces the strengthening effect, and that increasing the microspheres density and resin rigidity increases it. This agrees with the experimental observations of the impact of those parameters on the crush pressure (and therefore the amplification factor) on pure syntactic materials.

It also explains that macrospheres are not strengthened in composite foam because they are too thick (i.e., have a lower radius ratio than the microspheres). This means that making the macrospheres thinner would not reduce the composite foam strength as the macrosphere would be reinforced. Making macrosphere thinner would make them less dense and would provide better buoyancy. There is here a possible optimization of the composite foam to gain buoyancy without losing strength.

A numerical model of composite foam, based on the real structure of the material obtained from X ray tomography measurements, is proposed. It confirms that the main loading on the hollow inclusions in syntactic foam, during pressurization, is hydrostatic, and that the hollow spheres are the load bearer in the material.

Introduction

When used in buoyancy applications, syntactic foams are immersed in water under hydrostatic pressure for durations up to more than 20 years. Composite materials are made with a polymeric resin and those materials absorb water [START_REF] Fick | On liquid diffusion[END_REF]. This water absorption results in a weight gain, and it is admitted that this weight gain is the main cause of buoyancy loss in syntactic foams [START_REF] Poveda | Carbon nanofiber reinforced syntactic foams: Degradation mechanism for long term moisture exposure and residual compressive properties[END_REF] (this point will be experimentally checked here). But the water absorption mechanisms of syntactic foams at low temperatures (less than 25°C) and under hydrostatic pressure is not fully understood yet; this is the aim of this chapter.

In this section, an experimental study of the water absorption in syntactic foams will be made to characterize the influencing factors, try to understand the water absorption mechanisms and finally establish a prediction of the buoyancy loss due to water absorption. To do so, first, the water absorption in pure polymer resin is considered. Then, the water absorption of pure syntactics under hydrostatic pressure is characterized, revealing high water absorption in the material because of the filling of microspheres with water. To better understand the water absorption mechanisms, different parameters are studied. Here, a focus is made on the sample size, the hydrostatic pressure, and the type of microsphere and resin used. Two different mechanisms are identified and discussed, and a mathematical description is proposed. Finally, using the mathematical description of water absorption, a prediction of buoyancy loss is made and compared with experimental data to confirm the hypothesis that water absorption is indeed responsible for buoyancy loss in pure syntactic.

Water absorption in pure resin

The water absorption of the pure epoxy resins at 15°C is characterized on plate samples of 1 mm and 4 mm thickness. The experimental results, plotted versus the square root of time normalized by the sample thickness, are presented in Fig. 5.1. The 3 resins first show a linear increase of the water content, before reaching a saturation plateau. This is a typical Fickian behavior. The diffusion coefficient and saturation weight of the 3 resins is then calculated (Table 5.1), and the corresponding Fickian law describing their behavior is represented by the line in Fig. 5.1. From the values of the parameters presented in Table 5.1, the amine-based epoxy resin (E2) presents a slow water absorption rate with a high water content at saturation compared to the anhydride-based epoxy resins (E1 an E3) that present a faster water absorption rate but with a lower saturation weight.

To conclude, the experimental study of water absorption on the pure resin reveals that the amine-based epoxy resin absorbs more water than the anhydride-based one, but at a slower rate.

.1 Identified diffusion coefficient and maximal water uptake in neat resin at 15°C

The impact of water content on macromolecular mobility as well as mechanical properties has been considered: an increase in water content leads to a decrease in Tg as well as a decrease in the yield stress due to the well know plasticization process [START_REF] De'nève | Water absorption by an epoxy resin and its effect on the mechanical properties and infra-red spectra[END_REF][START_REF] Obeid | On the identification of the coefficient of moisture expansion of polyamide-6: Accounting differential swelling strains and plasticization[END_REF]. Due to the low amount of absorbed water, anhydride-based epoxy properties do not largely change in presence of water. However, the yield stress in tension of the amine-based epoxy decreases from 60 MPa to 30 MPa.

Since the material will be immersed in depth of several thousand meters, there is a possibility that their behavior changes because of the applied pressure. Characterization of the water absorption of the pure resin E3 at 15°C was made in deionized water under 45 MPa pressure, and compared with the results at atmospheric pressure (Fig. 5.2). Increasing the pressure leads to an increase of the maximal water absorption but did not change the behavior of the resin that stayed Fickian (Fig. 5.2). The water content at saturation increases from 0.8 to 1% when pressure is applied. Now that the pure resin water uptake is characterized, the pure syntactic water absorption will be studied.

Water absorption in pure syntactic

Comparison between pure syntactic and pure resin

The pure syntactics E2_GB38 and E3_GB38 were immersed in deionized water at 15°C under 45 MPa hydrostatic pressure for almost 3 years. Plate samples of 5 and 6 mm thickness (3 samples per thickness) and cubic sample of 25 and 50 mm size (1 sample per size) are used here. The water absorption was recorded almost every week and is represented in Fig. 5.3. The results show that the water absorption depends on the sample size. Considering similar immersion durations, the smaller the sample, the higher the water absorption. The water absorption of both materials does not stabilize after 900 days of immersion. Furthermore, the water absorption in E2_GB38 seems to increase linearly with the immersion time, while the water absorption in E3_GB38 is faster at the beginning, then seems to slow down. The water absorption of the plate sample is higher than 5%. It is higher than the saturation weight of the pure resin. It means that the water diffusion within the polymer matrix itself is not the only phenomenon responsible for water absorption in pure syntactics. To emphasize this conclusion a comparison of theoretical water absorption with experimental data is proposed below. If water absorption in a composite material is only due to water diffusion within the polymer matrix then water absorption of a composite is then evaluated theoretically by Eq. 5.1:

𝑊 𝑝𝑢𝑟𝑒-𝑠𝑦𝑛𝑡 = 𝑊 𝑟𝑒𝑠𝑖𝑛 . 𝑀 𝑟𝑒𝑠𝑖𝑛

Eq. 5.1

With Wpure-synt the water uptake in the syntactic foam, Wresin the water uptake in the resin, and Mresin the mass fraction of the resin in the composite.

The theoretical value of the water absorption at saturation in the pure syntactic from Eq. 5.1 is recorded in Table 5.2. It is compared with the water absorption of the pure syntactic measured after 900 days of immersion under 40 MPa (on plate samples of 5 mm thickness). The measured values are higher than the theoretical value, showing that the water absorption in the resin is not the only phenomenon occurring here.

Materials

Theoretical water uptake at saturation from Eq. 5.1 (%)

Water uptake of plate sample of 5 mm thickness after 900 days of immersion under 45 MPa (%) E2_GB38 2.2 5.9 ± 0.5 E3_GB38 0.6 6.5 ± 0.4 Table 5

.2 Comparison between experimental and theoretical value of water absorption in pure syntactic, considering that only the resin absorb water

It is now clear that water absorption in pure syntactic is not only due to water diffusion within the polymer resin. In order to understand the origin of this mass increase, X-ray microtomography is used in the next part to study the water absorption process in pure syntactic.

Highlighting the water filling the microsphere

To investigate where the water goes inside the pure syntactic samples, X-ray microtomography is used.

For that, the plate samples of 5 mm thickness are considered here. Samples of E2_GB15 pure syntactic are immersed in deionized water under 10, 19 and 25 MPa pressure at 15°C for durations up to 314 days.

One sample per condition is retrieved after 7, 21, 63 and 314 days of immersion. The weight gain of those samples is measured (Table 5.3), and then those samples were scanned using X ray microtomography. This highlights that during aging, water fills the microspheres. To the best of our knowledge, it is the first time that we can visualize the microspheres filled by water in pure syntactic material during immersion at low temperatures under hydrostatic pressure. This technique also allows the observation of the preferential pathways for water filling the material. It indeed appears that while water progresses in the material, it does not fill all the microspheres on its path and seems to follow the larger ones. Based on the X-ray tomography results it clearly appears that glass microspheres are filled with water when pure syntactic materials are placed in water under hydrostatic pressure. Let us now focus on the origin of this behavior.

Weight gain [%] Immersion time [days

Highlighting of the coupling effect of pressure and water

The first question here is, why do glass bubbles collapse during ageing? More especially, what is the role of water in this collapse? To answer this question a quantitative determination of the amount of collapsed microsphere during ageing of E2_GB15 at two pressures (19 and 25 MPa) is performed. Let us recall that the crush pressure of this pure syntactic is equal to 38 MPa so ageing is performed below the crush pressure of the material.

Two different kinds of samples are used here: those with no protection meaning that the material is in contact with water and those sealed in plastic bags under vacuum in which water cannot diffuse (Fig. 5.5) meaning that these samples are under pressure but not in contact with water. Damage measurements of the samples were made in the same way as those presented in 3.2.6.2. . The study is still ongoing, particularly to measure the repeatability of the measurements. However, early results are recorded in

Fig. 5.6 Evolution of number of broken microspheres in E2_GB15 with aging time in presence of water or not

As expected, whatever the pressure (19 or 25 MPa), when samples are under pressure and in contact with water, an increase in the number of broken microspheres occurs during the aging. However, when samples are under pressure but not in contact with water, the number of broken microspheres does not evolve much overtime. This means that the collapse of microspheres within pure syntactic materials is due to a coupling effect between water and pressure. Let us now focus on the potential origin of this collapse process considering three hypotheses.

Microsphere hydrolysis

It has been shown in literature that microspheres can undergo hydrolysis. This process leads to a decrease in wall thickness and so in hydrostatic strength [START_REF] Gimenez-Fouque | Vieillissement hydrolytique de mousses syntactiques époxydeamine/verre pour l'isolation thermique sous hautes pression: mécanismes de dégradation et simulation de la prise en eau[END_REF]. At the author's knowledge, the exact hydrolysis rate of microspheres has not been studied in literature at low temperatures. The hydrolysis rate largely depends on the glass composition, and it is not straightforward to propose a general rate from literature. However, to model the water uptake in his materials, Gimenez [START_REF] Gimenez-Fouque | Vieillissement hydrolytique de mousses syntactiques époxydeamine/verre pour l'isolation thermique sous hautes pression: mécanismes de dégradation et simulation de la prise en eau[END_REF] estimates the hydrolysis rate of the microspheres used in this study to be between 0.1 and 0.003 µm/year at 20 °C. Since the GB15 microspheres have a thickness of at least 0.9 µm (see 3.1.1. Fig. 3.4), one year of aging is required for the microspheres to lose 10% of its wall thickness at 20°C. This process seems too slow to explain the collapse of 86% of the microspheres under 25 MPa after 63 days of immersion (Fig. 5.6) To confirm or infirm this hypothesis, a study of the hydrolysis rate of microspheres is currently being pursued in the PrimeVerre analysis laboratory. For this analysis, GB38 microspheres are classified by size, and two classes of microspheres are considered: the first class contains microspheres with diameters between 20 and 40 µm while the second one focuses on microspheres with diameters between 40 and 63 µm. Then, the microspheres are placed in a water column, and the amount of Si ions released by the microsphere over time is measured. From the specific surface of the microspheres (that depends on their size) and the release speed, a hydrolysis rate can be estimated. The study is carried out at different temperatures (40, 50 and 60°C), to determine an activation energy of the process and estimate the hydrolysis speed of microspheres at 15°C, in deionized water. But the study is not over yet, and results cannot be presented here.

Hydrolysis of the coupling agent

Another hypothesis that may explain the collapse of microspheres is associated with a water-induced degradation of the interface between the microspheres and the resin. This aspect was studied by some authors to characterize its impact on composites materials mechanical properties [START_REF] Duigou | Exploring durability of interfaces in flax fibre/epoxy micro-composites[END_REF][START_REF] Ramirez | Evaluation of water degradation of vinylester and epoxy matrix composites by single fiber and composite tests[END_REF]. This interface degradation can change the local mechanical loading on a given microsphere and be a starting point for collapse [START_REF] Gimenez-Fouque | Vieillissement hydrolytique de mousses syntactiques époxydeamine/verre pour l'isolation thermique sous hautes pression: mécanismes de dégradation et simulation de la prise en eau[END_REF]. However, the study of this phenomenon is difficult, requiring, for example, more knowledge on the microsphere's coating, and the use of microspheres with different coatings. We tried to produce pure syntactic samples with microspheres where the silane coating was removed prior the process. Unfortunately, in the absence of coatings, many voids are formed during the pure syntactic manufacturing. Therefore, it was not possible to further investigate this hypothesis. A highly timeconsuming study is indeed required to fully understand and optimize the process parameters.

Swelling of the resin

Another hypothesis that can be formulated is related to the swelling of the resin induced by water absorption, that can generate strain fields in composite materials during aging [START_REF] Youssef | Stress-dependent moisture diffusion in composite materials[END_REF]. Such a process can create an increase of the local mechanical load on the microspheres, therefore making it collapse. To study this possibility, the swelling of the matrix must be evaluated.

Theoretically, the swelling of a polymer due to water absorption corresponds to the volume of water absorbed. This theory was already checked experimentally [START_REF] Obeid | On the identification of the coefficient of moisture expansion of polyamide-6: Accounting differential swelling strains and plasticization[END_REF][START_REF] Bordes | Etude du vieillissement de liaisons adhésives en milieu marin pour application offshore[END_REF], and so the swelling of the epoxy matrix can be expressed as a function of the weight gain using Eq. 5.2.

Using the water absorption at saturation values presented in Table 5.1, the maximal swelling of our material can be calculated and is given in Table 5.4. Now, knowing the swelling of the matrix, it is possible to estimate the stress inside a microsphere.

∆𝑉 𝑉 0 = 𝑊 𝑟𝑒𝑠𝑖𝑛 𝜌 𝑤𝑎𝑡𝑒𝑟 𝜌 𝑟𝑒𝑠𝑖𝑛 Eq. 5.2 With ∆V V 0
the volume expansion of the resin (%), W resin its weight gain (%), ρ resin its density (g/cm 3 ) and ρ water the water density (g/cm 3 ). To evaluate this stress in a simple way, one microsphere is considered, with an external radius b and an internal radius a. It is surrounded by a sphere of resin with a radius c as shown in Fig. 5.7 (same model as the one used in Chapter 4). The dimensions were chosen so that the ratio a/b corresponds to the mean radius ratio of the microsphere GB38 (calculated using Eq. 1.11). (b/c) 3 is the volume fraction of the microsphere in the pure syntactic (of 0.57 for E3_GB38). Then, a displacement ∆b is imposed at the microsphere's surface (Fig. 5.7), such that the volume variation of the resin sphere is equal to the swelling of an epoxy resin. The approach here, for simplification, considers a homogeneous displacement, and that the resin only swells from the inside. The detailed mathematical calculation of the displacement and the circumferential stress inside the microsphere can be found in Appendix B. For this calculation, the glass Young's modulus considered here is 74 GPa and its Poisson coefficient 0.2. The change in stress inside a microsphere induced by the swelling of the resin is given in Fig. 5.8.

Resin

Results show that the stress inside the microsphere shell significantly increases with swelling. When the matrix reaches a swelling value of 3%, the stress in the glass reaches 800 MPa. The glass compressive strength is around 1000 MPa, and largely vary depending on the glass (values found in literature between Furthermore, here the displacement is considered homogeneous, but when water diffuses in the resin, it reaches the microsphere from one direction, making this displacement due to swelling local, and the stress not purely compressive in the glass shell. In tension/flexion, glass is very weak [START_REF] Karabulut | Mechanical and structural properties of phosphate glasses[END_REF], therefore, local swelling seems to be able to make microsphere collapse. And finally, the calculation here is made without additional pressure, that would increase of the stress in the microsphere, making more probable the collapse due to swelling. This simple calculation illustrates that swelling should not be neglected in microsphere collapse mechanisms, and that further investigation should be made.

Fig. 5.8 Estimation of stress inside a microsphere due to the resin swelling

To conclude, it is revealed here that the water absorption in pure syntactic subjected to hydrostatic pressure is due to the filling of the microspheres with water. The coupling effect of pressure and water are responsible for the collapse of the microspheres. As a consequence, water fills the broken microspheres. Additionally, based on the hypothesis and explanations found in this section, it is possible that the collapse mechanism is due to the swelling of the matrix. Also, to confirm or infirm that hydrolysis is not the preponderant phenomenon, a study of the hydrolysis rate of microsphere is ongoing.

It should be mentioned that plasticization of the matrix due to water absorption might occur, leading to stress fields variations and microspheres collapse, but in our case only the resin E2 plasticize due to water, while E1 and E3 does not. So, plasticization is not a general phenomenon and cannot explain the filling of microspheres observed in all our materials.

The filling process of microspheres by water is complex. To try to increase our knowledge on this, from a practical point, it is necessary to study the parameters influencing those processes as well as the material's water absorption. All this will allow an estimation of the kinetics of water absorption in pure syntactic materials.

Investigation of water absorption kinetics in pure syntactic, and influencing parameters

Based on the imaging analysis and damage evaluation, the filling of microspheres by water during immersion has been witnessed for the pure syntactic. To understand how water fills a given microsphere, the parameters influencing the water absorption are studied. Here, the chosen parameters are the following: the influence of sample size, pressure, and finally microsphere size. First, we focus on the pure syntactic E3_GB38.

Influence of sample geometry on water absorption kinetics

It was shown in Fig. 5.3 that decreasing the sample size accelerates the water absorption in the sample.

For example, after 900 days in water at 45 MPa and 15°C, the 5 mm plate sample absorbs 5.9% whereas the 50 mm edge cubic sample absorbs 2.0 % of water. It is thus interesting to use smaller samples to reduce aging duration, but it means that we need to be able to extrapolate from small samples to larger ones.

To compare the influence of sample geometry on the water absorption of pure syntactic, the water absorption of pure syntactic E3_GB38 is plotted versus the square root of time, normalized by the geometry ratio S/V where S is the sample surface and V the sample volume (assuming Fickian behavior [START_REF] Fick | On liquid diffusion[END_REF]). After normalization, the water absorption curve presented in Fig. 5.3.b now becomes Fig. 5.9. During the studied immersion period, the water absorption of all samples appears linear with the square root of time and seems to follow a master curve that can be described with Eq. 5.3, represented by the dotted line on Fig. 5.9. The typical Fickian equations cannot be used here because the maximal water uptake is unknown.

𝑊 𝑝𝑢𝑟𝑒 𝑠𝑦𝑛𝑡 = 𝐴 𝑆 𝑉 √𝑡 Eq. 5.3
With W pure synt (no unit) the water absorption of the pure syntactic, S (mm²) the sample surface, V (mm3) its volume, t (hours) the immersion time, and A (mm/h 1/2 ) the absorption rate measured experimentally. This indicates that in those conditions, the water absorption of pure syntactic is governed by a diffusion process in the material. This result is significant, because it means that it is possible to predict the water absorption from small to large samples (used for the application), and this based on results obtained with samples tested over a short period of time (a few months for 5 mm thick samples). An example of the prediction made using this result is presented later in section 5.5. .

Influence of the type of water

Syntactic foams are used in seawater, so the impact of the type of water must be studied to ensure that they will exhibit the same behavior in oceans and in deionized water. To do that, plate samples (5 mm After 4 months of immersion, there is no difference in water absorption when considering seawater or deionized water (Fig. 5.10). Thus, the previous work performed in deionized water is usable for seawater applications.

Influence of pressure

Since pure syntactic as buoyancy materials are used under different hydrostatic pressures, the influence of pressure on water absorption must be studied. According to Watkins [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF], increasing pressure should accelerate the water absorption. This would therefore accelerate the aging process within the material.

It is thus important to better understand the water absorption mechanisms.

To study the influence of pressure on the water absorption of pure syntactic, plate samples of 5 mm thickness of the E3_GB38 material were used. These were immersed in deionized water at 15°C with From this, an increase in pressure from 30 to 60 MPa, i.e. doubling the pressure, leads to an increase in A by a factor of 1.9 (from 0.0007 to 0.0013 h 1/2 /mm).

Here data with GB38 microsphere have been presented. To be more general about pure syntactic materials, the water absorption of pure syntactic containing GB15 and GB25 microspheres is presented in the next section, alongside the influence of pressure on the behavior of those materials.

Influence of microsphere grade

To investigate the influence of the type of microsphere on the material behavior, the pure syntactic E1_GB15 is immersed at 15°C with an additional pressure ranging from zero to 30 MPa (between zero and 80 % of the material HCP of 38 MPa) for 2 months. The results of water absorption under some of those hydrostatic pressures are presented in Fig. 5.12, on two figures for better readability. Once again, as already observed with GB38, increasing the pressure increases the water absorption rate;

it can be thus considered as a general trend whatever the nature of the microsphere. However, a new phenomenon is highlighted here:

• At pressure between zero and 15 MPa, the water absorption within the material is low (lower than 6 % after 2 months of immersion). The water absorption is also linear with the square root of time as already observed before (Fig. 5.12.a).

• At pressures of 19 MPa and 21 MPa, the two plots still appear quite linear with the square root of time (Fig. 5.12.b). However, it will be seen later in 5.3.6.2. that it is not exactly the case.

Also, a clear increase in water absorption rate is witnessed compared with results at a pressure of 15.3 MPa. The material indeed reaches a water content higher than 12 % after two months of immersion.

• However, for pressure higher than 21 MPa, the plots tend to deviate from simple linear behavior and a clear increase in water absorption rate is observed (Fig. 5.12.b). Furthermore, this increase in water absorption rate is higher at 25 MPa than at 30 MPa. Also, the plots appear to reach water saturation at a value of around 80%.

The water absorption of the material with the GB25 microspheres, E1_GB25, at pressures between zero and 55 MPa (77% of the material HCP of 69.5 MPa), revealed the same behavior: at "low" pressures (between zero and 35 MPa), the water absorption is guided by water diffusion and can be described as a function of the square root of time (Fig. 5.13.a). At higher pressures (48 and 55 MPa), the occurrence of another mechanism leads to a significant acceleration of the water absorption rate (Fig. 5.13.b). It must be mentioned here that this acceleration of the water absorption has not been observed within the E3_GB38 presented previously in section 5.3.3. This difference can probably be explained by the fact that the applied pressure is only 70% of the HCP, not high enough for this acceleration to appear during the test duration. This specific point is discussed later in 5.3.6.2. At "low" pressures, (i.e., lower than 15 MPa for E1_GB15 and 35 MPa for E1_GB25), where the water uptake is linear with the square root of time, it is possible to determine the water absorption rate A. The increase in absorption rate with the applied pressure for both materials is presented in Fig. 5.14. As for the E3_GB38 material, the water absorption rate increases with the applied pressure. To compare the behavior of the 3 materials, the water absorption rate A of the three materials is normalized by the absorption rate at atmospheric pressure, A0, and plotted versus the normalized pressure (P/Pc) in Fig. 5.15. As observed here, the relative water absorption rate increases with the applied load level for all materials. However, the increase is faster for pure syntactics with GB15 microspheres than with GB25 microspheres. Those two are also faster than with the GB38 microspheres.

The impact of pressure on water absorption depends on the type of microspheres used

As presented in 1.2.2. Watkins proposed a way to describe the influence of pressure on the relative water absorption rate using Eq. 1.6 [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF]. This description is compared here with our experimental results in Fig. 5.15 (black dotted line). His proposal is not able to describe the influence of pressure of the three kinds of materials. The best way to describe the influence of pressure on the water absorption rate in a general way (for all kinds of microspheres) is the use of a linear function (colored dotted line on Fig.

5.15)

. The linear function used here is presented Eq. 5.4.

𝐴 = 𝐴 0 (1 + 𝐵 𝑃 𝑃 𝑐𝑟𝑢𝑠ℎ )

Eq. 5.4

With A (mm/h 1/2 ) the water absorption rate of the material, P (MPa) the applied pressure on the material, Pc (MPa) its crush pressure, A0 (mm/h 1/2 ) the absorption rate at atmospheric pressure, and B a constant that depends on the material

The values of B, A0 and Pcrush for each pure syntactic is given in Table 5.5. When Eq. 5.4 is used to describe our results, it is seen that the coefficient B, describing the increase of A with the applied pressure, decreases as the microsphere density and crush pressure increase. It means that using thicker and smaller microspheres, which are stronger, will reduce the impact of pressure on water absorption rate of the material. Therefore, for buoyancy application at a given depth, the use of stronger microspheres will increase the longevity of the material. Results presented here can be used to predict the water absorption rate over a large range of pressure for three types of pure syntactic materials. 5

.5 Values of A0 and B of pure syntactic with epoxy anhydride resin dependning on their crush pressure

Here, the influence of pressure on the water absorption of anhydride-based epoxy pure syntactic is investigated, revealing that at "low" pressure, the water absorption is linear with the square root of time, and the influence of pressure on the water absorption rate can be represented by a linear function. At higher pressure, another mechanism appears as the water absorption rate accelerates with time. The range of "low" pressure can be defined here for 3 different types of microspheres used in pure syntactic, based on the 2 months water uptake tests on plate samples of 5 mm thickness.

To investigate this possible second mechanism and generalize the understanding of the water uptake mechanisms to all types of pure syntactic epoxies, the influence of the type of resin is studied in the next section.

Investigation of the acceleration of water uptake in pure syntactic

In this section, the acceleration of the water absorption in pure syntactic is studied using results from the pure syntactic made with amine-based epoxy resin (i.e., E2 resin). The process behind this acceleration needs to be define first.

The water absorption, presented in Fig. 5.3.a, of the pure syntactic E2_GB38 under 45 MPa hydrostatic pressure at 15°C, is plotted versus the square root of time on Fig. 5.16. The water absorption is linear only at the beginning of the immersion (for √t.S/V < 10 h 1/2 /mm), before an acceleration of the water absorption appears. The difference between both types of resin, amine, and anhydride-based epoxies, is that the amine-based epoxy presents a slower water diffusion rate. The latter results in a slower diffusion of water in the pure syntactic matrix. This slow diffusion allows another phenomenon to appear, that induces an acceleration of water absorption overtime (when plotted versus the square root of time). As this phenomenon seems linear with time, we think that this second phenomenon is the same as the one observed in 5.3.3. , on E1_GB15 at pressures of 25 and 30 MPa, which is also linear with time.

To investigate the possibility of a damage behavior, the water absorption is considered here as a fluid that flows through the material, in the same way as water would flow through a porous medium. In this case, the flow of water can be described by Eq. 5.5, which describes the speed of a fluid flow through a surface [START_REF] Douglas | Fluid mechanics[END_REF].

𝑄 = 𝑞. 𝑆

Eq. 5.5

With Q the water flow (in m 3 /s), S the surface (m²) it goes through, and q (m/s) its speed.

Then, the mass displaced by this flow over time can be written:

∆𝑚 = 𝑄. 𝑡. 𝜌 𝑤𝑎𝑡𝑒𝑟

Eq. 5.6

With ∆m (kg) the displaced mass over the time t (hours) through the surface S (mm²) with a debit Q (mm 3 /h), and ρ water (kg/m3) the water density.

In our case, making the hypothesis that water "flows" through our sample, the mass variation of our pure syntactic can be express using Eq. 5.7.

𝑊 𝑝𝑢𝑟𝑒 𝑠𝑦𝑛𝑡 = ∆𝑚 𝑚 0 = 𝑄. 𝑡. 𝜌 𝑤𝑎𝑡𝑒𝑟 𝑚 0 = 𝑞. 𝑆. 𝜌 𝑤𝑎𝑡𝑒𝑟 . 𝑡 𝑉. 𝜌 𝑝𝑢𝑟𝑒 𝑠𝑦𝑛𝑡,0

Eq. 5.7

With W pure synt (no unit) the water absorption of the material, ρ pure synt,0 (kg/m3) its initial density, ρ water (kg/m3) the water density, t (hours) the immersion time, q the flow rate (mm/h), Q the debit (m 3 /h), and S (mm²) and V (mm3) the sample surface and volume.

To evaluate the possibility of this mechanism, the water absorption of the pure syntactic E2_GB38 is plotted versus the time normalized by the geometry ratio S/V on Fig. 5.17. At the beginning (t.S/V < 1000 h/mm), the water absorption is fast before slowing down (as seen on Fig. 5.17.). Then, the water absorption rate stabilizes and becomes linear with t.S/V. Furthermore, the proportionality is the same for all the different geometries. The mathematical equation presented in Eq. 5.7 allows the description of the water absorption of the material for t.S/V > 1000 h/mm (dashed line on Fig. 5.17) for all samples of E2_GB38 with different sizes using the same flow rate q. The water absorption of pure syntactics made with the amine-based epoxy matrix is similar to a flow of water infiltrating the material. Now that another mechanism has been identified and that a mathematical description is proposed, it will be used to analyze our experimental results.

Description of water absorption mechanisms in pure syntactics

Mathematical and schematic representation

Two different mechanisms of water absorption in pure syntactics have been identified, one guided by water diffusion, the other attributed to damage in the material. To represent those mechanisms, a mathematical description of the water absorption in pure syntactics is proposed, Eq. 5.8:

𝑊 𝑝𝑢𝑟𝑒 𝑠𝑦𝑛𝑡 = 𝑆 𝑉 . (𝐴. √𝑡 + 𝑞. 𝜌 𝑤𝑎𝑡𝑒𝑟 𝜌 𝑝𝑢𝑟𝑒 𝑠𝑦𝑛𝑡,0
. 𝑡)

Eq. 5.8

With W pure synt (no unit) the water absorption of the pure syntactic, A the diffusion coefficient (mm/h 1/2 ) and q a flow rate (mm/h), t the immersion time, S (mm²) and V (mm 3 ) the sample surface, volume, and ρ pure synt,0 (kg/m 3 ) the initial density of the material, ρ water (kg/m 3 ) the water density.

Since our measurements are performed at atmospheric pressure (the sample is taken out of the hydrostatic vessel, before being weighted), the water density ρ water is taken as the water density at 15°C and atmospheric pressure (1.00 g/cm 3 ).

The first mechanism, guided by water diffusion and the parameter A, can be represented using Fig. 5.18.

The proposition being made here is that first, the water diffuses in the resin. Then, because of the swelling of the resin, the microspheres collapse. Finally, the water fills the microspheres. The second mechanism, that can be related to damage development, is represented by the scheme in Fig.

5.19

, where the water flows from one microsphere to another. This representation comes from the observation made using X-ray microtomography (Fig. 5.4), where the water is seen filling the microsphere following preferential pathways, going from one neighboring sphere to another. The flowing of water from one microsphere to another is associated here to mechanical damage in the resin, creating microcracks and allowing the water to flow, or to microcracks in the microsphere in contact with each other (i.e., without resin between them). Those damage mechanisms have yet to be proved but are hypotheses that must be deeply studied in the future. The results presented in 5.3.4. and 5.3.5. show that those two mechanisms appear for all pure syntactic epoxies, but the rate at which they occur depends on the material and the applied pressure.

To illustrate that Eq. 5.8 can describe the water absorption of pure syntactic materials, the model is fitted to the data of water uptake of E2_GB38 under 45 MPa pressure. The values of the constants are A = 2.2.10 -4 mm/h 1/2 and q = 1.7.10 -6 mm/h. The experimental data and the model fit are compared in Fig. Two water absorption mechanisms in pure syntactics under hydrostatic pressure are identified and described analytically. It is shown that the rate of each mechanism depends on the type of resin used, but also on the applied pressure on the material. The damage mechanism is faster than diffusion at high pressures (Fig. 5.12). To further understand when each phenomenon prevails upon the other, the influence of the applied pressure on the flow speed q is investigated in the next section.

Influence of pressure on kinetics of damage mechanism

Here, the influence of pressure on the damage mechanisms is investigated. It is of particular interest because pure syntactics are immersed under hydrostatic pressure.

To study the influence of pressure on the parameter q of Eq. 5.8, the 2 months immersions performed on all the materials, at 15°C and under different hydrostatic pressure, are used. Unfortunately, only one sample size was characterized at those pressures, making an efficient fit using the least square method of the model on our data rather difficult. Because of that, the obtained values of A and q using this method are non-unique. This revealed the need to use two different sizes of sample when studying water absorption of pure syntactic. To overcome this lack of data but still make use of the available experimental results, other methods to estimate the values of A and q must be investigated. Two different methods are used and depend on the type of resin used in the pure syntactics.

Determination of q for anhydride-based pure syntactics.

The results from 5.3.4. showed that at low pressure, for anhydride-based pure syntactics (i.e., with E1 and E3 resin), the parameter A evolves linearly with the applied pressure. Hence, the approach used here is to assume that Eq. 5.4 describes the influence of pressure on A.

Then, to estimate the speed flow v related to the damage mechanism, the parameter A is fixed in Eq. 5.8. When A is fixed, it becomes possible to estimate the value of the flow speed q using measurements of water absorption of 1 sample geometry only (here, plate samples of 5 mm thickness).

As an example, the model fitted to our data is shown on Fig. 5.21 for E1_GB15 pure syntactic under different pressures. The model can describe the changes in water uptake in our material before saturation.

Determination of q for amine-based pure syntactics

To estimate the influence of pressure on both mechanisms for materials made with amine-based epoxies, the pure syntactic made with the E2 resin and GB15, GB25 and GB38 microspheres are also immersed for at least 2 months at different hydrostatic pressures in deionized water, using 5 mm plate thickness.

The water absorption of E2_GB15 at pressures of 5.2, 10 and 15.3 MPa is presented Fig. 5.22, plotted versus the square root of time. At those pressures, the water absorption of the material is not linear with the square root of time, as an acceleration appears, similar to what is observed for E3_GB38 under 45

MPa pressure (Fig. 5.16). This means that whatever the pressure, for materials made with the E2 resin, the damage mechanisms can never be neglected. Only one sample geometry is tested for each condition (plate sample of 5 mm thickness). To estimate both our parameters for the fit, the following procedure is applied. First, the constant q is taken as zero, and A is manually adjusted to fit the diffusion mechanism with the initial part of the curve (t 1/2 .S/V <10 h 1/2 /mm). When A is adjusted, q is manually changed to fit the model to the rest of the curve. The condition to respect the fit is that A must increase with pressure, as observed on the pure syntactic made with anhydride-based epoxy resin. Examples of the fit for E1_GB15 immersed under 5.2, 10 and 15.3 MPa (5 months immersion time) are plotted in Fig. 5.22 alongside the experimental data, revealing once again the capability of the model to describe our experimental results. The identification methods to determine the model constant present limits, as the value of A is arbitrary chosen, and more experimental results are needed to refine the identification of both parameters. But using the presented methods as a first approach, it is possible to estimate the flow rate q of all materials when immersed under different pressures. The influence of pressure on this mechanism can now be investigated.

Influence of pressure on the flow speed q for each pure syntactic

In this part, to have a first understanding of the damage mechanisms, the influence of pressure on the flow speed is studied. Fig. 5.23 presents the change in flow speed q of all the materials studied with the applied pressure (i.e., results from water absorption measurements on plate samples of pure syntactics with GB15, GB25 and GB38 microspheres, made with E1, E2 and E3 resin, immersed in deionized water at 15°C, under hydrostatic pressure between zero and 110 MPa). For all materials, at first the flow speed is extremely low but increases with the applied pressure. Then, when a certain pressure is reached, the flow rate increases drastically. The pressure at which the flow speed increases depend on the nature of the material. To efficiently compare all materials, the change in flow speed v is plotted versus the normalized load level P/Pc for all materials on Fig. 5.24. For both materials containing GB15 microspheres, the flow speed increases drastically when the applied load is 0.5 times the crush pressure. For both materials with GB25 microsphere, this increase appears at higher load levels, around 0.6 times the crush pressure, for both materials. For pure syntactic containing GB38 microsphere, this increase appears at load levels of 0.8 times the crush pressure. Those results illustrate that the damage mechanism is a pressure activated mechanism: at low load levels, its rate is low, while the phenomenon accelerates drastically when a critical pressure is reached. This critical pressure depends on the type of microspheres used, being 0.5 times the crush pressure for pure syntactics containing GB15 microspheres, 0.6 times for materials with GB25 microspheres, and 0.8 times for materials with GB38 microspheres.

Conclusion on the experimental study of water absorption of pure syntactics

With this experimental study of the water absorption of pure syntactics, that regroup data on 6 different materials immersed under more than 20 different aging conditions (nature of water, pressure, sample size), two water absorption mechanisms were identified: one that can be described by diffusion, the other by a flow of water representing a damage of the material. The influence of pressure revealed that each phenomenon accelerates with the increase of the applied hydrostatic pressure. It also reveals that a critical pressure exists, below which diffusion is preponderant, and above which damage mechanisms takes the lead. This critical pressure is 50% of the HCP of the pure syntactic when it contains GB15 microsphere, 60% when containing GB25 microspheres, and 80% for materials with GB38 microspheres. Using denser (stronger) microsphere increases the level of the critical pressure. A mathematical description of each phenomenon is proposed, and those can be either considered simultaneously or independently whether the hydrostatic pressure is below or above the critical pressure.

For materials presenting slow diffusion, the damage mechanism cannot be neglected, even at pressures under the critical pressure. The determination of the rate of both mechanisms must be done through water absorption measurements of samples with different sizes. Since the damage mechanism is proportional to time, for better longevity, pure syntactic should be used below the critical pressure at which the damage mechanism becomes preponderant.

The water absorption of pure syntactics was studied. It is admitted that buoyancy loss of pure syntactics is due to water absorption, but other mechanisms such has creep can appear. In the next section, we will show that creep of epoxy pure syntactic is negligible. After that, we will use our results from the water absorption study to make a prediction of the buoyancy loss in pure syntactic and compared with experimental measurements.

Creep of pure syntactic

Since creep can lead to volume variations, and so buoyancy variations, the possibility of creep happening in pure syntactic under pressure must be studied. For that, a cubic sample of 50 mm 3 of E2_GB38 was pressurized under 40 MPa at room temperature during 96 hours, and its volume strain during the test is recorded in Fig. 5.25. The volume strain during the plateau is nearly constant and does not evolve much, as the volume strain during the pressure plateau increases of only 0.00015. This is at least 10 times lower than the values observed for composite foam (that will be presented in Chapter 6), which can reach 0.003 during the test. The low volume strains measured here are attributed to slight temperature change during the test which impact the gauges measurements.

From this result, it appears that epoxy pure syntactics do not creep when loaded under hydrostatic pressure, in accordance with statements from literature [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF]. Now that we can neglect creep, in the next section we will make a prediction of buoyancy loss of pure syntactic using the results from our study of water absorption in pure syntactic.

Buoyancy loss prediction

To further confirm that water absorption is responsible for buoyancy loss of pure syntactics and also to validate a method to predict buoyancy loss in pure syntactics, a prediction of water absorption in pure syntactics is made by using the description of water absorption previously presented.

For that, the pure syntactic E1_GB25 is considered. For this study, several samples of different geometries are immersed under 20 MPa hydrostatic pressure in seawater for one year, at room temperature. 20 MPa being lower than the critical pressure (35 MPa) identified for E1_GB25, the water absorption mechanism considered here is the diffusion one, and damage mechanisms can be neglected.

Then the flow speed value is zero, and the water absorption within the material can be described by Eq. 5.3. After a year of immersion, the water absorption of the different samples is recorded and represented in Fig. 5.26, and the available data allows the determination of the diffusion coefficient A in these conditions using linear regression. In the same hyperbaric tank, a specific sample of E1_GB25 was immersed and attached to a weighing device. This sample is named a 'garland' sample and is shown in Fig. 5.27. It consists of 100 plate samples of 4 mm thickness, connected to each other by a Dyneema cable. By doing so, the total surface and volume of the sample increases, increasing the nominal water uptake of the garland. This allows the measurement of the buoyancy loss of the material with a weighing device, otherwise the buoyancy loss would be too low to be measured with the available equipment. Since all plate samples are attached together, the weighing device measures the sum of the buoyancy of all those samples. Then, the garland is pressurized at 20 MPa for one year, and its buoyancy loss is measured over time. The evolution of buoyancy loss with time is plotted in Fig. 5.28 (after subtracting the buoyancy loss due to elastic deformations). The measurement is then compared with the prediction of buoyancy loss (red line in Fig. 5.28) made using Eq. 5.9: ∆𝐹 = ∆m. 𝑔 = 𝑊 𝑝𝑢𝑟𝑒 𝑠𝑦𝑛𝑡 . 𝑚 0 . 𝑔 = 𝑚 0 . 𝐴 𝑆 𝑉 √𝑡. 𝑔 Eq. 5.9

With ∆F the buoyancy loss (N), m 0 the initial mass of the garland (kg), S its surface exposed to water (mm²) and V its volume (mm 3 ), W pure synt (no unit) the water absorption of the pure syntactic, A the absorption rate (mm/h 1/2 ), t the immersion time (hours) and g the gravitational constant (m/s²). The prediction almost perfectly matches the measurements, with less than 5% deviation, on the period considered. This confirms that buoyancy loss in pure syntactic only comes from water absorption. A prediction can be made from water absorption prediction/measurements.

Conclusion on the long-term behavior of pure syntactics

The long-term behavior of pure syntactics used for buoyancy applications was studied, under low temperature (15°C) and high hydrostatic pressures (up to 110 MPa). For the first time, as far as we know, the filling of microspheres with water was observed, and the collapse of microspheres was highlighted.

The investigation of the influencing parameters revealed that two mechanisms are occurring during water absorption of pure syntactic epoxies. Both of these induce a collapse of the microspheres due to the presence of water, that allows the filling of the microspheres with water. The difference between each mechanism is the way the water reaches the microsphere. The first one is controlled by water diffusion in the resin. The hypothesis proposed here is that the swelling of the matrix induces a collapse of the microspheres, allowing them to be filled with water. This water absorption behavior is a function of the square root of time. This phenomenon accelerates as the hydrostatic pressure increases, with a diffusion rate that increases linearly with the applied pressure.

The second mechanism is a function of time, following a mechanism that is still not fully understood.

Here, this mechanism is described using a water flow that goes from one microsphere to another. This phenomenon greatly accelerates when a critical pressure is reached, that depends on the material's hydrostatic strength. This indicates that pure syntactics as buoyancy materials should be used under this critical pressure, to prevent the damage mechanism to become preponderant and lead to rapid degradation of the material.

Finally, using the water absorption description and prediction, a prediction of the buoyancy loss of pure syntactic was made, and compared with experimental data. The prediction was excellent. Together with the creep tests made on our materials, it proved that in pure syntactic water absorption is the only mechanism responsible for buoyancy loss, and that viscoelastic deformations are negligible.

Introduction

In this section, the long-term behavior of composite foam (i.e., in presence of macrospheres) is investigated. Buoyancy loss is measured over time, and its origin is discussed, revealing that in composite foam creep appears alongside water absorption. The influence of temperature and pressure on creep is also investigated, and a description of creep and water absorption is proposed in order to predict buoyancy loss over long service life (1 or 20 years).

Buoyancy measurements and usual predictions

To study the buoyancy loss of composite foam, the Instrumented Buoyancy Loss (IBL) test presented in 2.2.4.3. 0 is used. Here, an example is presented using the composite foam made of microsphere GB38, the pure resin E3, and the large carbon macrospheres MC2 (material name: CF_E3_GB38).

During the test, the composite foam is pressurized up to 45 MPa (corresponding to 4500 m depth), for 60 days at 15°C in tape water. The buoyancy change, in N, of the sample is presented in Fig. 6.1. When the pressure is applied to the sample, a volume change occurs due to compressibility of the material. This results in a decrease of buoyancy. Then, when the test pressure is reached and becomes constant, the sample buoyancy continues to decrease. This provides clear evidence that a time dependent buoyancy loss exists. When the pressure is released, a rapid increase of buoyancy loss is observed, due to the change of water density. Then, the buoyancy loss decreases due to the viscoelastic behavior of the material. Finally, an irreversible buoyancy loss is observed.

F 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 = -100. ∆𝐹 𝐹 0

Eq. 6.1

With F 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 the relative buoyancy loss (%), ∆𝐹 (N) the buoyancy change recorded during the IBL test, and 𝐹 0 the initial buoyancy of the sample, recorded placing the sample in the pressure vessel.

In industry, the measured buoyancy change is used to calculate the relative buoyancy loss of the material, expressed in percent, with Eq. 6.1. Since the buoyancy change is negative, here the minus sign is used to consider the absolute value of buoyancy loss. The relative buoyancy loss of our sample tested here is plotted in Fig. 6.2. Then, to make predictions, the buoyancy loss plot is fitted by a logarithmic function and extrapolated to service life of 1 year for material designed for seabed exploration, and 20 years for material designed for production. The fit is made using the least square method, on all the data but removing the first 24 hours of the test (to not consider the elastic part when fitting the description) (Fig. 6.3). The test period considered in industry for the fit is 96 hours. However, there is a question about whether to increase test duration and its impact on the buoyancy loss prediction. To answer this question, the same prediction is performed here with test durations of 15 and 60 days. Buoyancy loss predictions made on both materials are presented in Table 6.1, depending on the time period considered for the fit. The material with the epoxy amine base resin (CF_E2_GB38) is tested at 40 MPa. The other one (CF_E3_GB38) is tested under 45 MPa hydrostatic pressure. There is no standard deviation for the prediction using 60 days period tests because only one sample of each material is tested for this duration. The predicted buoyancy loss of both materials is lower than 5%, in agreement with the recommendations of the standard API-16F. Using those predictions, the material can be considered to be well designed as they would pass the qualification criterion.

One can notice in Table 6.1 that the longer the test period considered, the higher is the predicted buoyancy loss. When making buoyancy loss predictions in this way, using a longer test period is more conservative. There is a compromise between the test period and the precision of the prediction.

Those prediction methods have been used for years with success based on feedback from service.

However, they are completely empirical. Since the aim of the industry is to go deeper and so increase the hydrostatic load on the materials, it must be verified that the material behaves in the same way, and that the tools used for life prediction are still valid.

For that, more knowledge on the material behavior is needed. Since time-dependent buoyancy loss is observed, to improve the understanding of the material behavior, the origin of this buoyancy loss is investigated.

Investigation of the origin of the buoyancy loss in composite foam

In this section, the origin of buoyancy loss is investigated. Since one phenomenon responsible for buoyancy loss in pure syntactic is water absorption (as presented in Chapter 5), the water absorption of composite foam is studied first.

Water absorption of composite foam

To compare the measured buoyancy loss with the water absorption of the material, a cyclic test is performed on a sample of composite foam made with anhydride-based pure syntactic and carbon macrospheres (CF_E3_GB38). A cycle consists of pressurizing the sample at 45 MPa for a duration of 96 hours or more, and then releasing pressure to 0 MPa for at least 72 hours to measure changes in buoyancy. Then the sample is removed from the pressure vessel to measure its mass to evaluate water absorption. Let us first focus on the red curve in Fig. 6.4 that shows buoyancy loss as a function of time. Here again, when pressure is applied an increase in buoyancy loss occurs due to compressibility of the material.

Then during the pressure plateau, a time dependent increase in buoyancy loss can be observed. When pressure is released a decrease in buoyancy loss occurs rapidly due to the elastic behavior of the composite foam and then a slow continuous decrease can be observed suggesting a recovery of the sample volume. Nevertheless, a clear nonreversible buoyancy loss is highlighted here: after 700 hours of recovery at the end of the test, the buoyancy loss does not decrease down to zero. As shown in Chapter 5, when immersed in water, syntactic foam absorbs water that leads to an increase in sample weight and so a decrease in buoyancy. The question here is, is the irreversible buoyancy loss observed here due to

water? To answer this question, the measured buoyancy loss is compared with water absorption measurements Table 6.2. The water absorption generates a buoyancy loss lower than what is measured during the test, representing less than 15% of the total buoyancy loss at the end of each cycle plateau, and less than 25% of the residual buoyancy loss at the end of each cycle. So, the water uptake is not the only mechanism responsible for the irreversible buoyancy loss observed at the end of each recovery plateau. It should be noted that the water absorption is greater during the first cycle, because of the presence of external defects of the sample (Fig. 6.5) that are easily filled with water. Water absorption only represents a small portion (less than 15%) of the total buoyancy loss measured (Table 6.2). This indicates that another phenomenon occurs during immersion. Let us now focus on volume changes of composite foam during immersion under hydrostatic pressure.

Number of cycles

Volume changes of composite foam during ageing

Since it has been shown that water absorption is not the principal reason that leads to buoyancy loss, it was decided to investigate creep phenomena during the tests.

A sample of composite foam CF_E3_GB38 (same material as presented in the previous cyclic test) was instrumented with strain gauges during an IBL test. The gauges were placed at mid height of the sample, with positions Ji (i between 1 and 8) represented Fig. 6.6. The gauge direction was perpendicular to the length axis to measure local circumferential strain during the test. The gauges, depending on their position, were positioned near to or far from a visible macrosphere (Fig. 6.6). This highlights the influence of pressure on buoyancy loss and will be discussed later in this work. Using strain gages during IBL tests, it has been possible to highlight that the volume of composite foam changes during ageing with a time dependent behavior. This leads to two questions:

• Where does creep come from and what are its mechanisms?

• How can it be measured/estimated to predict the buoyancy loss it will generate over longer periods? First, when the pressure increases, both macrospheres present volume strain which is linear with the applied pressure (Fig. 6.9). Those volume variations are due to the elastic deformations of the material.

Then, when pressure is held constant, both macrospheres show time-dependent volume strain, indicating that creep is appearing in both macrospheres. The creep rate of MC2 and the elastic deformations during pressurization are lower than for MC1 (Fig. Two physical phenomena that generate buoyancy loss in composite foam (i.e., water absorption and creep) have been identified. In the next section a description of those phenomena is proposed, with the final aim to make a prediction of buoyancy loss in composite foam under service conditions.

Water absorption description

Based on results presented above it appears that to propose a reliable description of buoyancy loss, it is necessary to consider the water absorption process within composite foam.

Water absorption of the composite foam CF_E3_GB38 under 45 MPa hydrostatic pressure is measured and is presented in Fig. 6.11 (blue dots on the graph). Some water absorption measurements were also made on the same material immersed in tap water at 15°C but without hydrostatic pressure. Those results are presented in Fig. 6.11 alongside the water absorption measurements under 45 MPa hydrostatic pressure (dots on the graph). Applying hydrostatic pressure on the material increases the water absorption, in a similar way to pure syntactic.

So, how can we predict the water ingress in composite foam? To do so and with the idea to use a simple characterization, a prediction of water absorption in composite foam is proposed based on results obtained with pure syntactic in Chapter 5.

To perform this prediction two hypotheses are made here:

• While the macrospheres can absorb water, when placed in composite foam, the water must first diffuse through the pure syntactic before reaching the macrospheres. Since this process is slow, to a first approximation if can be considered that the macrospheres, when placed in the composite foam, does not absorb water.

• The water absorption of composite foam due to filling of external defects is instantaneous.

Based on those hypotheses, it is possible to evaluate water absorption in composite foam using a pure syntactic water absorption description (see 5.3.6.1. ). For that, we consider a volume of pure syntactic of the same dimensions as the composite sample size (i.e., a cylinder of 300 mm diameter and 1 m height). Then, the water absorption model presented in 5.3.6.1. is used to describe water absorption in the sample. Since at atmospheric pressure and at 45 MPa, the pressure is lower than the critical pressure (which is higher than 110 MPa for E3_GB38), the damage mechanism of water absorption in pure syntactic can be neglected (flow speed q = 0). Then, since the absorption rate at atmospheric pressure and 45 MPa for this material is known from the water absorption measurements of plate samples (see 5.3.1. ), water absorption can be estimated using Eq. 6.2. The value of ∆ 𝟎 is chosen to be 0.02% to fit the estimation with the initial water absorption observed experimentally, and the value of A chosen from results of 5.3. Then, prediction of water absorption in the theoretical cylinder of pure syntactic is made and compared with the water absorption measurements of composite foam containing the same pure syntactic, in Fig. 6.11.

First, initial water absorption due to instantaneous filling of the sample default is seen, before the water absorption increases with the immersion time. The prediction of water absorption in the pure syntactic (dashed line) fits the experimental measurements.

It is thus possible to estimate the water absorption of composite foam from pure syntactic water absorption characterization. To consider all types of pure syntactic, the description proposed in 5.3.6.1.

by Eq. 5.8 is used, with the values of the absorption rate A and the flow rate q according to the pure syntactic used and the applied pressure on the material. Then the water absorption can be expressed as:

𝑊 = 𝑆 𝑉 (𝐴√𝑡 + 𝑞. 𝜌 𝑤𝑎𝑡𝑒𝑟 𝜌 𝑝𝑢𝑟𝑒 𝑠𝑦𝑛𝑡,0 𝑡) 𝑔 + ∆ 0
Eq. 6.3

With W (no unit) the relative weight gain of the hypothetical cylinder of pure syntactic, A a diffusion coefficient (mm/h 1/2 ) and q a flow speed in mm/h, t the immersion time (hours), S (mm²) and V (mm 3 ) the sample surface, volume, and ρ pure synt,0 (kg/m 3 ) the initial density of the material, ρ water (kg/m 3 ) the water density, and ∆ 0 the water absorption due to the filling of the external defects of the sample.

Now that a method is available to estimate the water absorption in composite foam, we can focus on implementing a description of creep to make predictions of buoyancy loss originating from creep.

Creep description

It was shown that the main phenomenon observed during IBL tests on composite foam is creep. In this section a description of creep will be proposed to estimate volume change of the material over time.

Global creep measurements

To describe creep, global volume measurements are needed. But it was shown in Fig. 6.8 that creep is heterogeneous in the material, and so gauges are not suitable to estimate the global volume change of the material. Another method to estimate volume change must be used.

Since a description of water absorption was proposed before, it is possible to estimate volume change from buoyancy loss measurements. It is possible to calculate composite foam volume using Archimedes' law:

𝑉 = 𝐹 0 + ∆𝐹 𝑔 + 𝑚 0 (1 + 𝑊) 𝜌 𝑤𝑎𝑡𝑒𝑟

Eq. 6.4

With V (m 3 ) the sample volume, F0 (N) and m0 (kg) its initial buoyancy and mass respectively, ΔF (N) the buoyancy loss measured during IBL tests, W (no unit) the estimated water absorption from Eq. 6.3, ρ water (kg/m 3 ) the water density, and g (m/s²) the gravitational constant. As shown in 1.4.2. , several models can be used to describe creep of composite material under various loading conditions [START_REF] Findley | Creep and relaxation of nonlinear viscoelastic materials[END_REF]. Since we observe a decreasing but non-null creep rate, the constitutive Kelvin Voigt and Maxwell models, and the models based on it such as series of Voigt or Maxwell elements, were put aside. While they can be used to describe creep very well over the time considered, they are not suited to extrapolate results over longer periods. Suitable model in our case, that describe decreasing but non-null creep rate, are power or logarithmic law, presented in 1.4.2. [START_REF] Findley | Creep and relaxation of nonlinear viscoelastic materials[END_REF]. Those kinds of models are both commonly used to describe creep of polymer and composite materials. Since the usual method in industry to predict buoyancy loss is to model it with a logarithmic law, this law was chosen to describe the creep we measure during our tests. To consider the elastic deformations in our model, the following model creep was chosen:

𝜀 𝑡𝑜𝑡𝑎𝑙 = 𝜀 0 + 𝐶 * log (𝑡 + 𝑡 0 )

Eq. 6.5

With ε total (no unit) being the total volume strain of the material, ε 0 (no unit) the elastic volume strain, t 0 (h) the time at the beginning of the pressure plateau, t (h) the creep time, and C (h -1 ) a coefficient determined experimentally, named creep rate, that depend on the material, pressure, and temperature.

The volume creep strain ε c in this case is:

𝜀 𝑐 = 𝜀 𝑡𝑜𝑡𝑎𝑙 -𝜀 0 = 𝐶 * log (𝑡 + 𝑡 0 )

Eq. 6.6

With ε c (no unit) the volume creep, ε total (no unit) being the total volume strain of the material, ε 0 (no unit) the elastic volume strain, t 0 (h) the time at the beginning of the pressure plateau, t (h) the creep time, and C (h -1 ) a coefficient determined experimentally, that depend on the material, pressure, and temperature.

The influence of pressure on creep rate can be described by an exponential law (dotted lines on Fig. 6.15). C0 and C1 are estimated for all our materials from experimental data and are given in Table 6.3.

Then, the creep model becomes Eq. 6.8.

𝐶(𝑃) = 𝐶 0 * exp ( 𝑃 𝑃 0 )

Eq. 6.7

With C (h -1 ) the creep rate at pressure P, C0 (h -1 ) and C1 (MPa) empirical constants. . This description allows us to extrapolate results from a pressure to another, knowing the constants C0 and P0. Based on the results it appears that it is possible to accelerate creep in syntactic foam by increasing the hydrostatic pressure applied to the material. Moreover, the impact of pressure on creep rate is now known, as an acceleration factor has been determined up to 45MPa. The remaining question concerns the maximal testing pressure that can be used to increase creep rate.

𝜀

During the IBL test on the composite foam made with the amine-based epoxy, CF_E2_GB38 at different pressures, presented in Fig. 6.16, a drastic volume loss was measured during the 50 MPa pressure plateau. This volume loss was due to the collapse of macrospheres at the sample surface. The collapse of macrospheres created holes at the sample surface, that ware filled by water and led to buoyancy loss, and so volume loss, of the material. The number of broken macrospheres during the period was 36 and did not seem to stop.

In this case, the pressure is high enough to damage the composite foam that leads to the collapse of macrospheres. Since this damage mechanism appears, the expression in Eq. 6.8 can no longer be used in those conditions. The pressure at which macrospheres start to collapse is a limiting pressure for the use of such materials as buoyancy modules, because it generates high buoyancy loss. The pressure at which our studied material exhibits macrospheres collapse is reported in Table 6.4, from observation during IBL tests at different pressures. Increasing pressure accelerates the occurrence of damage in the foam. However, no macrospheres collapse was seen during IBL tests at 40 and 45 MPa on both materials with carbon macrospheres. To try to predict macrospheres collapse in composite foam, it was decided to try to predict the time at failure of macrospheres when not surrounded by pure syntactic. Knowing the stress at break of our glass fiber macrospheres, those were chosen to be studied first. Macrospheres were loaded under pressure such that the circumferential stress in the macrospheres shell was equivalent to 70, 80 and 85% of the macrospheres stress at break (that was determined in 3.1.2. ). Macrospheres were loaded until failure, and their time to failure was recorded. Five macrospheres were tested for each load level.

The time at pressure, depending on the stress level in the shell, is represented Fig. 6.17. A trend is observed: the time to failure of the macrospheres decreases as the load level increases. But there is high scatter in the results, revealing that the collapse of macrospheres depends on other parameters, that are probably related to imperfections in their geometry. Due to the variability of the results, it was impossible to make a prediction of time to break of macrospheres at lower load levels. This is a subject for future work. As the time to failure of macrospheres is difficult to estimate, and since no macrospheres collapses were seen during IBL tests at service pressure, the macrospheres collapse will not be considered in our creep description, but the pressure at which macrospheres failure occurs will define the pressure limit of creep description.

Influence of temperature

All the tests on buoyancy loss and creep described above were made at 15°C, but in service conditions the water temperature is close to 4°C. To study the influence of this difference of temperature, IBL tests were made on CF_E2_GB38 and CF_E3_GB38 at 5°C under service pressure. The evolution of creep during those tests was recorded, Fig. 6.18. Changing the temperature did not affect the behavior of CF_E3_GB38 composite foam (Fig. 6.18.a).

The glass transition temperature of both the macrospheres and the pure syntactic are respectively 80°C and 110°C (see 2.1.4. 0), much higher than the temperature of the test, explaining why the behavior at 5°C and 15°C is similar.

For CF_E2_GB38, decreasing the temperature reduced the creep rate (Fig. 6.18.b). This time, the glass transition of the pure syntactic is 40°C. Because of that, decreasing the temperature increases the resin mechanical properties, so the pure syntactic rigidity is higher, and so the stress applied to the macrospheres (from stress estimations in 4.2. ) is lower. Since creep rate is related to the stress applied to the macrospheres, decreasing the temperature will decrease the composite foam creep rate. This decrease is more significant as the temperature is close to the material glass transition temperature.

In conclusion it appears that testing at 15°C is conservative. However, it should be noted that changing the water temperature will change its density, therefore impacting the buoyancy exhibited by the material, and should be taken into consideration. Now that a description of water absorption and creep behavior has been proposed for our epoxy composite foams, it is possible to evaluate buoyancy loss and make a new prediction over their service life.

Buoyancy loss description and predictions

By using the previous description of water absorption and creep for composite foam, it is possible to describe buoyancy of composite foam, based on the description of the phenomena that take place within the material when immersed for long durations in water.

Introducing in Archimedes' Law the creep and water absorption descriptions, that were developed in this work, the buoyancy of composite foams becomes Eq. 6.9: 𝐹 = [𝜌 𝑤𝑎𝑡𝑒𝑟 𝑉 0 ( (𝑃))) 𝑡] 𝑔 Eq. 6.9

With F (N) the material buoyancy, ρ water (kg/m3) the water density, S 0 (mm²), V 0 (mm3) and m 0 (kg) the initial surface, volume and mass of the sample, K (MPa) the composite foam bulk modulus, P (MPa) the applied pressure, C (h-1) the creep rate, t0 (h) the time at the beginning of the pressure plateau, A (mm/h1/2) the water absorption rate of pure syntactic, q (mm/h) the flow speed of the pure syntactic, ρ pure synt,0 (kg/m3) the pure syntactic density, ρ water the water density (kg/m3) t (h) the time and g (m/s²) the gravitational constant.

Here, the term ∆ 𝟎 representing the initial water absorption due to external defects, is not used. It is because from a practical point of view, the water absorption due to sample defects is instantaneous.

Therefore, it can be considered in the elastic deformations, such has ∆ 𝟎 =0 and the initial water absorption is considered in the bulk modulus K. The relative buoyancy loss can then be calculated:

F 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 = 100 𝐹(𝑃, 𝑡) -𝐹(0,0) 𝐹(0,0)

Eq. 6.10

With F relative the relative buoyancy loss (%), F(P, t) the buoyancy loss at pressure P after an immersion time t, using Eq. 6.9., and F(0,0) the initial buoyancy at atmospheric pressure.

Using Eq. 6.9, buoyancy loss predictions for buoyancy modules over their service life can now be made.

The module size is chosen as the real size of the module (i.e., height of 4.5 m, diameter of 1.5 m). The real module size must be considered because it affects the water absorption.

The predictions are made using the parameters presented in Table 6.5, that are estimated using the experimental measurements during IBL tests and water absorption tests on pure syntactic. Since epoxy anhydride systems are generally used for short term applications, the prediction of buoyancy loss of CF_E3_GB38 is made under 45 MPa over one year. For the other material, since amine-based systems are usually used for long term applications, its buoyancy loss will be predicted over 20 years under 40

MPa hydrostatic pressure. The initial buoyancy, needed to estimate relative buoyancy loss in percent, is taken as the buoyancy of the module at atmospheric pressure. The 1-year buoyancy loss prediction on the material CF_E3_GB38 is presented in Fig. 6.19. The black dashed line represents the buoyancy loss due to volume change, and the red curve the total buoyancy loss (considering also water uptake). Here, the predicted buoyancy loss after 1 year is 1.78 %, in the range of the prediction presented in 6.1. Furthermore, water is only responsible for 0.03 % of buoyancy loss. This is negligible in front of the 1.74 % of buoyancy loss due to creep and elastic deformations.

For applications of 1 year, making buoyancy loss prediction on composite foam using only IBL tests and incorporating water absorption in the volume measurements (i.e., considering mass constant during the test) is a good approach because the water uptake rate is negligible over this period, because of the large size (4.5 m height per 1.5 m diameter) of the buoyancy module. The 20-year buoyancy loss prediction on the material CF_E2_GB38 is presented in Fig. 6.20. The black dashed line represents the buoyancy loss due to volume change, and the red curve the total buoyancy loss. In Fig. 6.20, the predicted buoyancy loss is 2.17%. It is higher than the prediction presented in 6.1.

Here, after 20 years, the buoyancy loss from water absorption is of 0.16%. Furthermore, the buoyancy loss rate is higher than if only buoyancy loss due to volume variations were considered. In this case, water absorption should not be neglected. When using this type of material (i.e., composite foam with amine-based epoxy resin) for 20 years or longer periods, the characterization of both water absorption and creep are needed to make buoyancy loss predictions.

Conclusions

In this chapter, the buoyancy loss of composite foam has been studied and presented, as far as we know for the first time. The results reveal that buoyancy loss originates from two mechanisms, water absorption and creep that appears due to the macrospheres creep. Tests of longer duration than what is recommended in the industry were used, and more than 15 IBL tests, including cyclic tests, were performed to characterize the material behavior under different conditions (temperature, pressure).

While difficult to make because of the water, strain gauge measurements were used to highlight the material behavior for the first time to the best of the author's knowledge.

The main phenomenon observed during the IBL tests, usually employed to make buoyancy loss prediction, is creep. But considering only this phenomenon is not enough for buoyancy loss predictions of several years, as water absorption, while slower initially, will generate non-negligible buoyancy loss over the long term.

A description of creep and water absorption of composite foam has been proposed, that considers the influence of pressure on both phenomena. Since the phenomena are studied at temperature of 15°C, the buoyancy loss predictions will be conservative, knowing that the water temperature in service conditions is lower than 5°C, and that decreasing the temperature will slow down both creep and water absorption.

The collapse of macrospheres due to creep has been highlighted, but due to the scatter of the results, no estimation of the time at failure of macrospheres in composite foam could be made. This is a point that needs to be investigated in the future, since it can cause damage and buoyancy loss of the material.

General conclusions and perspectives

Syntactic foams are complex materials which are key elements in offshore structures due to their buoyancy properties. As buoyancy modules, they are used in a harsh environment (i.e., in sea water, at great depth with hydrostatic pressure), and their failure can lead to catastrophic consequences.

Therefore, the understanding and the prediction of their behavior in service is crucial. However, even if the suppliers and end-users have a significant knowledge of the behavior of these materials after years of use, the published literature is quite scarce. The purpose of this thesis is to investigate several problem areas that can be summarized as follows:

• Their mechanical behavior under hydrostatic pressure: how does the material volume change with the applied hydrostatic pressure? What is the material crush pressure?

• What is their water absorption rate that leads to direct buoyancy loss? Is it really the only factor leading to buoyancy loss over time, or can viscoelastic strains also make an important contribution?

• Are the current testing and qualifying methods adequate? The duration of the tests and qualification procedures (96 hours), is extremely short compared to the material service life (20 years or more).

The first part of this study is dedicated to the state of the art on these subjects. It is shown that epoxy syntactic foams are the main materials used for buoyancy applications and that their main advantages are their high strength and their good long-term behavior in a water environment under hydrostatic pressure. The epoxy foams usually present an elastic behavior until failure, and both characterization methods and predictive models for determining elastic properties are available. The damage behavior of syntactic foams has been studied and revealed collapse of microspheres inside the material. But the crush mechanisms have not been established and no predictive model exists. The origin of the reinforcing aspect of the epoxy resin on the microspheres has not been studied either, and the collapse mechanisms of microspheres, embedded in a resin or not, are still unclear. Several definitions of crush pressure could be applied, in this work it is defined as the pressure at which the material enters its densification region, and measured using the tangent method.

The knowledge on the long-term behavior of epoxy syntactic foams mostly concerns water absorption.

Descriptions of water absorption under high temperature and high pressure exist, but there is no identification of the mechanisms of water absorption when syntactic foams are immersed in cold (i.e., 25°C or less) water under hydrostatic pressure. Creep studies on epoxy syntactic foams do not exist, and buoyancy loss is assumed to only results from the water absorption. However, creep studies on polypropylene syntactic foams have revealed that creep under hydrostatic loading could lead to volume variations of the material, implying buoyancy loss. Finally, there is little published work concerning composite foams and hollow macrospheres, and what does exist is mainly concerned with the manufacturing process of such materials.

The second chapter has introduced the materials and the methods used in this study. For this work, model materials have been studied. Several compositions were available, allowing us to study the influence of each component on the material properties. The experimental methods for testing pure syntactics and composite foams, that already existed at the laboratory, have been improved to provide better results.

Some experimental procedures have also been developed during the thesis.

The third chapter focused on the hydrostatic strength of syntactic foams through experimental means. It revealed that the collapse of syntactic foams is governed by the collapse of the spherical inclusions in the material, with the weakest sphere (i.e., the largest and thinnest) collapsing first. For pure syntactic foams, the strength of the microspheres is enhanced by the surrounding matrix, while the strength of the macrospheres in composite foam is not improved. An empirical model was set up, that describes the influence of the microsphere volume fraction and resin rigidity on the crush pressure of the studied pure syntactic. For composite foams, since there is no reinforcing effect, their crush pressure can be estimated by knowing the collapse pressure of macrospheres. The macrospheres collapse due to their shell material failure in compression mode, and their collapse pressure can be estimated knowing the sphere dimensions and the maximal compressive stress of the shell material.

The fourth chapter focuses on the modelling of stress fields in syntactic foams. The stresses inside the hollow sphere wall thickness in syntactic foams, when subjected to hydrostatic loading were evaluated using an analytical model. This model indicates that while microspheres have all different sizes in the material, the resin reduces the stress in them, especially the weakest ones, explaining the origin of the strengthening effect observed experimentally in Chapter 3. This model predicts that increasing the microspheres volume fraction in the material reduces the strengthening effect, and that increasing the microspheres density and resin rigidity increases it, in agreement with experimental observations. It also reveals that macrospheres are not strengthened in composite foam because they have a lower radius ratio than the microspheres. This suggests that it is possible that making the macrospheres thinner would not reduce the composite foam strength, as the macrospheres would be reinforced, and it would make them less dense and provide better buoyancy. It would be interesting to test this hypothesis experimentally in the future. There is here a possible optimization of the composite foam to gain buoyancy without losing strength. For the first time, at the best of our knowledge, a numerical finite element model of composite foam based on the real structure defined by X ray tomography was set up, confirming that the main load on the inclusions in syntactic foams is hydrostatic, and that the hollow spheres are the load bearer in the material. Based on that model, it is possible to understand the volume change of syntactic foam under hydrostatic pressure.

The last two chapters are dedicated to the long-term behavior of the pure syntactic and the composite foam. The long-term behavior of pure syntactics used for buoyancy applications was studied, under low temperature (15°C) and high hydrostatic pressures (up to 110 MPa). The filling of microspheres with water in these conditions was observed, and the collapse of microspheres was highlighted (even if the applied pressure is below the HCP). The investigation of the influencing parameters revealed that two mechanisms are occurring during water absorption of pure syntactic epoxies, and that each can induce a collapse of the microspheres due to the presence of water, allowing the water to fill them. The difference between the two mechanisms comes from the way the water reaches the microspheres. The first one is controlled by water diffusion in the resin, a function of the square root of time, and accelerated by pressure. It leads to microsphere collapse, here supposed to be because of the resin swelling. The second mechanism is a function of time, following a mechanism that is still unclear. Here, this mechanism is described using a water flow that goes from one microsphere to another. This phenomenon greatly accelerates when a critical pressure is reached, that depends on the material's hydrostatic strength but largely below the HCP. This indicates that pure syntactics as buoyancy materials should be used below this critical pressure, to prevent the damage mechanism from becoming preponderant and leading to rapid degradation of the material. Using these results, a prediction of buoyancy loss was made, correlating very well with experimental results. It confirms, alongside creep tests, that in pure syntactics, creep can be neglected, and water absorption and buoyancy loss are correlated.

In Chapter 6, the buoyancy loss of composite foam has been studied and presented; as far as we know it is the first time that such a study is available in literature. The results reveal that buoyancy loss originates from two mechanisms, water absorption and creep of the macrospheres. The main phenomenon observed during the IBL tests on composite foams, which are usually employed to make buoyancy loss predictions, is creep. A description of creep and water absorption of composite foam has been proposed, that considers the influence of pressure on both phenomena. The water absorption being slow, it has been estimated considering that the material is entirely made of pure syntactic (this is one limitation of the proposed approach). Using this description, a prediction of buoyancy loss for composite foam can be made, revealing that when those materials are used for applications with short service lives (1 year), the water absorption can be neglected, and IBL tests are sufficient to characterize the buoyancy loss due to creep. The industrial method to extrapolate the curve and the method proposed here both give results in the same range, but the method developed here is more conservative, especially when using 96 hours IBL tests. The use of a longer test period, in both cases, improves the precision of the prediction. For long service life (20 years or more), water absorption becomes non negligible and must be considered, that must be studied through longer test periods (in this work, at least 2 months water uptake tests on pure syntactic samples of several sizes are used). The collapse of macrospheres due to creep has been highlighted, but due to the scatter of the results, no estimation of the time to failure of macrospheres in composite foam could be made.

This work addressed three main questions. Several answers have been found to each of those questions.

However, even though this study has allowed a better understanding of the material behavior under harsh environments, those answers have also brought new questions.

First, considering the mechanical behavior of syntactic foams, the collapse mechanisms of the microspheres in pure syntactic under hydrostatic loading have yet to be fully explained. In particular, the interactions between microspheres are not considered in this work, but can lead to stress concentrations in the material, therefore affecting the pure syntactic hydrostatic strength. The use of numerical modelling using finite elements should be a good approach to study those interactions. First versions of such models have been examined at IFREMER, in order to find suitable methods, but those are in the early stage and should be investigated in more detail in the future. The impact of the distance between the spheres, their size and thickness, and the surrounding resin, are all parameters that influence these interactions and must be considered. An approach using contact theory (such as the Hertz theory) might also be necessary, since some microspheres in the material are in contact with each other. The collapse of microspheres also leads to local stress field variations, that will affect the material collapse mechanisms. Here, the proposed model also considers the material solely in its elastic domain, but the use of less rigid matrix could lead to plastic strains in the material if the stress in the matrix becomes too high. This can be considered based on the analytical model presented here, using plasticization criteria in the stress estimations, and knowing the plastic behavior of the resin. Also, the impact of the loading speed was not investigated, but may be needed to further understand the damage mechanisms.

Concerning the empirical model, here the empirical constants must be determined each time a new type of microsphere is used. More data concerning the influence of the type of microsphere on the crush pressure of pure syntactic are needed to include this parameter in the model.

When not placed in the resin, the microsphere collapse mechanisms are still unclear: buckling or compressive failure of the glass? One of the issues of testing the microspheres using the presented method is that it does not totally prevent microsphere contact, and therefore the method can impact the microsphere collapse mechanism. Using approaches that tests microsphere one by one, and not by batch, would be better to understand their failure mode, but much more time consuming than testing by batch, as being made of glass a statistical approach may be needed. For macrospheres, it is proposed here to reduce their thickness to improve composite foam density while not reducing its strength by making the reinforcing effect appear, but this should still be verified experimentally.

Second, the water absorption in pure syntactic that comes from a diffusion mechanism is supposed to be related to the swelling of the resin, that induces stress concentration in the microspheres, leading to collapse. This hypothesis also needs more investigation. Once again, modelling seems an interesting way to investigate it, but a more detailed model than the one proposed here would be required to study the heterogeneous stress induced by the heterogeneous swelling in the material. Other hypotheses such as hydrolysis could explain the collapse of microspheres, but seem less likely at low temperature.

Nevertheless, hydrolysis of microspheres is not put aside yet and is currently being investigated. The impact of the interface between the microsphere and the resin should also be considered in the future, as SEM observations made during this project suggest that the microspheres are not always perfectly surrounded by the resin, creating small spaces for the water to enter, also changing the mechanical loading on the inclusions. The second phenomenon that generates water absorption in pure syntactic is described by water flowing within the material from microsphere to microsphere, and this point has to be confirmed in the future. Two methods could be envisaged. The first would be the use of X-ray tomography at higher resolution, for example nano tomography, to observe damage of microspheres that cannot be seen using microtomography because their thickness is around one micrometer. Another method would be the application of water absorption/drying sequences, as the presence of damage in the material might change the drying and absorption kinetics. Finally, more data are needed from samples of different size, to better estimate the absorption and flow rate in order to model the water absorption behavior with more accuracy.

Third, creep of composite foams was investigated, but the creep mechanisms should be studied to further improve their description. It would be interesting to dissociate the viscoelastic and plastic strains induced by creep, to predict the reversible and irreversible strains in the case of cyclic loading of buoyancy modules. Concerning cyclic use, it may also be a good approach to accelerate creep tests, though the main difficulty resides in the cycle frequency, which is difficult to increase due to the difficulty to increase the loading speed. The loading speed might also change the creep behavior, in particular accelerating the first stage, as in our case the loading speed is low, and therefore some creep might appear during the pressurization of the sample. Macrosphere collapse due to creep was highlighted, but the time to failure of macrospheres in composite foam should be studied to ensure that the material does not damage excessively during its service life. Increasing test temperature would accelerate the phenomenon and hence the study of creep and macrospheres collapse, either in composite foam or not.

One of the issues is that the time to failure of microspheres, when not in composite foam, strongly depends on geometric imperfections. It would be interesting to use X ray tomography to characterize those imperfections, or use a more statistical approach to predict microsphere lifetime. The mechanisms that transfer the creep of macrospheres to creep of composite foam are also still unclear. This creep in the composite material, might lead to shear stress in the pure syntactic, and experimental study of shear properties of pure syntactic should be considered. Numerical modelling would again be a useful tool, using the existing finite element model of the composite foam as a basis for future work.

Finally, the understanding of the water absorption in composite foam is still at an early stage. While it is here associated to water absorption in the matrix only, the water can also diffuse in the macrospheres, as they are made from reinforced epoxy material. This water absorption can lead to changes in the mechanical properties: some early studies performed at IFREMER, revealed that water absorption decreases the macrospheres strength. Eq. A. [START_REF] Saunders | Conversion of pressure to depth in the ocean[END_REF] 𝜎 𝑟𝑟 (HEM) = -𝑃 -4𝐺 (𝐻𝐸𝑀) 𝐻 𝑟 3

Eq. A.8

With A1,A2, B1, B2 and H constants depending on the boundary conditions, σrr and σθθ the stress fields (in MPa), K (MPa) the bulk modulus and G (MPa) the shear modulus, and (i), (m) and (HEM) indicating respectively the inclusion, matrix or homogenized medium .

The constants A1, A2, B1, B2 and H are determined using the null internal stress condition (Eq. A.9) and the continuity of normal stress (Eq. A.10 and Eq. A.12) and displacement (Eq. A.11 and Eq. A.13) from the hypothesis of perfect bonding. By replacing the stress fields with their analytical expressions, the boundary conditions become: These five equations can be represented, for better clarity, by the matrix system given Eq. A.19.

3𝐴 1 𝐾 (i) -4 𝐴 2 𝐺 (i) 𝑎 3 = 0 Eq. A.14 3𝐴 1 𝐾 (i) -4 𝐴 2 𝐺 (i) 𝑏 3 -3𝐵
( 3𝐾 (𝑖) -4 𝐺 (𝑖) 𝑎 3 0 0 0 3𝐾 (𝑖) -4 𝐺 (𝑖) 𝑏 3 -3𝐾 (𝑚) 4 𝐺 (𝑚) 𝑏 3 0 1 1 𝑏 3 -1 - 1 𝑏 3 0 0 0 3𝐾 (𝑚) -4 𝐺 (𝑚) 𝑐 3 4 𝐺 (𝐻𝐸𝑀) 𝑐 3 0 0 1 1 𝑐 3 - 1 𝑐 3 ) . ( 𝐴 1 𝐴 2 𝐵 1 𝐵 2 𝐻 ) = ( 0 0 0 -𝑃 - 𝑃 3𝐾 (𝐻𝐸𝑀))
Eq. A. [START_REF] Fick | On liquid diffusion[END_REF] The first equation Eq. A.14 gives us:

𝐴 2 = 𝐴 1 3𝐾 (i) 4𝐺 (i) 𝑎 3
Eq. A. [START_REF] Crank | The mathematics of diffusion[END_REF] Then, substituting the analytical expression of A2 in Eq. A.15 and Eq. A.16 (and dividing Eq. A.16 by b) we obtain the following equations:

3𝐴 1 𝐾 (i) (1 -( 𝑎 𝑏 ) 3 ) -3𝐵 1 𝐾 (𝑚) + 4 𝐵 2 𝐺 (m) 𝑏 3 = 0 Eq. A.21 𝐴 1 (1 + 3𝐾 (i) 4𝐺 (i) ( 𝑎 𝑏 ) 3 ) -𝐵 1 - 𝐵 2 b 3 = 0 Eq. A.22
The operations (Eq. A.21 + Eq. A.22 * 4G (m) ) is made, leading to the following equation:

𝐴 1 (3𝐾 (i) (1 -( 𝑎 𝑏 ) 3 ) + 4𝐺 (𝑚) (1 + 3𝐾 (g) 4𝐺 (i) ( 𝑎 𝑏 ) 3
)) -𝐵 1 (3𝐾 (𝑚) + 4𝐺 (m) ) = 0

Eq. A. [START_REF] Bardin | Durabilité des élastomères thermoplastiques pour applications marines[END_REF] Then we do the operation (Eq. A.21 -Eq. A.22 * 3K (m) ) to obtain the following equation:

𝐴 1 (3𝐾 (i) (1 -( 𝑎 𝑏 ) 3 ) -3𝐾 (𝑚) (1 + 3𝐾 (i) 4𝐺 (i) ( 𝑎 𝑏 ) 3
)) + 𝐵 2 𝑏 3 (4𝐺 (m) + 3𝐾 (𝑚) ) = 0 Eq. A. [START_REF] Arhant | A study of pure hydrolysis of carbon fibre reinforced polyamide 6 composites tested under mode I loading[END_REF] Both equations Eq. A.23 and Eq. A.24 can be reduced and give:

(4𝐺 (m) + 3𝐾 (𝑚) )𝐵 1 = 𝐴 1 (3𝐾 (i) (1 -( 𝑎 𝑏 ) 3 ) + 4𝐺 (𝑚) (1 + 3𝐾 (i) 4𝐺 (i) ( 𝑎 𝑏 ) 3
))

Eq. A.25

(4𝐺 (m) + 3𝐾 (𝑚) )𝐵 2 = 𝐴 1 𝑏 3 (-3𝐾 (i) (1 -( 𝑎 𝑏 ) 3 ) + 3𝐾 (𝑚) (1 + 3𝐾 (i) 4𝐺 (i) ( 𝑎 𝑏 ) 3
))

Eq. A. [START_REF]API Spec 16F Cementing Float Equipment[END_REF] Then, we divide each equation Eq. A.25 and Eq. A.26 by 3K (m) and divide Eq. A.26 by c 3 to obtain:

(𝛽 (𝑚) + 1)𝐵 1 = 𝐴 1 (𝛷(1 -𝜂 3 ) + 𝛽 (𝑚) (1 + 𝜂 3 𝛽 (𝑖) ))

Eq. A. [START_REF]API Spec 17L Specification for Flexible Pipe Ancillary Equipment[END_REF] (𝛽 (𝑚) + 1)𝐵 2 𝑐 3 = 𝐴 1 𝑓 (-𝛷(1 -𝜂 3 ) + (1 + 𝜂 3 𝛽 (𝑖) ))

Eq. A.28

With f = (b/c) 3 and 𝛷 = 𝐾 (𝑖) 𝐾 (𝑚) and 𝛽 (𝑘) = 3𝐾 (k) 4𝐺 (k) with (k) = (i) for the inclusion, and (m) for the matrix.

After that, we sum both equations Eq. A.27 and Eq. A.28 to obtain a new equation:

(𝛽 (𝑚) + 1) ( 𝐵 2 𝑐 3 + 𝐵 1 ) = 𝐴 1 (𝛷(1 -𝑓)(1 -𝜂 3 ) + (𝛽 (𝑚) + 𝑓) (1 + 𝜂 3 𝛽 (𝑔) )) Eq. A. [START_REF] Ruckebusch | Microsphères creuses de verre pour mousses syntactiques[END_REF] Now, we focus on Eq. A.18. We divide it by c, and multiply it by (1+β (m) ) to obtain:

(𝐵 1 + 𝐵 2 𝑐 3 )(1 + 𝛽 (𝑚) ) - H 𝑐 3 (1 + 𝛽 (𝑚) ) = - 𝑃 3𝐾 (𝐻𝐸𝑀) (1 + 𝛽 (𝑚) )
Eq. A. [START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF] After that, we substitute Eq. A.29 in Eq. A.30 to obtain:

𝐴 1 (𝛷(1 -𝑓)(1 -𝜂 3 ) + (𝛽 (𝑚) + 𝑓) (1 + 𝜂 3 𝛽 (𝑔) )) - 𝐻 𝑐 3 (1 + 𝛽 (𝑚) ) = -𝑃 (1 + 𝛽 (𝑚) )
3𝐾 (𝐻𝐸𝑀) Eq. A.31

Eq. A.31 can be reduce to:

𝐻 𝑐 3 (1 + 𝛽 (𝑚) ) = 𝑃 (1 + 𝛽 (𝑚) ) 3𝐾 (𝐻𝐸𝑀) + 𝐴 1 (𝛷(1 -𝑓)(1 -𝜂 3 ) + (𝛽 (𝑚) + 𝑓) (1 + 𝜂 3 𝛽 (𝑔) ))
Eq. A. [START_REF] Bratt | Mechanical testing of glass hollow microspheres[END_REF] Then we focus on Eq. A.17, in which we replace B1, B2 and H by their expression with A1 (obtained from Eq. A.27, Eq. A.28 and Eq. A.32):

3𝐾 (m) 𝐴 1 (1 + 𝛽 (𝑚) ) . (𝛷(1 -𝜂 3 ) + 𝛽 (𝑚) (1 + 𝜂 3 𝛽 (𝑖) )) - 4𝐺 (𝑚) 𝐴 1 (1 + 𝛽 (𝑚) ) 𝑓 (-𝛷(1 -𝜂 3 ) + (1 + 𝜂 3 𝛽 (𝑖) )) + 𝑃𝛽 (𝐻𝐸𝑀) + 𝐴 1 4𝐺 (HEM) (1 + 𝛽 (𝑚) ) (𝛷(1 -𝑓)(1 -𝜂 3 ) + (𝛽 (𝑚) + 𝑓) (1 + 𝜂 3 𝛽 (𝑖) )) = -𝑃 Eq. A.33
We divide Eq. A.33 by 3K (m) to obtain:

𝐴 1 (1 + 𝛽 (𝑚) ) (𝛷(1 -𝜂 3 ) + 𝛽 (𝑚) (1 + 𝜂 3 𝛽 (𝑖) )) - 𝛽 (𝑚) 𝐴 1 (1 + 𝛽 (𝑚) ) 𝑓 (-𝛷(1 -𝜂 3 ) + (1 + 𝜂 3 𝛽 (𝑖) )) + 𝐴 1 4𝐺 (HEM) 3𝐾 (m) (1 + 𝛽 (𝑚) ) (𝛷(1 -𝑓)(1 -𝜂 3 ) + (𝛽 (𝑚) + 𝑓) (1 + 𝜂 3 𝛽 (𝑖) )) = -𝑃 ( 1 + 𝛽 (𝐻𝐸𝑀) 3𝐾 (m) )

Eq. A.34

With 𝛽 (𝐻𝐸𝑀) = 3𝐾 (HEM) 4𝐺 (𝐻𝐸𝑀) This equation can be simplified to Eq. A.35: Eq. A. [START_REF] Wunderlich | Buckling behaviour of imperfect spherical shells[END_REF] Then simplifying again:

𝐴 1 (1 + 𝛽 (𝑚) ) (𝛷(1 -𝜂 3 ) (1 + 𝛽 (𝑚) 𝑓 + 4𝐺 (HEM) 3𝐾 (𝑚) (1 -𝑓)) + (1 + 𝜂 3 𝛽 (𝑖) ) (𝛽 (𝑚) (1 -𝑓) + 4𝐺 (HEM) 3𝐾 (𝑚) (𝛽 (𝑚) + 𝑓))) = -𝑃 ( 1 + 𝛽 (𝐻𝐸𝑀) 3𝐾 (m) )
Eq. A. [START_REF] Deriano | Physical and mechanical properties of a new borosilicate glass[END_REF] Now we have the expression of A1 in function of the applied pressure P and the constituent's elastic properties, and the expression of each unknown with P and A1. Thus, we can calculate the maximal stress in a microsphere, which is the stress inside the glass at the inner surface (r=a), which can be expressed:

𝜎 𝜃𝜃 (i) (r = a) = 3𝐴 1 𝐾 (i) + 2 𝐴 2 𝐺 (i) 𝑎 3 = 𝐴 1 9𝐾 (i) 2
Eq. A.37

By substituting Eq. A.36 in Eq. A.37, we obtain:

𝜎 𝜃𝜃 (i) (r = a) = -𝑃 ( 1 + 𝛽 (𝐻𝐸𝑀) 3𝐾 (m) ) 9𝐾 (g) (1 + 𝛽 (𝑚) ) 2 (𝛷(1 -𝜂 3 ) (1 + 𝛽 (𝑚) 𝑓 + 4𝐺 (HEM) 3𝐾 (𝑚) (1 -𝑓)) + (1 + 𝜂 3 𝛽 (𝑖) ) (𝛽 (𝑚) (1 -𝑓) + 4𝐺 (HEM) 3𝐾 (𝑚) (𝛽 (𝑚) + 𝑓)))
Eq. A. [START_REF] Chen | Radiation effects on structure and mechanical properties of borosilicate glasses[END_REF] Finally, the maximal stress in a microsphere can be written as: With the expression of A1, it is possible to estimate the expressions of the other constants, and therefore estimating the stress inside the resin surrounding the microsphere. Les matériaux développés pour répondre à ce besoin de flottabilité sont les mousses syntactiques [START_REF] Fonblanc | Mousses syndactiques: matériaux composites pour grandes profondeurs[END_REF].

Ce sont des matériaux composites constitués de sphères creuses de différentes tailles (de micromètres à centimètres) enrobées dans une résine polymère (Fig. Les principales qualifications visées sont les suivantes : les mousses syntactiques doivent avoir une résistance à la compression hydrostatique 1.25 à 1.5 fois leur pression de service, des faibles variations de volume durant la mise en service, et présenter une perte de flottabilité à la fin de leur durée de vie inférieure à 5%. Les prédictions de perte de flottabilité sont effectuées à partir de mesure de variation de flottabilité en caisson hyperbare, durant des essais de durée d'un minimum de 96 heures, qui sont ensuite extrapolées sur la durée de vie complète du matériau [START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF].

Ces qualifications soulèvent plusieurs problématiques qui doivent être étudiées :

• Quel est le comportement mécanique des mousses syntactiques face à la pression hydrostatique ? Comment le volume du matériau change sous la pression ? Comment définir sa pression de rupture ?

• Quelle est la vitesse d'absorption d'eau dans les mousses syntactiques, qui mène à une perte de flottabilité ? Est-ce seulement le seul facteur à l'origine de cette perte, ou des déformations viscoélastiques peuvent elles aussi y contribuer ?

Il est également nécessaire de s'assurer que les méthodes d'essai soient adaptées pour caractériser le comportement du matériau, basé sur les phénomènes physiques impliqués plutôt que sur des observations empiriques, pour améliorer la compréhension et se prévenir de phénomènes inattendus. De plus, aujourd'hui, les durées de tests pour prédire la perte de flottabilité des mousses syntactiques sont généralement de 96 heures, ce qui est dérisoire au regard des durées de vie de 20 ans ou plus. IL faut s'assurer de la pertinence de tels tests.

Le comportement mécanique des pures syntactiques (mousses syntactiques contenant uniquement des microsphères) a été étudié par de nombreux auteurs [START_REF] Jayavardhan | Quasi-static compressive response of compression molded glass microballoon/HDPE syntactic foam[END_REF][START_REF] Gupta | Compression properties of syntactic foams: effect of cenosphere radius ratio and specimen aspect ratio[END_REF][START_REF] Gupta | Comparison of compressive properties of layered syntactic foams having gradient in microballoon volume fraction and wall thickness[END_REF][START_REF] Gupta | Comparison of tensile and compressive characteristics of vinyl ester/glass microballoon syntactic foams[END_REF][START_REF] Gupta | Hygrothermal studies on syntactic foams and compressive strength determination[END_REF][START_REF] Gupta | Characterization of mechanical and electrical properties of epoxy-glass microballoon syntactic composites[END_REF][START_REF] Song | Dynamic compressive response and failure behavior of an epoxy syntactic foam[END_REF][START_REF] Huang | Elastic behaviour and failure mechanism in epoxy syntactic foams: The effect of glass microballoon volume fractions[END_REF][START_REF] Ale | Dynamic attenuation and compressive characterization of syntactic foams[END_REF][START_REF] Huang | X-ray microtomography and finite element modelling of compressive failure mechanism in cenosphere epoxy syntactic foams[END_REF][START_REF] Gall | Novel mechanical characterization method for deep sea buoyancy material under hydrostatic pressure[END_REF][START_REF] Liu | Novel test method for the hydrostatic properties of syntactic foams[END_REF], mais principalement en compression uniaxiale, qui n'est pas représentatif du chargement que subissent les modules de flottabilité. Les études sous chargement hydrostatique sont moins nombreuses, du fait de la mise en place plus complexe de ces méthodes. Néanmoins, des connaissances sur ce sujet existent. Elles révèlent que lors de la mise sous pression, les pures syntactiques époxy présentent un comportement élastique, puis un plateau de densification qui correspond à une perte de volume drastique du matériau (Fig. V) [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]. Ce plateau de densification correspond à la rupture du matériau. Enfin, il n'existe que peu d'étude concernant les mousses composites (avec macrosphères), et elles concernent principalement les procédés de fabrication de ces matériaux [START_REF] Wu | Preparation and characterization of three phase epoxy syntactic foam filled with carbon fiber reinforced hollow epoxy macrospheres and hollow glass microspheres[END_REF].

A partir de cet état de l'art, le cadre de l'étude a pu être établi. Il est présenté Fig. VI, et s'établi autour de deux axes :

• L'étude de la résistance à la compression hydrostatique du matériau, qui définit les profondeurs possibles d'utilisation des mousses syntactiques. Le but est de mieux comprendre la réponse face à la compression hydrostatique des matériaux, via l'étude des paramètres qui influencent leur résistance.

• L'étude du comportement à long terme des mousses syntactiques, des phénomènes à l'origine de la perte de flottabilité, pour proposer des prédictions fiables à partir de tests adéquats. Vient ensuite l'étude de la résistance des macrosphères, en les mettant, individuellement, sous pression jusqu'à rupture. La pression de rupture de chaque macrosphère est relativement variable, mais est plus importante pour les macrosphères en fibre de carbone qu'en fibre de verre. Les dimensions des macrosphères en fibre de verre furent mesurées par tomographie à rayon X. En estimant les contraintes dans leur coque à partir de leur dimension, il a été montré que la contrainte circonférentielle à rupture ne dépend pas de la géométrie de la sphère. Les macrosphères cassent donc à cause de la rupture en compression du matériau qui constitue la coque (le composite époxy/fibre de verre).

Après s'être concentré sur les sphères creuses, l'étude se concentre ensuite sur les purs syntactiques.

Ces matériaux sont mis sous pressions, jusqu'à rupture. Un comportement similaire à celui décrit dans la littérature [START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF] est observé sur les courbes pression/volume (Fig. La rupture des pures syntactiques est régie par la rupture des microsphères en son sein. Les estimations de contraintes à l'aide du modèle ont ensuite été transposées au cas des macrosphères dans les mousses composites. Les macrosphères au sein du matériau subissent des niveaux de contraintes équivalents au cas où elles ne seraient pas entourées d'une matrice pure syntactique. Puisqu'il n'y a pas de réduction de contrainte, il n'y a pas de renforcement. Cette réduction de contrainte n'apparait pas car le ratio géométrique des macrosphères est proche de 0.9 (Fig. Abstract: Nowadays, in the offshore industry, oil companies extract the petroleum from the supplies located at the sea floor. Those oil deposits are located at depth up to 3000 m, requiring the use of floating platform and pipelines of several kilometer length. Because of their length, those pipelines are heavy. It is necessary to add buoyancy to those pipelines to reduce their apparent weight and reduce their fatigue stress that can lead to failure. For that, buoyancy modules are attached on the pipelines. They need to be able to withstand the high pressure of the deep sea and keep their integrity over their lifetime of up to 27 years. The materials that developed in response of this need of buoyancy are syntactic foams. They are composite materials made of hollow spheres, surrounded by a polymer resin. One of the challenges of the oil companies is to access new oil supplies located at higher depth.

The aim of this study is to improve the knowledge and understandings of the behavior of syntactic foams, to ensure their usability at these new depths over long period, and eventually extend the service life of the existing buoyancy modules. The first part of this study focuses on the mechanical resistance of syntactics foams subjected to hydrostatic loading, through experimental and modelling methods. An empirical description of syntactic foam hydrostatic strength is proposed to predict it. The second part focuses on the durability of syntactic foams. The origin of buoyancy loss, that are water absorption and creep, are highlighted. Mathematical descriptions of these phenomena are proposed to make life expectancy predictions.

Fig. 0 . 1

 01 Fig. 0.1 West Africa Dalia Field (TotalEnergies) .....................................................................................................................

Fig. 0. 2 (Fig. 1 . 1 Fig. 1 . 2 Fig. 1 . 6

 2111216 Fig. 0.2 (Left) Buoyancy module (Right) Schematic of the location of buoyancy modules on pipeline: steepwave flexible riser configuration (Balmoral) .......................................................................................................................................................... Fig. 1.1 Composition of a syntactic foam ................................................................................................................................. Fig. 1.2 Typical curve obtained from IBL tests on syntactic foam [10]. Zone1: pressurization. Zone2: constant pressure. Zone3: Depressurization to atmospheric pressure [16]. ........................................................................................................................ Fig. 1.3 (left) SEM picture of hollow glass microspheres [29], (Right) Example of microspheres size distribution [28] ........ Fig. 1.4 Example of 3M TM microspheres grade and description ............................................................................................... Fig. 1.5 Compression curve from isostatic testing of microspheres. The percentage of survived microspheres is found from the difference between compression and collapse curve [30] ........................................................................................................ Fig. 1.6 Aspect ratio (here diameter/thickness) vs hydrostatic buckling pressure for individual microspheres [32] ................

Fig. 1 . 7

 17 Fig. 1.7 Typical stress/strain curve during uniaxial compression test on epoxy pure syntactics [42] .......................................

Fig. 1 . 8

 18 Fig. 1.8 Picture of different sample geometry of pure syntactic tested in uniaxial compression (left). Stress/strain curve of the different sample geometry tested in uniaxial compression (right) [51] ....................................................................................

Fig. 1 . 9 Fig. 1 .

 191 Fig. 1.9 Pressure/volume strain curves of hydrostatic compression tests on pure syntactic made of epoxy and GB38 microspheres (blue), named GSEP, and polyurethane and S38 microspheres (red) named GSPU [14] ................................... Fig. 1.10 Evolution of microstructure of epoxy pure syntactic before (left) and after (right) pressurization [14]. ................... Fig. 1.11 Evolution of microstructure of polyurethane pure syntactic before (left) and after (right) pressurization [14]. ........ Fig. 1.12 The three-phase model used to describe syntactic foam mechanical behavior [62] .................................................. Fig. 1.13 Example of a finite element mesh of a representative cell of syntactic foams [67] ................................................... Fig. 1.14 Example of a mesh cell of syntactic foam containing several microspheres [68]......................................................

Fig. 1 .

 1 Fig. 1.15 Water concentration curve following a classical Fick behavior (0) or other non-Fick behavior (1-4) [72] ...............

Fig. 1 .

 1 Fig. 1.16 The main directions of diffusion in a unidirectional composite fiber material [71] ..................................................

Fig. 1 .

 1 Fig. 1.17 Water concentration in epoxy syntactic foam for different filler contents: microspheres volume fraction = 0 (1), 0,1

  . ....................................................................................................................

Fig. 1 .

 1 Fig. 1.18 Water absorption VS hydrostatic pressure (psi) for epoxy pure syntactics [10] ........................................................

Fig. 1 .

 1 Fig. 1.19 Scheme of the water absorption model in syntactic foam under high pressure/temperature [86] ..............................

Fig. 1 .

 1 Fig. 1.20 The three stages of creep [86] ................................................................................................................................... Fig. 1.21 Creep and recovery of materials [87] ........................................................................................................................

Fig. 1 .

 1 Fig. 1.22 Kelvin-Voight model (left) and Maxwell model (right) ............................................................................................

Fig. 1 . 39 Fig. 1 . 40 Fig. 1 . 41 Fig. 2 . 44 Fig. 2 . 2 45 Fig. 2 . 3 46 Fig. 2 . 4 47 Fig. 2 . 5 47 Fig. 2 . 6 48 Fig. 2 . 7 48 Fig. 2 . 8 50 Fig. 2 . 51 Fig. 2 . 51 Fig. 2 . 52 Fig. 2 . 53 Fig. 2 . 54 Fig. 2 . 55 Fig. 2 . 56 Fig. 2 . 57 Fig. 2 . 58 Fig. 2 . 58 Fig. 3 . 1 63 Fig. 3 . 2 64 Fig. 3 . 4 64 Fig. 3 . 5 65 Fig. 3 . 6 65 Fig. 3 . 66 Fig. 3 . 8 67 Fig. 3 . 9 68 Fig. 3 . 69 Fig. 3 . 72 Fig. 3 . 72 Fig. 3 . 73 Fig. 3 . 75 Fig. 3 . 76 Fig. 3 . 77 Fig. 3 . 78 Fig. 3 . 79 Fig. 3 . 80 Fig. 3 . 82 Fig. 3 . 83 Fig. 3 . 84 Fig. 4 . 1 86 Fig. 4 . 2 95 Fig. 4 . 3 97 Fig. 4 . 4 98 Fig. 4 . 5 99 Fig. 4 . 6 101 Fig. 4 . 103 Fig. 4 . 9 Fig. 4 .Fig. 5 . 1 108 Fig. 5 . 2 108 Fig. 5 . 3 111 Fig. 5 . 5 112 Fig. 5 . 6 112 Fig. 5 . 7 114 Fig. 5 . 8 Fig. 5 . 9 Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 6 . 1 Fig. 6 . 5 Fig. 6 . 8 Fig. 6 . 9 Fig. 6 .Fig. 6 .Fig. 6 .Fig. 6 .Fig. 6 .Fig. 6 .

 1391401412442245234624472547264827482850251251252253254255256257258258316332643464356536653663867396836937237237337537637737837938038238338441864295439744984599461014103494511085210853111551125611257114585955555555555555561656869666666 Fig. 1.23 Hydrostatic creep test on polypropylene syntactic foam under different hydrostatic loads [91] ............................... 39 Fig. 1.24 Three phase epoxy syntactic foam, containing GB38 microspheres and carbon fiber macrospheres [94]. ............... 40 Fig. 1.25 Scheme of the scope of this project ........................................................................................................................... 41 Fig. 2.1 MEB observation of GB38 microspheres .................................................................................................................... 44 Fig. 2.2 Picture of a glass fiber macrosphere MG (left), carbon fiber macrosphere MC1 (middle), and glass fiber macrosphere cut in half (right) ...................................................................................................................................................................... 45 Fig. 2.3 Picture of a cylindrical composite foam sample .......................................................................................................... 46 Fig. 2.4 Picture of cubic and plate samples of anhydride-based epoxy E1 (left) and amine-based epoxy E2 (right) ............... 47 Fig. 2.5 Cylindrical samples of epoxy (left) and paraffin (right) for uniaxial compression tests.............................................. 47 Fig. 2.6 Mixture of microspheres/glycerol in a sealed plastic bag ............................................................................................ 48 Fig. 2.7 Picture of samples of epoxy pure syntactic ................................................................................................................. 48 Fig. 2.8 Set up for weight measurements in air and in water .................................................................................................... 49 Fig. 2.9 Left: composite foam sample with its ballast, attached using chains to the load cell. Right: composite foam sample being weighted in a water tank ................................................................................................................................................. 50 Fig. 2.10 Epoxy sample in the tensile machine with parallel plates for compression tests ....................................................... 51 Fig. 2.11 Particles moving through the spread parallel laser beam, creating diffraction [97] ................................................... 51 Fig. 2.12 (left) Schematic diagram of the hydrostatic compression test using a weighing device [14], (right) Picture of the 60 MPa pressure vessel ................................................................................................................................................................. 52 Fig. 2.13 Scheme of the hydrostatic compression test using a piston vessel [14] (left). Picture of the 200 MPa piston vessel (right) ....................................................................................................................................................................................... 53 Fig. 2.14 Hyperbaric vessel for compression tests with strain gauge measurements. The gauge cable passes through the hole in the middle of the vessel plug. ................................................................................................................................................... 54 Fig. 2.15 Picture of the 100 MPa pressure vessel for HCP tests on composite foam ............................................................... 55 Fig. 2.16 Composite foam sample surrounded by the metallic structure (left) and then by a bag before being tested ............. 56 Fig. 2.17 Pressure vessel for water absorption tests on pure syntactic (left) -Pressure vessels in water tank for temperature control (right) ........................................................................................................................................................................... 57 Fig. 2.18 Schematic diagram showing the principle of the IBL test (left), Sample of composite foam placed in the set-up (middle), picture of the measurements set-up (right) ................................................................................................................ 58 Fig. 2.19 Pressure vessel used for IBL tests and water absorption measurements on composite foam. 60 MPa pressure vessel (left) and 30 MPa pressure vessel (right) .................................................................................................................................. 58 Fig. 3.1 Pressure/volume loss curve of GB15 microspheres subjected to hydrostatic loading -21°C -1.5MPa/min ............. 63 Fig. 3.2 Microspheres size distribution by number................................................................................................................... 63 Fig. 3.3 (a) Image of the microstructure of pure syntactic through X ray tomography measurements. (b) Same image after grey level processing, to consider only the glass (in white) on the picture, from which thickness measurements of microspheres are made. ........................................................................................................................................................................................ 64

98 Table 4 . 6 99 Table 4 . 7 100 Table 4 . 8 103 Table 5 . 1

 984699471004810351 .................... 35 Table 2.1 Resins used in the study and their characteristics ..................................................................................................... 43 Table 2.2 Microspheres used in the study and their densities ................................................................................................... 44 Table 2.3 Macrospheres used in the study ................................................................................................................................ 45 Table 2.4 Pure syntactics and their composition ...................................................................................................................... 45Table 2.5 Composite foams constitution .................................................................................................................................. 46 Table 3.1 Microspheres collapse pressure. ............................................................................................................................... 63 Table 3.2 Hydrostatic strength of composite hollow sphere. .................................................................................................... 66 Table 3.3 Crush pressure of E1_GB15 pure syntactic at 21°C depending on the method used and the sample size ................ 69 Table 3.4 Uniaxial compressive modulus of the tested resins. ................................................................................................. 71 Table 3.5 Hydrostatic strength of pure syntactic with different kind of microspheres ............................................................. 76 Table 3.6 Value of the constant of the empirical crush pressure model or different type of microspheres .............................. 77 Table 3.7 Crush pressure values of composite foam depending on their composition ............................................................. 82 Table 4.1 Comparison between experimental results and model prediction of bulk modulus of pure syntactics ..................... 89 Table 4.2 Estimation of elastic properties of composite foam, compared with experimental measurements ........................... 89 Table 4.3 Estimated mean radius ratio and thickness for the considered microspherescomparison with thickness measurements ........................................................................................................................................................................... 95 Table 4.4 Parameters used for the estimations of stress in microspheres in E1_GB15 pure syntactic using the analytical model presented in this study .............................................................................................................................................................. 96 Table 4.5 Parameters used for the estimations of stress in microspheres in pure syntactic depending on the microspheres volume fraction and resin Young's modulus ......................................................................................................................................... Parameters used for the estimations of stress in microspheres in pure syntactic depending on the microspheres' average density ......................................................................................................................................................................... Parameters used for the estimations of stress in macrospheres in composite foam CF_E2_GB25 using the analytical model resented in this work .................................................................................................................................................... Parameters of the finite element model of composite foam CF_E2_GB25 ............................................................ Identified diffusion coefficient and maximal water uptake in neat resin at 15°C ................................................... 108 Table 5.2 Comparison between experimental and theoretical value of water absorption in pure syntactic, considering that only the resin absorb water ............................................................................................................................................................. 110 Table 5.3 Water absorption measurements of E2_GB15 immersed under different hydrostatic pressure, before being analyzed by X-ray tomography. ............................................................................................................................................................ 110

Fig. 0. 1

 1 Fig. 0.1 West Africa Dalia Field (TotalEnergies)

Fig. 0. 2 (

 2 Fig. 0.2 (Left) Buoyancy module (Right) Schematic of the location of buoyancy modules on pipeline: steepwave flexible riser configuration (Balmoral)

  The second chapter is devoted to the presentation of model materials used in this study (more than 150 different formulations) as well as experimental methods. Then Chapter 3 examines the hydrostatic resistance of each component of syntactic foam materials based mainly on experimental considerations, and an analytical model is proposed and compared to experimental observations in Chapter 4. Both Chapter 5 and Chapter 6 focus on the long-term behavior of syntactic foams. Chapter 5 considers water absorption in pure syntactic, especially the determination of mechanisms involved in this process based on experimental data. The last chapter focuses on the buoyancy loss of composite foams under hydrostatic pressure, with a discussion on the extrapolation of experimental data obtained in the laboratory. Finally, conclusions and perspectives of this work are proposed.

Fig. 1 . 2

 12 Fig. 1.2 Typical curve obtained from IBL tests on syntactic foam [10]. Zone1: pressurization. Zone2: constant pressure. Zone3: Depressurization to atmospheric pressure [16].

Fig. 1 . 3 (

 13 Fig. 1.3 (left) SEM picture of hollow glass microspheres [29], (Right) Example of microspheres size distribution [28]

Fig. 1 . 4

 14 Fig. 1.4 Example of 3M TM microspheres grade and description

Fig. 1 .

 1 5 provides an example of the two curves, which are used to determine the percentage of collapsed microspheres in the test.

Fig. 1 . 5

 15 Fig. 1.5 Compression curve from isostatic testing of microspheres. The percentage of survived microspheres is found from the difference between compression and collapse curve[START_REF] Carlisle | Structure and mechanical properties of micro and macro balloons: An overview of test techniques[END_REF]

Fig. 1 . 6

 16 Fig. 1.6 Aspect ratio (here diameter/thickness) vs hydrostatic buckling pressure for individual microspheres [32]

Fig. 1 . 7

 17 Fig. 1.7 Typical stress/strain curve during uniaxial compression test on epoxy pure syntactics [42]

Fig. 1 . 8

 18 Fig. 1.8 Picture of different sample geometry of pure syntactic tested in uniaxial compression (left). Stress/strain curve of the different sample geometry tested in uniaxial compression (right) [14]

Fig. 1 . 9

 19 Fig. 1.9 Pressure/volume strain curves of hydrostatic compression tests on pure syntactic made of epoxy and GB38 microspheres (blue), named GSEP, and polyurethane and S38 microspheres (red) named GSPU [14]

Fig. 1 .Fig. 1 .

 11 Fig. 1.10 Evolution of microstructure of epoxy pure syntactic before (left) and after (right) pressurization [14].

Fig. 1 .

 1 Fig. 1.12 The three-phase model used to describe syntactic foam mechanical behavior [62]

Fig. 1 .

 1 Fig. 1.13 Example of a finite element mesh of a representative cell of syntactic foams [67]

Fig. 1 .

 1 Fig. 1.14 Example of a mesh cell of syntactic foam containing several microspheres [68]

Fig. 1 .

 1 Fig. 1.15 Water concentration curve following a classical Fick behavior (0) or other non-Fick behavior (1-4) [72]

Fig. 1 .

 1 Fig. 1.16 The main directions of diffusion in a unidirectional composite fiber material [71]

Fig. 1 .

 1 Fig. 1.17 Water concentration in epoxy syntactic foam for different filler contents: microspheres volume fraction = 0 (1), 0,1 (2), 0,2 (3) 0,3 (4), 0,4 (5), 0,5 (6) et 0,6 (7) [83].

Fig. 1 .

 1 Fig. 1.18 Water absorption VS hydrostatic pressure (psi) for epoxy pure syntactics [10]

Fig. 1 .

 1 Fig. 1.19 Scheme of the water absorption model in syntactic foam under high pressure/temperature [86]

  It is usually described by 3 different stages presented in Fig. 1.20. First, an elastic strain response ε0 is observed. Then, creep occurs at a decreasing rate, it is the primary stage of creep. The secondary stage of creep corresponds to the part of the curve where the creep rate is almost constant. The third stage corresponds to the final acceleration of creep until the material failure. Depending on the stress applied to the material, the strain response might only present the first stage of creep, the first and secondary stages, or if the stress is high enough the three stages of creep [86].

Fig. 1 .

 1 Fig. 1.20 The three stages of creep [86]

Fig. 1 .

 1 Fig. 1.22 Kelvin-Voight model (left) and Maxwell model (right)

Fig. 1 .

 1 Fig. 1.23 Hydrostatic creep test on polypropylene syntactic foam under different hydrostatic loads [91]

Fig. 1 .

 1 Fig. 1.24 Three phase epoxy syntactic foam, containing GB38 microspheres and carbon fiber macrospheres [94].

  Experimental characterization of the mechanical behavior of such materials was developed over the years. The epoxy foams usually present an elastic behavior until failure, and both characterization methods and predictive models for determining elastic properties are available. The damage behavior of syntactic foams has been studied and revealed collapse of microspheres inside the material. But the crush mechanisms have not been established since damage quantification could not be done for epoxy syntactic foams. While parameters influencing the hydrostatic strength of syntactic foams have been highlighted, no predictive model exists. The origin of the reinforcing aspect of the epoxy resin on the microspheres has not been studied either, and the collapse mechanisms of microspheres, embedded in a resin or not, are still unclear.The long-term behavior of epoxy syntactic foams mostly concerns water absorption. Descriptions of water absorption under high temperature and high pressure exist, but there is no identification of the mechanisms of water absorption when syntactic foams are immersed in cold water. Creep studies on epoxy syntactic foams do not exist, and buoyancy loss is assumed to come only from the water absorption. But creep studies on polypropylene syntactic foams have revealed that creep under hydrostatic loading could lead to volume variations of the material, implying buoyancy loss. Finally, there is little published work concerning composite foams and hollow macrospheres, and what does exist is mainly concerned with the manufacturing process of such materials.

Fig. 1 .

 1 Fig. 1.25 Scheme of the scope of this project

Fig. 2 .

 2 Fig. 2.1 MEB observation of GB38 microspheres

Fig. 2 . 2

 22 Fig. 2.2 Picture of a glass fiber macrosphere MG (left), carbon fiber macrosphere MC1 (middle), and glass fiber macrosphere cut in half (right)

2. 2 . 1 . 1 .

 211 Resin sample preparationEpoxy resin samples for water absorption tests are plane (50*50*1 and 50*50*4 mm 3 ) or cubic (50*50*50 mm 3 ) samples (Fig. 2.4) cut with a wet saw.

Fig. 2 . 4

 24 Fig. 2.4 Picture of cubic and plate samples of anhydride-based epoxy E1 (left) and amine-based epoxy E2 (right)

Fig. 2 . 5

 25 Fig. 2.5 Cylindrical samples of epoxy (left) and paraffin (right) for uniaxial compression tests

Fig. 2 . 6

 26 Fig. 2.6 Mixture of microspheres/glycerol in a sealed plastic bag

Fig. 2 . 7

 27 Fig. 2.7 Picture of samples of epoxy pure syntactic

Fig. 2 . 9

 29 Fig. 2.9 Left: composite foam sample with its ballast, attached using chains to the load cell. Right: composite foam sample being weighted in a water tank

Fig. 2 .

 2 Fig. 2.10 Epoxy sample in the tensile machine with parallel plates for compression tests

Fig. 2 .

 2 Fig. 2.11 Particles moving through the spread parallel laser beam, creating diffraction [97]

Fig. 2 .

 2 Fig. 2.12 (left) Schematic diagram of the hydrostatic compression test using a weighing device [14], (right) Picture of the 60 MPa pressure vessel

2 . 13 .

 213 The piston is displaced with the help of a 200 KN tensile machine used in compression mode. Both pressure and piston displacement are recorded during the test.

Fig. 2 .

 2 Fig. 2.13 Scheme of the hydrostatic compression test using a piston vessel [14] (left). Picture of the 200 MPa piston vessel (right)

Fig. 2 .

 2 Fig. 2.14 Hyperbaric vessel for compression tests with strain gauge measurements. The gauge cable passes through the hole in the middle of the vessel plug.

Fig. 2 .

 2 Fig. 2.15 Picture of the 100 MPa pressure vessel for HCP tests on composite foam

Fig. 2 .

 2 Fig. 2.16 Composite foam sample surrounded by the metallic structure (left) and then by a bag before being tested

Fig. 2 .

 2 Fig. 2.17 Pressure vessel for water absorption tests on pure syntactic (left) -Pressure vessels in water tank for temperature control (right)

Fig. 2 .

 2 Fig. 2.18 Schematic diagram showing the principle of the IBL test (left), Sample of composite foam placed in the setup (middle), picture of the measurements set-up (right)

Fig. 2 .

 2 Fig. 2.19 Pressure vessel used for IBL tests and water absorption measurements on composite foam. 60 MPa pressure vessel (left) and 30 MPa pressure vessel (right)

  Adrien at the MATEIS laboratory (INSA Lyon), and imaging was made at different resolutions, with a voxel size of 25 μm 3 , 1.5 μm 3 and 0.3 µm 3 , to get scans of the sample microstructure. Scanning at 0.3 µm provides rough microspheres thickness measurements. Post processing of imaging was made by MATEIS to recover void volume fraction in the material.

3. 1 )

 1 . The volume change measured is a loss of volume, but for better readability the absolute values are plotted shown as positive. This representation in absolute values of strain under compression loading is made throughout the document.

Fig. 3 . 1

 31 Fig. 3.1 Pressure/volume loss curve of GB15 microspheres subjected to hydrostatic loading -21°C -1.5MPa/min

Fig. 3 .Fig. 3 . 4

 334 Fig. 3.3 (a) Image of the microstructure of pure syntactic through X ray tomography measurements. (b) Same image after grey level processing, to consider only the glass (in white) on the picture, from which thickness measurements of microspheres are made.

Fig. 3 . 5 3 Eq. 3 . 1 WithFig. 3 . 6

 3533136 Fig. 3.5 Evolution of the microspheres GB25 size distribution with the applied pressure when subjected to hydrostatic loading

Fig. 3 .

 3 Fig. 3.7 3D reconstruction of a glass macrosphere using X ray tomography, with on the right the scale of the thickness measurements

Fig. 3 . 8

 38 Fig. 3.8 Stress at break of a macrosphere with their thickness.

Fig. 3 . 9

 39 Fig. 3.9 Pressure/volume strain curve during HCP test with volume variations measurements on E1_GB15 pure syntactic -21°C -1.5MPa/minvolume fraction of microspheres: 0.56.

Fig. 3 .

 3 Fig. 3.10 Pressure/piston displacement curve during HCP test in piston pressure vessel, on E1_GB15 pure syntactic -21°C -1.5MPa/minvolume fraction of microspheres: 0.56

  Fluid injection 100 mm 3 cubic sample 37.7 ± NA Fluid injection 50 mm 3 cubic sample 38.0 ± 1.0 Piston displacement 50 mm 3 cubic sample 37.1 ± NA Piston displacement 50*50*5 mm 3 plate sample 38.5 ± 0.5 Table 3.3 Crush pressure of E1_GB15 pure syntactic at 21°C depending on the method used and the sample size

Fig. 3 . 4

 34 Fig. 3.11 (a) Stress/strain curve during uniaxial compressive tests on pure resins E1 and WAX at 21°C (b) Stress/strain curve during uniaxial compressive test on the pure resin E1 at different temperatures

Fig. 3 .

 3 Fig. 3.12 (a) Hydrostatic load test on E1_GB15 and WAX_GB15 pure syntactic at 21°C (a) and on E1_GB15 at different temperature (b) microspheres volume fraction of 0.56 in the foams

Fig. 3 .

 3 Fig. 3.14 Comparison between experimental results and model, crush pressure of pure syntactic E1_GB15 (a) and WAX_GB15 (b) depending on microspheres volume fraction, at 21°C.

  2.2. , at testing temperatures of 21, 25, 30, 35 and 50°C; results are presented in Fig.3.15. As already observed, an increase in testing temperature leads to a decrease in matrix stiffness as well as a decrease in collapse pressure of the foam.These experimental results (dots in Fig.3.15) are compared with the empirical prediction proposed here (line in Fig.3.15) and show a reasonable correlation. Only one point shows disagreement between model and experimental data, but this could be attributed to the test temperature, for this point, being 35°C, is close to the material Tg. This suggests that the proposed model can be used to describe the crush pressure of any pure syntactic made with GB15 microspheres, knowing both resin properties and microspheres volume fraction within the range considered here (i.e., compressive modulus between 5 Pa and 1600 MPa and volume fraction of microspheres between 0.3 and 0.6).

Fig. 3 .

 3 Fig. 3.15 Application of the numerical model to pure syntactic E2_GB15 (volume fraction: 0.59)

Fig. 3 .

 3 Fig. 3.16 Crush pressure prediction versus the crush pressure measured experimentally for pure syntactic with GB15 microspheres.

Fig. 3 .

 3 Fig. 3.17 Crush pressure prediction versus the crush pressure measured experimentally for pure syntactic with (a) GB25 microspheres and (b) GB38 microsphere.

Fig. 3 .

 3 Fig. 3.18 (a) In situ observation of the microstructure of E2_GB15 pure syntactic using X ray micro tomography (volume fraction of 0.60) (b) In situ observation of the same material after being pressurized up to 50 MPa.

Fig. 3 .

 3 Fig. 3.19 Evolution of the microspheres' density inside the material with the applied pressure (volume fraction of 0.6 in both pure syntactic).

Fig. 3 .

 3 Fig. 3.20 Evolution of the damage (number fraction of collapsed microspheres) in pure syntactic E2_GB15 and E2_GB25, with the applied load level (hydrostatic pressure -15°C -1.5 MPa/min)

Fig. 3 .

 3 Fig. 3.21 Pressure/volume strain curve during HCP test with volume variations measurements on a composite foam CF_E2_GB25 (with glass fiber macrosphere) -15°C -0.3 MPa/min

  surface. Sample with skins of pure syntactic of thickness of 10 and 20 mm were made. Those samples have the same dimensions as the classic composite foam samples (i.e., right circular cylinders of 1 m height and 0.3 m diameter). Those samples were tested to failure. The pictures of the collapsed samples are presented Fig. 3.22, and the volume strain/pressure curves are plotted Fig. 3.23.

Fig. 3 .

 3 Fig. 3.22 Pictures of composite foam samples with glass fiber macrosphere after HCP tests (a) classic sample (b) sample with skin of 10 mm of pure syntactic (c) sample with skin of 20 mm of pure syntactic.

Fig. 3 .

 3 Fig. 3.23 Pressure/volume strain curve during HCP test with volume variations measurements on a composite foam E2_GB25_Glass macrosphere with of pure syntactic of thickness 0mm (classic sample), 10 mm and 20 mm -15°C -0.3MPa/min

4. 3 . 1 . 4 . 5 .

 3145 Microspheres geometry description .................................................................................... 4.3.2. Estimation of stress inside pure syntactic ........................................................................... 4.3.3. Evolution of the stress depending on the pure syntactic microstructure parameters. ......... 4.3.4. Estimation of stress inside macrospheres in composite foam: origin of the lack of reinforcing effect in composite foam. ........................................................................................................... 4.3.5. Conclusion on the analytical model of stress fields in syntactic foams ............................ 4.4. Numerical estimation of stress inside syntactic foams ............................................................. 4.4.1. Numerical modelling of a composite foam ....................................................................... 4.4.2. Results of the numerical model ........................................................................................ Conclusions on the modelling of stress inside syntactic foams ................................................

Fig. 4 . 1

 41 Fig. 4.1 Schematic diagram of the three phases spherical model [63]

Fig. 4 . 1 [ 0 Eq. 4 . 1

 41041 Fig. 4.1 [104]. This energy correspondence can be expressed in spherical coordinates following Eq. 4.1 [104]:∫ [(𝜎 𝑟𝑟 (0) 𝑢 𝑟 (𝑐) + 𝜎 𝑟𝜃 (0) 𝑢 𝜃 (𝑐) + 𝜎 𝑟𝜑 (0) 𝑢 𝜑 (𝑐) ) -(𝜎 𝑟𝑟 (𝑐) 𝑢 𝑟 (0) + 𝜎 𝑟𝜃 (𝑐) 𝑢 𝜃 (0) + 𝜎 𝑟𝜑 (𝑐) 𝑢 𝜑 (0) )] 𝑑𝑆 𝑆

  b and c the dimensions (in mm) of the RVE presented Fig. 4.1.

23

 23 [START_REF] Lei | Kinetic studies on the rate of hydrolysis of N-ethyl-N'-(dimethylaminopropyl) carbodiimide in aqueous solutions using mass spectrometry and capillary electrophoresis[END_REF] and displacement (Eq. 4.21 and Eq. 4.23) from the hypothesis of perfect bonding.With σrr and σθθ the stress fields (MPa), ur the radial displacement (mm), a (mm) the internal radius of the inclusion, b its external radius (mm), and c the radius of the RVE (mm).

  no unit) the volume fraction of inclusion in the material, ( a b ) = η (no unit) the radius ratio of the inclulsion, K (MPa) and G (MPa) the bulk and shear modulus of the wall of inclusion (i), matrix (m) and homogenized material (HEM), P (MPa) the applied pressure

Fig. 4 . 2

 42 Fig. 4.2 Radius ratio population by volume for GB15, GB25 and GB38 microspheres, considering constant thickness. The dashed line represents the value calculated from the average density of the microspheres.

Fig. 4 . 3

 43 Fig. 4.3 Evolution of the maximal stress inside a microsphere with its radius ratio, with (black dashed lines) or without (continuous red) being surrounded by resin, for GB15 microspheres at hydrostatic pressure of 40 MPa.

Fig. 4 .

 4 4.a presents the evolution of the stress inside microspheres of varying radius ratio, depending on the volume fraction of microspheres in the material with a resin Young's modulus constant of 2.8 GPa. Fig. 4.4.b presents the evolution of the stress inside microspheres depending on the resin Young's modulus, at volume fraction constant of 0.55. The stress estimation was made using the parameters presented in Table 4.5. Since our measurements of resin Poisson's coefficient at different temperatures are in the range of 0.41-0.43, in this evaluation, it is considered constant with a value of 0.41 (Poisson's coefficient of the resin at 21°C

  Fig. 4.4 Stress inside microspheres GB15 of varying radius ratio, placed in pure syntactic, (a) for different volume fractions of microspheres in pure syntactic with a resin of Young modulus E = 2.8 GPa (b) for different Young modulus of resin (volume fraction of 0.55 in the pure syntactic) -applied hydrostatic pressure of 40 MPa

6

 6 Fig. 4.5 Evolution of stress inside a microsphere with its radius ratio, depending on the type of microspheres used in the pure syntacticapplied hydrostatic pressure of 40 MPa

Fig. 4 . 6

 46 Fig. 4.6 Evolution of stress inside a glass fiber macrosphere, surrounded by pure syntactic or not, with its radius ratio.

2 .

 2 Creating spherical void in the cylinder by cutting solid sphere, with the position of the solid sphere corresponding to the measured position of the macrospheres in the sample.3. Placing hollow sphere inside the void. 4. Meshing the hollow sphere with hexagonal elements, and the cylinder with tetrahedron elements. The meshed hollow sphere and cylinder are shown Fig. 4.8. The parameters of the model are presented in Table

Fig. 4 . 8 (

 48 Fig. 4.8 (Left) Mesh of the medium cylinder (Right) mesh of the spherical inclusions

Fig. 4 . 9

 49 Fig. 4.9 View cut of the modelled composite foam: values of equivalent Von mises stress inside the material. Von mises being positive values, red indicates high stresses, and blue low stresses.

Fig. 4 .

 4 Fig. 4.10 (a) Radial stress and (b) circumferential stress inside a macrosphere in our modelized composite foam. The macrosphere was isolated from the rest of the material for better visibility. Compressive stresses being negative values, red indicates low stresses and blue high stresses.

Fig. 5 . 1 Fick

 51 Fig. 5.1 Water uptake in the pure resins at 15°C in seawater (SW) and deionized water (DI), experimental data and Fick model based on results.

Fig. 5 . 2

 52 Fig. 5.2 Water uptake in the pure resins E3 at 15°C in deionized water at atmospheric pressure and under 45 MPa hydrostatic pressure

Fig. 5 . 3

 53 Fig. 5.3 Water uptake of pure syntactic E2_GB38 (a) and E3_GB38 (b) in deionized water at 15°C under 45 MPa hydrostatic pressure. Different sample sizes are considered.

Fig. 5 .

 5 Fig. 5.4 presents the tomography analysis made on the unaged material and on the sample aged for 63 days under 19 MPa. The aged sample was also scanned after being dried in air at room temperature for 7 days (Fig. 5.4.c). The analysis on the unaged sample (Fig. 5.4.a) allows the observation of the microsphere shell (in white), the epoxy matrix (in light grey), and the void inside the microsphere (in dark grey). The image of the aged sample (Fig. 5.4.b) reveals a change in color inside the microspheres during aging from dark to light grey. This observation can mean that microspheres are filled with water upon hydrostatic aging. To verify that, the same sample is tested after 7 days in dry air (Fig. 5.4.c). After being dried for 7 days in air, a change in color is witnessed again (from light to dark grey), meaning that the microspheres near the sample surface (previously filled with water) become empty again (Fig. 5.4.c).

Fig. 5 .

 5 Fig. 5.4 X-ray microtomography imaging of E2_GB15 (a) unaged (b) immersed 63 days in 15°C under 19 MPa pressure (c) the aged sample after 7 days drying in air.

Fig. 5

 5 Fig. 5.6.

Fig. 5 . 5

 55 Fig. 5.5 Picture of a sample of pure syntactic in a sealed bag, to prevent it from being in contact with water when immersed under hydrostatic pressure.

Fig. 5 . 7

 57 Fig. 5.7 Representation of the displacement ∆b imposed to a microsphere to represent the swelling of the resin

Fig. 5 . 9

 59 Fig. 5.9 Water uptake of pure syntactic E3_GB38 in deionized water at 15°C under 45 MPa hydrostatic pressure. Results of sample of different size, normalized by the geometry ratio S/V, in function of the square root of time.

  thickness) of E2_GB38 and E3_GB38 are immersed in seawater under 45 MPa pressure at 15°C. The immersion time is 4 months and is still ongoing. The results are plotted versus the square root of time in Fig. 5.10, within a range corresponding to 4 months. Then, these same results are compared to those obtained in deionized water under the same hydrostatic pressure and temperature. Only plate samples are considered here.

Fig. 5 .

 5 Fig. 5.10 Water absorption of E2_GB38 and E3_GB38 plate sample of 5 mm thickness under 45 MPa pressure -15°C -deionized water (DI) and seawater (SW).

  pressure ranging from zero to 110 MPa (between zero and 70% of the material HCP of 155 MPa) for a total duration of 2 months. Results are shown in Fig 5.9.

Fig. 5 .

 5 Fig. 5.11 (a) Water absorption of plate sample (5 mm thickness) of E3_GB38 pure syntactic at different hydrostatic pressure, 15°C and (b) evolution of the absorption rate of E3_GB38 with the hydrostatic pressure, at 15°C.

Fig. 5 .

 5 Fig. 5.12 Water absorption of plate sample (5mm) of E1_GB15 pure syntactic under different hydrostatic pressures -15 °C

Fig. 5 .

 5 Fig. 5.13 Water absorption of plate sample (5mm) of E1_GB25 under different hydrostatic pressures -15 °C

Fig. 5 .

 5 Fig. 5.14 Evolution of the water absorption rate for E1_GB15 (left) and E2_GB25 (right)with the applied pressure -15°C -deionized water

Fig. 5 .

 5 Fig. 5.15 Change in normalized absorption rate with the load level

Fig. 5 .

 5 Fig. 5.16 Water uptake of pure syntactic E2_GB38 in deionized water at 15°C under 45 MPa hydrostatic pressure. Results of sample of different size, normalized by the geometry ratio S/V, in function of the square root of time.

Fig. 5 .

 5 Fig. 5.17 Water uptake of pure syntactic E2_GB38 in deionized water at 15°C under 45 MPa hydrostatic pressure. Results of sample of different size, normalized by the geometry ratio S/V, as a function of the time (a) zoom in on t.S/V <2000 h/mm. (b) full test duration

Fig. 5 .

 5 Fig. 5.18 Scheme of the diffusion mechanisms (a) diffusion of water in the resin (b) collapse of microsphere because of matrix swelling (c) filling of the microsphere through the cracks induced by the microsphere collapse.

Fig. 5 .

 5 Fig. 5.19 Scheme of the damage mechanism (a) filling of the open microsphere at the sample surface or through the microcracks in the resin/microsphere due to the local pressure (b) collapse of the neighboring microsphere or failure of the resin due to local mechanical loading (c) filling of the neighboring microsphere

5. 20 .

 20 The model is clearly able to describe the water absorption of the different sample sizes of the material. But there is a need to characterize the water absorption of samples of different sizes to efficiently estimate both values of A and v.

Fig. 5 .

 5 Fig. 5.20 Comparison between modelling of water absorption using Equation60, and experimental measurements, on E2_GB38 sample of different geometries-45 MPa -15°C -deionized water

Fig. 5 .

 5 Fig. 5.21 Water absorption of plate sample of E1_GB15 immersed in deionized water at 15°C under different hydrostatic pressure, experimental data and fitted model.

Fig. 5 .

 5 Fig. 5.22 Water absorption of plate samples of E2_GB15 immersed in deionized water at 15°C under different hydrostatic pressures.

Fig. 5 .

 5 Fig. 5.23 Change in flow speed q with the hydrostatic pressure for different pure syntactic materials immersed in deionized water at 15°C

Fig. 5 .

 5 Fig. 5.24 Evolution of the flow rate q with the load level P/Pcrush for different pure syntactic materials immersed in deionized water at 15°C

Fig. 5 .

 5 Fig. 5.25 Creep test on E2_GB38 at room temperature under 45 MPa hydrostatic pressure (a) total volume strain (b) volume strain during pressure plateau

Fig. 5 .

 5 Fig. 5.26 Water absorption of samples with different geometries of E1_GB25 pure syntactic at room temperature under 20 MPa pressure. Determination of the diffusion coefficient using linear regression (dashed line)

Fig. 5 .

 5 Fig. 5.27 Picture of the garland of plate sample of E1_GB25 pure syntactic.

Fig. 5 .

 5 Fig. 5.28 Buoyancy loss of E1_GB25 at room temperature under 20 MPa hydrostatic pressure -Experimental data (without considering buoyancy loss due to elastic deformations) and prediction made from water absorption measurements.

Fig. 6 . 1

 61 Fig. 6.1 Buoyancy change measurements during IBL test on CF_E3_GB38 composite foam-15°C -45 MPa

Fig. 6 . 2

 62 Fig. 6.2 Buoyancy change measurements during IBL test on CF_E3_GB38 composite foam-15°C -45 MPa

Fig. 6 . 3

 63 Fig. 6.3 Fit of the logarithmic law on the experimental data that will be used for extrapolation. Here the period considered is 96 hours (without the first 24 hours). Elastic deformations are considered.

  The full test consists of 3 cycles of 96 hours at pressure and 72 hours of recovery time, 2 cycles of 172 hours at pressure and 120 hours of recovery time, and one last cycle of 172 hours at pressure and 800 hours of recovery time. The total immersion time is 100 days. The entire pressure cycle is represented by the dotted line Fig.6.4. alongside the recorded buoyancy loss in red, and the water absorption in blue dots. Water absorption is expressed in Newtons to be compared with the recorded buoyancy loss.

Fig. 6 . 4

 64 Fig. 6.4 Cyclic IBL test on CF_E3_GB38 composite foam, with weight measurements at the end of each cycle. The red curve represents the buoyancy loss, and the blue dots the water absorption of the sample. Test temperature is 15°C, and pressure is 45 MPa.

Fig. 6 . 5

 65 Fig. 6.5 Picture of the upper surface of a composite foam sample. Due to processing, the upper surface present defects that can be filled with water, and that are difficult to dry before making weight measurements. Those defects are only present on the upper sample surface.

Fig. 6 . 6

 66 Fig. 6.6 Left: position of the gauges around the composite foam sample perimeter, located near (middle picture) or far (picture on the right) from a visible macrosphere.

Fig. 6 . 7

 67 Fig. 6.7 IBL test on composite foam CF_E3_GB38 with incremental pressure, at 15°C.

Fig. 6 .

 6 Fig. 6.8 presents the evolution of local strain rate during the test. During the plateau, a time dependent strain appears at the sample surface. Local creep is occurring in the material. Moreover, it appears here that this creep is heterogeneous: some gauges exhibit decompression behavior, showing that this creep leads to deformation of the sample global shape. This highlights the difficulty of estimating global creep using gauges providing local measurements.

Fig. 6 . 8

 68 Fig. 6.8 Local strain measurements on CF_E3_GB38 composite foam during the IBL test with incremental pressure -15°C -Each curve represents the signal given by one gauge.

Fig. 6 . 9

 69 Fig. 6.9 Volume creep of carbon fiber macrosphere -15°C -45 MPa

6. 9 )

 9 . This is attributed to MC2 being thicker (4.37 ± 0.06 mm) than MC1 (3.95 ± 0.07 mm), reducing the circumferential stress for the same applied pressure, and increasing the apparent bulk modulus of the sphere. Increasing the sphere thickness increases the bulk modulus of the composite foam containing them, and should reduce the creep of the material, at the cost of increasing the sphere density and the composite foam density.During the creep recovery, a residual volume creep appears. Creep of macrospheres involves viscoelastic and plastic strain of the shell material during the pressurization. Creep tests were also made on a glass fiber macrosphere. One macrosphere MG was loaded at 25MPa (i.e., about 70% of the collapse pressure), and kept pressurized for period longer than 96 hours to try to see if the creep rate might change during the test. The measured volume strain of the macrosphere is presented Fig.6.10, revealing that glass macrospheres also exhibit volume creep. After 120 hours of pressurization, the gauge signal was lost due to the collapse of the macrosphere. Creep can cause failure of the macrosphere shell material.

Fig. 6 .

 6 Fig. 6.10 Volume creep of glass fiber macrosphere -15°C -25MPa.

  With W (no unit) the relative weight gain of the hypothetical cylinder of pure syntactic, A the absorption coefficient of the pure syntactic (mm/h 1/2 ), t (hours) the immersion time, S (mm²) and V (mm 3 ) the sample surface, volume, and ∆ 0 the water absorption due to the filling of the external defects of the sample.

Fig. 6 .

 6 Fig. 6.11 Water absorption of CF_E3_GB38 composite foam at atmospheric pressure (0 MPa) and under 45 MPa pressure. The dashed lines represent the estimated water absorption made using Eq. 6.2

Fig. 6 .

 6 Fig. 6.13 Creep volume strain (without elastic deformations) of CF_E2_GB38 during a 60 days IBL test -40 MPa -15°C

Fig. 6 . 4

 64 Fig. 6.16 Evolution of volume strain during incremental IBL test on composite foam CF_E2_GB38 at 15°C.

Fig. 6 .

 6 Fig.6.17 Time to break of glass fiber macrospheres (MG), depending on the load level applied.

Fig. 6 .

 6 Fig. 6.18 Creep strain at 15°C and 5 °C of (a) composite foam CF_E3_GB38 and (b) CF_E2_GB38

Fig. 6 .Fig. 6 .

 66 Fig. 6.19 Prediction of buoyancy loss of CF_E3_GB38 under 45 MPa pressure at 15°C for 1 year of service life

  (𝑟 = 𝑏) = 𝜎 𝑟𝑟 (𝑚) (𝑟 = 𝑏)

  (1 -𝜂 3 ) (1 + 𝛽(𝑚) 𝑓 + 𝜃𝛽(𝐻𝐸𝑀) (1 -𝑓)) + (1 + 𝜂 3 𝛽 (𝑔) ) (𝛽(𝑚) (1 -𝑓) + 𝜃𝛽(𝐻𝐸𝑀) (𝛽 (𝑚) + 𝑓))) no unit) the volume fraction of microsphere in the material, (a b) = η (no unit) the radius ratio of the microsphere, K (MPa) and G (MPa) the bulk and shear modulus of the glass (g), matrix (m) and homogenized material (HEM), P (MPa) the applied pressure

Fig. II (

  Fig. II (Gauche) Flotteur pour pipelines (Balmoral). (Droite) Configuration « steep-waves » du positionnement des flotteurs sur un pipeline (Balmoral).

  ). Ces matériaux possèdent une faible densité et une forte résistance à la pression hydrostatique (jusqu'à plus de 100 MPa en fonction de leur composition), et l'utilisation de résine époxy les rend stables en environnement marin. Dans ce cas d'application, les sphères creuses introduites dans la résine sont des microsphères de verres creuses (diamètres de 10 à 200 micromètres), ainsi que des macrosphères faites principalement en matériaux composites (diamètres de quelques centimètres). Les mousses syntactiques constituées uniquement de microsphères sont appelées « pure syntactique », et celle contenant également des macrosphères sont appelées « mousse composite » (composite foam en anglais) (Fig. F).

Fig. III Composition

  Fig. III Composition d'une mousse syntactique

Fig. IV Courbe

  Fig. IV Courbe de perte de flottabilité d'une mousse syntactique lors d'essais de caractérisation[START_REF] Watkins | Syntactic foam buoyancy for production risers[END_REF]

Fig. V

 V Fig. V Courbe pression/volume lors de compression hydrostatique de mousses syntactiques époxy (GSEP) et polyuréthane contenant des microsphères S38 (GSPU)[START_REF] Choqueuse | Experimental study and analysis of the mechanical behaviour of syntactic foams used in deep sea[END_REF]

Fig. VI Schéma

  Fig. VI Schéma du plan de la thèse

Fig. VII Courbe

  Fig. VII Courbe pression/volume lors de la compression hydrostatique de microsphères GB15

  ) : d'abord un comportement élastique, avant de présenter un plateau de densification. La pression de rupture est définie comme le point d'inflexion entre les deux zones, mesuré par la méthode des tangentes. La pression de rupture de la première mousse testée est 38 MPa, alors que celle des microsphères la constituant (microsphères GB15) est de 1 MPa. Cela illustre bien la présence du renfort des microsphères par la résine dans nos matériaux.

Fig. VIII Courbe

  Fig. VIII Courbe pression/volume d'une pure syntactique étudiée, et détermination de la pression de rupture par la méthode des tangentes

Fig. IX Evolution

  Fig. IX Evolution de la pression de rupture d'une mousse syntactique avec la rigidité de la résine

Fig. X

 X Fig. X Evolution de la pression de rupture d'une pure syntactique époxy (a) et paraffine (b) avec la fraction volumique de microsphères

Fig. XI Evolution

  Fig. XI Evolution de l'endommagement dans les pures syntactiques époxy avec la pression hydrostatique

Fig. XII Schéma

  Fig. XII Schéma de la représentation d'une mousse syntactique utilisée pour la détermination des contraintes

  ).Le modèle prédit qu'une réduction des contraintes pourrait apparaitre si les macrosphères avaient un ratio géométrique plus proche de 1 (Fig.XIII), comme pour les microsphères. Nous voyons ici une optimisation possible des mousses composites, car une augmentation du ratio géométrique des macrosphères impliquerait une diminution de leur densité, donc un gain de flottabilité. Aussi, en augmentant ce ratio, le renfort des macrosphères par la matrice pure syntactique apparaitrait, permettant au matériau de conserver une bonne résistance à la compression hydrostatique.

Fig. XIII Evolution

  Fig. XIII Evolution des contraintes dans une microsphère entourée de résine (gauche) et dans une macrosphère entourée de pure syntactique (droite) en fonction de son ratio géométrique. Comparaison avec le cas de sphère non enrobée.

Fig. XIV Modèle

  Fig. XIV Modèle élément fini d'une mousse composite : estimation des contraintes de Von Mises

Fig. XV Observation

  Fig. XV Observation par microtomographie à rayon X de l'avancée de l'eau dans une pure syntactique

Fig. XVI Schéma

  Fig. XVI Schéma du processus d'absorption d'eau, dans une pure syntactique, lié à la diffusion et au gonflement de la résine

Fig. XVII Schéma

  Fig. XVII Schéma du processus d'absorption d'eau, dans une pure syntactique, lié à l'endommagement dans une pure syntactique

Fig. XVIII Absorption

  Fig. XVIII Absorption d'eau d'une pure syntactique époxy à différentes pressions, et comparaison avec le modèle analytique

Fig. XX Prédiction

  Fig. XX Prédiction de perte de flotteurs constitués de mousses composites époxy sous 45 MPa de pression pendant 1 an (gauche) et sous 40 MPa de pression pendant 20 ans (droite)

Table 5 .

 5 4 Estimated swelling of the pure resin. ......................................................................................................................

Table 5 .

 5 5 Values of A0 and B of pure syntactic with epoxy anhydride resin dependning on their crush pressure .................

Table 6 .

 6 1 Buoyancy loss predictions on the studied composite foam.....................................................................................

Table 6 .

 6 2 Buoyancy loss and weight measurements during each cycle of the test presented Figure2. ...................................

Table 6 .

 6 3 Values of C0 and P0 of the studied composite foam ...............................................................................................

Table 6 .

 6 4 Pressure at which collapse of macrospheres was observed during IBL tests on composite foam ...........................

Table 6 .

 6 5 Parameters used for buoyancy loss predictionmeasured during IBL tests and water absorption tests at 15°C in deionized water under service pressure ..................................................................................................................................

	List of major Abbreviations, Symbols
		and Acronyms
	A	Absorption rate	Ri	Internal radius
	AUV	Autonomous Underwater Vehicle	Ro	Outer radius
	C	Creep rate	r	Radius
	D	Diffusion coefficient	S	Surface
	Dp	Damage parameter	SEM	Scanning Electron Microscopy
	DSC	Differential Scanning Calorimetry	T	Temperature
	d	Diameter	Tg	Glass transition temperature
	E	Young's Modulus	Tm	Melting temperature
	E comp	Compressive modulus	t	Time
	e	Thickness	u	Displacement
	F	Apparent weight in water or buoyancy	V	Volume
	F relative	Relative buoyancy loss	V0	Initial volume
	f	Volume fraction	RVE	Representative Volume Element
	G	Shear Modulus	W	Relative water uptake
	GB	Glass Bubbles (microspheres)	β	Compressibility
	g	Gravitational constant	ε	Strain
	HCP	Hydrostatic Crush Pressure	εc	creep strain
	HEM	Homogenized Equivalent Medium	εtotal	Total strain
	IBL	Instrumented Buoyancy Loss	ε0	Elastic strain
	K	Bulk modulus	ρ	Density
	M	Mass fraction	ρ g	Glass density
	m	Mass	ρ matrix	Resin density
	mwater	Weight in water	ρ micro	Microspheres density
	n/N	Integer Number	ρ pure-syntactic	Pure syntactic density
	Nc	Number of collapsed microspheres	ρ syntactic-foam	Syntactic foam density
	P	Pressure	ρwater	Water density
	P c,sphere	Collapse pressure of hollow sphere	ν	Poisson's Coefficient
	P b,sphere	Buckling pressure of hollow sphere	η	Radius ratio
	Pc	Crush pressure	ηo	Average radius ratio
	Q	Debit	σ	Stress
	q	Flow rate

Table of contents

 of Introduction ....................................................................................................................................... 1.1. Buoyancy: definition .................................................................................................................. 1.2. Syntactic foams .......................................................................................................................... 1.2.1. Generalities on syntactic foams ..........................................................................................

1.2.2. Syntactic foams for buoyancy applications: testing and requirements ............................... 1.3. Mechanical behavior of syntactic foams .................................................................................... 1.3.1. Mechanical behavior of hollow microspheres .................................................................... 1.3.2. Mechanical characterization of syntactic foams ................................................................. 1.3.3. Modelling of syntactic foams ............................................................................................. 1.4. Long term behavior of syntactic foams ...................................................................................... 1.4.1. Water absorption of syntactic foams .................................................................................. 1.4.2. Creep of syntactic foams .................................................................................................... 1.5. Syntactic foams containing macrospheres: composite foams .................................................... 1.6. Conclusions ................................................................................................................................ 1.7. Study overview ...........................................................................................................................

Table of contents

 of Introduction ....................................................................................................................................... 2.1. Materials ..................................................................................................................................... 2.1.1. Resins .................................................................................................................................. 2.1.2. Microspheres.......................................................................................................................

2.1.3. Macrospheres ...................................................................................................................... 2.1.4. Pure syntactic ...................................................................................................................... 2.1.5. Composite foams ................................................................................................................ 2.2. Methods ...................................................................................................................................... 2.2.1. Sample preparation ............................................................................................................. 2.2.2. Initial characterization ........................................................................................................ 2.2.3. Hydrostatic Crush Pressure (HCP) characterization ........................................................... 2.2.4. Long term behavior characterization .................................................................................. 2.2.5. Post-testing characterization ............................................................................................... 2.2.6. Modelling ............................................................................................................................ 2.3. Conclusions ................................................................................................................................

Table 2 .1 Resins used in the study and their characteristics

 2

	Name	Resin	Density* (g/cm 3 )	Glass transition temperature** Tg (°C)	Melting temperature** Tm (°C)
	E1	Anhydride based epoxy	1.122 ± 0.001	81 ± 1	-
	E2	Amine based epoxy	1.150 ± 0.002	42 ± 1	-
	E3	Anhydride based epoxy	1.201 ± 0.002	117 ± 7
	WAX	Paraffin	0.983 ± 0.003	-	59 ± 1
				*Measured by gas pycnometer (see 2.2.2.1. for details)
				**Measured by DSC (see 2.2.2.2. for details)

Table 2 .2 Microspheres used in the study and their densities

 2

	3 )	Measured nominal density* (g/cm 3 )

*Measured by glass pycnometer (see 2.2.2.1. for details)

Table 2 .3 Macrospheres used in the study

 2

1 3.96 ± 0.07 0.51 ± 0.01 85 ± 1 MC2 49.0 ± 0.1 4.37 ± 0.06 0.55 ± 0.01 *Measured by X ray tomography (see 0 for details) **Measured by Archimedes' principle (see 2.2.2.1. for details) ***Measured by DSC (see 2.2.2.2. for details)

Resin Microspheres grade Microspheres volume fraction* Tg** (°C) Tm** (°C)

	E1_GB15		GB15	From 0.3 to 0.6	91 ± 2
	E1_GB25	E1	GB25	From 0.5 to 0.55	94 ± 2	-
	E1_GB38		GB38	0.55 ± 0.01	88 ± 1
	E2_GB15		GB15	0.59 ± 0.01	38 ± 1
	E2_GB25	E2	GB25	0.59 ± 0.01	45 ± 1
	E2_GB38		GB38	0.54 ± 0.01	49 ± 1
	E3_GB38	E3	GB38	0.57 ± 0.01	113 ± 2
	WAX_GB15		GB15	From 0.3 to 0.6		56 ± 1
	WAX_GB25	WAX	GB25	0.55 ± 0.01	-	58 ± 1
	WAX_GB38		GB38	0.55 ± 0.01		58 ± 1

*Measured from density measurements by Archimedes' principle (see 2.2.2.1. for details) **Measured by DSC (see 2.2.2.3. for details)

Table 2 .4 Pure syntactics and their composition

 2

Table 2 .5 Composite foams constitution Fig. 2.3 Picture of a cylindrical composite foam sample

 2

	(Table 2.3)	Density* (g/cm 3 )

*Measured from density measurements by Archimedes' principle (see 2.2.2.1. for details)

Table of contents

 of Introduction ....................................................................................................................................... 3.1. Hydrostatic strength of hollow spheres ......................................................................................

3.1.1. Hydrostatic strength of hollow glass microspheres ............................................................ 3.1.2. Hydrostatic strength of macrospheres ................................................................................. 3.1.3. Conclusion .......................................................................................................................... 3.2. Experimental study of the hydrostatic strength of pure syntactic............................................... 3.2.1. Pure syntactic crush pressure .............................................................................................. 3.2.2. Impact of resin mechanical properties on pure syntactic hydrostatic strength ................... 3.2.3. Impact of microspheres volume fraction on pure syntactic crush pressure ........................ 3.2.4. Modelling hydrostatic strength of pure syntactic with GB15 microspheres ....................... 3.2.5. Influence of the type of microspheres ................................................................................. 3.2.6. Damage behavior of pure syntactic .................................................................................... 3.2.7. Conclusions on the hydrostatic strength of pure syntactic .................................................. 3.3. Hydrostatic strength of composite foam ..................................................................................... 3.4. Conclusions on the hydrostatic strength of syntactic foams .......................................................

Table 3 .1 Microspheres collapse pressure.

 3

Table 3 .2 Hydrostatic strength of composite hollow sphere.

 3

	Type of macrosphere	Macrosphere density (g/cm 3 )	Hydrostatic strength (MPa)	Number of tested macrospheres
	MG	0.45 ± 0.02	34.6 ± 3.7	20
	MC1	0.51 ± 0.01	58.4 ± 2.6	6
	MC2	0.55 ± 0.01	71.8 ± 3.4	6

Table 3 .5 Hydrostatic strength of pure syntactic with different kind of microspheres

 3 56 ± 0.1 in E1 pure syntactics, and 0.58 ± 0.1 in WAX pure syntactic). Results of crush pressure measurements are presented Table3.5.

			Pure syntactic crush pressure P crush (MPa)
	Microspheres	Microspheres collapse pressure (MPa)	E1 pure syntactic	WAX pure syntactic
			(v = 0.56)	(v = 0.58)
	GB15	1.0 ± 0.1	38.0 ± 1.0	7.1 ± 0.1
	GB25	2.9 ± NA	57.5 ± 0.5	11.1 ± 0.1
	GB38	31.8 ± NA	148.0 ± 1.7	35.1 ± 0.6

Table 3 .6 Value of the constant of the empirical crush pressure model or different type of microspheres

 3 .6.

	Microspheres	α (no unit)	δ (MPa)	R²
	GB15	0.0519	2.79	0.99
	GB25	0.0755	4.06	0.99
	GB38	0.1624	15.76	0.94

Table of contents

 of Introduction ....................................................................................................................................... 4.1. Estimation of syntactic foam elastic properties .......................................................................... 4.2. Analytical determination of stress inside hollow sphere in syntactic foams. .............................

4.3. Evaluation of stress inside hollow spheres inside syntactic foam: origin of the reinforcing effect of the resin .........................................................................................................................................

Table 4 .1 Comparison between experimental results and model prediction of bulk modulus of pure syntactics

 4

	Pure syntactic	Eresin (GPa)	Poisson ratio	𝜼 𝟎	Volume fraction	Predicted bulk modulus (GPa)	Experimental bulk modulus (GPa)	Deviation (%)
	E1_GB15	2.73 ± 0.1	0.41 ± 0.1 0.980	0.55 ± 0.1	2.16	2.05 ± 0.43	5
	E2_GB15			0.980	0.59 ± 0.1	1.98	2.10	6
		2.85 ± 0.1	0.39 ± 0.1
	E2_GB38			0.947	0.54 ± 0.1	3.71	3.72 ± 0.03	0.3

Inclusion: Glass fiber macrosphere Medium: Pure syntactic E2_GB25 Homogenized Bulk modulus (GPa)

	Material
		η0	E	ν	f	E	ν	Model estimation	Experimental measurements
	CF_E2_GB25	0.919	15.6	0.337	0.5	2.82	0.330	1.94	1.75 ± 0.04

.2.

The comparison between model prediction and experimental measurements of bulk modulus (made through the IBL tests), is given in Table

4

.2. The difference between the experimental measurements and the model is 11%. The homogenization method is once again able to give a good approximation of the material elastic properties.

Table 4 .2 Estimation of elastic properties of composite foam, compared with experimental measurements

 4

Table 4 .4 Parameters used for the estimations of stress in microspheres in E1_GB15 pure syntactic using the analytical model presented in this study

 4

	Parameter	Value
	Glass Young's modulus	74 GPa
	Glass Poisson's coefficient	0.20
	Epoxy Young's modulus	2.8 GPa
	Epoxy Poisson's coefficient	0.41
	Microspheres volume fraction	0.55
	Microspheres radius ratio range	0.96 -0.99
	Microspheres average radius ratio η0	0.980 (GB microspheres)
	Pressure applied to the homogenized material	40 MPa

Table 4 .8 Parameters of the finite element model of composite foam CF_E2_GB25

 4

	Parameter	Value
	Cylinder height	292 mm (to prevent macrospheres at the top and bottom to be cut)
	Cylinder diameter	300 mm
	Macrospheres composite fiber Young's modulus	15.6 GPa
	Macrospheres composite fiber Poisson's coefficient	0.337
	Pure syntactic Young's modulus	2.8 GPa
	Pure syntactic Poisson's coefficient	0.330
	Macrospheres volume fraction	0.50
	Macrospheres radius	20 mm
	Macrospheres thickness	1.75 mm
	Pressure applied to the homogenized material	40 MPa

Table 5 .3 Water absorption measurements of E2_GB15 immersed under different hydrostatic pressure, before being analyzed by X-ray tomography.

 5

	]	10 MPa	19 MPa	25 MPa
	7	0.3 ± 0.1	2.8 ± 0.2	12.3 ± 0.5
	21	0.9 ± 0.1	7.3 ± 0.2	35.6 ± 1.3
	63	2.5 ± 0.1	18.7 ± 0.2	76.4 ± 1.0
	314	10.6 ± 02	-	-

Table 5 .4 Estimated swelling of the pure resin.

 5

	Swelling [%]

Test period considered 20-years buoyancy loss (%) prediction at service pressure (40 MPa) 20-year buoyancy loss (%) prediction at service pressure (45 MPa)

		CF_E2_GB38	CF_E3_GB38
	96 hours	1.59 ± 0.07	1.65 ± 0.21
	15 days	1.62 ± 0.08	1.77± 0.28
	60 days	1.75	2.01

Table 6 .1 Buoyancy loss predictions on the studied composite foam

 6

Table 6 .2 Buoyancy loss and weight measurements during each cycle of the test presented Fig. 6.4.

 6

		Maximal time-			Proportion of	Proportion
		dependent buoyancy loss (at the end of the pressure plateau, elastic part not	Cumulated residual buoyancy loss (N)	Cumulated buoyancy loss due to water uptake (N)	water uptake in maximal time-dependent buoyancy loss	of water uptake in residual buoyancy
		considered) (N)			(%)	loss (%)
	1	1.50	0.80	0.20	13	25
	2	1.93	1.11	0.22	11	20
	3	2.23	1.32	0.28	13	21
	4	2.63	1.59	0.26	10	16
	5	2.92	1.81	0.32	11	18
	6	3.31	1.99	0.38	11	19

6.2.3. Origin of creep in composite foam: macrospheres creep

  In 5.4 it has been shown that the studied pure syntactic presents no creep when being pressurized. Let us therefore investigate the hypothesis that creep in composite foam comes from the macrospheres.To study this hypothesis, hydrostatic creep tests were made on glass (MG) and carbon (MC1 and MC2) macrospheres using hydrostatic compression tests with gauge measurements (see 2.2.4.2. ). Here, the presented tests are made on carbon macrospheres MG1 and MG2, that are pressurized at 40 MPa for 96 hours. Three gauges were placed on each macrospheres during the test, and the volume variations were estimated as presented in 2.2.3.3. . The volume variations of both macrospheres are presented in Fig.

	6.9.

Table 6 .3 Values of C0 and P0 of the studied composite foam

 6 With ε creep the volume creep, t 0 the time at the beginning of the pressure plateau, t the creep time, C0 (h -1 ) and C1 (MPa) empirical constants.

		𝑐𝑟𝑒𝑒𝑝 = 𝐶 0 * exp (	𝑃 𝐶 1	) * log (𝑡 + 𝑡 0 )	Eq. 6.8
	Material	C0 (h -1 )		C1 (MPa)
	CF_E1_GB25	6.21*10 -5	13
	CF_E2_GB25	4.38*10 -5	12
	CF_E2_GB38	2.00*10 -5	13
	CF_E3_GB38	5.75*10 -5	20

Table 6 .5 Parameters used for buoyancy loss prediction -measured during IBL tests and water absorption tests at 15°C in deionized water under service pressure

 6

	Material	CF_E2_GB38	CF_E3_GB38
	Service life	20 years	1 years
	Service pressure	40 MPa	45 MPa
	Bulk modulus [GPa]	2.1	2.2
	Material density [g/cm 3 ]	0.612	0.645
	Creep rate [h -1 ]	4.4 *10 -4	4.7 *10 -4
	Absorption rate [mm/h 1/2 ]	2.2 *10 -4	9.8*10 -4
	Flow speed [mm/h]	1.7 *10 -6	0
	Module dimensions	Right Cylinder of 4.5 m height, 1.5 m diameter
	Water density at service depth (deionized water) [g/cm 3 ]	1.017	1.019

  Therefore, water absorption, by decreasing the mechanical properties of the macrospheres, would accelerate creep failure, and reduce the composite foam lifetime.But while water absorption in macrospheres alone can be measured, in composite foam it is so slow that it is difficult to estimate experimentally the rate at which the water reaches the macrospheres, and saturate them. Increasing the test temperature, could accelerate the phenomenon. The possibility of the inner polystyrene part of the macrospheres being filled with water should be studied, because if macrospheres are filled with water it would be dramatic for the buoyancy of the material. But to date, this phenomenon has not been observed for the macrospheres aged in water.In summary, this work studied the hydrostatic strength of syntactic foam containing microspheres and macrospheres through experimental and modeling approaches, and proposes an empirical model to predict it. It also investigates the origin of buoyancy loss in those syntactic foams, and proposes mathematical descriptions to predict it during the material lifetime, based on experimental testing up to 2 month durations. Moreover, this study proposed several hypotheses to explain experimental observations, especially water absorption. More studies are needed to confirm those hypotheses, and improve the mathematical descriptions.

	𝜎 𝑟𝑟 (m) = 3𝐵 1 𝐾 (m) -4	𝐵 2 𝐺 (m) 𝑟 3

CHAPTER 6 Buoyancy loss of composite foams

Table of contents

From those volume measurements we can calculate the volume strain of our material. One example of is given in Fig. 6.12, which presents the volume strain of the material during the 2-month IBL test under 45 MPa on CF_E3_GB38 (that was presented in Fig. 6.1). It is seen in Fig. 6.12.a that during the pressurization part, volume strain evolves linearly with the applied pressure, due to elastic deformations (the volume strain is positive because the absolute value is represented here, but it means a decrease of volume). Then, when the pressure is maintained, the volume strain continues to increase over time (the sample continues to lose volume). We also observe in Fig. 6.12.a that there is an irreversible volume strain when the sample is brought back at atmospheric pressure, so plastic deformations or damage appear.

To examine in more detail the volume strain due to creep, volume creep of the material (without elastic deformations) is plotted on Fig. 6.12.b. The creep of the material almost stabilizes after 2 months of testing, but the plateau was not kept long enough to ensure that the creep rate becomes null. The creep rate is decreasing over time, meaning the creep behavior of the material can still be considered in the first stage.

To further illustrate that, a 2-month IBL test was performed on the composite foam made with the aminebased epoxy and carbon macrospheres (CF_E2_GB38) to estimate volume creep of the material. The same method was used (i.e., the water absorption estimated from water absorption in the pure syntactic was subtracted from the measurements), but the applied pressure was 40 MPa. The volume creep of the material is plotted in Fig. 6.13. It reveals that this material also loses volume over time, and so exhibit creep. After 2 months, the creep rate does not stabilize as the material continue to lose volume.

A coefficient C was determined using the least square method to fit our model to the experimental data.

Since the standard for testing buoyancy material recommends that samples should be tested for at least 96 hours, in a first approach the coefficient C was determined using the first 96 hours of our IBL tests.

To check the minimum duration to determine the constant C of the model, the 2-month IBL test on CF_E3_GB38, presented in Fig. 6.12, is used to fit our model with experimental data as well. The model is first fitted using the 96 hours of the test, then using the first 15 days, and finally using the whole testing period (60 days). Three different values of C were obtained, for each time considered. This method is similar to what was done in 6.1. , but this time the known elastic part ε 0 is removed from the results to consider only creep strain, and the first 24 hours of the test were kept for the fit.

Then, the 3 fits are used to describe creep of our material over a 60-day period and compared with experimental data in Fig. 6.14. The fit using the total period gave the best value of C for fitting the model with experiment (R² = 0.99, red curve in Fig. 6.14). The fitting using only the first 15 days, once extrapolated to 60 days, also gave a good description of the experimental creep measured (green curve in Fig. 6.14, R² = 0.97). However, when extrapolating the model with C determined using only the first 96 hours of the test revealed a deviation compared to the experimental data (blue curve in Fig. 6.14, R² = 0.89). This reveals that during the first 96 hours of the test, the creep has not fully entered the secondary creep stage. When using only 96 hours to fit our model, C is overestimated. The model then gives a conservative description of creep in our material. In this work, because time was a constraint to realize our tests, some creep test periods are only of 96 hours. It was therefore decided to always use the first 96 hours of every test to determine our constant model C. Since the final aim of this section is to make a prediction of buoyancy loss, using a conservative model is not a problem.

A description of creep is proposed here to describe creep of materials over 60 days of immersion.

However, this duration is still short in comparison with service life, and there is no indication of whether the material can enter the third stage of creep during its lifetime or exhibit a different behavior. To investigate the third stage of creep, the influence of pressure on the creep rate will be investigated to try to accelerate creep. Furthermore, since materials are used at different hydrostatic pressures, it is necessary to study the influence of pressure on creep.

Influence of pressure

To study the influence of pressure on creep, the cyclic test with CF_E3_GB38 presented in Fig. 6 On Fig. 6.15 it appears that increasing the applied pressure increases the creep rate for our four materials.

Furthermore, the creep rate of the materials is different depending on their composition. Both materials containing two types of macrospheres that have the same composition but different size (i.e., MC1 and MC2), present different creep rates. The size of the macrospheres impacts the creep rate of the material. Furthermore, the materials containing the glass macrospheres MC have different resins (E1 and E2) and present different creep rates. The choice of resin also has an impact on the creep rate, as both materials present different creep rate values. Furthermore, the creep rate of the material with E2 resin and glass macrospheres under 20 MPa is higher than the creep rate of composite foam with E2 resin and carbon macrospheres at the same pressure. The choice of macrospheres therefore influences the creep rate of the material.

Appendix A. Calculation of stress inside hollow inclusions of syntactic foams

In this section, the resolution of the analytical system presented in Chapter 4 is detailed to present how the maximal stress inside a microsphere in a pure syntactic is estimated. As a reminder, the configuration considered is given on First, let us recall the displacement and stress fields presented in Chapter 4:

Eq. A. [START_REF] Fine | Compressibility of water as a function of temperature and pressure[END_REF] 𝜎 𝑟𝑟 (i) = 3𝐴 1 𝐾 (i) -4 𝐴 2 𝐺 (i) 𝑟 3

Eq. A. 5 𝜎 𝜃𝜃 (m) = 3𝐵 1 𝐾 (m) + 2 𝐵 2 𝐺 (m) 𝑟 3

Eq. A.6

Appendix B. Stress estimation in microsphere due to resin swelling

This section presents the analytical description of stress inside a microsphere due to swelling of the resin, due to water absorption. As a reminder, the configuration considered is given Fig. B. The displacement induced by the resin swelling is supposed homogeneous, therefore the stress and displacement fields inside the microsphere, as done in Appendix A, can be expressed in spherical coordinates as follow:

Eq. B. [START_REF]Pazflor, one of the world's largest ever deepwater development[END_REF] 𝜎 𝑟𝑟 (i) = 3𝐴 1 𝐾 (i) -4 𝐴 2 𝐺 (i) 𝑟 3

Eq. B.3

This time, the boundary conditions are:

The first equation Eq.B.4. gives us:

Eq. B.6

And the second equation Eq. B.5 gives us:

Eq. B.7

It is thus possible to estimate the stress inside the microsphere shell knowing the displacement induces by the swelling of the resin. To estimate this displacement, let us express the resin volume before swelling:

Eq. B.8

The volume of resin after the swelling, considering that the swelling only occurs inward, is:

Eq. B.9

The swelling is expressed as:

This gives the expression of the displacement as a function of the swelling:

)

Then we obtain the expression of 𝐴 1 :

)

Knowing A1, the maximal stress in the microsphere shell, being the circumferential stress at the inner surface of the sphere, is:

Eq. B.13

)

Eq. B.14

Résumé en Français

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