Wlf Williams-Landel-Ferry

FDM Fused Deposition Modelling FEA Finite Element Analysis FFF Fused Filament Fabrication IR Infrared LS Level-set MFI Melt Flow Index PC Polycarbonate PCL Polycaprolactone PLA Polylactic Acid POM Polarised

Keywords:

Je commence ce manuscrit par la traditionnelle page des remerciements. Paradoxalement, c'est la dernière page que j'écris. C'est donc avec un léger pincement au coeur que je commence à réaliser que la thèse, c'est fini. Mais cette thèse n'aurait jamais pu s'achever sans l'aide de toutes les personnes qui m'ont soutenu pendant ces 3 années. En effet, écrire la thèse est une entreprise solitaire mais la réaliser, en revanche, a été possible grâce à de nombreuses personnes.

Je tiens à remercier, en premier lieu, toutes les personnes qui ont participé à l'encadrement de ma thèse. En effet, j'ai commencé avec 4 encadrants pour ne passer qu'à 1 puis à 2 encadrants. Merci donc à Jean-François, Michel et Séverine pour les débuts.

Je souhaiterais en particulier remercier Franck qui est la seule personne qui a bien voulu me supporter tout au long de ces 3 années. Merci pour ton soutien et ton accompagnement tant sur le plan scientifique qu'humain. Merci d'avoir respecté mes choix et de m'avoir permis d'étudier l'ensemble des sujets qui m'intéressait, même si tu n'étais pas toujours emballé par ce que je te proposais. Je crois quand même t'avoir entendu dire que les polymères n'étaient pas si inintéressants vers la fin ;).

Merci également à Yancheng qui a repris l'encadrement de la thèse en cours de route. Le temps que tu m'as dédié avant même d'être officiellement mon co-encadrant montre ta générosité naturelle et ta passion pour les sujets que tu étudies et la recherche en général.

Enfin, merci à Christophe, Guillaume, Gaby, Romain, Arnaud, Cyrille, Jean-Luc et à tous les permanents du Cemef qui m'ont aidé aussi bien à monter les manips, qu'à les comprendre, à exploiter les résultats et à répondre à toutes mes questions.

Après avoir remercié les personnes qui m'ont majoritairement aidé sur le plan scientifique, je souhaiterais remercier les personnes qui m'ont aidé sur le plan humain, qui est, je pense, au moins aussi important dans une thèse. Merci à Clément et à Rémi pour avoir fait l'avocat du diable lors de tous nos débats (à moins que ce soit nous les diables ?). Merci Mehdi pour ton rire qui arrive presque à couvrir la voix de Yacine, Totof alias Enguerrand pour le soutien musical en soirée et les sorties ski, Coraline pour avoir sponsorisé toutes mes sorties kites, et Charles pour les nombreux weekends à squouatter ton canap, et pour les jam sessions à devoir raisonner le vieux pat (pas étonnant que tu te sois cassé pour jouer à CFL :P).

Enfin, je peux remercier la chance qui m'a permis de me retrouver dans la même promo que les deux loustiques : Lulu et Yacine. C'est rare de s'entendre aussi bien et aussi vite avec des personnes, et si je dois remercier la thèse pour une chose, c'est de m'avoir permis de vous connaitre. Merci à toi, Yacine, pour ta bonne humeur au quotidien, (même si mon foie ne te remercie pas d'avoir entretenu mon alcoolisme) et aux nombreuses belles soirées et souvenirs (parfois partiels ?) vécu en ta compagnie. Je pense que l'on a encore de nombreuses aventures à vivre ensemble. Merci à toi, Lulu, pour ton code mtc. Non je déconne tu m'as apporté bien plus que ça au cours de ces 3 années : l'amour de la montagne, du kite, une conscience (jamais assez) écologique. On peut dire que la thèse ma changé et tu en as été l'un des acteur principal, et ça, ça n'a pas de prix.. Finalement, merci à tous les autres : Emile, Madu, PF, Marion, Victor, Cec, Alice et toutes les personnes avec qui je garderai un merveilleux souvenir de ces 3 années. Comme tout thésard qui se respecte, la thèse c'est des hauts, mais aussi des bas. C'est pourquoi avoir un groupe d'amis aussi fourni, qui permet de sortir la tête de l'eau quand on est submergé est, je pense, un soutien primordial pour mener à bien une thèse. i Et pour finir je veux remercier la personne qui a été la plus importante au cours de ma thèse, mais aussi dans ma vie. Celle qui m'a remonté le moral toutes les fois où j'étais découragé. Celle qui est resté à mes côté au cours de mes nuits blanches (alors que l'on connait ton amour pour les grasses mat). Celle qui m'accompagne à l'escalade, au kite, au ski. Celle qui partage ma vie et la rend meilleure chaque jour. Celle qui me rend heureux, partage mes peines et mon bonheur. J'ai bien sûr nommé. . . Pirouette ! Ahah je devais faire cette blague. Non, c'est à un autre petit chat que je pense. 我爱你小猫。

Résumé

Ce chapitre présente une introduction du procédé de fabrication additive par dépôt de filaments de polymères fondus, aussi appelé « Fused Filament Fabrication » (FFF), en anglais. Une description du procédé permet de mettre en lumière les différentes étapes de celui-ci. Une vue d'ensemble de la littérature sur les études expérimentales et numériques du procédé FFF est ensuite présentée.

La première étape du procédé est la fusion du polymère dans le liquéfacteur. Différents modèles numériques et analytiques sont présentés pour décrire la chauffe et l'écoulement du polymère ainsi que la perte de charge présent dans le liquéfacteur. Des protocoles expérimentaux sont également détaillés, permettant de mesurer la perte de charge, la température ainsi que de visualiser l'écoulement dans le liquéfacteur.

La seconde étape du procédé est l'extrusion et le dépôt du polymère fondu sur le substrat ou sur les couches précédentes. Des modèles numériques et analytiques ont permis de calculer la déformation du polymère lors de son extrusion. De plus certains auteurs se sont penchés sur les limites maximales de vitesses d'impression. Enfin, d'autres ont développé des simulations numériques modélisant la dynamique de dépôt alors que certains se sont intéressés au refroidissement du polymère une fois déposé. En effet, connaître l'évolution de la température une fois le polymère déposé permet de prédire certains paramètres tels que le degré d'adhésion, la cristallinité ou la déformation d'une pièce lors du refroidissement.

Certains auteurs se sont ainsi intéressés à développer des modèles analytiques et des protocoles expérimentaux pour caractériser l'adhésion entre les couches. En particulier la modélisation de la déformation du polymère lors de l'extrusion et du dépôt permet de connaître l'état de l'interface entre plusieurs cordons. Il permet également de connaître l'orientation du polymère ce qui joue un rôle sur la cristallisation. Des observations expérimentales de la cristallinité d'objets produits en FFF ont été réalisées par microscopie optique ou Raman. Enfin des simulations numériques et des observations par capteur optique ont permis d'étudier la déformation d'une pièce lors du refroidissement.

1.1. INTRODUCTION

Introduction

Additive manufacturing (AM) technologies are defined as the "process of joining materials to make objects from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies, such as traditional machining" by the ASTM: F2792 -12a (2012) standard. This technology has been widely developed in recent years since it enables to build objects with complex shapes. Moreover, it requires minimum needs for post processing and nearly no material wastes. It can be used with different materials, such as metals, ceramics and polymers [START_REF] Bikas | Additive manufacturing methods and modelling approaches: a critical review[END_REF]. Different techniques have been developed such as laser based processes, extrusion processes, material jetting, adhesive or electron beam. The present study focuses on the extrusion deposition process of polymers called Fused Deposition Modeling (FDM) or Fused Filament Fabrication (FFF). This process is commonly used as it is easy to operate, reproducible, low cost and is able to accept sustainable materials [2].

The FFF process consists in feeding a liquefier with a solid filament polymer via a pinch roller mechanism. The polymer is heated in the liquefier above its melting temperature and then deposited on a surface. The substrate on which the object is printed is usually heated for processing purposes [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. The printing nozzle can either move on the three directions of space or only on the horizontal x -y plane while the build stage is moving in the vertical z plane. This depends on the type of machine. A scheme of the process is depicted in Figure 1.1. Moving in the three directions of space enables to build objects with complex shapes. The desired object is obtained after solidification of the polymer. The major parameters influencing the strand shape are the gap distance g between the nozzle and the depositing surface, the extrusion diameter D of the nozzle, the extrusion velocity U and the nozzle velocity V [4]. The gap between the nozzle and the surface is tailored depending on the height ∆H of the deposited strand. Most 3D printers can reduce this gap down to 0.1 mm. The global geometry of the liquefier is common to the majority of the 3D printers. The liquefier can be divided into three zones depicted in Figure 1.1. Zone I is the first cylindrical part through which the neat filament is introduced. Zone II is the converging part linked to zone III which is the capillary part of the liquefier. Various nozzle designs exist depending on the machine. In particular, the nozzle angle ω can differ significantly.

Typical polymers used in this process are thermoplastic polymers. They are fed into the liquefier in form of filaments with a diameter of about 1.5 to 3 mm [5]. The heating of the filament occurs in the heated die which has a slightly larger diameter than the filament. For a filament of 1.75 mm, the heated die has a 2 mm diameter. Currently, the most common polymers used in FFF are Acrylonitrile Butadiene Styrene (ABS) and Polylactic Acid (PLA) [START_REF] Bikas | Additive manufacturing methods and modelling approaches: a critical review[END_REF]. The pinch rollers mechanism provides the force needed to extrude the polymer melt. The extrusion force is mainly dependent on the printing head geometry and the polymer melt viscosity. A thermistor is typically used in the liquefier to melt the polymer [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. Once extruded, the deposited strand adheres to the build surface. The first deposited layer also needs to be easily removable from the substrate after printing. The environment temperature should also be controlled to provide good adhesion between strands. Moreover, large thermal gradients between the melt and its environment can lead to warping and distortion of the final structure [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. The printed object can be post-processed by chemical (solvent vapors) or mechanical (abrasion) smoothing to obtain the desired surface finish.

The FDM process can be divided into three main stages. The first one is the heating and fusion of the polymer in the liquefier. The second is the deposition stage during which the polymer melt is extruded and deposited on the surface. The third is the healing stage dedicated to the solidification of polymer and adhesion between the strands. The polymer undergoes changes of conformation and different thermal phenomena during these three steps. This leads to different microstructure of the final object. The changes undergone by the polymer need to be better understood.

Understanding the liquefier heating behaviour and the dynamics of the melting polymer is critical to better control the FFF process [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. Different techniques have been used to measure the melt temperature and flow profile inside the liquefier. Analytical models have been developed to compute the heating and pressure drop in the liquefier [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF][START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF][START_REF] Osswald | Fused filament fabrication melting model[END_REF]. This is detailed in section 1.2.

The strand shape of the deposited polymer melt depends on the fluid dynamics and heat transfer. Knowing the size of the strand enables to better calibrate the deposition path depending on the geometry of the desired object. This has an influence on the porosity of the produced part [START_REF] Nouri | Structural imperfections in additive manufacturing perceived from the X-ray micro-tomography perspective[END_REF]. The extrusion of the molten polymer out of the nozzle also impacts the conformation and micro structure of the deposited polymer [START_REF] Mcilroy | A fundamental rule: Determining the importance of flow prior to polymer crystallization[END_REF]. This has an influence on the properties of the printed object upon cooling. Indeed, depending on the state of the melt, the adhesion with adjacent strands may be different [START_REF] Mcilroy | Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing[END_REF]. Analytical models and simulations have thus been put forward to understand the deposition behaviour [4,[START_REF] Xia | Fully resolved numerical simulations of fused deposition modeling. Part I: fluid flow[END_REF][START_REF] Xia | Fully resolved numerical simulations of fused deposition modeling. Part II -solidification, residual stresses and modeling of the nozzle[END_REF]. Different experimental and analytical studies are investigated to better understand the thermomechanical properties of the FFF process. This step is described insSection 1.3.

The cooling behaviour of single and multiple strands have a major impact on the properties of the printed object. The deposited strands undergoes multiple heating and cooling cycles during the different stages of the process [START_REF] Seppala | Infrared thermography of welding zones produced by polymer extrusion additive manufacturing[END_REF]. This has an influence on the solidification, microstructure, weld strength and porosity of the part [START_REF] Seppala | Weld formation during material extrusion additive manufacturing[END_REF]. The strength of the weld between the strands and the layers is particularly important as it is one of the main factors giving the mechanical properties of the object [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. Details about the post-deposition of the filaments is presented in section 1.4.

In the case of semi-crystalline polymers, the cooling rate has a major impact on the final crystallinity of the printed object. Multiple studies have thus focused on the measurement of the polymer cooling during this stage. Moreover, numerical models to simulate the various thermal 1.2. DYNAMICS OF THE LIQUEFIER phenomena occurring during this cooling stage give a better understanding on the distortion behaviour. The review of works on microstructure is presented in subsection 1.5.

Dynamics of the liquefier

Fused Deposition Modelling is a temperature driven process. The polymer experiences different transitions during the process. The first step to investigate is the liquefier where the polymer is heated above the melting temperature. Since polymers have poor thermal conductivity, temperature gradients may appear between the center of the liquefier and its rim. This can induce complex flows. Melt properties are non-linear functions of temperature and shear rate, which makes it difficult to study and model [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF].

The flow and temperature profiles are essentially controlled by the roller feeding mechanism (fluid velocity), heat transfer and pressure drop in the liquefier [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF].

Heat transfer in the liquefier

To better understand how the polymer is melted, the major heat transfer mode has to be identified. This provides information about the state at which it is extruded through the liquefier. Some authors [START_REF] Yardimci | Thermal analysis of fused deposition[END_REF][START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF] assume that the wall temperature of the liquefier cylinder is constant due to its high thermal conductivity. Other authors [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF][START_REF] Phan | Rheological and heat transfer effects in fused filament fabrication[END_REF] consider that the current-carrying heater provides a constant heat flux through the liquefier walls. The heat power can thus be calculated based on a simple heat balance as follows:

Φ p = ṁC p (T e -T i ), (1.1)
where ṁ is the mass flow rate of the polymer through the nozzle, C p the specific heat capacity of the polymer, T e and T i are the temperatures of the polymer at the exit and entrance of the liquefier, respectively. This relation assumes that the heat capacity is constant. It is not true since the polymer undergoes different phases during the printing such as the glass transition [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF] for amorphous polymers, and fusion for semi-crystalline polymers.

Osswald et al. [START_REF] Osswald | Fused filament fabrication melting model[END_REF] propose an analytical model to describe the melting of polymer in the liquefier.

The model considers melting of the polymer with a pressure flow removal. The polymer only melts at the bottom surface of the liquefier, in the converging region. The filament is pressed against the heated surface with a given force. A melt region thus forms in the cylindrical part, with a certain length starting from the converging surface. An illustration of the proposed model is displayed in Figure 1.2. The model predicts a decrease of the melt thickness from 80 to 35 µm when increasing the applied force from 20 to 100 N [START_REF] Osswald | Fused filament fabrication melting model[END_REF]. The study then investigates the influence of the heater temperature, initial filament temperature, and nozzle angle on the extrusion velocity for a given applied force. One interesting result underlines a decrease of the extrusion velocity when increasing the nozzle angle.

Anderegg et al. [START_REF] Anderegg | In-situ monitoring of polymer flow temperature and pressure in extrusion based additive manufacturing[END_REF] developed an experimental protocol to perform in-situ measurements of the polymer temperature and pressure in a liquefier. The measurements are done in the cylindrical region I of the liquefier. The study underlines that the temperature of the polymer decreases linearly with the flow rate. A maximum decrease reaches 6.5 • C at a flow rate of 5.5 mm 3 /s for the particular design used in reference [START_REF] Anderegg | In-situ monitoring of polymer flow temperature and pressure in extrusion based additive manufacturing[END_REF]. The viscosity of the polymer is also determined based on the pressure, apparent shear rates and thermal measurements. The result underlines higher viscosity compared to capillary measurements. Anderegg et al. [START_REF] Anderegg | In-situ monitoring of polymer flow temperature and pressure in extrusion based additive manufacturing[END_REF] concluded that isothermal condition in the cylindrical region is not reached. They suggest that local thermal gradients increase the melt viscosity. These results are in agreement with in-situ measurements of the polymer melt temperature in the liquefier done by Peng et al. [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF]. Temperature measurements are achieved by embedding ultra-fine thermocouples inside the polymer filament during fusion in the liquefier. The temperature profile is highly in-homogeneous in the liquefier. Indeed for extrusion velocities larger than 90 mm/min, the temperature at the center of the filament never reaches the targeted wall temperature.

Melt flow in the liquefier

To give information about velocity, shear rate and pressure drop, the rheology of the molten polymer has to be known. Usually, the shear thinning behaviour of the polymer melt is considered by writing the dynamic viscosity as follows [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF][START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF][START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF][START_REF] Yardimci | Thermal analysis of fused deposition[END_REF][START_REF] Phan | Rheological and heat transfer effects in fused filament fabrication[END_REF]]

η = K( γ) m-1 , (1.2)
where η is the dynamic viscosity, γ the shear rate, K the consistency index and m the shear thinning index. However the dependence of the viscosity with the temperature has to be taken into account as the melt temperature is not uniform in the section of the cylinder. Therefore some authors have introduced the shift factor a T , which gives the following expression of viscosity:

η( γ, T ) = η 0 [a T γ] m-1 a T . (1.3)
Authors such as Bellini [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF] and Ramanath et al. [START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF] have introduced a shift factor that follows an Arrhenius' law. Other authors [START_REF] Hofstaetter | Simulation of a downsized fdm nozzle[END_REF][START_REF] Coogan | In-line rheological monitoring of fused deposition modeling[END_REF] have used a Williams-Landel-Ferry (WLF) shift factor. The advantage of using a power-law model is its simplicity, however it does not account for all the phenomena occurring in the liquefier [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. According to Peng et al. [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF] the relation (1.3) of Bellini [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF] is only valid for semi-crystalline polymers above their melting temperature and amorphous polymers well above their glass transition temperature (T > T g + 100°K). In particular, the power law model is only valid for a limited range of shear rates. An improvement of the power 1.2. DYNAMICS OF THE LIQUEFIER Figure 1.3: Comparison between predicted flow profile (solid line) and pigment particle distribution (white and black circle) in the liquefier [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF].

law model is to take into account the Newtonian plateau at low shear rates by incorporating a Carreau-Yasuda or a Cross-WLF law. Coogan et al. [START_REF] Coogan | In-line rheological monitoring of fused deposition modeling[END_REF] have developed a 3D printing nozzle with in-line measurements of the polymer rheology during extrusion. The pressure inside the nozzle is measured to calculate the shear stress. The viscosity of the polymer is then calculated based on the shear rate. The in-line rheology measurements are fitted with a Cross-WLF law such as:

η( γ, T ) = η 0 1 + η 0 γ γcrit 1-m , (1.4)
with γcrit the critical shear rate at which the viscosity profile moves from Newtonian to shear thinning and m the power law index. In this model the zero shear viscosity η 0 accounts for the dependence of the viscosity with temperature such as:

η 0 (T ) = D 1 exp - A 1 (T -T ref ) A 2 + (T -T ref ) , (1.5)
with D 1 , A 1 and A 2 the fitting parameters and T ref a reference temperature. The in-line rheology tests show that this model accurately describes the rheological behaviour of the polymer in the liquefier.

Modelling the fluid dynamics and heat transfer can be tricky as polymer experiences different states, especially near the T g and thus behaves differently through the liquefier. Therefore, Peng et al. [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF] have focused on visualizing experimentally the melt flow in the liquefier. The experimental protocol managed to acquire in-situ measurements through the process. The melt flow is visualised by incorporating inorganic pigments in the section of the polymer filament at given distances. The liquefier is then cut off to take out the solidified polymer with their pigments. Sections of solidified melts are then examined using optical microscopy and X-Ray tomography.

The polymer temperature behaviour is also measured throughout the whole extrusion.

Peng et al. [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF] show a parabolic profile in zone I depicted in Figure 1.1 and a stretched profile in zones II and III highlighting a rapid acceleration at the center of the liquefier in these zones. However, the flow profile differs from the isothermal power law profile in the liquefier. In zone I, the shear rate is smaller near the center and a plug-flow profile is observed while an isothermal Newtonian model predicts a parabolic profile as illustrated in Figure 1.3. According to Peng et al. [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF] the plug flow profile is due to lower polymer temperature at the center of the filament CHAPTER 1. CONTEXT AND LITERATURE REVIEW explained in § 1.2.1. Lower temperature induces higher viscosity at the center and thus lower shear rates leading to this plug flow profile.

Pressure drop

The pressure drop governs the force and torque provided by the rollers to push the polymer filament through the liquefier [START_REF] Coogan | In-line rheological monitoring of fused deposition modeling[END_REF]. Difference of cross section through the liquefier influences the pressure drop and thus the polymer flow [START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF]. The geometry of the liquefier is approximately the same for most of actual 3D printers [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. It can be divided in three sections as illustrated in Figure 1.1.

Bellini et al. [START_REF] Bellini | Liquefier Dynamics in Fused Deposition[END_REF] and Ramanath et al. [START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF] determined the pressure drops in the three sections taking into account the temperature dependence. They solved the momentum balance using the Hagen-Poiseuille equation with a power law fluid coupled with an Arrhenius shift factor. The pressure drops in each section of the liquefier are given as:

∆P I = 2L 1 U i K 1/m m + 3 R m+1 i 1/m a T (1.6
)

∆P II = 2m 3 tan(ω/2) 1 D 3/m - 1 (2R i ) 3/m R 2 i (m + 3)2 m+3 1/m a T (1.7) ∆P III = 2L 2 U i K 1/m (m + 3)R 2 i (D/2) m+1 1/m a T (1.8)
with L 1 the length of the cylinder in region I, L 2 the length of the capillary, U i the inlet velocity, K the power law consistency, m the power law index, R i the radius of the cylinder at the inlet, D the diameter of the capillary, and ω the nozzle angle. The shift factor a T accounts for the dependence of the pressure drop with temperature and is defined by an Arrhenius law such as

a T = exp E a R 1 T - 1 T ref .
(1.9)

The total pressure drop in the liquefier is calculated by adding the partial pressure drops of all three sections such that ∆P = ∆P I + ∆P II + ∆P III .

(1.10)

The total pressure drop gives the force needed to extrude the polymer out of the liquefier. Ramanath et al. [START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF] have studied the impact of several parameters on the pressure drop by performing a Finite Element Analysis (FEA). The study focused on the influence of the velocity, nozzle angle and diameter. The FEA calculations were achieved for a steady-state regime and isothermal condition. Like most studies, the no-slip condition is assumed at the liquefier wall.

Varying the angle from 20 to 60°in zone II showed similar velocity profiles. The total pressure drop decreases when increasing the angle in zone II up to 60°and when increasing the exit nozzle diameter. Finally the simulations performed by Ramanath et al. [START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF] show smaller pressure drops compared to the mathematical model developed by Bellini et al. [START_REF] Bellini | Liquefier Dynamics in Fused Deposition[END_REF]. Anderegg et al. [START_REF] Anderegg | In-situ monitoring of polymer flow temperature and pressure in extrusion based additive manufacturing[END_REF] performed in-situ measurements of the pressure in the liquefier. The pressure measured in the liquefier is also compared with the analytical power law model provided in [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF]. The result shows higher pressure in the liquefier compared to the model.

Other authors [5,[START_REF] Phan | Rheological and heat transfer effects in fused filament fabrication[END_REF] have proposed analytical estimations of pressure drop in the liquefier. Phan et al. [START_REF] Phan | Rheological and heat transfer effects in fused filament fabrication[END_REF] have used a modified Cogswell model to express the pressure drop in zone II in order to account for the elongational viscosity. The model also defines the shear viscosity of 1.3. FILAMENT DEPOSITION the polymer as a power law fluid. By incorporating a shift factor that follows an Arrhenius law, the study links the pressure drop and the temperature at the exit of the nozzle. This model enables to determine the actual polymer melt exit temperature by measuring the pressure drop through the process. The pressure drop was measured by monitoring the power needed to drive the filament through the nozzle. Experiments have shown that additional pressure drop that were not taken into account in the model could be observed. According to Phan et al. [START_REF] Phan | Rheological and heat transfer effects in fused filament fabrication[END_REF], the additional pressure drop could be due to thermal resistance caused by fouling on the wall of the nozzle. The fouling would induce additional heat transfer effects which would decrease the actual wall temperature. Most 3D printer liquefiers are uneasy to clean efficiently and polymer fouling can be an issue. Serdeczny et al. [START_REF] Serdeczny | Experimental and analytical study of the polymer melt flow through the hot-end in material extrusion additive manufacturing[END_REF] have compared experimental measurements of the pressure drop with the models proposed by Bellini [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF] and Phan et al. [START_REF] Phan | Rheological and heat transfer effects in fused filament fabrication[END_REF]. The results showed differences between the experimental data and analytical models, especially for feeding rates U i > 100 mm/min and nozzle diameter D < 0.6 mm. According to Serdeczny et al. [START_REF] Serdeczny | Experimental and analytical study of the polymer melt flow through the hot-end in material extrusion additive manufacturing[END_REF], these discrepancies are due to anisothermal flow conditions and even incomplete melting of the polymer in the liquefier at high feeding rates which are not taken into account in the previous models.

Filament deposition

The velocity field of the flow changes throughout the process as it comes from a cylindrical die to a moving free surface. In the case where the nozzle is far from the substrate, the polymer melt expands when extruded due to the die swell phenomenon [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. In the most common case where the gap is small (around 0.1 to 0.2 mm from the surface), understanding the velocity field during the extrusion and deposition gives insights on the polymer conformation, limiting feed velocity and strand shape.

Extrusion of the melt

When the deposition is performed with high gaps, the free extrusion needs to be considered. It is the moment when the polymer melt is extruded out of the nozzle before touching the surface. During this step and according to Bellini et al. [START_REF] Bellini | Liquefier Dynamics in Fused Deposition[END_REF], the melt is only subjected to gravitational force and to surface tension. It is also only subject to heat convection with air in its environment.

Bellini [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF] has simulated the viscoelastic behaviour of the melt during the free extrusion step of the process. Swelling is observed during this step which is similar to the swelling observed experimentally. Similar results of swelling due to viscoelastic effects have been simulated by Xia et al. [START_REF] Xia | A numerical study of the effect of viscoelastic stresses in fused filament fabrication[END_REF] using a Maxwell model (Oldroyd B model). An increase of the swelling is observed when increasing the polymer relaxation time. Serdeczny et al. [START_REF] Serdeczny | Experimental and analytical study of the polymer melt flow through the hot-end in material extrusion additive manufacturing[END_REF] have studied the influence of the temperature, length of the capillary region and nozzle diameter on the swelling phenomenon. Measurement of the diameter of the strand is done with a digital camera during extrusion. The swelling ratio is obtained by dividing the strand diameter by the nozzle diameter. The swelling increases when increasing the extrusion temperature and feeding rate, or decreasing the nozzle diameter and the length of the liquefier.

McIlroy et al. [START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF] have proposed a model to describe the strain field during extrusion. They assumed that the polymer melt experiences a 90°turn from the exit of the nozzle to the substrate as illustrated in constraints due to convective flow. The model enables to study the polymer chain orientation after its 90°turn. Figure 1.4 illustrates the polymer chains as deformed coloured spheres. An increase in the velocity field leads to more stretching of the polymer, especially on the outer edge of the deposit. The same phenomenon is observed by Xia et al. [START_REF] Xia | A numerical study of the effect of viscoelastic stresses in fused filament fabrication[END_REF] during numerical simulations of strand deposition. The stretching of polymer chains increases at the nozzle wall where the shear rate is the highest. It then drops dramatically once it is deposited. The deformation of the polymer has an impact on the properties of the final product and will be further discussed in § 1.4.2.2. Bertevas et al. [START_REF] Bertevas | Smoothed particle hydrodynamics (SPH) modeling of fiber orientation in a 3D printing process[END_REF] have developed a numerical model to simulate the orientation of particles inside the polymer melt during extrusion and deposition. The results showed that decreasing the nozzle velocity leads to a more skin/core effect with more oriented particles along the deposition direction at the surface of the strand. The model also studies the impact of the different nozzle regions on the orientation of particles. An increase of fibre alignment is observed with the stream from the cylindrical to the converging region.

The feed velocity is an important parameter to obtain a good strand deposition. Indeed, if the feed velocity is too high, polymer filament will not spend enough time in the liquefier to be correctly melted. If the core of the filament does not reach the melting temperature, failure may occur during extrusion. Mackay et al. [5] have measured the failure feed velocity as a function of the nozzle temperature. Figure 1.5 shows the results of the failure feed velocity as a function of the nozzle temperature for different polymers. As expected, the failure feed velocity increases with increasing the liquefier temperature. The results shows that the failure is dependent on polymer type, flow condition and nozzle diameter. Mackay et al. [5] have developed a master curve to predict the failure feed velocity for different polymers. It enables to determine the minimum temperature for which viscosity of the polymer is too high for passing through the nozzle and thus causes failure.

Serdeczny et al. [START_REF] Serdeczny | Numerical modeling of the polymer flow through the hot-end in filament-based material extrusion additive manufacturing[END_REF] have also proposed a model to predict the failure feed velocity based on experimental observations. The model considers that the failure is only due to incomplete melting of the polymer filament in the liquefier. It is thus only based on the heat balance in the liquefier and is independent of the pressure drop. The maximum feeding velocity is thus given by:

U max e = h∆T π(2R e )L 1 [∆H f + C p (T e -T i ] ρA (1.11)
with h the heat transfer coefficient at the liquefier wall, ∆H f the latent heat of fusion, ∆T the mean temperature difference between the barrel wall and the melt, ρ the density and A the cross section area of the polymer melt. This model gives good prediction of the maximum feed velocity for a specific liquefier.

Strand shape

Several printing parameters have an influence on the strand shape. As previously explained, they are mainly the extrusion velocity, the moving nozzle velocity, the gap distance and the nozzle diameter [4]. Other factors can influence the shape of the strand such as its interaction with the tip of the head [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF] or the spreading of the bead due to similar surface energy with the solidified strands [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. Simulation of the melt flow and better understanding of the correlation between the nozzle speed rate and the extrusion flow rate can give insight on making more homogeneous strands.

A numerical model have been developed by Comminal et al. [4] to study the strand shape by modelling the flow field exiting the nozzle. The numerical model assumes a laminar and fully developed flow at the exit of the nozzle. The fluid is assumed incompressible and Newtonian. The viscosity is assumed sufficiently large to assume a creeping flow (Stokes regime). Comminal et al. [4] have studied the influence of two dimensionless parameters: the velocity ratio V /U and the normalised gap g/D on the shape of the strands. These results are illustrated in Figure 1.6.

The simulation shows that with a small gap and slow printing speed the polymer spreads more on the substrate and especially on the front side of the printing head. On the other hand, high printing speeds and large gaps show less flow on the side. The gap distance does not have an influence on the area of the strand cross section. Moreover, cylindrical strands are produced at fast printing speed and large gap for 0.8 ≤ g/D ≤ 1 and velocity ratio 0.75 ≤ V /U ≤ 1. According to Comminal et al. [4], the strand thickness is not always equal to the gap and varies between -10% to +35% of the gap. It is especially lower than the gap when the printing speed or gap is high. On the other hand when the printing velocity or gap is low, the extruded material can flow beside the nozzle tip as shown in Figure 1.6. The strand being no more restrained by the tip, its thickness can exceed the gap height. Moreover, the aspect ratio ∆W /∆H defined as the ratio between the width and the thickness of the deposited strand is studied. The aspect ratio increases when the printing velocity and gap distance decrease. According to Comminal et al. [4], this trend is similar to the idealised strand shape behaviour (which can be either ellipse, oblong or cuboid) but the simulation values are different. These simulations give a rough idea of the printing parameters needed to obtain good strand shapes. Bellini [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF] has simulated the deposition of the melted polymer to highlight the influence of the printing head on the strand shape. In the case where deposition is performed with a high gap from the surface, the strand should not have interactions with the printing head. The simulations thus describe the two cases of low and high gap deposition. When simulations are done without the interaction of a printing head, the road is not flat and present distortion. On the other hand when the printing tip is present, it provides flattening of the top surface as depicted in Figure 1.7.

Bellini [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF] have also described the velocity and temperature during single and multiple layer depositions. These simulations are in good agreements with experimental observations in terms of geometry and temperature distribution. They highlighted reheating of the first layer when depositing a second layer upon the previous one. It is especially noticeable at the interface. The simulated fluid dynamics also underlines that the deposition of a second layer does not have an Simulation of the first deposited strand shape with and without interaction of the printing head according to Bellini [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF] effect on the shape of the first layer. This statement can be argued as when higher flow rate and/or higher temperature were assumed, the position of the interface was affected.

Xia et al. [START_REF] Xia | Fully resolved numerical simulations of fused deposition modeling. Part I: fluid flow[END_REF][START_REF] Xia | Fully resolved numerical simulations of fused deposition modeling. Part II -solidification, residual stresses and modeling of the nozzle[END_REF][START_REF] Xia | A numerical study of the effect of viscoelastic stresses in fused filament fabrication[END_REF][START_REF] Xia | Simulations of fused filament fabrication using a front tracking method[END_REF] have also developed numerical models to simulate the deposition of multiple strands. In particular they highlighted the importance of the visco-elastic behaviour of the polymer during strand deposition. A comparison of the strand shape with and without visco-elasticity is investigated. An increase of the deposited strand height and a decrease of the strand width are found when the viscoelastic polymer is considered. According to the study, this behaviour is due to the swelling effect and to a decrease of the flowability due to visco-elasticity.

Temperature behaviour of the deposited strand

Understanding the thermal behaviour during printing is very important as it provides information for the post deposition analysis. It is especially essential when describing the bonding quality. During the FDM process, the polymer filament is subject to multiple heating and cooling. Indeed, after being melted in the liquefier the polymer is extruded on the substrate or on previously deposited layers. It then cools down and solidifies. Re-heating of the layer occurs if a new layer is deposited upon it. All these thermal cycles induce complex thermal and flow boundary conditions [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF]. Moreover, different heat transfer modes occurs whether the strand has just been deposited or not. When a strand is deposited on a layer, it is subjected to conduction with the preceding surface and adjacent roads and convection with the environment. The stacked rows however mainly experience conduction with neighbouring strands [2]. The different heat transfer phenomena depending on the stage of the process are illustrated in Figure 1.8.

Analytical and numerical models

Bellehumeur et al. [START_REF] Bellehumeur | Modeling of bond formation between polymer filaments in the fused deposition modeling process[END_REF] have developed an approximation model to describe cooling through the length of the strand. The analytical model assumes uniform temperature distribution throughout the cross section, semi-infinite filament length and constant heat transfer coefficients. This study focuses on the cooling of a single deposited strand. The model is thus unidimensional and considers the conduction heat transfer with the substrate as part of heat convection. The following heat transfer equation is considered,

V dT dx = k ρC p d 2 T dx 2 - P ρC p S h(T -T ∞ ), (1.12)
with T the section average temperature, x the direction following the strand axis, T ∞ the temperature of the strand for x → ∞, V the printing speed, k the thermal conductivity, P the filament cross section perimeter, S the filament cross section area and h the heat transfer coefficient taking into account the heat convection with air and heat conduction with the substrate. The analytical solution of equation (1.12) is then given by: .13) In this model, the conduction heat transfer with the substrate is dominant compared to convective heat transfer with air. The advantage of the model is its simplicity. It enables to determine the time during which a cooling polymer strand is maintained above T g . It allows to investigate the healing behaviour between adjacent strands as it will be further explained in §1.4. However the model only focuses on the cooling of a single strand. Therefore it does not consider the different cooling and heating cycles inherent to the process. These thermal phenomena are more tricky to study with an analytical model. Therefore, numerical simulations have been proposed by other authors.

T = T ∞ + (T 0 -T ∞ ) exp   - 1 + 4 k ρCpV P ρCpSV h -1 2 P ρCpSV h x   . ( 1
Costa et al. [2] have developed a numerical model to predict the cooling of a strand. It takes into account conduction between adjacent filaments but neglects heat conduction with the substrate. They incorporated a boolean variable to define if a deposited strand is in contact with other strands or not. Different heat transfer are considered depending on the existing contacts with other strands. Using this variable, Costa et al. [2] have computed the filament average temperature through the cross section of the strand over time. The temperature is given depending on the location x on the strand and the time during which the length x has been deposited. Experimental measurements were also performed by a thermography set-up. A good agreement between simulations and experiments is obtained for the first three stacked polymer filaments.

Zhang et al. [START_REF] Zhang | Linear-Time Thermal Simulation of As-Manufactured FDM Components[END_REF] computed the temperature field in a FFF printed product using a finitedifference method. Multiple contacts between printed strands are taken into account. The model thus takes into account the different heat transfer phenomena corresponding to these 1.4. POST DEPOSITION BEHAVIOUR contacts. Thermal computations are only performed on segments directly influenced by the deposition of a new segment. The overall temperature is predicted with minimum computation time.

Experimental measurements

As explained in §1.3.3.1, Costa et al. [2] have developed a model to describe the cooling during strand deposition. Experimental data of the strand temperature are conducted using an Infrared (IR) camera to determine the heat transfer coefficient during cooling using the Hilpert correlation [2]. The measured temperature is compared with the simulated temperature. Peel-like tests are done by pulling apart bonded roads to determine the adhesion strength. The cooling effect provided by the support in the first layer is more important than the heating effect provided by a new deposited strand. Moreover, the contact area between stacked roads is more important for bottom roads than upper roads. Decreasing the extrusion temperature showed a decrease in the bonding strength which highlights the importance of providing enough healing time and temperature.

Seppala et al. [START_REF] Seppala | Infrared thermography of welding zones produced by polymer extrusion additive manufacturing[END_REF] have also used IR thermography to study the temperature behaviour of the deposited strand during solidification. IR camera was preferred compared to thermocouples as matching the spacial and temporal scales with the process is more straightforward. This study focuses on the recording of the temperature profile of deposited ABS polymer strands. IR imaging of the heated nozzle without extrusion is used to subtract the reflection signal, which provides estimation of indirect radiative heating from the nozzle. This technique enables to determine the reflected intensity from the polymer. This leads to the calculation of the intensity emitted from the polymer. To convert the emitted intensity into the actual temperature, a temperature-IR signal curve is plotted by measuring the polymer emitted intensity at known temperatures. The temperature can then be calculated using the Planck's law.

According to Seppala et al. [START_REF] Seppala | Infrared thermography of welding zones produced by polymer extrusion additive manufacturing[END_REF], the temperature of a strand drops below the polymer glass transition temperature in less than 2 seconds after deposition. Re-heating of the previously deposited layer above the glass transition temperature occurs approximately 0.5 seconds after deposition. Deposition of a new layer induces re-heating of the two previously deposited layers, as shown in Figure 1.9 [START_REF] Seppala | Infrared thermography of welding zones produced by polymer extrusion additive manufacturing[END_REF]. The weld temperature is determined by using the average temperature of the two adjacent layers. The influence of the printing head temperature on previously deposited strands is studied. Heating from radiative heat transfer of the printing head can increase the strand temperature to less than 3°C. This study provides useful information as the time during which a strand is maintained above its glass transition temperature is an important parameter for the healing behaviour.

Post deposition behaviour

One key parameter for having good mechanical properties of the final part lies on the bonding between the solidified strands. Indeed the failure points of the parts made with FFF are most generally localised on the welding zones [START_REF] Seppala | Infrared thermography of welding zones produced by polymer extrusion additive manufacturing[END_REF]. The healing of the interface is achieved in four stages which can be described as: (1) surface contact, (2) neck growth or wetting, (3) diffusion, (4) randomisation as illustrated in Figure 1.10. The quality of the bonding essentially depends on the growth of the neck between the strands (wetting) and diffusion [START_REF] Bellehumeur | Modeling of bond formation between polymer filaments in the fused deposition modeling process[END_REF] which heals the interface. The temperature history of the deposited strand during the whole cooling and reheating process is a critical information for welding. Several authors have studied the adhesion quality based on the thermal models and measurements described in the previous section.

Neck growth

Bellehumeur et al. [START_REF] Bellehumeur | Modeling of bond formation between polymer filaments in the fused deposition modeling process[END_REF] did experiments on an ABS P400 polymer to determine the sintering process during bond formation. The goal is to determine the sintering rate, based on the heat transfer model described in § 1.3.3.1. Neck growth is studied under controlled environment. Indeed, in-situ observations of this phenomena during the FFF process is quite challenging. The coalescence behaviour is observed by a CCD camera coupled to an optical microscope. The sintering rate is then calculated using the following formula based on an approximation of neck growth for Newtonian fluids [START_REF] Bellehumeur | Modeling of bond formation between polymer filaments in the fused deposition modeling process[END_REF]:

∂ϑ ∂t = Γ a 0 η 2 -5/3 cos ϑ sin ϑ(2 -cos ϑ) 1/3 (1 -cos ϑ)(1 + cos ϑ) 1/3 , (1.14)
with ϑ = arcsin(y/a) with y the neck radius and a the particle radius as illustrated in Figure 1.10, η the viscosity, a 0 the initial particle radius, t the time and Γ the surface tension. Particles of ABS are heated under isothermal conditions to determine the surface tension of the material. Experiments achieved under anisothermal conditions were done to assess the developed model. According to the authors, the model slightly over predicts neck growth formation. It can however still be used for further studies as it is in acceptable agreements with experimental data.

POST DEPOSITION BEHAVIOUR

Using this model, the effect of different printing parameters are studied on the neck growth of the particles. The extrusion temperature has a bigger impact on the neck growth than the environment temperature. The same model is used by Chaunier et al. [START_REF] Chaunier | Fusion-bonding behavior of plasticized corn proteins in fused deposition modeling process[END_REF] for studying the bonding behaviour of plasticised corn protein in FFF. Sun et al. [START_REF] Sun | Effect of processing conditions on the bonding quality of FDM polymer filaments[END_REF] have experimentally evaluated this model by measuring the temperature at the bottom layer of a printed part and by measuring the variation in the neck growth between adjacent filaments. The analytical model underestimates neck growth between adjacent strands.

The model developed by Bellehumeur et al. [START_REF] Bellehumeur | Modeling of bond formation between polymer filaments in the fused deposition modeling process[END_REF] is useful to predict neck growth between polymer strands. It depends on polymer viscosity and surface tension. Prior knowledge of the dependence of viscosity and surface tension with temperature is thus needed. Lepoivre et al. [START_REF] Lepoivre | Coalescence in fused filament fabrication process: Thermo-dependent characterization of high-performance polymer properties[END_REF] have developed an experimental protocol to measure the polymer surface tension on a wide range of temperatures. The sessile drop method and pendant drop method are used for low and high temperature, respectively. The viscosity of the polymer is measured by rheology analysis and the dependence of the viscosity with temperature is considered using an Arrhenius law. Based on these material properties, the degree of coalescence is predicted as a function of time and temperature, assuming an initial cylindrical strand geometry. The model is then used under anisothermal conditions for the case of FFF. The temperature solution is simulated based on a 2D heat transfer model which enables to compute the degree of coalescence during cooling of a part. The results highlighted the importance of accounting for the dependence of the polymer surface tension with temperature. Moreover the results showed a predominant impact of the heating chamber temperature and the time between deposition of two filaments on the degree of coalescence.

Even though neck growth is an important stage of polymer healing, diffusion and randomisation are the most important stages. Indeed, according to Kim et al. [START_REF] Kim | A theory of healing at a polymer-polymer interface[END_REF] the characteristic strength of the material appears during these stages and is thus investigated in the next subsection.

Diffusion at welding zones

Global approach

The main factors required to obtain good adhesion are the welding zone temperature and the time during which it is maintained above T g . During this time, inter-diffusion of polymer macromolecules between the strands occurs and creates bonding between the roads. Several authors [2,[START_REF] Coasey | Nonisothermal welding in fused filament fabrication[END_REF][START_REF] Ko | Improving the filament weld-strength of fused filament fabrication products through improved interdiffusion[END_REF][START_REF] Yin | Interfacial bonding during multi-material fused deposition modeling (FDM) process due to inter-molecular diffusion[END_REF] have used the relation developed by Yang et al. [START_REF] Yang | Healing of Thermoplastic Polymers at an Interface under Nonisothermal Conditions[END_REF] to describe the adhesion behaviour in FFF based on the reptation theory [START_REF] De Gennes | Reptation of a Polymer Chain in the Presence of Fixed Obstacles[END_REF]. The polymer chains interpenetration at the interface between two strands depends mainly on thermal history. For a given temperature T > T g , a polymer chain can move across the interface over a certain distance. The maximal bond strength of the weld is achieved when the maximum interpenetration distance is reached. The welding time t w is the time for which the interpenetration distance reaches its maximum value. For a given temperature T > T g , the maximum of healing is thus achieved when the interface is maintained at T during a welding time t w . An increase of T allows further motions of polymer chains and the healing is reached faster. Yang et al. [START_REF] Yang | Healing of Thermoplastic Polymers at an Interface under Nonisothermal Conditions[END_REF] proposed a degree of healing written as follows

D h (t, T ) = 4 t 0 dt ′ t w (T ) . (1.15)
The properties of the interface reach the properties of the bulk material when D h = 1. The authors obtained the welding time at a given temperature T by performing lap shear tests. For a given polymer, the welding time is then fitted by an Arrhenius law (Eq. 1.9) to evaluate the welding time as a function of T . Coasey et al. [START_REF] Coasey | Nonisothermal welding in fused filament fabrication[END_REF] have proposed an alternate experimental protocol to determine the welding time in FFF based on rheological measurements. A generalised Maxwell model is used to determine the visco-elastic behaviour of the polymer. This enables to determine the relaxation time at different temperatures used to calculate the degree of healing. The dependence of the relaxation time with temperature is also given by an Arrhenius law. The degree of healing predicted by their model is then assessed with experimental bending tests at different temperatures. Yin et al. [START_REF] Yin | Interfacial bonding during multi-material fused deposition modeling (FDM) process due to inter-molecular diffusion[END_REF] have studied the impact of the substrate temperature, extrusion temperature and printing speed on the bonding strength. The bonding strength is measured by performing tensile tests on bonded specimens. The substrate temperature shows the most impact on the bonding strength. Moreover, an increase of 5.6% of the bonding strength is observed when the extrusion velocity increases from 8 to 12 mm/s.

Modelling polymer conformation

As explained in § 1.3.1, the extrusion process affects the polymer conformation and thus its microstructure as it experiences high shear rates during extrusion. McIlroy et al. [START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF] have developed a model for amorphous polymers that considers disentanglement induced by the flow. Indeed entanglement can be loss due to convection which has an impact on the welding. In this model the loss of entanglement is considered by the CCR parameter. The evolution of the entanglement fraction with time is also dependant on the reptation time of polymer chains. McIlroy et al. [START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF] have studied the asymmetric structure of the stand due to polymer deformation during extrusion. The polymer deformation depends on velocity and shear rate. The developed model shows changes in the polymer chains conformation during extrusion, when the flow experiences a 90°turn. According to McIlroy et al. [START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF], polymer chains are ultimately aligned with the flow direction during deposition. Computations also indicate that disentanglement, stretching and normal stress are more important at the outer edge during extrusion. This is due to the higher displacement field at this location, as represented in Figure 1.11. More disentangled polymer chains at the bottom of the strand after deposition is observed. This has an effect on the welding as more disentangle polymer chains induce better diffusion.

Indeed, according to McIlroy et al. [START_REF] Mcilroy | Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing[END_REF], the reptation time τ d depends on the temperature and shear rate such that:

1 τ d (T , γ) = 1 τ eq d (T ) + β CCR (K : A), (1.16)
1.4. POST DEPOSITION BEHAVIOUR Figure 1.12: Printed wall geometry [START_REF] Mcilroy | Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing[END_REF].

with τ eq d the equilibrium reptation time, β CCR the convective constrain release parameter, A the conformation tensor, K the velocity gradient tensor and : designating the double dot product. The degree of adhesion is then calculated based on the reptation time, in a similar way as in [START_REF] Yang | Healing of Thermoplastic Polymers at an Interface under Nonisothermal Conditions[END_REF]. The influence of the temperature and deformation profile induced by flow on the degree of healing can be investigated. Moreover this study considers mutual diffusion between layers. The diffusion of polymer in strand L p is also affected by diffusion of polymer in strand L p-1 . L p and L p-1 are illustrated in Figure 1.12. McIlroy et al. [START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF] have shown that polymer configuration is different on the upper part a p-1 than on the lower part b p of a strand, due to the extrusion process. Asymmetric structure of the strand has an effect on the diffusion of polymer. Polymer in the upper zone typically diffuses further across the interface than polymer from the bottom zone.

The effect of the printing velocity, entanglement number and CCR parameter on the degree of deformation and disentanglement were investigated. When the CCR parameter is equal to 1, the fluid is less shear thinning and the disentanglement due to deposition is more important. Conversely, reducing the CCR parameter slightly decreases disentanglement at the nozzle wall. Moreover increasing the printing velocity induces an increase of the deformation and disentanglement. According to McIlroy et al. [START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF], the disentanglement during the FFF process principally occurs during the deposition flow rather than during nozzle flow.

After determining the interpenetration distance for different cases, McIlroy et al. [START_REF] Mcilroy | Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing[END_REF] have studied the effect of different printing parameters on the weld thickness. It can be easily understood that thicker weld zones provide better mechanical performances of the final part. The study highlights that increasing the printing head speed provides thicker weld. Increasing the shear rate induces more disentanglement of the polymer chains which enhances mobility. Moreover, the increases of the print temperature facilitates diffusion at the interface and a better en-CHAPTER 1. CONTEXT AND LITERATURE REVIEW tangled network through the interface. Finally, higher molecular weight polymer reduces the entanglement network at the interface. Ko et al. [START_REF] Ko | Improving the filament weld-strength of fused filament fabrication products through improved interdiffusion[END_REF] have also demonstrated that incorporating plasticizer in the polymer can improve interpenetration and thus adhesion between strands.

Distortion of parts

After adhesion of the strands, the printed part cools and solidifies. As described in §1.3.3, the cooling of the part is highly heterogeneous. Distortion and warping may thus occur during the cooling step. Therefore, the knowledge of the impact of the different printing parameters on part distortion is important to reduce warping of the final object.

Armillotta et al. [START_REF] Armillotta | Warpage of FDM parts: Experimental tests and analytic model[END_REF] have developed an experimental protocol to measure part distortion during the cooling step of the process, once the object is printed. The measurements are done with a three dimensional scanner capture to measure point coordinates. The study investigates the influence of part dimension and layer thickness on the warping of the piece. The result shows major impact of the horizontal length of the object on the distortion. Other studies were able to measure distortion and residual strain during the printing of the part. Holzmond et al. [START_REF] Holzmond | In situ real time defect detection of 3D printed parts[END_REF] used image correlation to study the distortion during printing. The study uses a specific polymer with wood filler to obtain a high contrast speckled pattern. This enables to perform image correlation during the process without painting the object. The image correlation allows to determine the difference of part geometry in each layer compared with the CAD model. A reconstruction of the whole part is then done to know the localisation of defects. Kantaros et al. [START_REF] Kantaros | Fiber Bragg grating based investigation of residual strains in ABS parts fabricated by fused deposition modeling process[END_REF] used fibre Bragg grating to measure the residual strain during printing of a part. The fibre Bragg grating is embedded in the part during a printing pause of 3 s. The results show an increase of the residual strain when increasing the layer thickness.

Numerical models have been developed to predict the distortion in FFF. Several models have used an element activation method to simulate the deposition process [START_REF] Zhang | A parametric study of part distortions in fused deposition modelling using three-dimensional finite element analysis[END_REF][START_REF] Brenken | Development and validation of extrusion deposition additive manufacturing process simulations[END_REF][START_REF] Nath | Optimization of fused filament fabrication process parameters under uncertainty to maximize part geometry accuracy[END_REF]. The domain to be built is discretised into a given number of elements. Successive activation of the elements is done following a given deposition path to simulate the printing process. The dormant elements are not considered as solid domain until it is activated. Nath et al. [START_REF] Nath | Optimization of fused filament fabrication process parameters under uncertainty to maximize part geometry accuracy[END_REF] have considered an elastoplastic constitutive equation for which distortion is induced by thermal strain due to cooling. The model is used to calibrate the extrusion temperature, extrusion velocity and strand length to minimize the error of a targeted property. Brenken et al. [START_REF] Brenken | Development and validation of extrusion deposition additive manufacturing process simulations[END_REF] have also developed a numerical model to predict part distortion. The model considers the visco-elastic properties of the polymer using a generalised Maxwell model. Anisotropic properties are given to the element depending on the orientation of deposition. The distortion of the object is also induced by thermal strain rate. The crystallisation kinetics is computed to account for extra shrinkage upon crystallisation. The model is assessed by comparing the simulated part geometry with experimental measurements. The global geometry is measured using a 3D laser scanner. The model shows good predictions of the global final part distortion.

Microstructure

Crystallisation

Experimental studies

Most of the studies described so far have focused on amorphous polymers. Semi-crystalline polymers can be used on a wider range of temperature and globally have better mechanical properties [START_REF] Kishore | Additive manufacturing of high performance semicrystalline thermoplastics and their composites[END_REF]. The printing of semi-crystalline polymers can therefore improve the performances 1.5. MICROSTRUCTURE Figure 1.13: Schematic diagram of experimental setup for process line Raman spectroscopy [START_REF] Northcutt | Effect of processing conditions on crystallization kinetics during materials extrusion additive manufacturing[END_REF]. of printed objects. This is especially true if crystallisation occurs in the welding zone. This would significantly improve the bonding strength and thus the overall strength of the printed object [START_REF] Yan | Interfacial Crystalline Structures in Injection Over-Molded Polypropylene and Bond Strength[END_REF].

Northcutt et al. [START_REF] Northcutt | Effect of processing conditions on crystallization kinetics during materials extrusion additive manufacturing[END_REF] have mounted a Raman fiber optic probe after a printer head to investigate the impact of different printing parameters on the crystallisation of polycaprolactone (PCL), as illustrated in Figure 1.13. The study focuses on the impact of temperature and shear rate on the crystallisation behaviour. A range of extrusion velocities is applied at different temperatures. The study exhibits no impact of the shear rate on the final degree of crystallinity as all samples reached the average PCL crystallinity after sufficient time. However, increasing the wall shear rate at low temperature (90°C) demonstrates an increase in the crystallisation rate. This effect is not observed at high temperature (140°C). The study points out that it is difficult to conclude on the influence of the wall shear rate on the crystallisation kinetics. The latter is temperature dependent and the study did not measure the temperature profile inside the nozzle. It is thus difficult to know if the enhancement of the crystallisation rate is due to the flow motion. Indeed, the increased crystallisation kinetics at low temperature and high shear rate could be due to incomplete melting of the semi-crystalline entities in the liquefier. The presence of crystalline nuclei in the deposited strand could enhance the crystallisation kinetics due to a "self-seeding" effect [START_REF] Northcutt | Effect of processing conditions on crystallization kinetics during materials extrusion additive manufacturing[END_REF].

Spoerk et al. [START_REF] Spoerk | The consequences of different printing chamber temperatures in extrusion-based additive manufacturing[END_REF] have studied the effect of the printing environment temperature on the crystallisation of polypropylene (PP). They analysed the localisation of spherulites and their size depending on the environment temperature. The temperature exhibits no impact on the localisation of the spherulites. The spherulites do not particularly appear at the welding zones. It does however have an impact on the size of the spherulites. Indeed, increasing the chamber temperature increases the spherulite size. In the case of this analysis, increasing the environment temperature leads the deposited strand to be kept at a temperature near the PP maximum growth rate temperature. This has for effect to create fewer but bigger spherulites, as displayed in Figure 1.14. Increasing the chamber temperature also induces lower warpage and lower elongation at break of the final part. On the other hand, at temperatures below the PP maximum crystal growth rate temperature, spherulites are smaller but more numerous. Spoerk et al. [START_REF] Spoerk | The consequences of different printing chamber temperatures in extrusion-based additive manufacturing[END_REF] also demonstrate orientation of the crystalline phase in the direction of flow deposition. This has for effect to increase the axial thermal conductivity which also affects the tensile, bending and toughness properties of the final part.

Numerical modelling

Based on a similar approach as described in 1.4.2.2, McIlroy et al. [START_REF] Mcilroy | Modelling flow-enhanced crystallisation during fused filament fabrication of semi-crystalline polymer melts[END_REF] and McIlroy [START_REF] Mcilroy | A fundamental rule: Determining the importance of flow prior to polymer crystallization[END_REF] have developed an analytical model to study the effect of flow on the crystallisation behaviour. This model is based on the principle that stretched polymer creates additional nuclei. The nucleation rate is thus enhanced by the flow such that:

Ṅ (T , Λ) = Ṅq (T )(1 -Υ(Λ 4 -1)), (1.17)
with Ṅq the quiescent nucleation rate at a given temperature, Υ a fitting parameter that governs the total number of additional nuclei created by flow, and Λ the polymer stretching which is defined as:

Λ = trA 3 . (1.18)
Based on the computation of the nucleation rate, and by determining the growth rate, the degree of crystallinity is calculated by solving the Schneider rate equations [START_REF] Schneider | Non -isothermal Crystallization of polymers[END_REF]. The conformation tensor is computed by considering the visco-elastic effects of the polymer with a Rolie-poly equation. The problem is decomposed into two steps. First, the flow trough the nozzle and deposition is calculated at constant temperature, without crystallisation. This step enables to compute the deformation of the polmyer due to flow deposition. The second step considers that the flow is null and the crystallisation is computed based on the conformation of the polymer calculated at the previous step. McIlroy et al. [START_REF] Mcilroy | Modelling flow-enhanced crystallisation during fused filament fabrication of semi-crystalline polymer melts[END_REF] show that the flow enhanced nucleation only affects the surface of the strand as depicted in Figure 1.15. Indeed, at the center of the strand, the relaxation time is reached before the crystallisation temperature. This results in having a higher density of spherulites at the surface. This can induce lower adhesion between the strands as an increase of the spherulites may lower the reptation time and thus decrease diffusion. The increased stretching at the surface combined with higher cooling rates compared to the center of the strand may induce faster crystallisation kinetics.

Brenken et al. [START_REF] Brenken | Development and validation of extrusion deposition additive manufacturing process simulations[END_REF] have also developed a numerical model including the crystallisation kinetics.

The model is based on Avrami theory and modified by Velisaris et al. [START_REF] Velisaris | Crystallization kinetics of polyetheretherketone (PEEK) matrices[END_REF] for non isothermal conditions. The study only accounts for the crystallisation to study the distortion of polymer parts during cooling. The impact of the process on the crystallisation kinetics is not investigated.

Porosity

Porosity is an important defect in FFF as it weakens the mechanical properties of the final object. When strands are not completely bonded, it creates porosity inside the material. These 1.5. MICROSTRUCTURE Figure 1.15: Cross sectional view profile of the total number of Nuclei. The quiescent saturation limit has been set at 10 1 2. Only nucleation at the surface is thus affected by flow. [START_REF] Mcilroy | Modelling flow-enhanced crystallisation during fused filament fabrication of semi-crystalline polymer melts[END_REF] Figure 1.16: Cross sectional view of ABS part built with 0/90°orientations -set with no air gap between the strands [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF].

zones are more subjected to failure. The final object is unable to carry important stress through the different layers or roads if porosity exists [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF]. A lack of full adhesion between strands is a very common phenomenon in FFF, especially for adjacent strands of the same layer, as illustrated in Figure 1. [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF]. Therefore knowing what tends to create porosity and how to avoid it is important.

Figure 1.17: 3D view of the porous network for a printing angle θ = 45° [START_REF] Nouri | Structural imperfections in additive manufacturing perceived from the X-ray micro-tomography perspective[END_REF].

Nouri et al. [START_REF] Nouri | Structural imperfections in additive manufacturing perceived from the X-ray micro-tomography perspective[END_REF] have used X-ray micro tomography and Finite Element Analysis (FEA) to understand the impact of various process parameters on the creation of porosity. Changes in path direction and deceleration at the edges of each layer increase the porosity. Successive layers are more likely to create connected porosity in the vertical direction. The presence of porosity is however less important in the vertical direction. Indeed, less porosities are observed when the strands are stacked than when they are adjacent. At the outer faces of the material, the lack of cohesion between the strands increases the roughness of the surface. The study investigates the impact of the deposition path on the porosity content and pore connectivity. A 3D view of the strand network of a part printed at 45°can be observed in Figure 1.17. The CHAPTER 1. CONTEXT AND LITERATURE REVIEW strands printed at 45°show the largest porosity level, highest porosity fluctuation but lowest pore connectivity. This results lead to investigate the relation between porosity level and pore connectivity. The results indicates that increasing pore level does not necessarily increases pore connectivity. Nouri et al. [START_REF] Nouri | Structural imperfections in additive manufacturing perceived from the X-ray micro-tomography perspective[END_REF] have then studied the stress field of the printed material. The results show that porosity is acting as a stress contractor and that crack initiation is more likely to happen at pore location. Stress distribution is heterogeneous due to heterogeneous location of porosity. However, all printing directions show approximately the same stress distribution.

Conclusion and Ph.D. purposes

The present introduction is an overview of the heating, deposition and post deposition studies done on the Fused Filament Fabrication process. This review paves the way for new investigations to grasp a deeper understanding on the process.

The studies done on the liquefier show the importance of the heat transfer, fluid dynamics and pressure drop in FFF. All these phenomena have an important impact on the deposition dynamics. However, most studies are based on simple assumptions considering either Newtonian or power law fluids and few have coupled the heat transfer and fluid flow. Most numerical models only consider the capillary end of the nozzle during extrusion. Moreover, the prediction of the strand shape based on printing parameters is a key point to manufacture objects with desired properties. However, the experimental study of the deposition dynamics can be tricky because of the small scale of the strand. The first purpose of this Ph.D. is to perform experimental and numerical investigations on the deposition dynamics in FFF. The development of a numerical model coupling the fluid dynamics in the liquefier with the deposition dynamics of the melt is implemented. Heat transfer phenomena are introduced to describe the state changes in the polymer melt. An experimental protocol is developed to investigate the impact of different printing parameters on the strand shape.

Several studies have modelled the effect of flow on the conformation of the polymer chain in the deposited strand. The polymer conformation has an impact on the adhesion properties and crystallisation kinetics of the printed part. However, few experimental studies have been done to investigate the impact of the printing parameters on the crystallinity of the produced part. An experimental study is therefore done to see the impact of the extrusion velocity, cooling kinetics and part geometry on the crystallisation behaviour of objects produced by FFF. The goal is to investigate crystal orientation and microstructures produced by the process.

One major drawback of CFD models is its inability to compute the deposition of more than few strands without dramatically increase the computation time. The simulations are thus limited to a very small time range of the process. Moreover, models that do not account for the fluid dynamics use an element activation method to simulate the process of successive deposition. This method is highly non effective for simulating objects with important size. Indeed, fine mesh is needed in the whole domain to accurately capture the thermal or mechanical phenomena. Finally, very few studies have modelled the global crystallisation kinetics of parts produced with semi-crystalline polymers. Therefore, the second objective of the Ph.D. is to develop a numerical model to predict the cooling kinetics of objects with real dimensions. A re-meshing technique is used to maintain the number of elements as low as possible. The computation of the crystallisation kinetics is then implemented based on a heat transfer model.

Finally, heterogeneous cooling of the part induces residual stresses, warping and delamination. This is especially true for semi-crystalline polymers that usually have higher thermal expansion coefficients than amorphous polymers [START_REF] Spoerk | Polypropylene Filled With Glass Spheres in Extrusion-Based Additive Manufacturing: Effect of Filler Size and Printing Chamber Temperature[END_REF]. The experimental protocols developed to measure

Introduction

The goal of this chapter is to investigate the deposition of an amorphous polymer during the FFF process. A numerical model focusing on the scale of a single strand is implemented. As mentioned previously, several authors have developed numerical models to simulate the deposition of strands [START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF]4,[START_REF] Xia | Fully resolved numerical simulations of fused deposition modeling. Part I: fluid flow[END_REF]. All of them are only considering the last cylindrical part of the extruder. The present model also takes into consideration the converging part of the extruder. Therefore it is possible to study the strain rate throughout the process, from the insertion of polymer to the extrusion on the substrate. Moreover, heat transfer are also taken into account. Therefore the dependence of the viscosity with temperature and strain rate is possible. An experimental protocol is also developed to perform measurements of the strand shape. The goal is to accurately predict the influence of the printing parameters on the strand dimensions. Indeed the strand shape has an important impact on the final properties of the printed object. The deposition path is set depending on the strand dimensions. This influences the porosity as well as the bonding quality and thus the mechanical properties of the object. The development of tools to predict the strand shape without experimental measurements is thus of primary interest. Finally, experimental measurements on single deposited strands are sometimes tricky due to the small scale. This is especially true for temperature measurements. Due to the nature of the process, in situ measurements inside the liquefier is also difficult. The development of a numerical model thus allows a more straightforward study of the temperature and deposition behaviours of the extruded polymer.

Modelling of fluid dynamics 2.2.1 Constitutive equation

The goal is to model the extrusion of polymer out of the liquefier and its deposition on the substrate. An amorphous polymer is considered in the present study. The polymer is extruded at an extrusion temperature T ext and cools on the substrate. The polymer solidifies when its temperature cools below its glass transition temperature T g . To take into account the solidification process due to cooling, the dynamic viscosity of the polymer is given by a Carreau-Yasuda's law [START_REF] Agassant | Mise en forme des polymères: Approche thermomécanique de la plasturgie[END_REF]:

η(T , γ) = η 0 a T [1 + (a T λ γ) a ] 1-m a , ( 2.1)
where η 0 is the Newtonian plateau viscosity, λ a time constant dependent of the polymer, m the power law parameter, a a parameter describing the transition between the Newtonian plateau and the power law regime and γ the generalized shear rate defined as:

γ = √ 2 ε : ε, (2.2)
with the symbol ":" describing the double dot product operator and ε the strain rate tensor. It is defined such that:

ε = 1 2 ( - → ∇ - → u + t - → ∇ - → u ) (2.3)
with -→ u the fluid velocity. The coefficient a T in Equation (2.1) is a shift factor defined by an

Arrhenius' law such that:

a T = exp E a R 1 T - 1 T ref , (2.4)
with E a the activation energy, R the ideal gas constant, T the temperature and T ref a reference temperature. The shift factor induces an exponential increase of the viscosity for a decreasing CHAPTER 2. FLUID DYNAMICS OF A DEPOSITED STRAND

temperature. An increase of the activation energy induces a stiffer increase of the viscosity when T decreases. The use of Equation (2.1) enables to simulate the solidification process of polymer cooling. Indeed, below the glass transition temperature, the shift factor leads to a very high viscosity. By penalisation, the strain rate tensor becomes equal to zero. As it is well known, the cancellation of the strain rate tensor lead to a solid body motion of the fluid domain [START_REF] Aris | Vectors, Tensors and the basic equation of fluid mechanics[END_REF].

Level-set approach

The deposition of polymer on the substrate is simulated by considering a two-phase domain. A level-set (LS) method [START_REF] Osher | Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[END_REF] is used to track the interface between polymer and gas and study the fluid dynamics. The level-set function ϕ is a signed distance function where ϕ = 0 corresponds to the interface between two fluids. No mass transfer are considered between the two fluids. The level-set function represents at any point of the domain the distance from the interface. The polymer domain corresponds to the positive distance of the level-set function. The outward unit normal at the interface is defined as:

- → n = - - → ∇ϕ ∥ - → ∇ϕ∥ . ( 2.5)
The level-set method is used to implicitly track the evolution of the polymer interface. This is done by convecting the interface (ϕ = 0) with the velocity field - → u . The corresponding transport equation is given as

Dϕ Dt = 0. (2.6)
It is mentioned that Dϕ Dt describes the material derivative such that

Dϕ Dt = ∂ϕ ∂t + - → u • - → ∇ϕ.
(2.7)

The property of the level set function is that ∥ -→ ∇ϕ∥ = 1. This property should be ensured at all time. However it can be lost when convecting ϕ. Therefore reinitialization of the level-set function is incorporated after the transport step [START_REF] Ville | Convected level set method for the numerical simulation of fluid buckling[END_REF]. Usually, the reinitialisation of the level-set function is done by solving the Hamilton-Jacobi equation :

∂ϕ ∂τ + s(ϕ)(∥ - → ∇ϕ∥ -1) = 0, (2.8)
with τ a virtual time and s(ϕ) the signed function of ϕ. According to Ville et al. [START_REF] Ville | Convected level set method for the numerical simulation of fluid buckling[END_REF], the transport and reinitialisation steps are achieved in a unique transport equation with an additional source term given by ∂ϕ ∂t

+ ⃗ u • ⃗ ∇ϕ = µs(ϕ) 1 -|| ⃗ ∇ϕ|| , (2.9)
with µ a parameter chosen as a function of the mesh size and the time step. For more details, see reference [START_REF] Ville | Convected level set method for the numerical simulation of fluid buckling[END_REF].

The level-set function allows to track the two fluids using characteristic functions defined with the Heaviside function H(ϕ). According to Kataoka [START_REF] Kataoka | Local instant formulation of two-phase flow[END_REF], a two-phase flow can be written as a single fluid but the balance equations are in this case written in the sense of the theory of distributions [START_REF] Schwartz | Méthodes Mathématiques pour les Sciences Physiques[END_REF]. Numerically, the Heaviside and Dirac functions have to be regularised. A 2.2. MODELLING OF FLUID DYNAMICS transition zone of length ϵ is thus defined to have a continuous evolution of the properties. A Heaviside function H ϵ is defined in the domain such that:

H ϵ (ϕ) =        0 if ϕ ≤ -ϵ 1 2 1 + ϕ ϵ + 1 π sin πϕ ϵ if ϕ ∈ [-ϵ, ϵ] 1 if ϕ ≥ ϵ.
(2.10)

The properties of the air and the polymer can thus be portion in their respective domain. Let's consider a material property X. The polymer property X p and the gas property X g needs to be attributed in their corresponding domain. To do so, the Heaviside function H ϵ is used such that:

X = X p H ϵ (ϕ) + X g [1 -H ϵ (ϕ)]. (2.11)
The material property X is then continuously defined in the whole domain. The same method is used to define every material property.

Governing equations

The momentum and mass conservation equations are solved to simulate the fluid dynamics. The polymer is considered as an incompressible fluid. This hypothesis is valid for amorphous polymers as the variation of the density with temperature is relatively low. Surface tension forces are also taken into account at the interface between the fluid and the air. Gravity forces are neglected in comparison to the viscous forces. The Navier-Stokes equations are thus written as follows:

- → ∇ • - → u = 0, (2.12
)

ρ D⃗ u Dt = - - → ∇P + - → ∇ • [2η ε] + Γκδ(ϕ)∥ - → ∇ϕ∥.
(2.13)

with ⃗ u the velocity, t the time, Γ the surface tension, κ the average curvature at the interface, ϕ the level-set function and δ(ϕ) the Dirac function of the level-set function. The average curvature at the interface is defined as

κ = - → ∇ • ( - → ∇)ϕ = - - → ∇ • - → n , (2.14)
with -→ n the normal unit at the interface. Finally, the heat transfer equation is solved to compute the temperature field in the whole domain. Previous studies have calculated the viscous dissipation in the liquefier for different extrusion velocities [START_REF] Pigeonneau | Heating and flow computations of an amorphous polymer in the liquefier of a material extrusion 3D printer[END_REF]. The result shows that it is always negligible in the liquefier. The maximum shear rates in our problem are encountered in the liquefier. It is therefore neglected in the present study. Moreover, the computational domain is open and the energy linked to the variations of pressure in the gas is also neglected. Finally, the energy induced by convection of the LS function is neglected. The heat transfer equation is therefore given by the following equation:

ρC p DT Dt = - → ∇ • (k - → ∇T ), (2.15)
with ρ the density, C p the specific heat at constant pressure and k the thermal conductivity.

Normalisation of the equations

The governing Equations (2.12), (2.13)

= ρ ρp Cp = Cp Cp p η = η ηp Θ = T -T air Text-T air t = tV g k = k kp ū = u V x = x g P = P g η 0 V
Θ = T -T air T ext -T air . ( 2.16)
The normalisation of all the variables is summarised in Table 2.1. The dimensionless variables are written with an over-bar. This leads to the following governing equations:

- → ∇ • - → ū = 0, (2.17) Re D - → ū D t = - - → ∇ P + - → ∇ • [2 η ¯ε] + 1 Ca κδ(ϕ)∥ - → ∇ϕ∥, (2.18)
ρ Cp

DΘ D t = 1 P e - → ∇ • ( k-→ ∇Θ), (2.19)
with Re the Reynolds number, Ca the capillary number and P e the Péclet number. The dimensionless numbers are defined by Eq. 2.20 with the orders of magnitude relevant with the characteristic properties of materials and process.

             Re = ρpV g η 0 ∼ 10 -6 Ca = η 0 V Γ ∼ 10 3 P e = ρpCp p V g kp ∼ 10 2
(2.20)

Implementation of the model

The numerical solution of the previous governing equations is done in the specific case of FFF.

The definition of the model to simulate the deposition of a single polymer strand on a substrate is detailed here.

Boundary conditions

The simulation focuses on the extrusion and deposition of a single strand. Therefore, only the tip of the nozzle is considered in the numerical domain. It is composed of the end of the first cylindrical region I, the convergent region II and the capillary region III of the nozzle, as displayed in Figure 1 A non-slip condition is considered on the substrate and nozzle boundaries. The nozzle is fixed in the computational domain. Therefore to simulate its motion, a constant velocity is applied on the substrate. The substrate velocity equals 1 in the x direction. As the nozzle moves in the x -direction, a convective flow of air is considered. A flow of air equal to the nozzle velocity V is applied on the boundary Ω in air depicted in Figure 2.1. No condition is applied along the x axis on the boundary surface ∂Ω out . On the other air boundaries, the fluid velocity is considered null on the outward direction. The other directions are let free. We consider that the flow of air has the same temperature as the initial air temperature. Therefore this air boundary Ω in air has an imposed temperature Θ = 0. The substrate boundary ∂Ω sub has an imposed temperature Θ sub = T sub -T air Text-T air . Two different temperatures are thus imposed at the interface between ∂Ω in air and ∂Ω sub . This can cause computation instabilities due to high temperature gradients. Therefore the temperature is imposed as:

Θ = Θ sub 1 -exp - x ϵ , (2.21)
where ϵ is a coefficient to configure how fast the temperature changes from 0 to Θ sub . At the inlet polymer boundary, a Poiseuille velocity profile is imposed such that:

ū(r) = -2 U i V 1 - r R i 2 , ( 2.22)
with U i the inlet average velocity, R i the inlet cylinder radius and r the radius coordinate. The inlet velocity U i is linked to the extrusion velocity U such that:

U i = U R e R i 2 , ( 2.23)
with R e the outlet cylinder radius. The inlet polymer has an imposed temperature of Θ = 1. All boundary conditions are summarised in Table 2.2.

Mesh adaptation

As explained previously, the level-set method allows to study the fluid dynamics. The simulated strand shape is studied by tracking the interface throughout the simulation. At each increment, the interface is moved based on the velocity field. It thus needs to be well defined at any time. A mesh adaptation is therefore performed after each increment to track the interface with precision. It is performed using a re-meshing tool available in the Cimlib library. It enables to modify the value of the length between each nodes and thus refine the meshing in the areas of interest. The remeshing tool is briefly explained in the present section. Readers are invited to read [START_REF] Coupez | Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing[END_REF][START_REF] Coupez | Solution of high-Reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing[END_REF][START_REF] Shakoor | Three-dimensional numerical modeling of ductile fracture mechanisms at the microscale[END_REF] for further details.

∂Ω noz - → ū = 0 Θ = 1 ∂Ω air / ∈ ∂Ω in air ∪ Ω out - → ū • - → n = 0 ∂Ω in air - → ū = 1 • - → e x Θ = 0 ∂Ω sub - → ū = 1 • - → e x Θ sub 1 -exp -x ϵ ∂Ω pol ū(r) = -2 U i V 1 -r R i 2 • - → e z Θ = 1
The discontinuities or gradients of the solution is highly directional when tracking the interface. Therefore anisotropic meshing is chosen as it enables to limit the increase of elements when re-meshing. A metric tensor field is defined to set the aimed mesh size. It is defined at each node of the domain. In the orthogonal basis, the metric tensor field M of node i is defined as:

M i (x, y, z) =     1 h 2 x 0 0 0 1 h 2 y 0 0 1 h 2 z     , (2.24)
with h the mesh size. The aimed mesh metric tensor of node i is created based on the targeted length of edges connecting this node. Therefore a length distribution tensor first needs to be created. The length distribution tensor enables to know the distance from any given element node. It is a statistical representation of the length distribution function of all nodes connected 2.3. IMPLEMENTATION OF THE MODEL to a given node. For a given node i, it is denoted X i and is defined as

X i = 1 |τ (i)| j∈τ (i) X ij ⊗ X ij , ( 2.25)
with τ (i), the set of nodes connected to the node i, and X ij = X j -X i the edge vector made of nodes i and j sharing at least one element. The optimal mesh metric is obtained when all the node edges have a unit length. For a node i, it can be computed by minimizing the following function:

M i = arg min M   j∈τ (i) MX ij • X ij -|τ (i)|   2 . (2.26)
An approximate solution is given by:

M i = 1 d (X i ) -1 , (2.27)
where d is the number of non aligned edge vectors. Let's consider a regular scalar field f . In order to compute the desired mesh metric based on this field, the actual values of f in the whole domain needs to be known. As the mesh metric field is defined on the nodes, interpolation of the elements values needs to be done. The value of the interpolation of f on a given mesh is written

F i and F ij = F j -F i .
The interpolation is done by projecting the gradient of f on the element edges. It is thus only defined with respect to the nodes. The global gradient field is calculated by minimizing the function G i such that:

G i = arg min G   j∈τ (i) |G • X ij -F ij | 2   . (2.28)
It is then possible to define the interpolation error such as:

e ij = |G ij • X ij |. (2.29)
The error depends on the edge length. A stretch factor is therefore introduced to link the variation of interpolation error with edge length. This enables to calculate the total interpolation error based on the total number of edges in the domain. The user is then enable to control the total interpolation error based on a single parameter. The variation of this parameter will impact the global mesh size. In the present example, the targeted mesh metric is calculated based on the gradient of a specific scalar field. The mesh adaptation procedure is not limited to a single scalar field. Multiple fields can be attributed to construct the metric. Each variable has its proper mesh metric. Intersection of all the mesh metrics results in a unique global mesh metric.

The resulting metric should fulfil all the mesh size constraints given by the different metrics.

The intersection is done by constructing a unique multi-component error vector. This vector contains all the errors from the metrics constructed with the different variables. Considering an intersection of n metrics based on n variables, the corresponding error vector is written

- → e ij = {e 1 ij , e 2 ij , ..., e n ij } (2.30)
The total error is then given by:

e ij = k (e k ij ) 2 1/2 . (2.31)
In practice, the vector of error source is constructed by the normalised variables considered for re-meshing. The normalisation is done to prevent any absolute values from dominating

CHAPTER 2. FLUID DYNAMICS OF A DEPOSITED STRAND ρ C p k T g kg/m 3 J/(kg K) W/(m K) °C 1010 2100 0.21 105
Table 2.3: Properties of the ABS taken in the simulation. The values of C p and k is taken from [5].

the error estimator. A certain weight can then be attributed to each variable depending on their importance. In our case the main factor to construct the metric is the level-set function.

Therefore it is the scalar field that has the main weight in the error estimation. The adaptation also takes into account the temperature gradient and polymer velocity in a lower manner.

Simulation procedure

The different simulation steps are detailed in the present subsection. At the initial state, the polymer domain is first immersed in the computation domain. It is done by incorporating a mesh that has the same geometry as the nozzle. A level-set function is then created based on the immersed domain to create the polymer domain. This enables to have a polymer domain fitting the interior of the nozzle. The temperature is then imposed in the polymer and air domain. As mentioned previously, the temperature of the polymer and gas domain is 1 and 0, respectively. Based on the level-set function, the polymer and air properties are attributed to their respective domain. The viscosity of the polymer is calculated based on its temperature.

At each time increment, the polymer viscosity is calculated before the resolution of the Navier-Stokes equations. It is calculated based on the temperature field and shear rate of the previous increment. The shear rate is calculated based on the velocity field at the previous increment. The material properties are attributed to their corresponding domain. The attribution is made based on the level-set function computed at the previous increment. The Navier-Stokes equations are then solved based on the velocity field at the previous increment and the material properties.

The heat transfer equation is then solved. Finally, the interface of the level-set function is transported based on the computed velocity field. Reinitialization of the level-set function is followed by re-meshing after its transport. The computation finishes when the final time step is reached. The model flow chart in Figure 2.2 summarises the different steps of the model.

Polymer properties

The simulation is performed for an amorphous polymer. The thermal and rheological properties of the polymer needs to be known to perform the simulations. The chosen polymer is an acrylonitrile butadiene styrene (ABS), which is a commonly used polymer in FFF. The ABS is purchased as neat filaments from Grossiste3D ® . The density is given by the supplier and is equal to 1010 kg/m 3 . The glass transition temperature of the ABS is measured by Differential Scanning Calorimetry (DSC) on a DSC4000 from PerkinElmer ® and is equal to 105°C. The heat capacity and conductivity of the ABS is taken from [5]. The properties of the ABS is summarised in Table 2

Strand shape and model assessment

An experimental protocol is developed to measure the strand shape as a function of the printing conditions. This is done by measuring the strands height and width. Indeed, these parameters play a major role when setting the deposition path to print an object. Moreover, the fluid dynamics simulation of the deposition of polymer strands needs to be assessed. This is done by comparing the experimental measurements with simulations. The same printing conditions are used in the experiments and simulations. The simulated shapes of the deposited strand are then compared with real strand shapes.

Experimental protocol

Single strand deposition is done using a Delta 2040 Turbo from the brand Wasp ® . The strands are printed at different ratios of U /V for different nozzle velocities V . All samples have a length of 100 mm. The strand is then cut at approximately one half of its length. The strand shape is then measured using an optical microscope Olympus ® PMG3 in reflection mode. For a given ratio U /V , around 10 printed strands are measured. The average width and height is calculated for a given set of printing velocities. An example of strand measurements is illustrated in Figure 2.5.

Influence of the printing parameters

Hebda et al. [START_REF] Hebda | A method for predicting geometric characteristics of polymer deposition during fused-filament-fabrication[END_REF] have studied the influence of the deposition gap on the strand width. They have shown that the ratio ∆W /D for different printing ratios U /V yields the following empirical relation

∆W D = B U V , (2.32)
with B a pre-factor and ∆W the width of the strand. This relation is assessed by measuring the strand width for different nozzle diameters and printing velocities U /V . The measurement is done on a range of printing ratio U /V ∈ [1.5; 5] for different nozzle velocities V . Indeed, it has been shown in [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF] that the relation of Hebda et al. [START_REF] Hebda | A method for predicting geometric characteristics of polymer deposition during fused-filament-fabrication[END_REF] is only valid for U /V lower than 5. Figure 2.6 depicts ∆W /D as a function of √ U /V for two diameters and four gaps.

The results show that the relation of Hebda et al. [START_REF] Hebda | A method for predicting geometric characteristics of polymer deposition during fused-filament-fabrication[END_REF] is only valid for a certain set of parameter.

For g = 0.2, D = 0.4 mm and for g = 0.15, D = 0.3 mm the evolution is linear and the values of ∆W /D is similar in both cases. However for other gaps, the relation is not valid any more. This highlights that the gap also has an important impact on the strand size. In particular, it is the ratio g/D that has a predominant impact on the strand width. Indeed, for a given ratio g/D, the evolution of ∆W /g is independent of the actual gap size and nozzle diameter. This is highlighted in Figure 2.7, where ∆W D/g 2 is plotted as a function of U /V .

The results show that for a given ratio g/D, the evolution of ∆W D/g 2 falls into a master curve. This is true for different nozzle velocities, gaps and nozzle diameters. Therefore, an empirical relation is proposed based on the one of Hebda et al. [START_REF] Hebda | A method for predicting geometric characteristics of polymer deposition during fused-filament-fabrication[END_REF], such that:

∆W D g 2 = B U V + C, (2.33)
with C a constant determined by linear regression. The fit of the former relation by the experimental measurements is also displayed in Figure 2.7 for both cases. The fitting parameters are shown in Table 2.5 for this particular ABS. Agassant et al. [START_REF] Agassant | Flow analysis of the polymer spreading during extrusion additive manufacturing[END_REF] have developed a 2D analytical model to estimate the dimension of the strand width. The model is based on the solving of the Stokes equations along the direction of deposition. It shows that the strand width can be expressed only as a function of D, U /V , g/D and the total nozzle tip diameter. This is in accordance with the present empirical relation. The main parameters are the gap g, and ratios g/D and U /V . The deposited strand cross section can be assimilated to an oblong shape. Indeed it has been shown in [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF] that when considering an oblong shape, the predicted evolution of the strand cross section with ratio U /V presented the best match with experimental measurements. The average strand cross section A is given by the following relation:

A = πD 2 4 U V . (2.34)
Assimilating the strand cross section to an oblong shape, Equation (2.33) and (2.34) gives a relation linking the strand width and height. It is given by the following quadratic equation:

-∆H 2 - π 4 + ∆Hg D 2 B U V + C - πD 2 4 U V = 0. (2.35)
Solving the previous equation gives an estimation of the strand height as a function of the ratio U /V . The predictions are compared with the actual strand height for different printing conditions. The predictions are done at different ratios g/D for different nozzle diameters and nozzle velocities. The results are displayed in Figure 2.8. The prediction gives a good first approximation of the strand height. They are more accurate for g/D = 0.8. Indeed it is underestimated for g/D = 0.5. However for each ratio U /V , the predicted strand height is in the same order of magnitude as the measured height. Knowing the strand width and height is of primary interest when setting the deposition path to build an object with the desired infill. These relations can thus assist the users to estimate the strand cross section without performing actual measurements. The empirical relations provided in this study can give a first estimation of the aimed set of parameters. They have been improved compared to the ones provided in [START_REF] Hebda | A method for predicting geometric characteristics of polymer deposition during fused-filament-fabrication[END_REF][START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF]. Indeed it provides insight on a larger set of parameters. This relation can be used for different nozzle gap as well as ratios g/D and U /V .

Model assessment

The numerical model is first assessed by studying the extrusion velocity field. The velocity field at the exit of the nozzle is plotted along its radius. The velocity profile at the inlet polymer boundary condition was assumed as a Poiseuille flow. Therefore the outlet velocity profile is compared with a Poiseuille flow in Figure 2.9. The evolution of the fluid velocity along the radius shows a more plug-flow profile compared to the Poiseuille flow. This is due to the plasticizing effect of the considered constitutive model. Indeed the Carreau-Yasuda law is a pseudo plastic law where the exponent m accounts for plasticity. This results in having a lower maximum velocity of the polymer compared to a Poiseuille flow.

Conservation of the fluid flow rate is also ensure in the model. This is done by comparing the polymer flow rate between the inlet and the outlet of the nozzle. The velocity profile is assumed axisymmetric in the nozzle. The extrusion flow rate Q is thus calculated by integrating the velocity profile u along the radius such that,

Q = Re 0 2πru(r)dr. (2.36)
The inlet flow rate is also calculated in the same manner. The comparison is done for an average extrusion velocity U ∈ [2; 5] m/min. The results are shown in Table 2.6, with Q i and Q e designating the inlet and extrusion flow rate, respectively. The deviation between the inlet and extrusion flow rate shows a good conservation. Indeed, the deviation is lower than 0.5% in all cases. This shows that the amount of polymer added in the system is well conserved in the model.

An assessment of the pressure field in the nozzle is also investigated. The comparison is done based on the work of Anderegg et al. [START_REF] Anderegg | In-situ monitoring of polymer flow temperature and pressure in extrusion based additive manufacturing[END_REF]. The study has experimentally measured the maximum pressure in the nozzle. For a printing flow rate of 0.702 mm 3 /s at 230 °C, the maximum pressure measured in the nozzle is 3580 kPa. A simulation done at the same extrusion flow rate and printing temperature shows a maximum pressure of 1412 kPa. A 61% deviation is thus found between the experimental measurement and the simulation. This difference can be explained by the actual polymer temperature in the experiment. Indeed the experimental study showed that the average temperature did not reach the wall temperature. This can be explained by the location at which the pressure is measured. Indeed the pressure is measured at approximately 5.5 mm from the end of the nozzle output. In the present simulation the polymer temperature is assumed equal to the wall temperature in the nozzle. Moreover, the rheological behaviour of the ABS polymer used in the experimental study may differ from the one used in our simulation.

Finally the geometry of the nozzle may not be exactly the same. This is especially true for the nozzle angle in the converging region. These parameters could explain the difference found in the maximum pressure. It can however be argued that the maximum pressure in both simulation and experimental measurements are in the same order of magnitude. Moreover the study of [START_REF] Anderegg | In-situ monitoring of polymer flow temperature and pressure in extrusion based additive manufacturing[END_REF] mentions that a printing failure happens when the pressure exceeds 6900 kPa. Experiments done with our 3D printer show failure for extrusion velocities U larger than 10 m/min. Simulation done at U = 10 m/min shows a maximum pressure of 5275 kPa. This corresponds to a 24% difference with the pressure measured in [START_REF] Anderegg | In-situ monitoring of polymer flow temperature and pressure in extrusion based additive manufacturing[END_REF]. Both printers are not the same and the feeding mechanism is different. It can therefore be concluded that the computed pressure field is in fairly good agreements with previous works done on the measurement of the pressure in the nozzle.

The model is then assessed by comparing the simulated strands height and width with experimental measurements. The assessment is done on the same printing conditions between the experiment and the simulation. The studied polymer is the ABS which properties are depicted in Subsection 2.3.4. The comparison is done for different ratios U /V . The nozzle velocity V is equal to 1 m/min. The substrate temperature, the extrusion temperature and the air temperature are taken equal to T sub = 94°C, T ext = 230°C and T air = 40°C, respectively. The strand deposition is assumed to be done at the start of the printing. The air near the substrate is heated by it. This explains the choice of taking this value for the air temperature. Several studies have shown that for an inlet velocity U i ≤ 180 mm/min, the polymer average temperature at the exit of the nozzle is equal, or very close to the nozzle's wall temperature [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF][START_REF] Pigeonneau | Heating and flow computations of an amorphous polymer in the liquefier of a material extrusion 3D printer[END_REF]. The maximum inlet velocity in our simulation is U i = 200 mm/min. For computation simplicity, the polymer temperature inside the nozzle is thus considered equal to the wall temperature at the initial state. The nozzle diameter is D = 0.4 mm. The comparison is done for 2 different gaps g = 0.5 and g = 0.65 mm. The surface tension is taken equal to 0.029 N/m from Bellehumeur et al. [START_REF] Bellehumeur | Modeling of bond formation between polymer filaments in the fused deposition modeling process[END_REF]. This leads to the Reynolds, Capillary and Péclet numbers summarised in The results show accurate prediction of the strand size between simulations and experimental measurements. This is especially true for the width of the strand. Indeed, the average deviation between the simulated strands width and the experimental measurements is around 3% for both gaps. The deviation ranges between 0.3% and 7.6%. The simulated strands height is however slightly less accurate. Indeed the deviations ranges from 1 to 8% with an average deviation of 5.9%. The predicted strand height is globally overestimated for g = 0.65 mm. However the predicted strand size shows less than 8.1% deviation for all the studied cases. This highlights the predictive accuracy of the developed model. Moreover the predicted global shape of the strand is in accordance with what can be seen in reality. For a given ratio g/D, the strand shape is more ellipsoidal at low ratio U /V . When the ratio U /V increases, the shape of the strand becomes more like an oblong. This trend has been shown in [4,[START_REF] Serdeczny | Experimental validation of a numerical model for the strand shape in material extrusion additive manufacturing[END_REF] and is confirmed in the present study. An example of the numerical and experimental strand shape is compared in Figure 2.11 for different ratio U /V . As the extrusion velocity increases, the strand spreads more on the side of the nozzle. The height of the strand is restrained by the nozzle tip which leads to an oblong shape. The global shape of the simulated strand is thus in accordance with measurements.

(a) Experiment -

U /V = 2. (b) Experiment -U /V = 4. (c) Simulation -U /V = 2. (d) Simulation -U /V = 4.

Influence of the printing parameters on the deposition.

The last section have focused on the strand shape. The experimental measurements have allowed to predict the strand dimensions based on an empirical relation. It also enabled to confirm the predictive value of the numerical model. The prediction of the strand shape is a useful tool to configure the printing of a piece. Another interesting aspect of the numerical model is its ability to study the different phenomena occurring during the deposition of a strand. Indeed, the printing of a strand is made on the scale of a few hundreds of micrometres. Therefore precise experimental measurements are often tricky to obtain. The use of the present numerical model can be a more convenient way to study the general trends.

Nozzle geometry

Nozzle angle and capillary length

The impact of the angle in the converging part of the nozzle is investigated. The comparison is done on 2 nozzles with different angles. One is based on the E3D-nozzle V6 with an extrusion diameter D = 0.4 mm. The dimensions of the nozzle is in open source at e3d-online.com. The angle ω in the converging region is equal to 60°. A second simulation is done with the spitfire HT cartridge from WASP ® with the same nozzle diameter. The angle is twice larger than the first one with ω = 120°. A comparison of the resulting pressure field in the nozzle is displayed in Figure 2.12. The comparison is done at the end of the converging region. The results show that decreasing the angle ω results in increasing the pressure drop at the end of the converging region. Decreasing ω from 120 to 60°results in a 16% increase of the pressure drop at the exit of the converging part. More accurately, the pressure drop decreases from 0.7 to 0.59 MPa when doubling the angle from 60 to 120°. This trend is in agreement with previous studies of the pressure drop in the liquefier [START_REF] Ramanath | Melt flow behaviour of poly-epsiloncaprolactone in fused deposition modeling[END_REF]. The decrease of the nozzle angle increases the length of the converging region. Increasing the nozzle angle should therefore be beneficial for the extrusion of the polymer. Indeed as the pressure drop decreases, the maximum force needed to extrude the filament also decreases. It is thus expected that increasing the nozzle angle enables to extrude at higher extrusion rates.

The impact of the length of the capillary region on the pressure drop is also investigated in Figure 2.13. The pressure field is plotted along the velocity streamline. This is done to compare the velocity profile in next paragraph. As expected, the pressure drop increases when increasing the length of the last cylindrical region. The nozzle with an angle ω = 120°has a capillary of 1 mm. The nozzle with an angle ω = 60°has a capillary of 0.6 mm. The result shows that the total pressure drop in the liquefier is higher when the capillary is 1 mm. Indeed the pressure drop is equal to 1949 and 2566 kPa for ω = 60 and ω = 120°, respectively. The capillary length has a predominant impact on the pressure drop compared to the nozzle angle. Nozzle with short capillary should therefore be favoured for extruding at high flow rates.

The influence of the nozzle angle on the velocity profile is also investigated. The comparison of the velocity streamline is done in Figure 2.13 for both angles. It shows that when increasing the angle ω, the fluid has naturally a more abrupt change of the flow direction. This is especially true near the wall of the nozzle. A comparison of the velocity profile is also done at one half of the converging region for both cases as illustrated by the black dotted lines in Figure 2.13. The results are shown in Figure 2.14. A more parabolic profile is seen when decreasing the angle ω. On the other hand when increasing the angle, the velocity is lower near the wall. At approximately 0.3 mm from the center of the nozzle, the velocity rapidly increases. The same trend has been observed experimentally by Peng et al. [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF] in the converging region. Having a higher angle ω can be detrimental on the extrusion. Indeed the velocity profile highlights regions of low velocities in the liquefier. This can cause problems when extruding the polymer. This is especially true if particles are embedded in the polymer. Indeed regions with low velocity are more inclined in accumulating particles and therefore clog the nozzle. The accumulation of polymer in these region can also cause problem when changing the extruding material. Mixing of older polymer with new polymer may happen in the liquefier. Moreover having a too much abrupt angle can cause extrusion defaults due to recirculation of the polymer. This is especially true for branch polymers. Beyond a certain critical shear rate, vortices at the entry of the capillary may be created. This can cause extrusion defects such as helical and chaotic defects [START_REF] Agassant | Mise en forme des polymères: Approche thermomécanique de la plasturgie[END_REF].

The evolution of the viscosity in the nozzle is also studied depending on the nozzle angle. In both cases, the viscosity decreases in the capillary region. This is due to an increase of the shear rate in this region. The lowest viscosity is equal to 0.3 Pa.s in both cases. The evolution of the viscosity is studied at the center of the nozzle, along the vertical z axis. The focus is made in the cylinder and in the converging region. The results are displayed in Figure 2.15. The nozzle exit is located at z = 0.65 mm. The distance z = 5.2 mm corresponds to the highest location of the nozzle in the computation domain. The converging region starts at z = 2.64 mm and z = 2.05 mm for ω = 60 and ω = 120°, receptively as displayed by the dotted lines. The results show that the viscosity behaves approximately the same way in each case until z is approximately equal to 2.6 mm. A more rapid decrease of the viscosity is then observe for ω = 120°. It is due to the more rapid increase of the shear rate at the center of the nozzle for this angle, as shown in Figure 2.14. As the shear rate increases, the shear-thinning effect is more dominant and the viscosity decreases. Lowering the viscosity should be beneficial for the extrusion.

Overall the results show that the nozzle angle should be increased while maintaining a short capillary length to improve extrusion. Indeed the pressure drop should be as low as possible and the capillary length has a predominant impact on the pressure drop. Moreover increasing the nozzle angle enables to have a higher shear rate. This has a positive impact on the polymer viscosity. It is expected that extrusion can be done at higher flow rate without failure. Indeed printing failure primary occur due to buckling of the filament. It can however be argued that the nozzle angle should not be excessive to limit the recirculation effect. The present model can be used to tailor the most adequate nozzle geometry for a given purpose.

Nozzle diameter

The impact of the nozzle diameter on the extrusion and deposition of the melt is investigated. Two simulations are done with a nozzle diameters D equal to 0.3 and 0.4 mm. The nozzle angle is the same in both simulations with ω = 60°. The gap is chosen so that both cases have a ratio g/D = 1.25. This results in a gap g = 0.375 and g = 0.5 mm for D = 0. Both strands show very similar shapes. They exhibit nearly the same size, especially for the width. Indeed the width is 1 mm and 1.01 mm for D = 0.3 mm and D = 0.4 mm, respectively. It is worth mentioning that all distances are normalised by the gap distance between the substrate and the nozzle. As mentioned earlier the gap is different for both simulations. The actual strand size is therefore different for both cases. The simulation however points out that for a given ratio g/D, the ratio ∆W /g depends only on the velocity ratio U /V . These findings were already shown by experimental measurements in § 2.3.4. The simulation validates the empirical relation described in Equation (2.33). A more important difference of height is seen between both cases. This confirms that the ratio ∆H/g cannot be described only by the ratios g/D and U /V . Considering the strand as an oblong is therefore useful to experimentally predict the strand height.

Gap distance

Three cases with different g are simulated with the same nozzle geometry. The nozzle has a 0.4 mm diameter and an angle ω = 60°. As already mentioned, the dimensions of the nozzle is in open source at e3d-online.com. The different gaps are chosen based on the diameter of the nozzle. The ratio g/D is 0.75, 1.25 and 1.625, respectively for the three cases. The cases are chosen to have a gap that is lower, in the same order and higher than the diameter, respectively. The comparison is done once the deposition reaches a steady state regime. For all the studied cases, the nozzle velocity V = 1 m/min and the velocity ratio U /V = 3. The comparison is done in Figure 2.17. In each case, the strand cross section is displayed following the (x, z) plane. The spreading of the strand on the substrate is also displayed. The grey values correspond to the interface between the polymer domain and the air domain.

The results show that the polymer spreads more on the front of the nozzle when the gap decreases. As the gap decreases, the extruded polymer is more restrained by the nozzle. It thus spreads more in the radial direction with respect to the nozzle. The same trend is observable in the width of the strand, along the y axis. At the exit of the nozzle, the extruded polymer spreads in a more circular way when decreasing the gap. At g/D = 0.75, the spreading in front of the nozzle even exceeds the nozzle length. Deposition issued can occur if too much polymer accumulates in front of the nozzle. The gap should therefore be chosen according to this parameter. Agassant et al. [START_REF] Agassant | Flow analysis of the polymer spreading during extrusion additive manufacturing[END_REF] have proposed an analytical solution to predict the length L of polymer spreading in front of the nozzle. It also enables to estimate the dimension of the strand. The length is defined as:

L = -R n + (R n -D) 2 + π U D 2 V g (R n -D/2) 2 , (2.37)
with R n the total nozzle radius, comprising the metal tip. This gives the following relation to estimate the width ∆W and height ∆H of the strand:

       ∆W = 2 D 2 + L , ∆H = πD 2 4W U V .
(2.38)

INFLUENCE OF THE PRINTING PARAMETERS ON THE DEPOSITION.

The measurement of the length L is experimentally tricky. Indeed it requires measurements during the actual printing of a strand. Therefore the relation is assessed by our numerical model. The results show a relatively good prediction of the length L with less than 7% deviation for g/D = 0.75 and g/D = 1.25. When the gap is further increased to g/D = 1.625, the analytical results show a 31% deviation. The analytical solution assumes perfect contact between the polymer and the nozzle surface. As shown in Figure 2.17, this assumption is only valid for values of g ≤ D. This will be further detailed in the paragraph. When the polymer is not in contact with the nozzle any more, the relation is not valid. This explains why the deviation increases when increasing the gap distance. However for gaps lower or close to D, this relation is valid. The resulting strand width and height is thus compared with our numerical computation for g/D = 0. When looking downstream to the deposition, the gap distance also has an impact on the height of the strand. Indeed when decreasing the gap to a ratio g/D = 0.75 mm, the polymer sticks to the nozzle wall. This results in having a strand height that is very close to the gap. The resulting strand shape has a more planar upper surface due to this phenomenon. On the other hand for higher gaps, the nozzle does not impact the strand shape. It should be argued that measurements of the strand height shows higher values of the gap in some cases. Indeed in Figure 2.10, the strand height exceeds the gap for U equal to 5 m/min. This could be due to swelling due to viscoelastic effects at the exit of the nozzle. The viscoelasticity is not taken into account in the present model and this feature cannot be seen.

The simulations show that for all the studied gaps, the spreading in the y direction is higher downstream the deposition. At the exit of the nozzle, the spreading in the y direction is impacted by the motion of the nozzle. The nozzle velocity has however less impact on the spreading when decreasing the gap. Indeed the increase of the width is of 48, 18 and 10% for g/D = 1.625, 1.25 and 0.75, respectively. This is due to the pressure applied by the polymer at the exit of the nozzle. Indeed, the pressure increases at the exit when the gap decreases. This may also have an impact on the adhesion of the polymer with the preceding layers. If the pressure increases, the layers will be more stacked and the surface contact area between layers will increase. This should be beneficial for the adhesion between layers. This trend is observable with the value of the level-set function. In Figure 2.17, the zero iso-value of the level set function is white. It corresponds to the surface of the polymer. The polymer domain corresponds to the positive value of the Level-Set function. The distance between the positive value of the level set function in contact with the substrate and the nozzle exit is investigated. Increasing the gap leads to longer distances to reach positive values of ϕ. When extruding at g/D = 0.75, the positive value of ϕ is found from the exit of the nozzle. Nevertheless, this distance does not give information about the adhesion as no criterion is implemented in the simulation. It however gives insight about how the polymer spreads and how the contact with previous layers behaves.

Thermal behaviour

Heating in the liquefier

A focus on the heating of the polymer melt in the liquefier is detailed here. The goal is to investigate the maximum extrusion velocity at which the 3D printer fails to provide an extrudate Figure 2.18: L iso-Tg (mm) as a function of P e for the bisphenol-A polycarbonate and the acrylonitrile butadiene styrene. [5]. A modification of the model is performed to only account for the liquefier domain. The domain is adjusted to account for the whole length of cylindrical region I (see Figure 1.1). The boundary condition is modified so that the inlet polymer temperature is equal to Θ = 0 on the boundary ∂Ω pol . A comparison is done with two different amorphous polymers. One is the ABS polymer presented in this chapter and the other is a Bisphenol-A polycarbonate (PC) presented in [START_REF] Pigeonneau | Heating and flow computations of an amorphous polymer in the liquefier of a material extrusion 3D printer[END_REF]. Roughly, a failure in the extrusion is encountered when the polymer temperature has not reach T g before the exit of the liquefier. The evolution of the location along the length of the nozzle where the temperature is equal to T g is thus investigated as a function of the extrusion velocity. The minimum polymer temperature is found at the center of the liquefier and the comparison is therefore made on this position. The location in the liquefier at which T = T g is designated by L iso-Tg in the following. The comparison is done as a function of the Péclet number defined as P e = ρCpU 2R i k . Moreover, the simulation is done for three different extrusion temperatures for the PC. Further information on the numerical modifications of the model are detailed in [START_REF] Pigeonneau | Heating and flow computations of an amorphous polymer in the liquefier of a material extrusion 3D printer[END_REF].

Figure 2.18 provides the behaviour of L iso-Tg as a function of Péclet number for the PC at three liquefier temperatures and for the ABS with T ∞ = 230 • C. Whatever the heating condition and the polymer, L iso-Tg increases linearly with the Péclet number. The result arises directly from the scaling of the Graetz problem with an imposed temperature on the wall. Indeed, according to Bejan et al. [START_REF]Heat transfer handbook[END_REF], heat transfer can be written in reduced variables for which the axial characteristic length is proportional to P e. Even if this scaling is established for a Newtonian fluid, it does not change for a generalized Newtonian fluid. Consequently, it is expected to find a distance to reach the glass transition temperature scaling linearly with the Péclet number.

For the PC polymer, the slope of L iso-Tg as a function of P e increases with the reduction of the liquefier temperature T ∞ . The decrease of T leads to a larger distance to reach T g in the liquefier for the polymer. To see the effect of the heating, the inlet temperature and also the nature of the polymer (various T g ), L iso-Tg divided by θ g = (T g -T air )/(T ∞ -T air ) is plotted as function of the Péclet number in Figure 2.19. All data merge in a master curve. A linear fitting gives that L iso-Tg can be written as follows

L iso-Tg = 4.16 • 10 -3 θ g 2R i P e.
(2.39)

This result can be exploited to provide the correct working conditions of the 3D printer liquefier. Indeed, replacing L iso-Tg by the entire distance of the liquefier L in equation (2.39) gives the threshold of the Péclet number above which the polymer is not completely heated above T g . From the definitions of the reduced variables, the limit of the inlet velocity can be written as follows

U max i = 2.41 • 10 2 T ∞ -T 0 T g -T 0 k ρC p D 2 L 8R 4 i . (2.40)
This relation gathers working parameters such as the inlet and heating temperature but also various characteristics of the polymer and geometry of the liquefier.

Cooling of a single strand

The cooling kinetics of the polymer melt is an important parameter in FFF. It has an impact on the adhesion quality and microstructure. The comparison is first done at different gaps, for the same printing velocity U /V = 3. The chosen gaps are g = 0.3 mm and g = 0.65 mm. The temperature profile of the cross section is investigated in each case. The comparison is done at 3.23 mm from the exit of the nozzle. The results are presented in Figure 2.20. The temperature behaviour along the z axis (at y = 0) is also plotted for each case. The comparison is done at the center of the strand.

The temperature principally evolves over the z axis. The temperature is relatively uniform along the width of the strand for a given height. Indeed for the case g = 0.3 mm, the temperature at the center of the strand varies between 162.4 and 164.2 °C. The convective cooling still plays a minor role as the temperature is lower near the edge of the strand for a given height. However, this trend shows that the strand is principally cooled through conduction with the substrate. This is principally due to the fact that the air is heated by the nozzle. Therefore the air surrounding the strand downstream the deposition has a temperature T larger than T sub = 94°C. As shown in Figure 2.20c, the maximum temperature in the strand is lower for the strand with the lowest height. The convective cooling is principally driven by the surface area in contact with air.

On the other hand the conductive cooling with the substrate is mainly driven by the height of the strand. The temperature behaviour over the x axis is thus compared for strands with different heights and perimeters. The comparison is done from the exit of the nozzle to the end of the computation domain. The perimeter P for each strand is calculated based on the width and height of the strand. The results are shown in Figure 2.21a. No correlation between the perimeter of the strand and the cooling kinetics can be done. However, the cooling rate increases as the strand height decreases. The cooling of the first deposited strand is mainly driven by the substrate temperature. This result is confirmed by comparing the impact of the air and substrate temperature on the cooling kinetics. The temperature behaviour is compared for three cases in Figure 2.21b. The comparison is done along the x axis for different substrate temperature and initial air temperature. The results show that the substrate temperature have a predominant impact on the cooling kinetics. Indeed changing the air temperature shows nearly no impact on the cooling kinetics. On the other hand decreasing the substrate temperature shows a faster cooling kinetics.

It should however be argued that this phenomenon is enhanced by the numerical simulation. Indeed the computation domain is focusing on the nozzle. Therefore the air temperature surrounding the nozzle increases as the time increases. However in real processing conditions, the convective cooling should be more predominant as the nozzle moves in the printing chamber. Moreover the strands that are deposited far from the nozzle should also be more impacted by the convective cooling. Indeed as the nozzle moves away from the deposited strand, it does not heat the air surrounding it any more. Therefore the air temperature decreases and convective cooling plays a more important role. The present simulation only focuses on the first 0.4 s of the process. Different thermal phenomena occur after this time range. This is why other simulations on larger scale should also be done.

Cooling of multiple strands

The previous model is modified to study multiple strands. The model is changed such that the nozzle moves in the domain. The nozzle coordinates are set as a function of time. This enables to control the deposition path and speed. In this case, the Galilean framework does not move with the printer head. Therefore, the substrate velocity is set equal zero. No more air is introduced in ∂Ω in . The nozzle velocity is also imposed in the polymer domain inside the nozzle. This is done to take into account the motion of the nozzle in the domain. The rest of the boundary conditions described in § 2. The temperature is mainly dependent on the z axis for the bottom strand. Indeed for a given height z, the temperature is uniform along the width of the strand. This is in accordance with what is seen for the single strand deposition model. This results in having a polymer viscosity that also mainly depends on the z location for the bottom strand. Indeed looking on the xy plane, the viscosity is relatively uniform along the x axis for a given height z. The top strand has a different temperature field. The temperature is more homogeneous in the whole strand. The substrate has a high impact on the cooling kinetics of the first deposited strand. The temperature at the interface between the two strands is higher than the substrate temperature.

The top strand thus experiences a lower cooling kinetics which results in a more homogeneous temperature field. This also results in having a more homogeneous viscosity of the top strand after deposition.

The total height of the first deposited strand is 0.413 mm. Therefore the numerical sensor placed at z = 0.375 mm is near the interface between the two strands. The temperature behaviour with time shows that the deposition of a new strand only affects the temperature near the interface. Indeed no increase of the temperature is seen for the sensors below 0.375 mm. The deposition of a new strand is thus unable to completely heat the first deposited strand. This statement should however be qualified by the simulated processing time. Indeed only 0.5 s of the process is computed. The bottom strand has a temperature T above 140°C when the second strand is deposited. Therefore the temperature gradient is low which may reduce the impact of the top strand. The deposition of the second strand however shows an impact on the viscosity of the bottom strand. Indeed the whole strand shows a decrease of the viscosity when a new strand is deposited. The decrease of the viscosity is more prominent near the interface. The deposition of a new strand thus impacts the fluid dynamics of the bottom strand. As the viscosity decreases, the shape of the bottom strand also changes. The impact of the deposition of the top strand on the shape of the bottom strand can thus be studied by the present model.

Conclusion

A numerical model has been developed to simulate the deposition of single or multiple strands in FFF. The model simulates the extrusion and deposition of the polymer melt out of the nozzle. The fluid dynamics is modelled by a level-set approach coupled to the heat transfer equation. This enables to investigate the strand cooling kinetics and solidification process. Solidification is simulated by considering the dependence of the polymer viscosity with temperature and shear 2.6. CONCLUSION rate. An experimental protocol have also been developed to measure the shape of the strands. The impact of the printing velocities U /V and ratio g/D on the strand shape is studied. The strand width is only dependent on these two ratios. The numerical model have assessed this statement. An empirical relation has thus been proposed to estimate the strand dimensions based only on these printing parameters.

The experimental protocol has also assessed the numerical model. A comparison of the simulated strand shape with experimental measurements is done for different ratios U /V . Accurate predictions of the strand shape is found with the model. The numerical tool is thus used to study the influence of the nozzle geometry, gap distance and printing velocities on the the deposition behaviour. The pressure drop depends strongly on the length and the diameter of the capillary tube at the end of the nozzle. Increasing the nozzle angle also decreases the pressure drop and viscosity. The simulations have highlighted the influence of the gap distance on the polymer spreading and surface contact with the substrate. The model can be used to tailor the nozzle geometry and printing parameters to obtain strands with targeted shapes. The thermal study in the liquefier gives a limit for the inlet velocity at which extrusion fails. The cooling behaviour of a single strand shows the predominant impact of the substrate temperature on the cooling kinetics. This highlights the influence of the conductive cooling on the first deposited strand. A modified model has been implemented to simulate the deposition of multiple strands. Stacking multiple strands principally shows an influence on the viscosity of the first strand. This model can be further used in the future to study the stacking dynamics.

The model fully couples the fluid dynamics in the liquefier with the deposition dynamics of the polymer melt. Multiple studies have already simulated the deposition behaviour in FFF. However most models have only considered the capillary part of the nozzle [4,[START_REF] Serdeczny | Experimental validation of a numerical model for the strand shape in material extrusion additive manufacturing[END_REF][START_REF] Xia | Fully resolved numerical simulations of fused deposition modeling. Part I: fluid flow[END_REF][START_REF] Xia | Fully resolved numerical simulations of fused deposition modeling. Part II -solidification, residual stresses and modeling of the nozzle[END_REF][START_REF] Xia | A numerical study of the effect of viscoelastic stresses in fused filament fabrication[END_REF][START_REF] Xia | Simulations of fused filament fabrication using a front tracking method[END_REF][START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF]. To the author knowledge, it is the first time that the full nozzle is considered in a model. This model is a first step to fully understand the melting and solidification behaviour of the polymer in FFF. New implementation of the model where the heating of the polymer in the liquefier is considered should be implemented. Such model have already been developed in [START_REF] Pigeonneau | Heating and flow computations of an amorphous polymer in the liquefier of a material extrusion 3D printer[END_REF] and discussed in §2.5.3.1. A coupling between this model and the one presented in this chapter is needed. This model would consider all the thermal transitions of the polymer from the melting of the neat filament to the solidification of the melt.

Moreover, one limitation of the model is the computing time. Indeed the computation of the fluid dynamics and re-meshing is computationally heavy. The simulation of the deposition of a 6.5 mm strand on 20 cores of a 1.2GHz Intel Xeon cluster takes around 25 hours. The model focuses on the extrusion and deposition out of the nozzle. The simulated processing time is thus limited by the size of the computation domain. Therefore the actual processing time is restricted to around 0.5 s. Increasing the processing time results in increasing the computation domain and thus the computation time. This limitation has been especially highlighted on the temperature behaviour. The thermal phenomena occurring out of this time range also needs to be grasped. The following chapters will therefore focus on the process on a more global point of view. Another numerical model is developed to consider the process on a longer time scale. This enables to study the impact of the printing parameters on the cooling kinetics of parts with real dimensions.

Finally, the simulation has focus on an amorphous polymer. A visco-elastic model could be implemented to study the orientation of the polymer chains due to extrusion. This model would be particularly interesting for semi-crystalline polymers. Indeed the crystallisation behaviour is highly impacted by polymer orientation. Therefore, the next chapter focuses on the crystallisation behaviour of semi-crystalline polymers in FFF.

Introduction

Previous studies [START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF][START_REF] Mcilroy | Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing[END_REF][START_REF] Mcilroy | A fundamental rule: Determining the importance of flow prior to polymer crystallization[END_REF] have modelled the impact of the extrusion and deposition on polymer conformation and orientation. Orientation of polymer chains and shear rate have a predominant impact on the crystallisation behaviour. An experimental investigation is therefore needed to quantify the impact of the printing parameters on crystallisation. Indeed, the use of semicrystalline polymers in FFF has been less developed and studied than amorphous polymers. One of the major drawback in using semi-crystalline polymer is its shrinkage during cooling. However, its use is interesting as it can produce parts with better mechanical properties and better adhesion between the layers. The presence of transcrystalline layers in adhesion zones strongly increases the interfacial bonding strength [START_REF] Yan | Interfacial Crystalline Structures in Injection Over-Molded Polypropylene and Bond Strength[END_REF]. This is an important feature as most mechanical failure in FFF happens at welding areas [START_REF] Seppala | Infrared thermography of welding zones produced by polymer extrusion additive manufacturing[END_REF]. Another advantage of semi-crystalline polymers is its ability to be used on a wider range of temperature compared to amorphous polymers [START_REF] Kishore | Additive manufacturing of high performance semicrystalline thermoplastics and their composites[END_REF].

Several authors have highlighted the presence of the β phase of PP in FFF [START_REF] Petersmann | Process-induced morphological features in material extrusion-based additive manufacturing of polypropylene[END_REF][START_REF] Spoerk | Polypropylene Filled With Glass Spheres in Extrusion-Based Additive Manufacturing: Effect of Filler Size and Printing Chamber Temperature[END_REF][START_REF] Wang | Effect of fused layer modeling (FLM) processing parameters on impact strength of cellular polypropylene[END_REF]. The formation of β phase is usually observed when crystallisation occurs with thermal gradient or with orientation of the polymer melt [START_REF] Mollova | Crystallization of isotactic polypropylene containing beta-phase nucleating agent at rapid cooling[END_REF]. The addition of β-phase nucleating agent can also promote its formation [START_REF] Bai | Influence of Annealing on Microstructure and Mechanical Properties of Isotactic Polypropylene with β-Phase Nucleating Agent[END_REF][START_REF] Mollova | Crystallization of isotactic polypropylene containing beta-phase nucleating agent at rapid cooling[END_REF]. Varga [START_REF] Varga | β-MODIFICATION OF ISOTACTIC POLYPROPYLENE: PREPARATION, STRUCTURE, PROCESSING, PROPERTIES, AND APPLICATION[END_REF] has studied the influence of the β phase on PP.

The study showed that β PP has lower Young modulus and yield stress than α PP but higher ultimate tensile strength and strain. The presence of β phase in PP improves the heat deflection temperature, impact strength and fracture toughness [START_REF] Farahani | An approach for prediction optimum crystallization conditions for formation of beta polypropylene by response surface methodology (RSM)[END_REF][START_REF] Mollova | Crystallization of isotactic polypropylene containing beta-phase nucleating agent at rapid cooling[END_REF][START_REF] Bai | Influence of Annealing on Microstructure and Mechanical Properties of Isotactic Polypropylene with β-Phase Nucleating Agent[END_REF]. The presence of β phase thus impacts the mechanical properties of the final part. Understanding how it is formed would thus allow to better achieve targeted properties. However to the author knowledge, no studies have been focusing in understanding how this phase is created and what parameters influence its formation in FFF.

The properties of semi-crystalline polymers are tailored by their crystallinity. Therefore getting more in depth grasp on the crystallisation behaviour is of primary importance for their use in FFF. The goal of this chapter is to understand the impact of the extrusion velocity and the part geometry on crystallisation. The possible flow induced orientation of the crystalline phase is also investigated. The study focuses on the use of polypropylene. A comparison between two very different PP is done to understand the influence of polymer inherent properties on crystallisation. The formation of different crystalline phases induces different polymer properties. Indeed, PP can form three different phases, namely the α monoclinic, β hexagonal and γ orthorhombic. The formation mechanism of the β modification of PP is investigated in this Chapter.

Materials and methods

Materials

Two different kinds of PP are used in the present study. One PP is an isotactic homopolymer made for injection moulding under the reference PPH 3250 from Atofina. Its melting temperature is equal to 165°C. Its weight and number average molecular weight is M w =213 g/mol and M n =42.5 g/mol, respectively. Its melt flow rate MFI = 25 g/10min and isotactic index = 0.97. The crystallisation kinetics of the PPH3250 has already been studied in a previous work [START_REF] Boyer | Crystallization of polypropylene at high cooling rates: Microscopic and calorimetric studies[END_REF]. The result showed a crystallisation temperature equal to 110°C for a cooling rate of 10 K/min. The other PP studied in the present work is formulated for additive manufacturing and is provided by the brand FormFutura ® . It is called the PP centaur and its fusion temperature is around 110°C which is much below what is classically found for other PP grades made for extrusion processes [START_REF] Calhoun | Polypropylene[END_REF]. Comparatively, its crystallisation temperature is equal to 65°C for a CRYSTALLISATION BEHAVIOUR cooling rate of 10 K/min. The actual formulation of this PP is not communicated in the literature. The PP Centaur is received in form of 3D printable filaments. The PPH 3250 is received in form of pellets. A 1.75 mm diameter filament is made from the PPH3250 pellets with the filament maker Composer 350 from the brand 3devo ® . The pellets are extruded through a single screw extruder and experience a series of 4 plateau temperatures (180-200-190-175°C) before extrusion and cooling by a dual fan at the extruder exit.

Samples

Sample characterisation is either made from the neat filament before printing or from printed samples. When printing an object, the deposited polymer experiences multiple heating and cooling as adjacent strands are deposited [START_REF] Seppala | Infrared thermography of welding zones produced by polymer extrusion additive manufacturing[END_REF][START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF]. Two kinds of samples are thus printed to understand the impact of these thermal cycles. Single printed strands are studied to understand the influence of the printing velocities and substrate temperatures on crystallisation. The influence of the deposition of multiple strands on the microstructure is studied by printing small walls as depicted in Figure 3.1. The wall's length and height is 100 and 10 mm, respectively. Its width is composed of 4 or 6 strands and thus varies depending on the number and strand size. All printings are made with the Delta turbo 2040 3d printer from WASP ® . The printing parameters are set in a G-code file read by the printer. It enables to set the extrusion velocity U , the nozzle velocity V , the deposition path as well as the extrusion and substrate temperature. For all samples, the substrate temperature is set to 70°C and 105°C for the PP Centaur and PPH3250, respectively.

Differential Scanning Calorimetry (DSC)

DSC measurements are performed on a DSC 4000 from PerkinElmer ® connected to a PerkinElmer ® Intracooler to control the temperature. The latent heat of fusion is measured by performing a constant heating rate of 20 K/min for all the samples. The sample mass is around 1 to 2 mg and is encapsulated in a 40 µL aluminum pan from PerkinElmer with ∆H the latent heat of fusion of the sample and ∆H 100 the latent heat of fusion of a 100% crystalline PP. The latent heat of fusion is calculated by integrating the fusion peak with the Pyris manager software. The peak integration is performed on approximately the same range of temperature for a given polymer to ensure repeatability of the measurements. The enthalpy of fusion of a 100% crystalline PP ∆H 100 is taken equal to 207 J/g [START_REF] Gee | Thermal properties of melt and solution crystallized isotactic polypropylene[END_REF]. For single deposited strands, cuts are made at one half of the length of the strand. For the printed walls, different samples are picked up as depicted in Figure 3.1. For each studied condition, the result is calculated by averaging at least three measurements. The measurement error is given by the mean absolute difference.

Wide Angle X-Ray Scattering (WAXS)

The analysis of the crystalline phases are performed using X-Ray measurements with an X'pert Pro MRD from the brand PANalytical operating in reflection mode. The device is equipped with a copper anticathode that uses the K α2 ray from copper with a wavelength of 1.54 Å. Measurements are carried out with an angle 2θ between 5 -32.5°. Peak deconvolution is performed using the HighScore software. The characteristic (hkl) indexes of the α, β and γ crystalline phases of PP are summarised in table 3.1 with the corresponding peak positions. Only the peaks present in this study are shown in the table. The amount of a specific crystalline phase can be determined using the relation of Turner-Jones:

C 1 = H 1 H i (3.2)
with H 1 the peaks heights corresponding to a given crystalline phase 1 and H i the heights corresponding to every crystalline peaks present in the WAXS pattern. The orientation of the crystalline phases is studied by performing 2D Debye-Scherrer rings using the flat-film camera technique under vacuum at ambient temperature. For these measurements, a plate of 1 strand height and 20 strands width is printed. The X-ray beam in transmission is sent normal to the plate surface. Additionally, WAXS measurements in transmission and in reflection are performed on certain characteristic positions 2θ while varying the azimuthal angle.

The "CrysRay" method enables to carry out WAXS measurements while controlling the heating rate and temperature of the sample. The sample is put on a glass plate controlled in temperature with the CrysRay software. A first set of temperature measurements are conducted to calibrate the sample heating. This is done by measuring the sample temperature at given plate temperatures with a T-type thermocouple. A single calibration test is performed for a given polymer. WAXS measurements are done during the cooling and crystallisation of the sample. The sample temperature is also measured during this step. The cooling is performed at ambient air temperature. Quick identification of the presence of the α and β phases has to be CRYSTALLISATION BEHAVIOUR performed during cooling. Therefore measurements are carried out with an angle 2θ between 14.6 -18°to identify the presence of the (040) and (300) diffracting plans from the α and β phases, respectively.

For single deposited strands and neat filaments, WAXS measurements are made by positioning the strands or filaments alongside. WAXS measurements are only performed on the bottom or upper surface of a printed wall. For all specimens, special care is taken to ensure performing measurements on the same surface area when changing the sample orientation.

Polarised Optical Microscopy (POM)

Measurements are performed using the Leica DMRX microscope coupled with the CCD camera DFC 420C also from Leica Microsystems. The PPH 3250 samples are cut at ambient temperature using an LKB 4801A microtome. The glass knife enables to perform slices of the order of magnitude of 1 to 10 µm and thus allows transmission of the polarised light through the sample.

A waveplate is added to observe the birefringence sign of the different crystalline phases. As the PP centaur is too flexible at ambient temperature, samples are made by performing slices under liquid nitrogen cooling using the Leica Ultracut UCT cryomicrotome. The slices are performed at -60°C and have a thickness of around 2 µm. Measurements of the size of the crystalline phases are made using the Image J software. For all samples, slices are made perpendicular to the direction of deposition. The slices performed on wall samples are made on the entirety of the cross section.

Comparison of the crystallisation of the two printed PP

Microstructure of the two PP

The microstructures of the PPH 3250 and PP Centaur are compared using POM observations. The samples are taken from printed walls, at the surface and core areas as illustrated in Figure 3.1. Microstructures for a wall printed with the PPH 3250 are depicted in Figure 3.2 with, on the left, the sample taken close to the surface and, on the right, the sample taken in the core of the wall. For both locations, two phases are clearly identified. The α phase which is the predominant crystalline PP phase [START_REF] Varga | β-MODIFICATION OF ISOTACTIC POLYPROPYLENE: PREPARATION, STRUCTURE, PROCESSING, PROPERTIES, AND APPLICATION[END_REF] is clearly visible mainly on the surface of each strand.

Recall that the α phase is characterised by a negative birefringence. The β phase which is characterised by bright spherulites, is identified in the core of each strand. Figure 3.3 presents the microstructures of two samples printed with the PP centaur taken close to the surface and in the core of the wall. It is noteworthy that only the α phase appears. WAXS measurements will detail the nature of the phases present in each samples.

Two types of microstructures are observable for both polymers. They are namely spherulites and transcrystalline layers. The latter are seen when a high germination density occurs at the surface of the polymer. A competition of the spherulites growth leads to a one directional growth of the crystalline entities perpendicular to the surface [START_REF] Freire | Effets de vitesse et de transcristallinité dans l'analyse de la cristallisation de polyamides 66 d'architectures différentes[END_REF]. These structures are promoted by thermal gradients and shear stresses [START_REF] Yan | Interfacial Crystalline Structures in Injection Over-Molded Polypropylene and Bond Strength[END_REF]. It is thus located close to the surface of the strands. Measurements of the size of the different crystalline entities are performed using Image J. The spherulite size is measured considering an ellipse shape. The length of the transcrystalline layers is measured as illustrated in Figure 3.3a. The strands are expected to have different cooling kinetics between the core and the surface areas of the wall. The size of the crystalline entities is thus compared between these zones. Moreover, the transcrystalline layers in contact with air are expected to undergo higher cooling rates than in the core of the printed wall. A comparison of the length of the transcrystalline layer between the adhesion zones and the "surface zone" is thus performed. For each case, an average over ten measurements is performed. The results are displayed in Table 3.2.

Table 3.2 shows that the microstructure is globally homogeneous in the whole printed wall for both polymers. The transcrystalline layer length at the adhesion zone is comparable between the core and the surface area. Moreover the α phase spherulites have approximately the same size in the whole wall. However, the transcrystalline structure in the adhesion zone is 50% higher than in the surface area. A difference of spherulite size is observable between the PPH 3250 and the PP Centaur. The size of α spherulites of the PP Centaur is roughly one half smaller than for the PPH 3250. For both polymers, the length of the transcrystalline layer is larger than the size of the spherulites. However, the PP centaur shows a higher transcrystalline layer's length compared to its spherulite size. Indeed the transcrystalline layer is 1.5 and 1. the same for both polymers, i.e. T ext =250 • C, the PP Centaur is printed at a temperature much larger than its melting temperature. Consequently, the diffusion at the interface between strands is more efficient than for the PPH 3250.

Degree of Crystallinity

The influence of the size of the wall on its degree of crystallinity is studied for both PP. Table 3.3 gathers the wall dimensions, number and width of the strands for the different experiments.

For the PP Centaur, three cases have been considered. The two firsts are achieved with an extrusion velocity U = 2 m/min and a nozzle velocity V = 1 m/min and by varying the number of strands from 4 to 6. The last case is done by multiplying the extrusion velocity by a factor two and with 4 lines. By this way, the width of each strand is twice larger than for the two previous cases as it is indicated in Table 3 3.3. The degree of crystallinity is larger in samples printed with the PPH 3250 than with the PP Centaur for the same printing conditions. This trend can be correlated to the size of spherulites which are larger for the PPH 3250 than for the PP Centaur. Recall that the crystallisation temperature of the PPH 3250 is larger than T c of the PP Centaur, see § 3.2.1. As both polymers are printed at the same temperature T ext = 250°C, the PPH 3250 spends less time above its crystallisation temperature during cooling. This could result in a lower degree of crystallinity, especially at the top area where both polymers are subject to the same cooling conditions. To understand this difference, measurements of the spherulites growth rate G is done by POM for the PP Centaur.

The average spherulite diameter of the α phase is measured while cooling the sample at a constant rate from 210 to 50°C. An example of the shots are displayed in Figure 3.5a. The growth rate is then calculated as a function of the average temperature between two measurements. The results have been compared with the growth rates of the PPH 3250 measured by Boyer et al. [START_REF] Boyer | Crystallization of polypropylene at high cooling rates: Microscopic and calorimetric studies[END_REF] in Figure 3.5b. For the same range of temperature, the PPH 3250 shows a growth rate that is more than 2 decades higher than for the PP Centaur. Therefore, the polymer growth rate seems to have a predominant impact on the final degree of crystallinity compared to the crystallisation temperature.

The evolution of the degree of crystallinity inside the wall shows similar trends for all printing conditions. The degree of crystallinity is more important at the bottom than at the top area. An average increase of 10% crystallinity is observed between the bottom and the top areas. The samples collected at the bottom area are in contact with the substrate whereas the ones collected at the top layers are the last deposited strands of the wall (Fig. 3.1). As the substrate is heated, Figure 3.6: Influence of the strand's and wall's width on the degree of crystallinity between the shell and core areas. The samples are collected on a single layer at 0cm from the wall the strands in contact with it have a lower cooling kinetics than the ones directly subject to convective cooling with air. Crystalline entities thus have more time to grow resulting in higher degree of crystallinity. Moreover, the substrate is heated at 70°C and 105°C for the PP Centaur and PPH 3250, respectively. The deposited strands are thus maintained at a temperature near their crystallisation temperature which also promotes crystallisation. This will be further detailed with for the PPH3250 in the next section.

The shell areas have a lower degree of crystallinity than the core of the wall for all samples printed with the PP Centaur. This is also explainable by the difference of cooling kinetics. The shell zones are directly subject to convective cooling with air. The core of the part thus has a lower cooling rate resulting in a higher degree of crystallinity. The difference in χ between the surface and the core is however less prominent than between the bottom and top layers. Consequently, for thin walls the major factor controlling the degree of crystallinity of a strand is the height at which it is printed. The shell and core samples presented in Figure 3.4 have been collected on multiple layers. The result is thus averaged by the number of layer. The difference of degree of crystallinity on a single layer is displayed in Figure 3.6.

Increasing the width of the wall from 2 to 4 mm results in an increase of the core crystallinity of around 15%. As the total width of the layer increases, the core is more protected from convective cooling with air resulting in a higher degree of crystallinity. Therefore increasing the width of the wall induces a more heterogeneous cooling kinetics inside a layer which also induces a more heterogeneous crystallinity.

As already shown in Section 2.4.2, the strand size increases with the extrusion velocity. Experiments achieved at U = 4 and V = 1 m/min have the largest stand width, see Table 3.3. At the shell area the total width of the layer has less impact on the cooling kinetics. The two samples with the same strand width have approximately the same degree of crystallinity whereas the one with a larger strand width has a higher crystallinity. Previous studies in reference [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF] have shown that the characteristic cooling time of a single strand is mainly driven by its cross section area. As the strand cross section increases, the cooling kinetics decreases which allows more growth of the crystalline entities. Consequently, increasing the strand size promotes the crystallisation in parts printed with semi-crystalline polymers.

Crystal orientation

2D Debye-Scherrer patterns are studied to observe the possible crystal orientation in both PP.

The influence of the extrusion velocity is only investigated on the PP Centaur. Unfortunately, it was impossible to print the PPH 3250 for an extrusion velocity larger than 2 m/min. Previous studies showed that the FFF process induces crystalline orientation [START_REF] Petersmann | Process-induced morphological features in material extrusion-based additive manufacturing of polypropylene[END_REF][START_REF] Shmueli | In Situ Time-Resolved X-ray Scattering Study of Isotactic Polypropylene in Additive Manufacturing[END_REF][START_REF] Shmueli | In-situ X-ray scattering study of isotactic polypropylene/graphene nanocomposites under shear during fused deposition modeling 3D printing[END_REF]. The oriented crystalline structures was found for PP printed at the same range of extrusion velocities as in the present study. Therefore WAXS measurements in reflection mode is achieved to ensure no crystal orientation when printing an object. The different peak positions are identified by performing a first scan for a Bragg angle 2θ from 5 to 40°. Measurements of the peak intensity as a function of the azimuthal angle are then performed for the two main peaks of the α phase for both PP. These peaks correspond to the hkl indexes 110 and 040. For the PPH 3250 and the PP Centaur, the peak corresponding to the β and γ phases, respectively is also studied. The crystalline orientation of both polymer is compared for samples printed with the same conditions in Figure 3.10. The azimuthal angle 0°is parallel to the direction of deposition.

Small variations of the orientation are observable for the crystalline phases studied in Figure 3.10.

For the PPH 3250, α -110 and β -300 peaks exhibit their intensity maxima at around 0 and 180°which is the direction of deposition. This means that the a axis of the oriented α crystallites is parallel to the strand direction of deposition. The peaks are relatively broad, meaning that the polymer is lowly oriented. The α -040 peak shows lower intensities along the direction of deposition (0-180°). Consequently, the b axis of the α phase has a slightly preferred orientation perpendicular to the direction of deposition. The c axis corresponding to the polymer chain axis [START_REF] Kumaraswamy | Shear-enhanced crystallization in isotactic polypropylenePart 2. Analysis of the formation of the oriented "skin[END_REF] thus tends to orient perpendicular to the flow direction. For transcrystalline growth, the α lamellae are known to grow based on a parent-daughter model [START_REF] Kumaraswamy | Shear-enhanced crystallization in isotactic polypropylenePart 2. Analysis of the formation of the oriented "skin[END_REF][START_REF] Assouline | Lamellar twisting in α isotactic polypropylene transcrystallinity investigated by synchrotron microbeam X-ray diffraction[END_REF][START_REF] Petersmann | Process-induced morphological features in material extrusion-based additive manufacturing of polypropylene[END_REF]. In this model, the parent lamellae has a preferred orientation perpendicular to the flow direction. The daughter lamellae grows epitaxially on the parent surface with its a axis parallel to the flow direction. The b axis of both parent and daughter crystallites are perpendicular to the flow direction. The small changes of orientation in reflection could thus be attributed to the daughter lamellae of the transcrystalline layers observed in Figure 3.2. The 300 peak shows that the β phase has its a axis that is less inclined to be oriented perpendicular to the direction of deposition.

The PP Centaur printed with the same printing velocity as the PPH 3250 shows similar trends. Indeed, the maximum intensities are obtained at 0 • for the α-110 while the maximum intensities for α-040 are observed perpendicular to the flow direction. Consequently, the PP Centaur polymer chains have a slightly preferred orientation along the flow direction. For all samples printed with the PP Centaur, the γ phase shows exactly the same trends as the α -110. The γ phase is known to grow epitaxially on the α phase with an angle of 40° [START_REF] Lotz | Single crystals of γ phase isotactic polypropylene: combined diffraction and morphological support for a structure with nonparallel chains[END_REF][START_REF] Kalay | The occurence of the γ-phase in injection moulded polypropylene in relation to the processing conditions[END_REF]. According to Lotz et al. [START_REF] Lotz | Single crystals of γ phase isotactic polypropylene: combined diffraction and morphological support for a structure with nonparallel chains[END_REF], the γ phase chains are parallel to the 110 plan explaining the similar evolution of orientation in Figure 3.10 and 3.11.

It is noticeable that the α -110 and γ -117 peaks of the PP Centaur show a single intensity maximum around 0°. To explain this trend, the same measurements are achieved for different extrusion velocities in Figure 3.11. Increasing the extrusion velocity leads to an intensity maximum around ±45°from the flow direction. The single peak around 0°seen in Figure 3.10b is in fact composed of the two tighten broad peaks. The increase in the extrusion velocity from U = 2 m/min to U = 6 m/min results in an evolution in the crystal orientation. This is especially observable on the α -040 of the PP Centaur. Indeed at U = 2 m/min, intensity minima are in the flow direction contrarily to the α -110. The increase of the extrusion velocity leads to the occurrence of the intensity maxima at ±45 • for both α-040 and α-110 planes. Assouline et al. [START_REF] Assouline | Lamellar twisting in α isotactic polypropylene transcrystallinity investigated by synchrotron microbeam X-ray diffraction[END_REF] have studied the orientation of transcrystalline layers formed at the surface of a fibre. A twist in the orientation of the transcrystalline layer can be observed in their study. The α -110 plan would rotate from the meridian to the equatorial plane while the α -040 rotates from the equatorial to the meridian plane. This rotation happens gradually as a function of the distance from the fibre. In their study, orientation at ±45°is observable at 16 -32 µm from the surface of the fibre. This could explain the variation of intensity as a function of the azimuthal angle observed in our study. The increase of the extrusion velocity would imply more rotation of the transcrystalline layers at ±45°resulting in more distinct peaks. Dean et al. [START_REF] Dean | Matrix molecular orientation in fiber-reinforced polypropylene composites[END_REF] have also proposed a model considering the twisting of the transcrystalline layers. The proposed model For both polymers, no orientation is observable in transmission mode and very moderate orientation is seen in reflection mode, meaning that the spherulites do not have preferred localisation or orientation inside the strand. The variation of intensities observed in reflection mode seems to be attributed to the transcrystalline layers seen in Figure 3.2. The measurements are conducted on the whole deposited layer and is an average orientation of all crystalline entities. The absence of orientation inside each strand may weakens the orientation observed due to the transcrystalline layers [START_REF] Petersmann | Process-induced morphological features in material extrusion-based additive manufacturing of polypropylene[END_REF]. Table 3.2 shows that the length of the transcrystalline layer is relatively small compared to the rest of the strand. A pole figure would need to be done to study with precision the orientation of the crystallites in the transcrystalline layer. The variation of intensities is slightly higher for the PPH 3250 compared to the PP centaur at the same printing conditions. The PPH 3250 is thus more inclined to flow induced orientation than the PP Centaur.

Behaviour of the crystalline phases

WAXS measurements are also studied to understand the evolution of the crystalline phase before and after printing. The WAXS measurements of a sample printed with the PP Centaur are compared to the neat filament in Figure 3.12a. Seven peaks are observable, which correspond to the seven Debye rings observed in Figure 3.8. Figure 3.12a shows that the neat PP centaur filament does not exhibit the γ peak at 2θ = 20°. Consequently, the γ phase appears after printing of the sample. As reported by Lotz et al. [START_REF] Lotz | Single crystals of γ phase isotactic polypropylene: combined diffraction and morphological support for a structure with nonparallel chains[END_REF], the γ phase grows as two rows beside an elongated central α lath. This resembles to the transcristalline layers observed in Figure 3.3. The formation of the γ phase would thus happen mainly in the transcrystalline layer.

According to Foresta et al. [START_REF] Foresta | Competition between α and γ phases in isotactic polypropylene: effects of ethylene content and nucleating agents at different cooling rates[END_REF], the formation of the γ phase is favored by the presence of ethylene comonomer or/and the presence of nucleating agents. The specific formulation of the PP Centaur is not communicated by the supplier. However, the γ phase is usually known to grow under high pressure environment or during crystallisation of low molecular weight material [START_REF] Kalay | γ Phase in injection moulded isotactic polypropylene[END_REF][START_REF] Kalay | The occurence of the γ-phase in injection moulded polypropylene in relation to the processing conditions[END_REF]. It is thus likely that the PP Centaur exhibits the presence of ethylene or nucleating agent. According to Foresta et al. [START_REF] Foresta | Competition between α and γ phases in isotactic polypropylene: effects of ethylene content and nucleating agents at different cooling rates[END_REF], the growth of the γ phase enhances when decreasing the cooling rate. This is in accordance with the results found in Section 3.3.2.

The PPH 3250 crystalline phases are displayed in Figure 3.12b before and after printing. The presence of the β phase along with every characteristic peak of the α phase is observable in both WAXS patterns. The β phase is observable by the presence of a peak at 2θ = 19.74°c orresponding to the hkl index 300. The amount of β phase content is calculated for both cases using Equation 3.2. The amount of β phase between the neat filament and printed sample increases from 13% to 35% of the total amount of crystalline phase. The growth of the β modification of PP is promoted when crystallisation occurs at high temperature gradient and/or under important shear rate [START_REF] Varga | β-MODIFICATION OF ISOTACTIC POLYPROPYLENE: PREPARATION, STRUCTURE, PROCESSING, PROPERTIES, AND APPLICATION[END_REF]. Both polymers are printed at the same extrusion temperature and velocity. Moreover the PP Centaur is printed on a cooler substrate. However the PP Centaur does not exhibit the presence of the β phase. The formation mechanism of the β phase in FFF is therefore studied in the next section.

Mechanism of the formation of the β modification of PP in FFF

The investigation is focused on the PPH 3250 as it is the only one exhibiting the presence of β phase. Moreover the PPH 3250 has not been formulated for FFF. Studying its crystallisation behaviour can thus serve as a reference for upcoming researches. This study has been focused on a user point of view. Users generally receive the neat filament directly and do not have control on the processing conditions to produce the filament. The processing conditions at which the polymer is transformed from pellets to filaments is not therefore studied in this section.

Characterisation of the β content in the process

The influence of the cooling kinetics on the formation of the β phase is first studied. The enthalpy of fusion is measured by DSC after cooling the neat PPH 3250 filament at different rates in Figure 3.13. The neat filament is heated from 20 to 250°C at 20 K/min and then maintained at 250°C for 3 min prior to cooling.

The first peak between 142 -146°C (depending on the sample) corresponds to the fusion of the β crystal [START_REF] Varga | β-MODIFICATION OF ISOTACTIC POLYPROPYLENE: PREPARATION, STRUCTURE, PROCESSING, PROPERTIES, AND APPLICATION[END_REF] while the major peak between 159 -165°C corresponds to the fusion of the α phase. A peak of less intensity is observable at around 152°C which is attributed to the β ′ crystal that forms due to recrystallisation during the DSC heating cycle [START_REF] Spoerk | Polypropylene Filled With Glass Spheres in Extrusion-Based Additive Manufacturing: Effect of Filler Size and Printing Chamber Temperature[END_REF]. The presence of β crystal is observable in the neat filament but with lower intensity compared to the samples cooled at different cooling rates. This confirms the WAXS observation in Figure 3.12b. The fusion peak intensity of the β phase increases when decreasing the cooling rate. Consequently, decreasing the cooling rate promotes the formation of β phase for the neat filament. This is an unusual observation as the formation of β phase is normally promoted with high undercooling [START_REF] Bai | Influence of Annealing on Microstructure and Mechanical Properties of Isotactic Polypropylene with β-Phase Nucleating Agent[END_REF][START_REF] Varga | β-MODIFICATION OF ISOTACTIC POLYPROPYLENE: PREPARATION, STRUCTURE, PROCESSING, PROPERTIES, AND APPLICATION[END_REF][START_REF] Calhoun | Polypropylene[END_REF]. The β phase is not thermodynamically stable and it thus arises when the polymer is rapidly cooled under its crystallisation temperature which prevents the α crystal to form. As the β phase is already present in the neat filament, the increase of the β content with decreasing cooling rate may be attributed to annealing of the β phase. The fusion of the neat filament would not totally erase the crystalline history. Growth of the β phase would thus arise during cooling. Decreasing the cooling rate provides more time for growth, explaining the increase of the β content.

WAXS measurements with the Crysray method explained in section 3. 3.14a shows that the occurrence of the β phase at 2θ = 16.1°appears when the polymer reaches 105°C. An increase of the β -300 intensity with time is observed in Figure 3.14b. This increase is predominant for the β phase compared to the α phase at 2θ = 16.87. This shows that annealing of the β phase happens when the PPH 3250 is maintained at the substrate temperature.

To see the influence of the cooling rate on the formation of the β phase, two measurements of the β -300 and α -040 are done. Both measurements are achieved at slow heating from ambient temperature to approximately 220°C at 10 K/min followed by an isotherm step of 15 min. One . This shows that the amount of β phase is predominantly attributed to the way the polymer is heated. When the heating rate is increased, the amount of β entities present in the neat filament is less melted. If more β nuclei are present upon cooling, the resulting β content will also be larger. Northcutt et al. [START_REF] Northcutt | Effect of processing conditions on crystallization kinetics during materials extrusion additive manufacturing[END_REF] have reported this effect as a "self-seeding" mechanism. This confirms the conclusion drawn by DSC.

Moreover, Figure 3.14b shows that the amount of β -300 is predominant compared to the amount of α -040. This is not the case when the sample is heated at a low heating rate. This can be due to αβ-recrystallisation during cooling. Indeed Varga [START_REF] Varga | β-MODIFICATION OF ISOTACTIC POLYPROPYLENE: PREPARATION, STRUCTURE, PROCESSING, PROPERTIES, AND APPLICATION[END_REF] have reported the formation of α nuclei formed during annealing within the β phase. The slow heating cycle in these measurements may contribute to annealing the sample. The β to α transition could occur which causes a more important α peak after cooling. Figure 3.15a shows no evolution of the α and β contents after polymer crystallisation as the peak intensity remains the same for 1000 s. The samples cooled at high rates however show a slower crystallisation kinetics. Indeed an increase of the crystalline phases is observe up to 300 s after it has reached the substrate temperature. The substrate temperature is close to the crystallisation temperature of the PPH 3250. Annealing may thus occur, explaining the gradual increase of the β phase.

The effect of multiple heating cycles have been studied by performing 2 cycles with the same thermal conditions. The heating and cooling conditions presented in Figure 3.15b have been reproduced in a second cycle in Figure 3.16. The formation of the β phase is not observable even after 300 s at 105°C. At these cooling rates, few or no new nuclei of the β phase appear. However the presence of pre existing β content is not completely melted during the first heating cycle. Multiple cycles is thus needed to completely erase the initial microstructure of the neat filament.

All the measurements presented at high and low heating rates are slower than the heating rates encountered in FFF. Indeed for an extrusion velocity of 90 mm/min, a polymer reaches the inlet temperature in less than 5 s [START_REF] Pigeonneau | Heating and flow computations of an amorphous polymer in the liquefier of a material extrusion 3D printer[END_REF] which corresponds to a heating rate of approximately 3600 K/min. At these fast heating rates, the polymer does not have enough time to erase its crystalline history. The presence of β phase spherulites in the deposited strand arises from incomplete melting of the crystalline domain prior to deposition. Annealing thus occurs after deposition of the polymer. The "memory effect" of the presence of β spherulites is persistent in FFF. Consequently, at these fast heating rates, the β phase remains always present in FFF. This is true if β spherulites are already present in the neat filament. The microstructure of the neat filament thus plays a major role in the part final microstructure.

However, a few studies have highlighted the creation of β phase induced by the FFF process [53, [START_REF] Petersmann | Process-induced morphological features in material extrusion-based additive manufacturing of polypropylene[END_REF] have seen the β phase only for specific printing conditions. Several authors have measured the cooling kinetics of a single deposited strand on a substrate. These authors have shown that the strand cools at a speed ranging from 250 K/min to over 1000 K/min until it reaches a plateau temperature [START_REF] Seppala | Infrared thermography of welding zones produced by polymer extrusion additive manufacturing[END_REF]2,[START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF]. Moreover in real processing conditions, other parameters like the shear rate have an impact on the crystallisation behaviour. The combination of rapid cooling with shear rate can result in the formation of β nuclei. Shear rates and high cooling rates conditions could not be attained by the CrysRay method. It is thus interesting to understand if the formation of new β nuclei also occurs and has an impact on the growth of the β content or if it is negligible compared to the amount of pre existing β content from the neat filament. The impact of the cooling rate and shear rate on the β phase content in real processing conditions is thus investigated in the next subsection.

Impact of the printing parameters on the formation of β phase.

The impact of the cooling rate on the crystallisation behaviour of the β phase is studied by printing strands at different substrate temperatures. Single strands of PPH 3250 are printed at U = 2 m/min and V = 1 m/min on a substrate heated at a temperature T sub ranging from 70 to 105°C. Figure 3.17 summarises the DSC measurements obtained for four substrate temperatures. The first endothermic peak is attributed to the fusion of the β phase. The maximum peak amplitude is reached at around 150 °C for the sample printed with the highest substrate temperature. This peak is shifted to smaller temperatures when decreasing the substrate temperature of the printed samples. Moreover, the amplitude decreases gradually with the substrate temperature. It is noteworthy that for T sub equal to 70 • , the β phase is completely absent.

The strands studied in DSC are cut with a microtome and observed on POM. These observations are gathered in Figure 3.18. The β spherulite sizes are measured using imageJ and the results are shown in Figure 3.18d. The size of the β phase spherulites enhances with the substrate temperature. Therefore, the increase in the β content is likely due to an increase of the growth of the spherulites. The relatively high measurement errors are due to the important dispersion of the spherulite size, especially for the case T sub = 105°C. However, the global trend is confirmed as spherulites printed at T sub = 105°C are more than 5 times bigger than when printed at T sub = 90°C.

POM and DSC measurements show that the size and amount of β content increase with the substrate temperature. This is in agreements with the results found in subsection 3.4.1. As the substrate temperature decrease, the strand spends less time around the annealing temperature estimated at 105°C. The growth of the β phase is thus limited as the cooling rate increases. In the literature, thermal gradient promotes the formation of β phase nuclei. However in our case, the formation of new β phase nuclei upon cooling has a negligible impact on the final β content.

The predominant parameter to promote the final amount of β phase is the time at which the polymer stays near its annealing temperature. This amount is also linked to the proportion of initial β spherulites in the neat filament.

The impact of the extrusion velocity U on the formation of the β phase is also studied by POM and DSC. Single strands are deposited at different extrusion velocities on a substrate heated at T sub equal to 105°C. The latent heat of fusion of the strands are measured by DSC. Their microstructure is also observed by performing microtome cuts on the strands. The β spherulites sizes are measured and compared between the different printing velocities. The moving nozzle velocity V is also reduced to ensure the deposition of sufficiently uniform strands for low extrusion velocities. the amplitude of the endothermic peak attributed to the β phase decreases strongly when the extrusion velocity decreases from 2 to 1 m/min. The difference is less visible when the extrusion CRYSTALLISATION BEHAVIOUR velocity decreases from U = 1 m/min to U = 0.5 m/min. The residence time of the polymer in the liquefier decreases with the extrusion velocity. As shown in previous section, the amount of β phase in the printed strand is directly proportional to its amount in the neat filament. Consequently, the polymer is less inclined in erasing the neat filament microstructure when increasing the extrusion velocity. Moreover, as it is established in Chap. 2 and [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF], the strand cooling kinetics decreases when increasing the ratio U /V . As shown in 3.18and 3.20, the β spherulites are located in the core of the strand. The β nuclei in the core of the strand have thus more time to grow when the extrusion velocity increases. Moreover, Figure 3.20 shows that the number of β spherulites in the strand decreases with the extrusion velocity. The literature shows that the number of created nuclei increases with shear rate [START_REF] Mcilroy | Modelling flow-enhanced crystallisation during fused filament fabrication of semi-crystalline polymer melts[END_REF]. The increase of the extrusion velocity may lead to the formation of new β nuclei. According to McIlroy [START_REF] Mcilroy | A fundamental rule: Determining the importance of flow prior to polymer crystallization[END_REF], flow enhanced crystallisation is only possible for Weissenberg number W i larger than one. Considering a Newtonian fluid in the capillary and a reptation time of 1.10 -2 s, this limit is reached for U larger than 0.3 m/min. In conclusion, the increase of the extrusion velocity promotes the formation of β phase in a printed object

Conclusion

Two PP with different properties have been studied to understand the importance of the process parameters as well as the polymer characteristics on crystallisation. A combination of POM observations, DSC and WAXS measurements are achieved. This material is a very common polymer used in many different fields. However it is still rarely employed in FFF.

Both polymers showed transcrystalline layers at the adhesion and surface areas. For both polymers, the transcrystalline layer is more important in the adhesion zone. The microstructure is homogeneous in the rest of the part. Choosing a polymer with lower melting temperature allows more diffusion at the interface and thus longer transcrystalline layers. This should result in parts with better bonding quality. The WAXS measurements highlighted the presence of moderate orientation inside the printed parts. For both polymer, the observed orientation is essentially due to the transcrystalline layers. The shear rates induced by the extrusion velocities do not induce orientation of the crystalline entities in the core of the strand.

The use of two different PP enabled to study all of the three main crystalline phases of polypropylene, i.e. the α, β and γ phases. As the PPH 3250 is not formulated for FFF, it is taken as a reference of the crystallisation of PP. WAXS measurements during thermal cycles are done to mimic the thermal history encountered in FFF. The neat filament initial crystallinity has a predominant impact on the printed object crystallinity. The rapid heating in FFF never completely erase the initial crystallinity of the polymer. It is thus possible to tailor the printed part microstructure based on the initial neat filament microstructure. Consequently, the formation of the β phase is promoted by the decrease of the cooling rate since the growth is achieved over a larger range of time. The formation of the β phase also enhances with the extrusion velocity. Further investigations should be done to know which parameter is the most prominent between shear rate and cooling kinetics. Finally, the formation of β phase is not promoted by the cooling rate. Indeed, the lower growth of β has a predominant negative impact compared to the potential creation of β nuclei.

A map of the printed parts degree of crystallinity is done by DSC. The degree of cristallinity of the printed object is strongly controlled by the thermal cooling rate. The polymer growth rate has a predominant impact on the final degree of crystallinity. The latter is also mainly affected by the the dimension of the part and the size of a strand. The degree of crystallinity enhances with the number and size of the strands. The crystallinity is more homogeneous when decreasing 3.5. CONCLUSION the width of the wall. Major differences of the crystallinity are found between the bottom and the top of the wall. The difference of degree of crystallinity in a printed object is mainly due to heterogeneous cooling rates. This highlights the importance of getting more knowledge on the crystallisation kinetics of parts produced by FFF. A numerical model is thus developed in the next chapter to simulate the thermal behaviour during the printing of an object.

Résumé

Un modèle numérique est développé afin de simuler le procédé à plus grande échelle. La première étape est de simuler l'évolution thermique lors de la construction d'un mur de plusieurs centimètres de hauteur. Une méthode Level-set est également utilisée afin de discrétiser le domaine en fractions de couche, et ainsi permettre la construction d'un mur en changeant les propriétés de la fraction de couche, de l'air au polymère. Cela permet de simuler l'addition successive de matière, propre au procédé FFF, en choisissant l'ordre à laquelle les fractions de polymère sont ajoutées. La température est calculée dans tout le domaine en imposant une température d'extrusion à la fraction venant d'être déposée. Une étape de remaillage est également effectuée après le dépôt de chaque couche afin de permettre un calcul fidèle de la température dans les zones de forts gradients thermiques. Cela permet également de discrétiser la couche en fraction de géométrie précise.

Le modèle thermique est validé en comparant l'évolution de la température avec des mesures expérimentales de température à l'aide de thermocouples. La comparaison est effectuée sur des murs imprimés avec des paramètres identiques, dans une zone similaire. Une bonne corrélation entre les prédictions numériques et les mesures expérimentales est observée pour des cas avec une taille de cordons différents. Le degré d'adhésion entre les couches est ainsi calculé, basé uniquement sur le champ de température.

Le modèle thermique est ensuite adapté pour le cas d'un polymère semi-cristallin. Le but est de pouvoir prédire la cinétique de cristallisation basée sur le refroidissement de la pièce. Un modèle fondée sur la théorie d'Ozawa, modifié pour le cas de refroidissement non constant, est implémenté dans le modèle thermique. Un calcul de la fraction transformée de sphérolites basé sur la température et la vitesse de refroidissement est ainsi possible. On remarque que la prise en compte de remaillage en fonction du front de cristallisation dans le cas de pièce de hauteurs supérieures à 1 cm est importante pour avoir un calcul fidèle de la température. La hauteur à laquelle une couche est déposée a une influence prépondérante sur sa cinétique de cristallisation. Le modèle est ensuite utilisé pour comprendre les différences de degré de cristallinité observées dans le chapitre précédent. Enfin un cas pratique est étudié dans le cas où on veut avoir une cinétique de cristallisation la plus homogène dans la totalité de la pièce imprimée. Cela met en évidence l'intérêt du modèle développé dans le choix des paramètres d'impression.

INTRODUCTION

Introduction

One of the main objective of the PhD is to develop a macroscopic numerical model, which is able to predict the thermal and mechanical behaviours of a part printed with FFF. In order to predict the mechanical behaviour of the part, a well established thermal model needs to be implemented. The first step is therefore to develop and assess a thermal model which accurately predict the temperature behaviour of the polymer once deposited. This model is first developed for an amorphous polymer as it is simpler to model. Indeed semi crystalline polymers induce more complex heat phenomena when they crystallise. Therefore the assessment of the model is conducted for the ABS amorphous polymer. Part of this study have already been published in [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF]. Thermal numerical computations are performed based on the model developed by Zhang et al. [START_REF] Zhang | Macroscopic thermal finite element modeling of additive metal manufacturing by selective laser melting process[END_REF] and implemented in the C++ library CimLib. Initially developed for the selective laser melting process, the numerical procedure is modified to match with the FFF process.

Numerical finite element computation 4.2.1 Procedure of the wall printing

A wall with a length L and a height H set to 10 and 1 cm, respectively, is printed with different printing velocities. The wall is composed of four polymer filaments meaning that its thickness depends on the printing velocities. The thermal behaviour depending on the printing conditions is studied during the printing of the wall. The temperature is recorded using T-type thermocouples. This technique is chosen instead of the infrared thermography since a thermocouple allows to directly measure the temperature inside a layer. Moreover, few tries using infrared thermography have shown that measurements are disturbed by thermal radiation of the printer's hot-end. The T-type thermocouple is introduced during a printing pause of eight seconds when the wall height is equal to 5 mm. The knowledge of the exact position of the thermocouple is a fastidious task as it is impossible to put markers on the printed object. However, the thermocouple is always introduced during the printing of the second line, at one-third of L. The location of the thermocouple in the inserted layer is illustrated in Figure 4.2. The arrows depict the printing direction of the nozzle. The same deposition path is reproduced for all tested conditions. For each tested condition, a minimum of four experiments are achieved. An average of all data is then computed to know the average temperature behaviour for a particular set of parameters.

Numerical procedure to print a wall

The FFF technique is an incremental process where new layers are added upon deposited layers. The numerical process needs to follow this principle. A level-set function ϕ is used to define the building front of the deposited polymer. Each layer has a given height ∆H. The zero value of ϕ corresponds to the position z = H = p∆H where the p layer is deposited. The deposition layer is thus comprise in the range ϕ p ∈ [-∆H; 0]. The level-set function ϕ is reinitialised at each layer increment p as follows ϕ p = ϕ p-1 -∆H. (4.1)

A rectangular domain of width W , length L and height H is numerically printed. The computational domain is initially composed of air. The layer where the polymer is about to be deposited is then defined with the level-set function ϕ p . The layer is then divided into fractions of layer. The wall width W is divided into lines of same size ∆W representing strands of polymer. A single strand of length L is divided into fractions of length ∆L. The geometry of the strand is assumed rectangular in the numerical computation. The polymer is deposited by the introduction of a series of fractions within one layer. The deposition sequence is based on a G-code file. Each fraction of strand is successively added after a time step dependent on the nozzle velocity V . A single strand line is deposited during a time t l defined as follows

t l = L V . (4.2)
The deposition time of a strand fraction t f depends on the number of fractions n of a strand defined by

t f = t l n = ∆L V . ( 4

Meshing of the numerical domain

The goal is to have a mesh size that is sufficiently fine to have an accurate thermal resolution. On the other hand, the total number of element should be as low as possible to be computationally efficient. Therefore the mesh size needs to be as coarse as possible. The new deposited layer is where maximum temperature gradient is found. Indeed the polymer is deposited at the extrusion temperature upon an already cooled layer. The deposited layer is directly subject to convective cooling with air and thus has a high cooling kinetics. Moreover the strand dimension needs to be correctly defined in the deposition layer. The strand dimension is variable but can be of the order of magnitude of 100 µm depending on the extrusion velocity. For these reasons the mesh size needs to be the finest in the deposition layer. Thermal gradients decreases as the distance increases from the deposition layer. Therefore the mesh size can be coarser as the distance from the deposition layer increases. An anisotropic mesh metric is thus defined in the whole domain.

The mesh metric is defined as a 3 × 3 matrix denoted M. At each node, M defines the aimed mesh size in each direction. The deposition layer's height increases throughout the process. Therefore re-meshing needs to be performed at each layer increment. The aimed mesh metric is communicated to the re-meshing procedure to update the node position and mesh topology. This is done without changing the global domain geometry. The re-meshing strategy used in CimLib is detailed in [START_REF] Coupez | Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing[END_REF][START_REF] Coupez | Solution of high-Reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing[END_REF][START_REF] Shakoor | A new body-fitted immersed volume method for the modeling of ductile fracture at the microscale: Analysis of void clusters and stress state effects on coalescence[END_REF]. The goal is to define the appropriate aimed mesh size in the whole domain before re-meshing.

Different mesh sizes are set in the domain depending on the location in the part and the printing layer. The finest metric M f is defined with respect to the zero value of the level set function ϕ p . The mesh size is only dependent on the distance on the vertical z axis from ϕ p = 0. It increases with the distance from ϕ p = 0. M f is defined as:

M f =      1 h 2 f x 0 0 0 1 h 2 f y 0 0 0 1 h 2 f z      (4.4)
with h f x , h f y and h f z the mesh sizes along the x, y and z direction, respectively. The mesh size is the same in the horizontal x -y plane. A distance ϵ is defined such that the mesh size becomes isotropic when |ϕ p | ≥ ϵ. The aimed mesh size is defined by a specific method in Cimlib. Three different parameters are used in this method which are namely h grad , the minimum mesh size in the z direction, h orth the minimum mesh size in the x -y plane and h iso the mesh size outside the distance ϵ. The mesh size of the M f matrix is then defined as follows:

for |ϕ p | ≤ ϵ      1 h f x = 1 h f y = 1 h orth + |ϕ p | ϵ 1 h iso -1 h orth 1 h f z = 1 -|ϕ p | ϵ 1 h grad + |ϕ p | ϵ 1 h iso for |ϕ p | > ϵ 1 h f x = 1 h f y = 1 h f y = 1 h iso . (4.5)
M f is only defined on a distance range ϕ p ∈ [-2∆H; +∆H] as illustrated in Figure 4.5. More details of the definition of the aiming values of the mesh size can be found in [START_REF] Zhang | Linear-Time Thermal Simulation of As-Manufactured FDM Components[END_REF][START_REF] Shakoor | Three-dimensional numerical modeling of ductile fracture mechanisms at the microscale[END_REF]. The creation of M f enables to create very fine mesh in the deposition layer and gradually increase its size in the layers just above and below the deposition layer. This allows to accurately capture the thermal behaviour in the deposition layer p -1 as well as in the air above layer p.

Another mesh metric M sub is defined to capture the thermal behaviour at the contact area between the substrate and the printing part. A level set function ϕ sub is defined to know the distance from the intersection between the part domain and the substrate domain. The zero value of ϕ sub comprises every nodes that are part of these 2 domains. The mesh size gradually increases as the distance from ϕ sub = 0 increases. The mesh metric is defined in the same manner as in (4.4) and (4.5). The mesh size is around 8 times the mesh size of M f . M sub is defined on a distance range ϕ sub ∈ [-3; 3] mm. Finally, a global isotropic background mesh metric M g is defined in the rest of the domain. An isotropic mesh size 370 times larger than the finest M f is defined in the rest of the substrate (ϕ sub > 3 mm). In the printing part another distance function ϕ g is defined as

ϕ g (0) = ϕ p (0) -3∆H. ( 4.6)
The meshing M g is defined with respect to the distance from the zero value of the level-set function ϕ g . The mesh size gradually increase as the distance on the vertical z axis increases from ϕ g = 0. The mesh metric M g is defined as

M g = diag ζ |ϕ g | + ζh crit 2 . ( 4

.7)

With h crit being the critical value of the mesh size when ϕ g = 0 and ζ an arbitrary value. Therefore, the mesh size increases as the distance from ϕ g = 0 increases. The zero value of the distance function ϕ g is located at ϕ p = 3∆H. This value of ϕ g (0) is taken to enable small refinement of the mesh size upstream to the building front. This enables to have a smoother transition of the meshing from the very fine mesh of M f to the relative coarser mesh of M g . This facilitates the re-meshing of the deposition layer at each increment. The intersection between the different metrics is managed by taking the finer mesh when 2 metrics are competing such that

M = max(M f , M sub , M g ). (4.8)
The different meshing zones are illustrated in Figure 4.5.

Heat transfer equation and boundary conditions

The temperature field in the object obeys the unsteady heat transfer equation given by

ρC p ∂T ∂t = - → ∇ • (k - → ∇T ) + Φ, (4.9)
in which T is the temperature, t the time, ρ the density, C p the specific heat capacity and k the thermal conductivity. The source term Φ will be detailed below. A perfect contact is assumed between the polymer strands and between the part and the substrate. This is a strong hypothesis as the presence of porosity is practically always seen in FFF. For numerical simplicity the porosity in the part is not taken into account in the model. However the thermal conductivity of the polymer is modified to take into account the volumetric porosity. This will be detailed in Subsection 4.3.1.

As previously explained, the deposition of polymer fractions is performed by successively changing the properties of the domain from air to ABS polymer. Each fraction is deposited at a uniform temperature equal to the extrusion temperature T ext . The substrate is heated at a given temperature equal to T sub . The air surrounding the building object has a constant temperature equal to T air . The surrounding air is not meshed in the domain but taken into account through heat transfer coefficients.

Two boundary conditions are defined to take into account the heat transfer with the surrounding air. Along the surface of the solid domain, denoted as boundary condition 1 in Figure 4.6, a Fourier boundary condition given by the following relation

-k - → ∇T • - → n = h surf (T -T air ) (4.10)
is used with -→ n the unit outward normal, h surf the heat transfer coefficient and T air the air temperature assumed constant.

The convective heat transfer between the top surface of the printing object and the surrounding air has also to be taken into account. This surface is represented as boundary condition 2 in Figure 4.6. Since this boundary is immersed in the computational domain, the boundary condition is seen as a heat loss Φ introduced in equation (4.9). This sink term is applied in the whole layer in printing written as follows

Φ = h top (T air -T ) ∆H , ( 4

Degree of healing

The welding quality between adjacent strands is studied using a healing model developed by Yang et al. [START_REF] Yang | Healing of Thermoplastic Polymers at an Interface under Nonisothermal Conditions[END_REF]. The model have already been detailed in Subsection 1.4.2. Recall that the degree of healing is given as follows

D h (t, T ) = 4 t 0 dt t w (T )
, (4.12) with t w the welding time which depends on temperature. For a given polymer, the welding time is described by an Arrhenius law as follows

t w (T ) = A arr exp E a R 1 T - 1 T ref , ( 4.13)
with A arr a constant, E a the activation energy, R the universal gas constant and T ref a reference temperature. Knowing the temperature history of an object printed with the FFF process, the degree of healing between two strands can be easily determined using equation (4.12). At each time increment, the welding time is computed based on the temperature field. The degree of healing is then computed based on a trapezoidal rule such that:

D h (t) = t 0 1 2 1 t w (t) + 1 t w (t -1) ∆t 1/4 , ( 4.14)
with t w (t) the welding time at increment t, t w (t -1) the welding time at the previous increment and ∆t the time step. This relation is used to compute the welding quality anywhere in the printed object.

Numerical computation and comparison with experimental results

In the following section, comparisons between experimental measurements and numerical simulations are achieved to assess the numerical model. The first case is studied to determine the numerical parameters minimising the errors with experimental results. The same set of parameters is then used on different cases to assess the validity of the model.

Numerical computations

The model simulates the printing of an amorphous ABS polymer. The shape of the printed object is rectangular. Numerical computations are achieved for three cases characterized by the printing parameters provided in Table 4.2. The strand dimensions depend on the velocities U and V . Results reported in [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF] have shown that the strand size is a predominant parameter of the cooling kinetics. Computations are thus conducted for a smaller and a larger strand size to validate the thermal parameters found in case 1. Special attention is taken to ensure mass conservation. The strand is assumed rectangular in the numerical computations. Therefore the mass balance between the flow in the extruder and the motion of the strand gives

∆H = πD 2 4∆W U V , (4.15)
with D the diameter at the exit of the extruder. Completed by optical microscopy measurements, ∆H and ∆W are also provided in Table 4.2.

The strand morphology is assumed rectangular for all numerical computation. However, Xu et al. [START_REF] Xu | Thermal analysis of the fused filament fabrication printing process: Experimental and numerical investigations[END_REF] have shown that the oblong geometry better describes the geometry of the strand. The volume loss when considering an oblong compared to a rectangle is assumed as porosity.

Considering that the volumetric porosity is the same as the surface void ratio, it is defined as follows:

V p = H 2 1 -π 4 W H (4.16)
Therefore the overall thermal conductivity of the printed part is modified to take into account porosity in the medium. According to Bejan et al. [START_REF]Heat transfer handbook[END_REF], the thermal conductivity of the medium can be estimated as follows:

k m = (1 -V p )k s + V p k f , (4.17)
with k m the thermal conductivity of the medium, k s the thermal conductivity of the solid, in our case the ABS and k f the thermal conductivity of air. The thermal conductivity of the medium as well as the volumetric porosity is provided in Table 4.2 for all cases. The thermal conductivity of the substrate is considered constant and its value is given in The successive simulated heating and cooling are in good agreement with experimental measurements. Nevertheless, the numerical results seem more sensitive to adjacent depositions of new hot filaments. Indeed, heating is numerically observed when deposition of the adjacent strand occurs. The interlayer reheating due to the deposition of a new layer is strongly reduced once it is covered by more than three layers. After an extended period of time, the steady-state cooling observed experimentally is numerically very well reproduced.

Comparison with experimental results

To observe a good agreement with experimental results in case 1, h surf and h top are set equal to 13 and 8 Wm -2 K -1 , respectively. These values can be compared with the data given by the heat transfer literature. For natural convection of external flows over a horizontal surface, Bejan et al. [START_REF]Heat transfer handbook[END_REF] provides an empirical correlation depending on the Rayleigh number based on the air properties. If the characteristic length is taken equal to the width of the wall and by taking the properties of air at T air =50 • C, the heat transfer coefficient is approximately equal to 34 Wm -2 K -1 . Bellehumeur et al. [START_REF] Bellehumeur | Modeling of bond formation between polymer filaments in the fused deposition modeling process[END_REF] studied the influence of the convective heat transfer coefficient on the strand temperature. For T air =70 • C, the heat transfer coefficient is taken equal to 30 Wm -2 K -1 to compute the temperature profile of a single deposited strand. To match with their experimental measurements, Costa et al. [2] set a convective heat transfer The strand size of case 2 is larger than case 1 and 3 explaining lower sensitivity. It is noteworthy that the modified thermal conductivity calculated with our method allows good prediction of the thermal behaviour. This shows that our method enables to capture the influence of porosity on cooling kinetics.

The time for which the polymer temperature reaches T g is one of the major criteria to characterize the interlayer adhesion. According to McIlroy et al. [START_REF] Mcilroy | Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing[END_REF], the inter-diffusion between strands occurs until the welding temperature is lower than T g . Below T g , the molecular motions are too restrained to enable diffusion at the interface. For all tested cases, the time to which the polymer reach T g is very well reproduced with numerical simulation. A cooling rate is calculated to compare numerical computations with measurements. The cooling rate is defined as follows

Ṫref = ∆T ∆t = |T g -T 0 | t Tg -t 0 (4.18)
with T 0 the interlayer temperature when the polymer is deposited at t 0 . The time for which the interlayer temperature reaches T g is defined by t Tg . The experimental and numerical cooling rates computed are summarised in table 4.3. Good agreement is found between the experimental and numerical results. This criterion does not give information about how the temperature behaves during the cooling. It however shows that the overall cooling rate is very well described by the model.

For the three cases, the maximum temperatures seen during the two first deposited layers match well with the thermocouple measurements. The overall numerical temperature behaviour matches relatively well with the thermocouple measurements. It is noteworthy that the heat transfer coefficients determined from the 1st case are also acceptable for the two other cases. The same set of thermal parameters can thus be used, irrespective of strand dimensions.

Degree of healing

As explained in subsection 4.2.5, the degree of healing is computed using eq. (4.12), given page 91. Ko et al. [START_REF] Ko | Improving the filament weld-strength of fused filament fabrication products through improved interdiffusion[END_REF] gave the welding time parameters of eq. ( 4.13) for a PC-ABS copolymer. The parameters found by Ko et al. [START_REF] Ko | Improving the filament weld-strength of fused filament fabrication products through improved interdiffusion[END_REF] are given in The glass transition temperature of the PC-ABS copolymer used by Ko et al. [START_REF] Ko | Improving the filament weld-strength of fused filament fabrication products through improved interdiffusion[END_REF] is equal to 107.9 • C which is very close to the T g of our polymer. Moreover, the printing temperature and thermal properties are similar to the ABS polymer of our study. Consequently, the parameters written in Table 4.4 are used to estimate the degree of healing.1

The degree of healing is computed using the temperature profiles plotted in Figures 4.8 For case 1, the degrees of healing determined from the numerical and experimental results are very similar. A sharp increase of the degree of healing is observed shortly after deposition while the increase is least after an extended period of time. This is a consequence of the power-law of the equation giving the degree of healing.

The degree of healing obtained from the numerical results is slightly overestimated for case 2. This result is mainly due to the re-heating produced by the adjacent deposition of strands observed in the numerical computation. Nevertheless, the disagreement on D h stays moderate since the steady-state cooling is very well reproduced numerically. The results show that good predictions of the maximum temperature is very important to predict the adhesion between the strands. Indeed most of adhesion occurs when maximum temperature is reached, i.e. when the layer is deposited and when it is covered by a new layer. The degree of adhesion can be thus calculated with our model as the maximum temperatures are well described.

Numerical printing of a complex object

Numerical simulation is done on an object with a more complex geometry. Temperatures, T ext , T sub and T air and heat transfer coefficients h surf and h top are the same as previous computations.

The printed material is also an ABS polymer with the thermal properties provided in Table 4.1.

The temperature field is determined by solving equation (4.9) completed by the boundary condition (4.10) and the source term (4.11). The degree of healing is computed using parameters given in Table 4.4.

The size of each fraction of polymer used to print the object is ∆L = 12.5, ∆W = 1.5 and ∆H = 0.67 mm. The length of the object is set to 50 mm. As seen in Figure 4.11, the object is designed with two holes. One is large with thin walls of 1.5 mm width while the other is small The influence of the object geometry on its cooling behaviour is studied. After the complete printing of the object, the study focuses on the first 10 printed millimetres. Figure 4.12 depicts the temperature field in this area. As seen in Figure 4.12a, the temperature difference is around 7 • C between the thin and thick walls. As expected, the thin walls are colder than the thick walls. This is also seen in Figure 4.12b in which temperature is lower next to the outer surfaces of the object. It is noteworthy that the thin wall has a higher temperature close to its top and bottom surfaces. The bottom surface is heated by the substrate at 94 • C explaining the relatively constant temperature of this surface. For thin walls, the top surface is the closest location to the latest deposited fraction.

The previously explained procedure is also conducted to investigate the influence of the part geometry on its degree of healing. As shown in Figure 4.13b, the degree of healing does not exceed 41% in the whole object. The bottom outer surface of the large hole has a lower temperature and yet a higher degree of healing than layers above it (see Figure 4.12b and Figure 4.13b). As this area is close to the heated substrate, the polymer takes more time to cool below T g . Therefore the degree of healing is higher in this area. This highlights that slow cooling is beneficial to the healing of the printing object. For the two profiles, D h decreases close to the boundaries while it is quite homogeneous in the bulk of the object. The spatial variation of D h is larger over the thick wall than over the thin wall. It can be explained by a more pronounced thermal gradient in the thick wall.

Figure 4.14 highlights a difference in D h between the thick and thin walls. The cooling kinetics of the thick wall is slower than the thin wall as its core is less subject to convective cooling with air. This explains the higher value of degree of healing. Nevertheless, the difference of D h between the two lines stays relatively moderate. As shown in Figure 4.10, the healing occurs mainly just after the polymer deposition. During this first stage of the process, the temperature reaches its maximal value. Moreover, the temperature is quite homogeneous in the area of deposition. Consequently, the first stages of the process have a critical impact on welding. Special care has to be taken to ensure that a sufficient adhesion quality is reached during the early stages of the process.

The computation of the degree of healing based on a simple model of the welding time is promising for the design of complex objects. The thermal computations coupled to the determination of the degree of healing becomes a useful tool to quantify the quality of the printing.

Development of the model for semi-crystalline polymers

A first thermal numerical model have been developed to predict the temperature behaviour of a part printed by the FFF technique. The temperature behaviour has been assessed for an amorphous polymer. New implementations are performed to enable the use of this model for semi-crystalline polymers. The crystallisation kinetics is calculated based on the temperature behaviour. The evolution of the temperature is also impacted by the latent heat of crystallisation.

It is therefore taken into account in the heat transfer equation. The implemented model is used to better understand the differences of degree of crystallinity encountered in the parts printed in Section 3.3.2.

Modeling the crystallisation kinetics

The crystallisation of semi-crystalline polymers is an exothermic process. The variation of temperature due to the heat released by crystallisation is taken into account by adding a term in the energy equation. The modification of the energy equation is usually described in industrial CHAPTER 4. THERMAL MODELLING OF THE PRINTING OF A WALL processes by the following equation [START_REF] Janeschitz-Kriegl | Polymer solidification by crystallization under heat transfer and flow conditions[END_REF][START_REF]Handbook of polymer crystallization[END_REF][START_REF] Billon | Modelling of the Cooling of Semi-crystalline Polymers during their Processing[END_REF]:

ρ C p ∂T ∂t + ∂H ∂t = - → ∇.(k - → ∇T ) (4.19)
with ρ the density, C p the specific heat capacity, k the thermal conductivity and H the enthalpy per mass unit. In the latest equation the reaction is considered as isochoric. The variation of enthalpy with time can be defined as

∂H ∂t = -∆H 100 ∂χ ∂t , ( 4.20)
with ∆H 100 the latent heat of crystallisation of a 100% crystalline polymer and χ the degree of crystallinity of the polymer over time. χ(t) can be defined as a function of the final degree of crystallinity χ f at the end of the process and the transformed volume fraction of crystalline entities ξ(t) at a given time t, by the following expression:

χ(t) = ξ(t)χ f . (4.21)
Moreover the final degree of crystallinity is defined as:

χ f = ∆H f ∆H 100 , (4.22)
with ∆H f the total heat of crystallisation of the polymer at the end of the process. Equation (4.19) can thus be written as the following equation:

ρC p ∂T ∂t = k - → ∇ 2 T + ρ∆H f ∂ξ ∂t . ( 4

.23)

The latest equation is practical to use as the final heat of crystallisation ∆H f can be easily determined by DSC at the end of the process. However the transformed volume fraction of crystalline entities ξ(t) has to be determined. It is usually described by Avrami theory such as [START_REF] Avrami | Kinetics of Phase Change. I General Theory[END_REF][START_REF] Avrami | Kinetics of Phase Change. II Transformation-Time Relations for Random Distribution of Nuclei[END_REF][START_REF] Avrami | Granulation, Phase Change, and Microstructure Kinetics of Phase Change. III[END_REF]:

ξ(t) = 1 -exp -ξ(t) , (4.24)
with ξ the "extended" volume fraction of crystalline entities. It is the sum of the volume of all crystalline entities divided by the total volume of the sample. It takes into account the formation of crystalline entities created in the already solidified regions. It also neglects the overlapping conflicts between spherulites upon growth by considering that all entities reach their final size.

For spheres growing at a radial growth rate G(t), the extended volume fraction of spherulites is given by [START_REF]Handbook of polymer crystallization[END_REF][START_REF] Janeschitz-Kriegl | Polymer solidification by crystallization under heat transfer and flow conditions[END_REF]:

ξ(t) = 4π 3 t 0 t τ G(u)du 3 dN e dτ dτ , (4.25)
with dNe dτ the "extended" nucleation rate [START_REF]Handbook of polymer crystallization[END_REF] and τ the time at which the corresponding entity appears. The extended nucleation rate considers that all potential nuclei can be activated and is therefore defined as

dN e dt = - dN a dt = -qN e dt, ( 4.26)
with N a the number of activated nuclei and q the activation rate. Integration of the previous equation leads to with ψ(T ) a function that depends only on the temperature and n the Avrami exponent. For a given polymer, the temperature function ψ(T ) varies depending on its nucleation and growth rate. The Avrami exponent gives insight on the type of nucleation of the polymer. Namely n depends on the morphology of the crystallites. In general, three shapes are commonly observed corresponding to spheres, parallel disks or parallel rods. Table 4.5 summarises the value of the Avrami exponent depending on the nucleation type [START_REF] Billon | Modelling of the Cooling of Semi-crystalline Polymers during their Processing[END_REF].

N e = N 0 exp - t 0 q(u)du , ( 4
For the case of constant cooling rate, Ozawa [START_REF] Ozawa | Kinetics of non-isothermal crystallization[END_REF] proposed a relation based on the Avrami model for the extended volume fraction of crystalline entities. It is defined as

ξ(t) = ψ(T ) | Ṫ | n , (4.29)
with Ṫ the cooling rate. This relation assumes a constant nucleation rate, as well as a constant nucleation type (n = Cst). It also assumes that the growth rate only depends on temperature and is only valid for primary crystallisation [START_REF] Chabert | Etude rar DSC de la cristallisation non isotherme de matrices PA66 en presence de fibres de verre[END_REF]. This relation has been modified by Billon et al. [START_REF] Billon | Modelling of the Cooling of Semi-crystalline Polymers during their Processing[END_REF] in the case of non constant cooling rate. Billon et al. [START_REF] Billon | Modelling of the Cooling of Semi-crystalline Polymers during their Processing[END_REF] discretized the temperature evolution in time steps of constant cooling rates where the Ozawa's expression is valid. The extended volume fraction of spherulites for non constant cooling can then be written [START_REF]Handbook of polymer crystallization[END_REF]

ξ(t) = t 0 dψ 1/n (T ) dT ds n . (4.30)
The latest expression is used to model the crystallisation kinetics of objects printed with a semi crystalline polymer. It will be detailed in the following sections.

Determining the crystallisation kinetics of a specific PP

The only step to model the crystallisation kinetics using Eqs (4.33) and (4.34) is to determine the temperature function ψ(T ) for a given polymer. This temperature function can be determined by performing DSC measurements at different crystallisation rates. The temperature function describes the way the polymer crystallises on an overall scale. It depends on the growth and nucleation rate of the polymer but no actual measurements of these two entities are needed. These entities can be difficult to measure experimentally especially for the nucleation density [START_REF] Santis | Nucleation density and growth rate of polypropylene measured by calorimetric experiments[END_REF]. Therefore using a more trivial protocol such as calorimetric experiments to describe the modified Ozawa's model is very convenient. Based on these measurements, the temperature function ψ(T ) and the Avrami exponent n can be determined as follows

ln -ln(1 -ξ(T , Ṫ ) = ln ψ(T ) -n ln(| Ṫ |). (4.31)
The Avrami exponent is estimated by plotting the variation of ξ as a function of the cooling rate Ṫ for several temperatures. The temperatures are selected so that a value of the transformed volume fraction can be determined for at least three different cooling rates. The slope of the linear curves gives the value of the Avrami exponent as displayed in Figure 4. [START_REF] Peng | Complex flow and temperature history during melt extrusion in material extrusion additive manufacturing[END_REF]. It shows that the PP Centaur has an exponent n ranging from 3 to 4 meaning that the crystalline entities are mainly spherulites.

For the rest of the modelling, the nucleation is considered as instantaneous and therefore the Avrami's exponent is taken equal to 3. For each cooling rate, the second term of the right hand side of equation ( 4 parameters values of the PPH 3250 is taken from [START_REF] Boyer | Crystallization of polypropylene at high cooling rates: Microscopic and calorimetric studies[END_REF] and the evolution of the temperature function is also displayed in Figure 4.17. The PPH 3250 is a commonly used grade in extrusion processes and can thus be considered as a model PP. DSC measurements of the PPH 3250 show a larger crystallisation temperature than the PP Centaur. For a cooling rate of 20 K/min, the crystallisation temperature is around 59°C and 111°C for the PP Centaur and PPH 3250, respectively. For the temperature range displayed in Figure 4.17, the temperature function of the PPH 3250 is higher than the PP Centaur. This shows that the transformed volume fraction of spherulites approaches 1 at a higher temperature for the PPH 3250 than for the PP Centaur. This is in accordance with the DSC measurements conducted on both polymers. 4.6: Fitted function parameters for the PP Centaur and PPH 3250. The fitting parameters of the PPH 3250 is taken from [START_REF] Boyer | Crystallization of polypropylene at high cooling rates: Microscopic and calorimetric studies[END_REF]. For both polymers less digits are displayed than the actual fit for clarity.

Implementation in the numerical model

The goal is to calculate the transformed fraction of spherulites ξ(t) at each time step. This way, the crystallisation kinetics of the polymer can be studied throughout the process. It also enables to consider when the exothermic reaction of crystallisation happens. This gives insight on how the whole object crystallises in a FFF process.

As shown in Equation (4.24), the extended volume fraction of spherulites ξ(t) must be calculated. The cooling of a deposited strand is non constant and ξ(t) is calculated based on Equation (4.30). The temperature behaviour is descretised by the simulation time steps. At each each time step, the temperature behaviour is considered as constant cooling or heating. Therefore Equation (4.30) can be written as follows for an isothermal time step:

ξiso (t) = - ∂ψ 1/n (T i ) ∂T (t k -t i ) n , (4.33)
and for an anisothermal step:

ξaniso (t) =   p j=k+1 1 | Ṫj | |ψ 1/n (T j ) -ψ 1/n (T j-1 )|   n , (4.34)
where the period (t k -t i ) is an isothermal step and the time steps from k to p are either constant cooling or heating steps. In practice a boolean variable is created to determine if the temperature behaviour is isothermal or not. If the temperature variations between any time step is lower than 1 • 10 -4 K/s, then it is considered as isothermal.

As previously explained, the temperature function ψ(T ) has been determined for the PP Centaur. The crystallisation kinetics of this polymer is calculated as a function of the temperature and thermal history in the particular case of FFF. At each time increment, the temperature function ψ(T ) is calculated based on the fitted parameters of the cubic function given in Table 4.6. The derivative of this function is also calculated in the case of isotherm steps. The computations are made based on the temperature field obtained numerically at the previous time step in the polymer domain. The corresponding extended volume fraction of spherulites ξ(T , Ṫ ) is calculated based on equation (4.33) and (4.34). At each time step i, the sum

ξ(t) = t i=0 ξi (4.35)
is calculated based on the previous time steps, in the whole polymer domain. This leads to the computation of ξ(t) based on equation (4.24) at each time step.

Numerically, a heat source term is added to the energy equation to take into account the heat released by crystallisation. A constant latent heat of crystallisation ∆H f is considered in the The evolution of the volume fraction of spherulites as well as the enthalpy of crystallisation is plotted for the case where crystallisation is taken into account.

whole polymer domain. This is a strong assumption as a difference of around 10% crystallinity can be observed in a part printed with this particular polymer. However it is impossible to determine the final degree of crystallinity at each location prior to deposition. Actual measurements is needed after printing of the part. The average degree of crystallinity of the part is thus considered uniform in the whole polymer domain. At each time step, the heat source Φ introduced in the energy equation is calculated as follows

Φ = ρ ∂H ∂t = ρ∆H f ξ(t) -ξ(t -∆t) ∆t . ( 4.36)
To study the evolution of the volume fraction of spherulites until ξ = 1, a modification of the boundary conditions is made. A cooling step is implemented once the final height of the object is reached. In practice once the printing of the part is done, the substrate stops heating and the part cools down until it reaches the ambient air temperature. Therefore, the substrate temperature is no more imposed once the printed part has reached its final height. The boundary conditions at the surface of the part described in Subsection 4.2.4 remain the same.

For a given set of parameters, a comparison of the temperature behaviour is made when considering or not crystallisation. The comparison is made on the simulation of the printing of a part with the dimensions: W = 6 mm, L = 100 mm, H = 10 mm. Each layer is composed of four adjacent strands. In both cases, a numerical sensor is placed at the same location to record the temperature and crystallisation behaviour as displayed in Figure 4.18a. Figure 4.18b shows the evolution of the value of the crystallisation enthalpy throughout the process and its impact on the temperature behaviour.

Figure 4.18a shows that an increase of the temperature is noticeable when crystallisation occurs.

The result shows that crystallisation occurs at around 76 °C. The local increase of the temperature during crystallisation does not affect the global cooling behaviour. Indeed, the exothermic reaction could have increase the overall temperature. The shape of the evolution of the enthalpy of crystallisation describes a Gaussian function when crystallisation occurs. The evolution of ξ is also well described as it has the expected shape. Both ξ and dH/dt shows a relative continuous evolution throughout the process. This shows that the calculation of the crystallisation kinetics is well described and in accordance with expectations.

Adaptive meshing as a function of ξ

For amorphous polymers, re-meshing is only performed in the deposition layer. It is the main position where there is the most thermal gradients. As we move further from the deposition layer, the temperature decreases more slowly and the mesh size can consequently be increased. When considering the latent heat of crystallisation upon cooling for a semi-crystalline polymer, this is not valid any more. Indeed crystallisation also occurs far from the deposition layer. Remeshing is thus needed to accurately capture the heat source due to crystallisation. A level-set function is created based on the value of ξ(t). The zero value of the level-set function is defined as

ϕ(ξ) = ξ(x, t) - max (ξ) 2 . (4.37)
Re-meshing is performed around ϕ ξ (0) as described in Subsec. 4.2.3. The mesh size increases as the distance from ϕ ξ (0) increases. For a given set of parameters, the meshing when considering or not crystallisation is displayed in Figure 4. 19. Figures 4.19a and 4.19b show that the meshing is focused in the deposition layer (the last deposited layer) and where a transition of ξ from 0 to 1 occurs. On the other hand Figure 4.19c shows that fine meshes are only focused on the deposition layer and that the mesh size gradually increases below the deposition layer. The temperature is higher in the area where crystallisation occurs when re-meshing is considered.

The adaptive meshing as a function of ξ is important as the crystallisation kinetics is better caught. A comparison of the evolution of ξ is made when re-meshing as a function of ξ is 4.4. DEVELOPMENT OF THE MODEL FOR SEMI-CRYSTALLINE POLYMERS performed or not. When no adaptive meshing of ξ is considered, the same mesh metric as for amorphous polymer is applied. Both cases are compared with the case of a fine mesh as the reference. For the case with adaptive meshing as a function of ξ, a mesh size h equal to 1 mm in the x direction and h equal to 0.5 mm in the y, z direction is applied at the transition front. For the fine mesh metric case, an isotropic fine mesh of size h equal to 0.5 mm is applied. The same printing and thermal conditions are applied in all the studied cases. A comparison of the three cases is done for a wall of 20 mm height in Figure 4.20. When ξ adaptive meshing is applied, the transition front is better captured than with a coarse mesh. However when comparing the ξ adaptive mesh with the fine mesh, the transition front seems to be slightly over estimated. This is especially true at the edge of the wall where the ξ transition is stiffer. The crystallisation and temperature behaviour are compared at specific locations in Figure 4.21. For the coarse mesh case, the mesh size at the bottom layer increases with the height of the wall. Two cases are thus compared with the printing of walls of final height H = 10 mm and H = 20 mm. The comparison of the temperature and crystallisation behaviour are made by introducing a numerical sensor at the same position for all cases. The temperature and crystallisation behaviour are relatively similar for the three cases when H equals 10 mm. At z = 5 mm, the coarse mesh better describes the evolution of ξ than when the ξ adaptive meshing is applied. This is especially true at the end of crystallisation. When re-meshing is performed, interpolations of the different solutions are made which can cause slight jumps in the value of ξ. This explains why the ξ adaptive meshing can differ from the fine mesh case. At z = 1 mm, the evolution of ξ is better described when re-meshing is taken into account. Indeed, the slope of ξ as well as the onset of crystallisation are closer to the fine mesh case than the coarse mesh case. This is especially true at the end of crystallisation. The temperature and crystallisation behaviour are also better described when adaptive meshing is taken into account for the case where H equals 20 mm. A more accurate computation of the temperature and crystallisation behaviour is thus observed for the case with adaptive meshing when increasing the distance from the deposition layer. The wall is printed at V = 1.2 m/min followed by a cooling stage of 500 s. This corresponds to an overall process time of 900 s. The computing time for the coarse mesh size, the ξ adaptive meshing and the fine mesh size are 1h39, 2h11 and 3h08, respectively. This shows that the computation time increases when adaptive meshing as a function of ξ is considered compared to the coarse mesh case. This is understandable since re-meshing and finer mesh at the transition front result in more computing time. However the use of adaptive meshing is very interesting compared to the fine background mesh metric as the computation time is reduced by around 1h. The use of coarser mesh in the areas where no transition of ξ occur allows to limit the computation time while keeping a good accuracy of the solution. This is especially true when the height of the wall increases. Indeed for a part printed at 20 mm, the 4.5. IMPACT OF THE PRINTING PARAMETERS ON THE CRYSTALLISATION KINETICS use of adaptive meshing over fine meshes allows to reduce the computation time by 2h12. The computation time when using fine mesh and adaptive meshing is 5h42 and 3h30, respectively. When printing walls of more than 10 mm height, the evolution of ξ is well enough described with regards to the reduction of the computation time. The use of adaptive meshing as a function of ξ is thus interesting. However for walls printed with a height lower than 10 mm, the use of ξ adaptive meshing is not worthwhile. Indeed, the size of the coarse mesh describes well the ξ transition while keeping a low computation time.

Impact of the latent heat of crystallisation

The assumption of considering a constant latent heat of crystallisation in the domain is assessed. DSC measurements at different locations of a wall printed with the PP Centaur is conducted.

The latent heat of crystallisation varies between 40 to 60 J/g. Two numerical simulations with the same printing conditions but different latent heat of crystallisations are done to see the impact of such variation. The printing of a wall with L = 100 mm, H = 10 mm and W = 3 mm with 6 strands in each layer is simulated. The printing velocities taken for both cases are U = 2 m/min and V = 1 m/min. The evolution of the crystallisation kinetics as well as the temperature behaviour are shown in Figure 4.22 at 3 different positions. The x, y, z coordinates refer to the scheme in Figure 4.4.

An increase of the final latent heat of crystallisation has an impact on the temperature behaviour upon crystallisation. The increase of the temperature is logically higher during crystallisation when increasing ∆H f . The local increase of the temperature upon crystallisation does not have a significant impact on the neighbouring strands. The exothermic reaction of crystallisation could have impacted the overall temperature behaviour. However it is not the case and therefore the beginning of crystallisation occurs approximately at the same time for both cases and for all locations. The crystallisation kinetics is relatively slower when increasing the latent heat of crystallisation. This is easily understandable as increasing the temperature results in a decrease of the temperature function ψ(T ) value. This has for effect to slow the increase of ξ and thus the crystallisation kinetics. However, the difference of crystallisation kinetics is relatively low between the two studied cases for all positions. The rest of the study will mainly focus on the impact of the different printing parameters on the crystallisation kinetics. Taking a constant latent heat of crystallisation is thus considered as a valid simplification. Indeed the change of the latent heat of crystallisation has minor impact on the evolution of ξ. It is also valid when studying the temperature behaviour before crystallisation.

Impact of the printing parameters on the crystallisation kinetics 4.5.1 Polymer properties

The impact of the printing parameters on the crystallisation kinetics of walls printed by FFF is investigated. The study is done by considering the PP centaur from FormFutura ® . Its thermal properties have been detailed in subsec. 3.2.1. Its density is given by FormFutura ® as equal to 0.9 g/cm 3 . Its temperature function ψ(T ) has been determined in the previous section through DSC analysis. This allows to calculate the transformed volume fraction of spherulites ξ(t) at each time step. Its thermal conductivity have been taken as the average thermal conductivity found for classical PP in [START_REF] Patti | Polypropylene -Polymerization and Characterization of Mechanical and Thermal Properties[END_REF]. It is taken equal to 0.165 Wm -1 K -1 . The dependence of the heat capacity with temperature C p (T ) is also determined by DSC measurements. The calibration is done with a sapphire plate. Three samples are tested and the average C p is calculated. The

Evolution of the crystallisation in a printed part

The impact of the height of a part on its crystallisation kinetics is studied for 2 cases. Both cases are done with the same printing conditions U = 4 m/min, V = 1 m/min. The wall is composed of four strands and its length and width is equal to L = 100 mm and W = 4 mm, respectively. The former is printed to a height H equal to 10 mm while the latter has a height equal to 20 mm. A comparison of the evolution of the front of crystallisation with time is shown in Figure 4.24. The comparison is made only upon cooling when the final height of the wall is reached. For both cases, crystallisation has only occurred at the bottom of the wall when the wall has finished printing. This is easily understandable as they are the oldest deposited layers. The transition front is higher at t = 0 for the wall printed at z = 20mm since more time has past when beginning cooling. When new layers are deposited the previously deposited layers are heated. The deposition of new layers maintains a high temperature value in the neighbouring layers. The neighbouring layers are thus cooling down relatively slowly. A certain distance from the deposition layer is needed to observe a sufficiently important cooling leading to crystallisation of the polymer. This is why the transition front of ξ is relatively horizontal in the center of the part. However, the transition is faster near the edge of the part. This is understandable as the edges are more subject to convective cooling. Indeed, the crystallisation kinetics increases with the cooling rate. This is even more observable from t = 250s in both cases. The corners of the wall are more subject to convective cooling as more surface areas are in contact with air. Therefore crystallisation first occurs at the corners of the part.

The crystallisation of a layer mainly depends on the height at which it has been deposited. At the end of the printing, the wall is only subject to the convective cooling and no more heating occurs. The same heat transfer coefficient is applied on the whole lateral boundary. However the whole lateral surface does not crystallise at the same time. The evolution of the ξ transition at the center cross section of the part is shown in Figure 4.25. The crystallisation front first moves vertically along the z axis. The front is quasi-horizontal with a slight effect of the lateral boundaries. The last crystallising layers are not located at the middle height of the part. They are located near the last deposited layer. However, the last deposited layers have more surfaces in contact with air and crystallises faster. The crystallisation of the wall is highly inhomogeneous and it mainly depends on the deposition path. The time at which the layer is deposited, its height and the distance to the last deposited layer plays a role on the crystallisation kinetics. The crystallisation front follows three stages. During the printing of the wall, the front follows the deposition layer at a certain distance. When the wall has finished printing, a new front appears at the top of the part. Finally the crystallisation ends at the core of the wall. The use of this model is therefore useful to predict the localisation where the part crystallises the last. This has an impact on the final degree of crystallinity, residual stresses and warpage.

Impact of the crystallisation kinetics on the degree of crystallinity

A comparison of the crystallisation kinetics in specific locations of a wall is made. The comparison is done once the strand is deposited until it fully crystallises. A comparison with the final degree of crystallinity of the part measured in Subsec. Small variations of the crystallisation kinetics are found depending on the location on the horizontal x axis. The crystallisation rate is compared between all cases in the steady regime when the slope of ξ(t) is approximately constant. The steady regime is considered on the range of ξ ∈ [0.2; 0.8]. The crystallisation rate is quite similar in all 3 locations as its value is approximately 1 • 10 -2 s -1 for all cases. However, the crystallisation onset starts first near the edge of the wall. This is in agreement with the results observed in Figure 4.24. The final degree of crystallinity should therefore slightly decrease near the edge of the part. Indeed, increasing the crystallisation rate decreases the time for the crystalline entities to grow which leads to a decrease of the final degree of crystallinity. Figure 4.26b shows that the crystallisation onset starts at the same time between y = -1.25 mm and y = -0.25 mm but the crystallisation rate is higher at y = -1.25 mm. This is understandable as the strand located at y = -1.25 mm is in direct contact with air and thus cools faster. The samples taken from the surface are around 2% less crystalline than the samples taken from the core of the wall. The difference of crystallisation rate is relatively low (≈ 2 • 10 -4 s -1 ) explaining moderate variations of the final degree of crystallinity. Moreover, a higher variation of the onset crystallisation temperature is seen between y = -1.25 mm and y = 1.25 mm. Both numerical sensors are placed in the strands located at the outer surfaces of the wall. The same crystallisation kinetics could be expected as they are both subject to the same cooling environment. The same deposition path is made in each layer which is represented in Figure 4.2. The first strand is deposited at y = -1.25 mm following the x axis and the last strand is deposited at y = 1.25 mm. In each layer, the strand at y = -1.25 mm is heated when the strand at y = -0.25 mm is deposited. As the stand at y = -1.25 mm is the last deposited strand of the layer, no heating occurs until a new layer is deposited. The surface at y = 1.25 thus has a higher cooling rate than the surface at y = -1.25 mm. This shows that the deposition path has a predominant impact on the cooling and crystallisation kinetics.

Figure 4.26c shows that the crystallisation kinetics is predominantly dependent on the location on the vertical z axis compared to the other directions. The crystallisation rate between z = 5 mm and z = 9mm is quite similar. The slope of ξ(t) in the steady regime is approximately 9.9 • 10 -3 s -1 and 11 • 10 -3 s -1 for z = 5 mm and z = 9 mm, respectively. The difference of crystallisation kinetics between z = 5 mm and z = 9 mm is thus essentially attributed to the time at which crystallisation starts. The crystallisation kinetics at z = 1 mm is however different. This layer is deposited near the bottom surface of the wall which is in contact with the substrate. Therefore, the cooling rate is first relatively fast. It then stabilises once the temperature is close to the substrate temperature (at around 80°C). The cooling rate is then relatively low and constant compared to the other cases. The onset of crystallisation is similar to the layer at z = 9 mm. However the substrate maintains the polymer to a temperature above the crystallisation temperature for a longer period of time. The heating due to the deposition of new layers also plays a role in decreasing the cooling rate of the layer. The crystallisation rate in the steady state regime is approximately 5 • 10 -3 s -1 , which is lower than the other locations. The difference of slope of ξ in the steady state regime is approximately 6 • 10 -3 s -1 between the bottom and the top layers. The layer at z = 1 mm is approximately 10% more crystalline than the layer at z = 9 mm and 5% more crystalline than the layer at z = 5 mm. Decreasing the slope of ξ(t) thus has a positive impact on the final degree of crystallinity. The crystallisation rate has a predominant impact on the degree of crystallinity compared to the onset time of crystallisation.

The numerical results show that the predicted crystallisation kinetics are in accordance with the final degree of crystallinity measured experimentally. The onset point of crystallisation and the slope of ξ(t) have an impact on the final degree of crystallinity. The model can be used to give a first approximation of the variations of χ in an object printed with the FFF process.

IMPACT OF THE PRINTING PARAMETERS ON THE CRYSTALLISATION KINETICS

Impact of the wall geometry on the crystallisation kinetics

The influence of the wall geometry is studied by varying the strands dimensions and the total width of the printed wall. Three cases are simulated and compared with the final degree of crystallinity measured by DSC in Subsec. 3.3.2. The extrusion temperature is taken equal to T ext = 250°C, the air temperature is equal to T air = 50°C and the substrate temperature Table 4.7: Dimensions of the wall and strands depending on the printing conditions. H, W and L represents the height and width and length of the wall, respectively. ∆H and ∆W represents the height and width of a single strand, respectively.

T sub = 70°C.
Figure 4.27a shows that increasing the width of the wall results in decreasing the crystallisation kinetics at its core. The slope of ξ is also stiffer as the width decreases. The crystallisation rate in the steady state regime is equal to 1 • 10 -2 , 8.6 • 10 -3 and 6.4 • 10 -3 s -1 for U2V1 4L, U2V1 6L and U4V1 4L, respectively. This trend is in accordance with the results of degree of crystallinity shown in Subsec 3.3.2. Indeed, the degree of crystallinity increases with the width of the wall. The average degree of crystallinity at the core of the wall for U2V1 4L, U2V1 6L and U4V1 4L is 23.5, 27.2 and 29.2%, respectively.

The influence of the strand dimension on the crystallisation kinetics is studied in Figure 4.27b. The comparison is done at the surface of the wall since it is less impacted by the total width of the wall. Both samples printed at U = 2 m/min, V = 1 m/min have a crystallisation rate of approximately 1 • 10 -2 s -1 while the sample printed at U = 4 m/min, V = 1 m/min has a crystallisation rate of approximately 7 • 10 -3 s -1 . The samples printed at U = 2 m/min, V = 1 m/min have approximately the same crystallisation rate as they have the same strand dimensions. However, the onset of crystallisation is different for both samples. An increase of W by 1 mm results in delaying the onset of crystallisation by approximately 80s. The sample with the largest width (U2V1 6L) is approximately 2% more crystalline. This highlights once again the importance of the onset of crystallisation on the final degree of crystallinity.

The onset of crystallisation is very similar for all samples in Figure 4.27c. The slope of ξ is however different as it is equale to 9.1 • 10 -3 , 5.6 • 10 -3 and 3.1 • 10 -3 s -1 for U2V1 4L, U2V1 6L and U4V1 4L, respectively. The crystallisation rate is nearly 3 times faster for U2V1 4L compared to U4V1 4L. This results in having a difference of crystallinity of nearly 15% between 6L. The sample U4V1 4L is maintained at a high temperature for a shorter period of time compared to the sample U2V1 6L. Two more strands are printed in each layer for U2V1 6L which explains why this sample starts cooling later. Each layer is heated for a longer period of time. This highlights that the same crystallisation time can be achieved with different printing conditions depending on the deposition path. The crystallisation rate in all cases range from 1 • 10 -2 to 1.3 • 10 -2 s -1 . The resulting degree of crystallinity is also very similar for all cases. However, the trends are different when comparing the simulated and experimental results. This highlights limitation of the model when the difference of crystallisation rate is lower than 3 • 10 -3 s -1 .

The numerical model showed its ability to accurately predict the global trends of degree of crystallinity. The numerical model can therefore be used as a comparative tool before actual 4.5. IMPACT OF THE PRINTING PARAMETERS ON THE CRYSTALLISATION KINETICS printing of FFF parts.

Impact of the printing temperature on the crystallisation kinetics

The impact of the substrate and extrusion temperature on the crystallisation kinetics is investigated in the present section. It should provide the necessary information to better tailor the printing temperatures. A method to control the slope of ξ is done in the case where a control of the ambient temperature is possible. The model simulates the printing of a wall with dimensions equal to W = 4 mm, L = 100 mm and H = 10 mm. The wall is printed at U = 4 m/min and V = 1 m/min. The only varying parameters are the extrusion and substrate temperature.

The evolution of ξ(t) at t = 660 s is compared in Figure 4.28 for three extrusion temperatures. The substrate temperature is set at T sub = 50°C. Increasing the extrusion temperature delays the overall crystallisation of the wall. This is understandable as the time to reach the crystallisation temperature decreases with the extrusion temperature. Moreover, the right side of the wall crystallises faster than the left side for extrusion temperatures below 250°C. This is explainable by the deposition path of each layer illustrated in Figure 4.28e and 4.28f. The two figures are the top view of the x, y horizontal plane. For a given position illustrated by the red and blue crosses, maximum heating happens when an adjacent strand is deposited at the same x position. It is referred as a "pass" in Figure 4.28e and 4.28f. When the polymer is deposited at the left edge of the wall, the heating cycles are longer. In the example given in Figure 4.28, the x position is heated twice after a 9 s cycle. At the right edge of the wall the heating cycles are shorter. Decreasing the time between two passes results in an increase of the cooling rate. This is observable in Figure 4.28d where the temperature and crystallisation behaviour along the x axis is compared for an extrusion temperature equal to 230°C. At x = 45 mm, the variations of temperature are more important leading to a lower temperature after each cycle. This effect has an impact on the overall cooling of the part throughout the process. The right edge of the part thus cools and crystallises faster. This effect is not observable when printing at an extrusion temperature equal to 250°C. Depending on the extrusion temperature and the deposition path, the crystallisation kinetics may not be symmetric with respect to the center of the part.

The impact of the extrusion temperature on the crystallisation kinetics is also studied at specific locations. Increasing the extrusion temperature results in decreasing the crystallisation kinetics in most locations of the wall. However at the center of the part, the crystallisation kinetics can be very similar between two different extrusion temperatures. Figure 4.29a shows the temperature and crystallisation behaviour at x = 0, y = -1.5 and z = 5 mm for different extrusion temperatures. Crystallisation starts at the same time when printing at T ext = 250°C and T ext = 230°C. The crystallisation rate is higher for T ext = 250°C. The slope of ξ(t) is 6.9 • 10 -3 and 7.7 • 10 -3 s -1 for T ext = 230°C and T ext = 250°C, respectively. This can be explained by the latent heat of crystallisation. Figure 4.29b shows that when printing at 230°C, the cooling rate decreases when crystallisation is considered. The exothermic reaction of crystallisation thus has an impact on the global cooling kinetics. This is not observed when printing at 250°C.

The impact of the substrate temperature T sub on the evolution of ξ(t) is studied in Figure 4.30.

The comparison is made at a time increment t equal to 660 s. The extrusion temperature is set at T ext = 250°C and the air temperature at T air = 50°C for all cases. As expected, the crystallisation kinetics increases when decreasing the substrate temperature. The crystallisation front is similar when setting the substrate temperature to 50 or 70°C since the substrate temperature is set below or around the crystallisation temperature. Crystallisation first occurs at the bottom layers which corresponds the oldest printed layers. However, when the substrate temperature is set above the crystallisation temperature (T sub = 90°C), a total absence of crystallisation is observed during the full printing of the wall. The air temperature is lower than the substrate temperature which causes the top layers to crystallise before the bottom layers. The use of the appropriate substrate temperature allows to configure the global crystallisation kinetics.

Tailoring the printing parameters to enable homogeneous crystallisation of the part is interesting to lower the residual stresses [START_REF] Agassant | Mise en forme des polymères: Approche thermomécanique de la plasturgie[END_REF]. Let's consider that the polymer behaves like a visco-elastic solid once it crystallises. If the crystallisation kinetics is too heterogeneous in the part, contact can occur between solidified and melted polymer. The melted polymer will be constraint by the solidified strands upon crystallisation. This may result in residual stresses which can induce warping of the part. The numerical model is used to estimate the best thermal conditions for homogeneous crystallisation. The goal is to have a crystallisation that occurs approximately at the same time in the whole part. The best conditions are obtained when printing the part at T ext = 250°C, T sub = 80°C and T air = 50°C. The substrate stops heating once the part reaches its final height. The surrounding air temperature is then changed to T air = 90°C and cools to T air = 25°C at an applied cooling rate of 10 K/min. The evolution of ξ(t) in the part upon cooling is displayed in Figure 4.31. The result shows that most of the part crystallises in around 2 min. Figure 4.31e shows the temperature behaviour at the actual time when the polymer is deposited on the numerical sensor. This way the global time of crystallisation in the whole part can be compared. The comparison is only done on the vertical z axis as the main differences of crystallisation kinetics are found in the height of the part. The bottom of the part does not crystallise before the end of the printing. The half time of crystallisation for all the sensors happens in a 51 s range. This shows that the numerical model can be used to limit the variations of crystallisation time in a printed object.

Conclusion

A thermal model have been developed to simulate the Fused Filament Fabrication process. This model is used to predict the temperature behaviour of an object printed with this technique. The polymer temperature in FFF is by nature of the process very variable. Indeed, it is always in transient mode during the different steps of the process. Knowing the temperature behaviour is primordial to obtain a part with the targeted properties. The adhesion quality and crystallisation kinetics are driven by the polymer temperature.

The numerical thermal model is developed for the case of an amorphous polymer. An ABS polymer is chosen to limit the possible thermal transitions occurring during the process. Thermal The different degree of crystallinity measured by DSC in Subsec. 3.3.2 is explained by the thermal model. Studying the time at which a strand start crystallisation as well as the crystallisation rate gives insight on the final degree of crystallinity. The dimensions of the object and the strand has an impact on the cooling kinetics and thus crystallisation. The model has highlighted the importance of the deposition path and the height at which the polymer is deposited on the cooling rate. The transient aspect of the process induces heterogeneous crystallisation of parts printed by FFF. The model is used to determine where the part crystallises faster and therefore where modifications of the process parameters are needed. The numerical model is used to limit the differences of crystallisation time in an object. It shows that the simulation can be applied on a concrete case to obtain a specific need. The use of 3D printers which better control the ambient air and substrate temperature can be linked with the numerical model. This will allow the use of high performance semi-crystalline polymers such as PEEK which is still restricted in many areas [START_REF] Geng | Effects of extrusion speed and printing speed on the 3D printing stability of extruded PEEK filament[END_REF]. The predictive aspect of the model should not be taken as immutable.

The numerical tool should be used before actually printing a specific part.

Introduction

As highlighted in the previous chapter, the cooling of parts made with FFF is accompanied with high temperature gradients. This may induce residual stresses and warping of the final part [START_REF] Armillotta | Warpage of FDM parts: Experimental tests and analytic model[END_REF]. One considerable interest of FFF is its ability to quickly produce objects with complex geometries. A major improvement of the technique is therefore to avoid deviation of the part geometry from the CAD model. Empirical calibration of dimensions by trial and error method is costly and time consuming [START_REF] Brenken | Development and validation of extrusion deposition additive manufacturing process simulations[END_REF]. The present chapter thus focuses on the investigation of part distortion in FFF. An experimental protocol is presented to preform in-situ measurements of the strain field during printing. The development of an FEM model allowing to give insights on the part distortion is also implemented. The study focuses on the use of a semi-crystalline polymer as they present the most limitation in terms of part distortion upon crystallisation [START_REF] Amado | Simulation of warpage induced by non-isothermal crystallization of co-polypropylene during the SLS process[END_REF]. PP is a widely used polymer as it has good properties in terms of stiffness, creep and chemical resistance at a relatively low cost [START_REF] Zhou | Effects of temperature and strain rate on the tensile behavior of unfilled and talc-filled polypropylene. Part I: Experiments[END_REF]. However, it is less developed in the field of AM technologies which is why it has been chosen in the present study. The adopted model has been first developed for the additive manufacturing powder bed fusion technique with metal. The Finite Element method has been implemented in the C++ library CimLib and is described by Chen [START_REF] Chen | Thermomechanical numerical modelling of additive manufacturing by selective laser melting of power bed: Application to ceramic materials[END_REF]. It is adapted for the specific case of the polymer FFF technique.

Modelling of mechanical distortion in a printed part

Constitutive equation

The properties of the polymer are highly temperature dependant. It behaves differently in its fluid or solid state. Therefore 2 different constitutive equations are applied depending on the temperature. A critical temperature T c is defined to specify if the polymer is in its solid or fluid state.

Solid state -Elasto-viscoplasticity

An elastic-viscoplastic constitutive equation is applied to the deposited polymer below the critical temperature T c . The strain rate tensor ε is decomposed into an elastic, a viscoplastic and a thermal part such that ε = εel + εvp + εth .

(5.1)

The thermal strain rate takes into account expansion and shrinkage due to the temperate changes. It is defined as

εth = α T ∂T ∂t I, (5.2)
with α T the linear thermal expansion coefficient and I the identity tensor. In the elastic domain, the stress tensor σ behaves linearly as a function of strain. The elastic stress is defined by the Hooke's law such that σ = C el : ε el , (

with C el the 4th order elasticity tensor. The elastic strain rate can thus be written

εel = C -1 el σ + ∂C -1 el ∂t σ.
(5.4)

We make the assumption that the material is homogeneous and isotropic. The elasticity tensor can thus be defined only as a function of the Young modulus E and the Poisson's ratio ν. We 123 CHAPTER 5. EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF PART DISTORTION consider E and ν constant with time. The elastic strain rate is then defined as

εel = 1 + ν E σ - ν E tr( σ)I. (5.5)
We consider a threshold stress σ 0 below which the material has an elastic behaviour. Consequently, σ 0 is defined as the initial yield stress. Above this threshold, the non reversible part of the strain rate is considered viscoplastic. The relation between the non-reversible strain rate and the stress deviator is defined in [START_REF] Rappaz | Modélisation numérique en science et génie des matériaux[END_REF] and modified to consider strain hardening as:

εvp = √ 3 2σ eq σ eq -[σ 0 + J (1 -exp(-φε eq ))] K √ 3 1 m
s.

(5.6)

with K the viscoplastic consistency, m the strain rate sensitivity coefficient, J is the saturation value of dynamic recovery, φ a constant associated to dynamic recovery, ε eq the von Mises equivalent strain and s the stress deviator. The term J (1 -exp(-φε eq )) accounts for strain hardening. The brackets in the latter equation translate the presence of the yield strength such that ⟨x⟩ = max(0, x). The von Mises equivalent strain rate εeq and stress σ eq are defined as:

εeq = 2 3 εvp : εvp (5.7)
and

σ eq = 3 2 s : s (5.8)
This leads to the corresponding von Mises scalar relation between stress and non reversible strain rate. The equivalent von Mises stress σ eq and strain rate εeq is defined by an additive law such that σ eq = σ 0 + J (1 -exp(-φε eq )) + K( √ 3) m+1 ( εeq ) m .

(5.9)

Inserting Equation (5.9) into (5.6) simplifies into εvp = 3 εeq 2σ eq s.

(5.10)

Fluid state -Viscoplasticity

A purely viscoplastic constitutive equation is defined when the temperature is above T c . The strain rate tensor ε is split into a viscoplastic and a thermal part such that ε = εvp + εth .

(5.11)

The thermal part of the strain rate is defined as in Equation (5.2). The viscoplastic part of the fluid strain rate is defined by the Norton-Hoff law:

εvp = 1 2K ( √ 3
εeq ) 1-m s.

(5.12)

Applying the equivalent von Mises stress and strain rate yields the well known power law:

σ eq = K √ 3 m+1 εm eq .
(5.13)

It can be noted that Newtonian fluids are a particular case of this viscoplastic law. Indeed this law defines a Newtonian fluid when taking m = 1 and K = η l with η l the dynamic viscosity of the Newtonian plateau.

MODELLING OF MECHANICAL DISTORTION IN A PRINTED PART

Finite element formulation

The solid mechanics is modelled by the conservation of momentum equation. The problem is considered as quasi static and inertia is neglected. The following equilibrium equation is given by: with P the hydrostatic pressure. From Equation (5.1), the trace of the strain rate tensor is expressed as tr

- - → ∇ • σ = - → F v , ( 5
( ε) = - → ∇ • - → u = tr( εel ) + tr( εth ), (5.16)
since tr( εvp ) = 0. From Equation (5.5) and (5.15), the trace of the elastic strain rate is defined as

tr( εel ) = - Ṗ 3(1 -2ν) E .
(5.17)

Combining Equation (5.14) and (5.15) with (5.16) gives the strong formulation to solve the problem of velocity and pressure. The equilibrium and mass conservation equations are defined as

- → ∇ • s - - → ∇P + F v = 0 - → ∇ • - → u -Ṗ 3(1-2ν) E -3α T Ṫ = 0 (5.18)
The transformation of these equation to the weak form and its computation by a Finite Element method is detailed in [START_REF] Chen | Thermomechanical numerical modelling of additive manufacturing by selective laser melting of power bed: Application to ceramic materials[END_REF]. The boundary conditions of Equation (5.18) is given as

- → u = 0 on ∂Ω sub , (5.19)
with ∂Ω sub the surface boundary of the substrate.

Implementation of the model

As described in Chapter 4, the numerical simulation of adding matter to build a desired object remains the same. The computation domain is divided into fractions to mimic the successive deposition of matter. The flow deposition is not simulated in this model. Fractions of polymer with predefined geometries are successively deposited. The mechanic resolution of the problem is computed in the whole domain. The constitutive equations described previously are considered in the "solid domain" while the Newtonian behaviour is considered in the gas domain. The mechanic problem is solved after numerical computation of the heat transfer equation. with -→ P (x,y,z) (t i ) designating the coordinates of a specific point at a given time increment. RAdaptation re-meshing is done after each time increment to modify the nodes coordinates. The displacement field is tracked throughout the process to study the zones where most distortion occurred. It is calculated after deposition of the matter such that

- → U i = i n=0 - → u n ∆t n , (5.22)
with -→ U i the displacement vector at increment i and -→ u n the velocity vector at each increment.

The strain field is also calculated at each time increment such that

ε i = i n=0 εn ∆t n = i n=0 ( εel n + εvp n + εth n )∆t n , (5.23)
with ε i the strain tensor at increment i. The coupling of the thermal and mechanical resolution is done successively. The mesh metrics given for re-meshing during the HAdaptation steps are detailed in Subsec. 4.2.3. The determination of the fraction dimensions depends on the printing velocities and the number of steps to deposit a single strand. The dimensions of the fractions along with the printing velocities determine the value of the time step. Increasing the number of steps allows more accurate solution of the thermal and mechanical behaviour. However it also increases the computation time. Once a new fraction is added to the solid domain, the properties of the polymer are applied on the corresponding elements of the part.

The temperature is only imposed on the newly deposited fraction. The heat transfer equation is solved based on the temperature field at the previous time step and the boundary conditions. The transformed volume fraction of spherulites is calculated based on the temperature field at the previous time step (Chap. 4). The latent heat of crystallisation calculated by the volume fraction of spherulites is accounted to solve the heat transfer equation. The constitutive law parameters are determined based on the temperature field. The thermal expansion is calculated based on the variation of temperature between the two last time steps. The solid mechanics is solved based on the temperature field of the present and previous increment, as well as the velocity, pressure, stress and equivalent plastric strain of the previous increment. The resolution of solid mechanics gives the velocity and pressure field as well as the stress and stain rate tensor. The velocity field is used to perform RAdaptation at each increment. Numerical computations of the thermal and mechanical problems are maintained after the complete deposition of the building object. This enables to calculate distortion during the cooling of the part. No HAdaptation is performed during the cooling step. RAdaptation is performed at each time step after the numerical computation of the mechanical problem. The different steps of the simulation are summarised in a flow chart in Figure 5.1.

Mechanical properties of the Polymer

Tensile specimens preparation

The chosen semi-crystalline polymer is the PP Centaur which has been described in previous chapters. The mechanical properties of the PP need to be characterised before implementation in the numerical model. The constitutive equation parameters for this particular polymer are determined by tensile and rheology tests. This polymer is received in form of 3D printing filaments. Specimens thus need to be created. The filament is cut into small pieces of few millimetres length and put into a mould to make traction specimens. The mould's dimensions are depicted in Figure 5.2. The mould is heated at 160°C and maintained at this temperature for 10min in a Carver hydraulic press. Pressure is then applied for 10 min on the mould. A wedge of around 3 mm is set to obtain specimens with approximately the same thickness. The Carver is then cooled with water until it reaches room temperature. The pressure is maintained during the whole cooling process. For rheology tests, the pellets are put in a cylindrical mould of 25mm diameter and 1mm height. The same heating and cooling procedure as for tensile specimens is applied in the Carver. A pressure of around 2t is applied on the mould.

Digital Image Correlation (DIC)

The strain field of the tensile tests are measured by digital image correlation. This technique consists in measuring the displacement at the surface of a specimen. The goal is to make a stereoscopic system allowing to numerically track the deformation of the surface by triangulation [START_REF] Aris-Brosou | Soudage de polymères semi-cristallins utilisés dans l'isolation de pipeline offshore. Approches thermiques, rhéologiques et mécaniques[END_REF]. We consider an object which undergoes mechanical solicitations inducing a material transformation Ψ m . It is assumed that the image of this transformation is similar to the real transformation. The image of the initial state of the object is a greyscale function f (X) with X = (x, y) the Cartesian coordinates of the surface. Due to mechanical solicitations, the transformations of the specimen also induces modifications of the images. The transformed image is also a greyscale function that can be defined as:

g(Ψ m (X)) = f (X). (5.24)
The transformation Ψ m of the object with time is followed through a series of images. The evolution of the image greyscale function can be approximated by a polynomial function Ψ p . This polynomial function describes a transformation that accounts for several motions such as translations and rotations. The correlation function C corr is defined as follows

C corr (Ψ p ) = [f (X) -g(Ψ p )] 2 dX.
(5.25)

The determination of the transformation Ψ m is obtained by minimizing C corr (Ψ p ). The correlation function is integrated in the whole domain to get the displacement field. A complete description of the technique can be found in [START_REF] Schreier | Image Correlation for Shape, Motion and Deformation Measurements[END_REF][START_REF] Garcia | Mesure de formes et de champs de déplacements tridimensionnels par stéréocorrélation d'images[END_REF].

The software used to correlate the image is VIC ® 3D snap 8. Two cameras are placed on a same plane in front of the studied surface. A pair of images is taken at each time increment to track the displacement field. The cameras used are 4 Megapixels' Pike from Allied Vision with 25 mm Schneider Kreuzwach lenses. The cameras are connected along with the testing machine to the acquisition computer. A patterned plate is used to calibrate the stereoscopic system. The pattern plate is composed of plane black dots and three empty black dots, as displayed in Figure 5.3. The positions of the empty black dots are known by the software. Images of the patterned plate are captured while it is translated and rotated in the measurement plane. This allows to calibrate the distances and to offset possible errors in the camera alignment.

The surface needs to be speckled in a random manner to follow the deformation of the surface. Indeed if a repetitive pattern is applied, the software is not able to distinguish in which zone the surface is transformed. The correlation of the transformation function would then not be possible. The speckle is performed by painting random black droplets onto a white painted surface. The use of black and white allows better contrast for the determination of the greyscale function. The quality of the speckle is important to obtain a good correlation. It needs to be done few minutes before the test to avoid peeling the paint off during the test. The speckle also needs to be quite fine such as depicted in Figure 5.4a. The initial image is divided into sub domains called "subset". The subsets are the domains where the correlation function is integrated. The size of the subset is defined by the user. The displacement vector is calculated from the center of the subset. The correlation is made by taking points at regular distances on the whole domain. The distance between 2 points are called steps. Decreasing the step thus induces a higher density of calculation points. The density of these points has an impact on the calculated deformation. It has been shown that decreasing the step results in increasing the final value of deformation [START_REF] Aris-Brosou | Soudage de polymères semi-cristallins utilisés dans l'isolation de pipeline offshore. Approches thermiques, rhéologiques et mécaniques[END_REF]. A third domain called "filter size" is used in the software. The filter size determines the number of correlation points taken into account to calculate the deformation. The different software parameters are displayed in Figure 5.4b.

The Hencky strain tensor, also known as true strain is calculated in VIC ® 3D. Let's consider the initial position of a material point x 0 . In Lagrangian description the displacement -→ U is defined

MECHANICAL PROPERTIES OF THE POLYMER

The DIC measurements give values of the vertical and transverse strain (ε yy and ε xx on the surface of the sample). The deformation of the cross section of the sample is not directly measured. The sample is considered incompressible, and the strain in the thickness of the sample is defined as

-ε yy = ε xx + ε zz .
The stress is thus calculated as follows:

σ yy = F w 0 e 0 exp -ε Log yy , ( 5.34)
with F the force. An example of the resulting stress-strain curve at 50°C is displayed in Figure 5.5. No real trend is found when testing at a strain rate of 10 -4 or 10 -3 s -1 . The difference of behaviour between these two strain rates falls into the dispersion of the measurement. This may arise from discrepancies in the sample preparation. All samples are not cooled exactly at the same cooling rate as it is not controlled. Moreover variation of the applied pressure may occur during sample preparation. Different microstructures may arise and influence the mechanical response. The same trend is observed for all temperatures. This means that the strain rate does not have a significant impact on the mechanical response for ε < 10 -3 s -1 . Therefore only the tests conducted at 10 -3 s -1 are considered. An average of the stress-strain curves is done to take into account the dispersion of the results for each temperature. The elastic domain is considered by fitting a linear function at small strain values. This allows to determine the Young modulus E and the initial yield stress σ 0 . A python script is created to calculate the theoretical stress-strain curve in a 1D traction test.

The parameters in Equation (5.9) are optimised for each temperature to fit with the corresponding traction test. For all tested temperatures, the average stress-strain curve is plotted with the corresponding fit. The results are displayed in Figure 5.6. The optimised parameters are summarised in Table 5.1 for all temperatures. DISTORTION Table 5.1: Constitutive equation parameters described by Equation (5.9).

Fluid domain

The constitutive law parameters in the fluid domain are determined by rheology tests. A MCR-302 rheometer from Anton Paar operating in oscillating mode is used. The rheometer applies a shear stress on the sample and measures the resulting angular displacement. The measurements are performed using a 25PP/TG measurement plate from Anton Paar. A 1 mm gap between the 2 plane plates is imposed for the measurements. A frequency scan is applied from 0.1 to 100 rad/s at a constant strain of 1%. The measurements are conducted at different temperatures to see the dependence of the viscosity with temperature. The bottom plate is equipped with a thermocouple. The sample temperature is thus assimilated to the measured plate temperature. The measurements are carried out under nitrogen to avoid sample oxidation. A minimum of two measurements is done for each temperature to assess the reproducibility. The rheometer measures the magnitude of the complex viscosity | η * | as a function of the angular frequency. According to Cox-Merz assumption [START_REF] Cox | Correlation of dynamic and steady flow viscosities[END_REF], these two entities are equivalent to the steady state shear viscosity η and shear rate γ, respectively. As explained in Section 5.2.1, the shear thinning behaviour of the fluid is described by a power law. The dependence of viscosity with shear rate is thus described as η = K γm-

Thermal expansion coefficient

The linear thermal expansion coefficient needs to be determined for the PP Centaur at different temperature. The tensile test machine has a controlled temperature environment. It is therefore possible to measure the deformation of the sample during heating at a constant rate. The DIC set up is similar to what is used for tensile tests. The sample is maintained by only one pneumatic clamp to avoid compression during thermal dilatation. The sample is heated from 25 to 110°C at a constant heating rate of 1°C/min. The temperature cannot be increased above 110°C as the sample starts melting and the measurement is impacted by gravity. The DIC cameras are set to take an image every 10 seconds. The strain evolution with temperature is measured and DISTORTION the thermal expansion coefficient is calculated at each temperature step such that α T = ∆ε ∆T .

(5.37)

Two measurements of the strain evolution are done as displayed in Figure 5.8a. The dependence of the thermal expansion coefficient with temperature is then averaged between these two tests.

The results are displayed in Figure 5.8b. The thermal expansion coefficient is assumed constant above 110°C in the numerical model. Figure 5.8a shows that the evolution of the strain is nearly linear above 100°C for both tests.

Experimental investigation of part distortion

An experimental protocol is set-up to study the part distortion during printing. In-situ measurements are performed during the printing of the part to grasp knowledge on the distortion and strain rate occurring in FFF. The measurements are also used to assess the numerical model.

Experimental set-up

The evolution of the part distortion throughout the process is investigated by DIC. The cameras used are the same as for the traction tests. Two cameras are placed in front of the 3D printer, in the same horizontal and vertical plane. A calibration of the DIC system is performed with a pattern plate. A wall is printed at U = 4 m/min and V = 1 m/min. The wall is composed of four strands in each layer. Dimensions of the wall are H = 10 mm, W = 4 mm and L = 100 mm. The random speckle needed for DIC is performed during a printing stop of 20 s. The image correlation is therefore only performed in the area where the speckle is done. No measurements of the strand strain are possible for new deposited strands. The influence of new layers depositing upon the speckled area can however be investigated. The speckle is performed up to different heights to study the behaviour of the part distortion at different times of the printing. The printing continues once the speckle is done. Three different tests are done after a printing stop at z = 5 mm, z = 7.3 mm and z = 10 mm. The latter is the final height of the wall. The three mentioned speckle areas are displayed in Figure 5.9. In each case, the measurements are

Global strain field

The global strain field obtained by DIC is displayed in Figure 5.11 for the 3 different tests. The study is performed on the (x, y) plane at the surface of the wall. The edges thus refer to the left and right side of the surface. For all experiments, the global strain field is negative in both horizontal and vertical directions, meaning that only contraction happens in the wall. This is understandable as the printed object is mainly subject to cooling. The warping of the part does not induce dilatation anywhere. It is seen that the ε Log xx field shows similar trends for all experiments. At the bottom centre, the contraction is lower than at the top of the sample. The deformation gradually increases towards the upper edges. This result must be weighted by the CHAPTER 5. EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF PART DISTORTION time at which the measurement starts. Indeed the bottom layers are first deposited. Therefore it has more time to cool before the painting of the speckle, which corresponds to the beginning of the measurement. The bottom layers are thus in a more contracted state at the beginning of the measurement. The strain field is not homogeneous in a single layer following the x direction. A certain symmetry is observed along the x axis, with lower contraction at the center of the wall and higher contraction on the edges. For the test done with a speckle up to y = 10 mm, the contraction is more predominant on the right edge of the part. This phenomenon can be explained by the cooling kinetics shown in Subsec. 4.5.5. Depending on the deposition path, the right edge (positive x) can have a higher cooling kinetics than the left edge of the wall. This can result in higher contraction which could explain the result shown in Figure 5.11e. The maximum contraction observed in all tests in the x direction is around 4%. The strain field can be linked with the cooling kinetics studied in Sec. 4.5. The cooling kinetics is higher at the last deposited layers than at the bottom layers. Consequently, the contraction is less pronounced at the bottom than at the top of the part. Figure 5.9 and 5.11 show that the bottom half layers have a relatively rectilinear shape. Higher contraction at the top layers results in warping. This warping effect is also due to the strain in the y direction. Similar trends of ε Log yy are observed for experiments done with a speckle up to y = 5 mm and y = 7.3 mm. For the same reasons of cooling kinetics, the last deposited strands show higher contraction than the first layers. This also leads to bending the wall towards the negative y direction.

In the case where the speckle is done up to y = 5 mm and y = 7.3 mm, the maximum strain in the y direction is found at the upper edge of the speckle area. The maximum absolute strain value for these tests is approximately 7%. In the case where speckle is done up to y = 10 mm, the maximum absolute strain value is approximately 4%. The difference between Figure 5.11f and the two other tests is the time at which the measurement starts. For the experiment done with a speckle up to 10 mm, the strain field is only measured during the cooling of the wall, when the total wall has been printed. Comparing with the two other tests, most shrinkage in the y direction occurs before the end of the printing. Figure 5.11f shows that the maximum contraction in the y direction is found at the bottom edge of the wall. This is an unexpected result as it 5.4. EXPERIMENTAL INVESTIGATION OF PART DISTORTION has lower cooling rates which should result in lower contraction. Around 4% absolute strain value ε Log yy is found at the bottom of the wall for all the tests. At first glance, most shrinkage at the bottom of the wall happens after its total deposition. Indeed, the substrate is no longer heated when the wall is totally printed. The layers near the substrate have a temperature that stabilises near the substrate temperature during printing. A second cooling step thus occurs when the substrate is no more heated. This could explain why in Figure 5.11f the maximum contraction is found at the bottom of the wall. This will be further detailed in § 5.4.3.2.

An unexpected result is found when studying the strain field in x direction. The maximum contraction is found in an area at approximately 6 mm from the left edge of the wall and 10 mm from the right edge of the wall. The contraction then decreases on the edge of the wall. The maximum contraction is expected in areas of high cooling rates. It should therefore be close to the edge of the wall. The evolution of ε Log xx with time is depicted in Figure 5.12. The maximum contraction is first localised on the edges of the wall. As the wall cools down, the edges more rapidly reach a steady state while the area at approximately 10 mm from the edge continues to contract. This can be explained by the deposition path. When the printing head reaches the end of the length, it performs a 90°turn in the z direction to begin printing the adjacent strand. Since this motion is fast and the distance is less than 1 mm, polymer is still deposited. Consequently, small strands in z axis are linking adjacent strands on the edge of the wall. These small strands do not contract the same way as the strands deposited in the x direction. This has an impact on the shrinkage of the wall in the x direction. The deposition path has an important impact on the strain field of the printed object.

Strain in specific locations

Evolution on a single test

The DIC technique allows to study the evolution of the strain with time in specific locations. Numerical sensors are placed at specific points on the wall. The displacement of the sensor due to distortion phenomenon is followed by image correlation. The behaviour of the strain from an initial x 0 , y 0 position is thus tracked throughout the process. The global strain field gives value of the distortion without taking into account the time at which the strand is deposited. Three tests are performed for each condition. The average evolution of the strain for each position is then calculated.

The study focuses on the case where the speckle is done up to y = 10 mm. In Figure 5.13, ϵ Log xx is plotted as a function of time for four locations along the x axis. The comparison is made on the layers y = 5 mm and y = 8 mm. In both layers, the evolution of ε Log xx is nearly identical at x = 5 CHAPTER 5. EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF PART DISTORTION mm and x = 95 mm. A certain symmetry is thus observed at the edge of the wall. However, the center of symmetry is not completely localised at x = 50 mm as contraction is also observed at this location. Moreover, the strain increases in absolute value from the middle to the edge of the wall. As shown in Sec. 4.5, the cooling rate is slightly higher near the edge than at the center of the wall which results in a higher strain rate at x = 5 mm and x = 95 mm. The strain rate is obtained by the derivative of the strain with respect to time. At the position y = 8 mm, it is around -6 • 10 -5 s -1 close to the middle of the wall while it reaches -10 -4 s -1 close to the edges. At y =5 mm, the strain rate is around -8.5 • 10 -5 s -1 in the middle and -4.5 • 10 -5 s -1 close to the edges of the wall.

Comparison on multiple tests

Performing the speckles at different heights enables to compare the strain fields of the material along the y axis. Indeed in a single test, the layers at the bottom of the wall are older at the start of the measurement. A comparison along the y axis is thus impossible as the strand life time depends on its height. Therefore, three tests are done to compare the strain field at equivalent time after deposition. The layers chosen for comparison are positioned at y = 2.5, y = 5 and y = 7.5 mm for speckles performed up to y = 5, y = 7.3 and y = 10 mm, respectively. In this manner, the time between deposition and measurement is around 120 s for the three cases.

As depicted in Figure 5.14a, the maximum absolute value of ε Log xx increases with the deposition height. It shows that increasing the deposition height results in increasing the absolute strain rate. Indeed the strain rate is around -10 -4 , -4.5 • 10 -5 and -2.7 • 10 -5 s -1 for the cases with a speckle up to 7.5, 5 and 2.5 mm, respectively. As explained previously this is due to the cooling kinetics. It is shown in Sec. 4.5 that the cooling kinetics increases with the deposition height. The strain rate thus increases consequently. Moreover, it is noticeable that for both x and y directions, the time to reach a steady-state regime is not homogeneous in the wall. The steady state is reached later at the bottom of the wall. It corresponds to the position with the lowest cooling kinetics. Decreasing the cooling rate thus allows the polymer to deform for a longer period of time.

The comparison of the behaviour of ε Log yy with time is shown in Figure 5.14b for different heights. The absolute value of the final strain decreases when increasing the wall height. This result is unexpected since the layers localised at the top of the wall undergo higher cooling rates. The absolute value of the strain final state is thus expected to increase with the deposition height.

Studying the strain rate shows that the upper layers have indeed a higher absolute strain rate than the bottom layers at the beginning of the measurement. The maximum absolute strain rate at y = 7.5 mm is 3.4 • 10 -4 s -1 whereas the maximum absolute strain rate for y = 5 mm and y = 2.5 mm is 1.2 • 10 -4 s -1 . However, at y = 7.5 mm, the strain reaches the steady state more rapidly, after around 60 s. For y = 2.5 mm and y = 5 mm, the steady state regime is reached at around 600 s. Increasing the cooling rate leads the top layers to crystallise more rapidly. Once the polymer temperature is below the crystallisation temperature, the strain due to thermal shrinkage decreases. This is what is highlighted in Figure 5.14 at x = 30 mm and y = 7.5 mm. The strand crystallises after around 60 s and the strain reaches a steady state. For the layers below, the cooling kinetics is lower. More time is thus given for the layers to shrink. At the final state, the resulting strain is therefore more contracted at the bottom of the wall. This explains the global strain field in the y direction found in Figure 5.11.

To accredit this claim, a numerical simulation of the thermal problem is done at the same printing conditions. The behaviour of ε Log yy at x = 30 and y = 8 mm is compared to the numerical solutions of the temperature and crystallisation rate as depicted in Figure 5.15. The computed crystallisation starts 40 s after the beginning of the steady-state regime observed in ϵ Log xx . This corresponds to the same time scale. The dynamics of the strain decreases when the temperature is close to 83 • C. This temperature is in the range of temperatures where the polymer solidifies. Indeed, the rheology measurements underline an increase of the elastic modulus in the same range of temperatures.

To conclude, increasing the cooling rate results in a higher strain rate but the strain reaches a steady state more rapidly. Decreasing the cooling rate allows shrinkage on a longer period of time. This leads to higher absolute values of the strain final state in the y direction. The strain state is higher in the y direction than in the x direction. The maximum distortion is around 2% in the x direction while it reaches 4% in the y axis. The strain rate thus has a more important effect on the final strain state in the x direction. Moreover, the difference of strain state can also be due to the cooling kinetics. It has been shown in Sec. 4.5 that the major differences of cooling kinetics are found in the vertical direction of the wall. Higher temperature gradient results in higher strain state. Indeed, when an object cools in a non symmetrical way, the layers DISTORTION that cool first will constraint the layers that cool later [START_REF] Agassant | Mise en forme des polymères: Approche thermomécanique de la plasturgie[END_REF]. Residual stresses can arise and lead to warping.

Model assessment

Global displacement field

The experimental protocol described in the previous section is used to measure the displacement field -→ U of a printed wall. The printing of a wall with exactly the same process parameters is also numerically solved using the model described in §5.2. The parameters used for this numerical simulation are summarised in Table 5.3. In both cases, the polymer is the PP Centaur. The experimental displacement field is measured by DIC technique after performing a random speckle on the surface of the wall. The simulated displacement is extracted at different times. This is done to compare the experimental and numerical results on an equivalent range of time. It is calculated as described in Equation (5.22), with n the time increment corresponding to the time when the speckle is done. Moreover, the experimental displacement is only measured in the speckled area. The simulated displacement field is therefore only shown in the area where experimental correlation is performed. The global displacement field at the end of the cooling for the three experimental tests is compared. As seen in Figure 5.12, a systematic detachment of the wall from the substrate occurs at the right edge of the part. To mimic this phenomenon, a gap of air is replaced upon cooling at the interface between the wall and the substrate. The length of the gap is measured experimentally with ImageJ and the corresponding length is replaced in the numerical simulation. The gap is introduced along the x axis for x ∈ [35; 50] mm.

The global displacement field in the x direction -→ U x is compared in Figure 5.16. The simulated global evolution matches well with the experimental measurements. In both cases the maximum displacement value is found in the top corners of the wall. The displacement value then decreases from the edge to the center of the wall. The global shrinkage is thus well described in the numerical simulation. Nevertheless, disagreements can be underlined. In all the cases studied by the numerical method, the displacement strongly decreases between edges and the middle of the wall. Conversely, displacements observed experimentally decreases in the middle of the wall but more gradually. This phenomenon will be further studied in the following section. The value of the maximum displacement is compared between the experimental measurements and simulations. The comparison is done for all tests and the relative deviation is calculated in Table 5.4. The maximum deviation is found for the case with the speckle done up to 5 mm. The displacement on the left edge of the wall is overestimated by 21%. Apart from this test, all maximum displacements match well with the measurements. The maximum displacement is localised at the right area and its value is acceptable compared to experimental measurements. The model thus enables to accurately predict the maximum displacement in the x direction.

A comparison of the global displacement field in the y direction U y is shown in Figure 5.17. For the case where the speckle is done up to y = 7.3 mm and y = 10 mm, the global evolution of the simulated and experimental displacement have similar trends. The maximum displacement in the negative y -direction is located on the last deposited strands. The maximum displacement in the positive y + direction is located at the bottom right corner of the wall. The resulting warping of the wall is observed experimentally and numerically. However, the experimental displacement field shows that the maximum displacement in the y -direction is more localised at the center of the wall. It then gradually decreases towards the edge of the wall. In the simulation, the displacement is more homogeneous in each layer. Approximately the same value of y displacement is found in each layer from x = 5 to x = 95 mm.

For all tests, the experimental values of displacement at the bottom layer is always moving towards the positive y + direction. Indeed, the minimum -→ U y value at the bottom layer is 5 • 10 -3 , 6.6 • 10 -2 and 0.22 mm for speckles done up to z = 10, z = 7.3 and z = 5 mm, respectively. This phenomenon is not observed in the simulation. When the wall shrinks, the top and bottom surfaces tend to distort towards the center of the piece. In the experimental procedure, adhesion forces between the first layer and the substrate are present leading to lower displacement on the bottom layers. Detachment from the substrate can occur, explaining why only a positive y + DISTORTION (a) Experimental -→ U x for a speckle up to y = 10mm.

(b) Simulated -→ U x calculated after deposition at y = 10mm.

(c) Experimental -→ U x for a speckle up to y = 7.3mm.

(d) Simulated -→ U x calculated after deposition at y = 7.3mm.

(e) Experimental -→ U x for a speckle up to y = 5mm.

(f) Simulated -→ U x calculated after deposition at y = 5mm. displacement is seen in the case where the speckle is done up to 5 mm. In the simulation, perfect contact is considered between the part and the substrate. It is therefore impossible for the bottom layers to move towards the y + direction. The comparison of the maximum displacement value is summarised in Table 5.5 for all tests. The deviation of maximum displacement in the y + direction is approximately 70% in all cases. The simulated displacement in the y -direction is overestimated compared to the experiment.

Comparison of the displacement behaviour

The displacement behaviour is studied as a function of time in specific locations. Numerical sensors are placed at specific locations in the simulation. These sensors are moved during RAdaptation to follow the initial point throughout the process. The same protocol is done for the experimental measurements. The VIC software is used to follow the evolution of displacement at specific locations. The comparison is done in the case where the speckle is performed up to 10 mm. The wall is thus only subject to cooling. The displacement as a function of the height of the wall is studied. For the displacement in the x direction, the comparison is done at x = 95 mm for different heights. For the displacement in the y direction, the comparison is done at x = 50 mm and is gathered in Figure 5.18. Illustration of the sensors position is also displayed. For both cases the global trends are reproduced by the numerical simulation. The absolute value of displacement at the final state increases with the height of the wall. More displacement DISTORTION can be captured at the top layers as the measurement starts just after the deposition. It can however be seen that the final value of displacement is not well predicted by the simulation. For the displacement in the x direction, the deviation from the measurement ranges from 57 to 69% for the sensors located at y = 7.75 to y = 2.25 mm, respectively. All the predictions are underestimated in the simulation. This trend is similar to what have been seen in the simulated global displacement field. The absolute value of displacement decreases rapidly from the edges to the center of the wall. The displacements in the y direction are overestimated for all numerical simulations. Disagreements of 45 and 70 % are found between the experiment and the simulation for y = 5 mm and y = 7.75 mm, respectively. The overestimation is nearly 6 times the experimental displacement at the bottom layer.

The behaviour of the displacement with time is also compared. Both experimentally and numerically, -→ U x in x = 95 mm is negative meaning that the wall is in contraction. The magnitude of displacement roughly behaves similarly over the first 200 seconds. Nevertheless, changes of the behaviour arise for larger times. Experimentally, the contraction enhances sharply after 200 s. The contraction then continues smoothly. For t larger than 700 s, the steady state regime is reached. Numerically, the behaviour is different. Even if the wall is in contraction, the amplitude is less than the experimental result. Globally, the steady-state is observed at approximately an equivalent time. This difference can be explained by the crystallisation of the polymer. Indeed, in the simulation the crystallisation is only taken into account in the heat transfer equation. Since the crystallisation is an exothermic reaction, heat is released in the medium. This explains why the contraction decreases in the time range for which the crystallisation occurs. As displayed in Figure 5.19, the absolute decrease of displacement is not seen when no crystallisation is considered. In reality crystallisation happens when polymer chains adopt the correct conformation. It is linked to localised density fluctuation of high amplitude [START_REF] Zhang | A Review on Polymer Crystallization Theories[END_REF]. An increase of the density is thus observable upon crystallisation [START_REF] Olmsted | Spinodalassisted crystallization in polymer melts[END_REF]. The rapid increase of the experimental displacement towards the center of the wall is caused by crystallisation. As the density increases, the strand is more compact and rapid shrinkage is observed. However in our model, the change of density due to crystallisation during cooling is not taken into account. The measurements of the thermal expansion coefficient is conducted during the heating of the polymer. These changes of density do not occur at the same temperature and will not behave the same way whether the polymer is heated or cooled. Moreover, measurements of the thermal expansion coefficient is only conducted until the beginning of the fusion. The determination of the coefficient from the melt state to the crystallisation can therefore improve the accuracy of the predictions. It can however be noted that the crystallisation occurs at the same time between the experiment and the simulation. This shows that the calculated transformed volume fraction of spherulites upon cooling is relatively accurate.

It can be observed in the y direction that a decrease of the absolute displacement value is seen approximately at the same time as when the crystallisation occurs at x = 95 mm. The displacement is however slower than at x = 5 mm. Moreover, Subsec 4.5.2 shows that the crystallization kinetics is different between the edge and the center of the wall. Moreover, all displacements are in the same direction, even though the sensors are placed at different heights. This leads to think that the variation of displacement in the y direction is not due to local crystallisation of the polymer. The crystallisation occurring close to the edge of the wall has an influence on the displacement at the center of the wall. Crystallisation happens first from the edge to the center of the wall. The progression of the crystallisation front leads to a slow increase of the compression towards the center. This can result in displacements towards the y + direction as the wall is constrained by the substrate. This can also explain the positive displacement field found experimentally in Figure 5.17e. This phenomenon is not observed in

Improvement of the model

Numerical and experimental results point out different behaviours in the global and local displacements. It highlighted that perfect contact between the substrate and the wall is a strong assumption. The same simulations are done with an air gap at the interface between the substrate and the wall. This air gap is imposed during the cooling of the wall, when the whole object is deposited. No significant differences are found at the right edge of the wall as the air gap was already present. However, Figure 5.20 shows an improve of the simulated displacement at x = 5 mm.

When the air gap is taken into account, the steady-state regime is reached more slowly than in the case without air gap. When no air gap is considered the displacement reaches a plateau after 500 s. When the air gap is considered, the asymptotic value is reached after 800 s. The final displacement values are better predicted. An average decrease of 17% of the deviation between experiments and simulations is found at x = 5 mm in the x direction. It is however difficult to know exactly the time at which detachment of the wall from the surface occurs. From image observation it seems that the detachment from the substrate does not occur at the same time for the whole wall. The time step at which detachment of the wall from the substrate is performed in the simulation is chosen arbitrary. A criterion to know the minimum stress or energy needed for the wall to detach could be implemented. It would however need to be characterised experimentally which is not a trivial procedure. It involves a better knowledge of the interactions between the deposited polymer and the substrate. Moreover the displacement field in the y direction is always found to be negative in the simulation, even when the air gap is considered. This shows that the air gap alone is not sufficient to account for all the phenomena taking place during the process.

The simulated results have shown better predictions in the x direction compared to the y direction. The numerical model considers isotropic properties of the polymer. On the strand scale, this hypothesis is reasonable in our case. Indeed it was shown in Subsec. 3.3.3 that no orientation of the crystalline phase is seen during the process. Some authors [START_REF] Mcilroy | Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing[END_REF][START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF] have however considered disentanglement of the polymer chains during the process. Orientation of the crystalline DISTORTION phase can also be seen in some cases [START_REF] Petersmann | Process-induced morphological features in material extrusion-based additive manufacturing of polypropylene[END_REF][START_REF] Shmueli | Simultaneous in Situ X-ray Scattering and Infrared Imaging of Polymer Extrusion in Additive Manufacturing[END_REF]. This hypothesis should therefore be assessed depending on the printing conditions. Perfect contact between the layers and the strands is considered in the simulation. Therefore the printed object also behaves in an isotropic manner. This is a stronger hypothesis. Indeed several authors have highlighted the anisotropy of parts produced by FFF [START_REF] Lee | Measurement of anisotropic compressive strength of rapid prototyping parts[END_REF][START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF][START_REF] Das | Advances in modeling transport phenomena in material-extrusion additive manufacturing: Coupling momentum, heat, and mass transfer[END_REF]. The creation of porosity is inherent of the process [START_REF] Nouri | Structural imperfections in additive manufacturing perceived from the X-ray micro-tomography perspective[END_REF]. Parts produced with FFF have better tensile properties in the direction of deposition [START_REF] Shaffer | On reducing anisotropy in 3D printed polymers via ionizing radiation[END_REF]. Moreover the bonding quality between the strands has a predominant impact on the mechanical behaviour of the final part [START_REF] Turner | A review of melt extrusion additive manufacturing processes: I. Process design and modeling[END_REF]. If adhesion between the strands is not complete, the shrinkage of the part does not behave the same way in every direction. The DIC measurements are performed on the surface of the wall. The strands are deposited in the x directions. Therefore the shrinkage of the wall in the x direction is closer to the isotropic contraction. However in the y direction, the shrinkage is more influenced by the inter-layer bonding quality. This could explain why more deviation of the numerical model is found in the y direction. A better characterisation of the bonding quality for these set of printing parameters is thus needed. Knowing how the strain behaves in all direction could improve the numerical model.

Conclusion

The distortion happening during the printing of parts produced by FFF has been studied. The study focuses on the use of a semi-crystalline polymer, namely a polypropylene. A numerical model is developed to consider the mechanical behaviour of parts during the printing process. The resolution of solid mechanics is implemented based on the thermal model developed in Chapter 4. Two different constitutive equations are considered for the polymer behaviour in its fluid and solid state. An experimental protocol is developed to perform in-situ measurements of the strain field. The use of DIC at different times during the process is used to reconstruct the distortion behaviour at different positions. This protocol is relatively easily operable and can be used for different part geometries. The impact of parameters such as the temperature and the printing velocities on the strain field can be studied with this method. These measurements are also used to study the displacement field during the process. It is applied to assess the developed numerical model.

The goal is to implement a relatively simple model enable to describe the distortion of parts printed by FFF. As a first approach, a phenomenological constitutive equation is considered to describe the polymer behaviour. The results showed that the model allows an accurate description of the displacement trends in a printed wall. The model can be used to predict where the part shrinks the most and in which manner. The final warping in the simulation matches well with the corresponding part. The maximum displacement value is accurately predicted by the simulation in the horizontal x direction. This is an important parameter for FFF. Indeed most AM technologies are used to produce prototypes with complex shapes. Having a tool enable to better define a part geometry before its printing is thus interesting. Improvement of the model is however still needed to better predict the distortion in the vertical y direction.

Considering the anisotropy in the produced part could significantly improve the model. In particular, implementing different mechanical behaviours at the adhesion zones can improve the predicted displacement field in the y direction. Considering a perfect contact between the substrate and the part is also a strong hypothesis. Indeed this leads to underestimations of the displacement in the x direction. A criterion of part detachment from the substrate can improve the predictions accuracy. A characterisation of the polymer behaviour during cooling is also needed. Indeed the variation of density due to crystallisation has an impact on the displacement.

Finally, a better characterisation of the constitutive equation parameters can improve the model. Performing relaxation during traction tests can help get more accurate viscoplastic parameters. Moreover, a better understanding of the liquid-solid transition of the polymer is needed. Indeed in the solid domain, linear interpolation is considered between two fitted data set. It is a strong simplification as polymers are highly temperature dependent. Polymer characterisation can be done through Dynamic Mechanical Analysis (DMA) in the solid domain and rheology tests in the fluid domain. A better transition of the polymer from its fluid to its solid state can be applied. Based on these tests, the implementation of a visco-elastic constitutive equation in the solid domain can be done [START_REF] Maurel-Pantel | A thermo-mechanical large deformation constitutive model for polymers based on material network description: Application to a semi-crystalline polyamide 66[END_REF][START_REF] Careglio | An implementation of the generalized Maxwell viscoelastic constitutive model[END_REF]. A Carreau-Yasuda constitutive equation could be considered in the fluid domain [START_REF] Pigeonneau | Heating and flow computations of an amorphous polymer in the liquefier of a material extrusion 3D printer[END_REF].

Chapter 6

General conclusion and perspectives

Conclusions

This project aimed in better understanding the additive manufacturing process of the fused filament fabrication technique. This process can be studied and described at different scales. The printing of objects is made from addition of matter with a typical size of few hundreds of micrometres. Two models have thus been developed to describe this process on two scales. The Ph.D. has focused on confronting experimental measurements and numerical modelling. On a general framework, the experimental measurements have been used to measure and study different aspects of the process, as well as to assess the numerical models. On the other hand, the numerical models have been used to investigate phenomena that are more tricky to measure experimentally. It is also expected to be used as a predictive tool for further investigations. Several achievements produced during this research project are summarised here.

Deposition of a single strand

The first step of the project aimed at the scale of the deposition of single or multiple strands. Experimental measurements are done to develop an empirical model to predict the strand shape. This model is only based on the printing parameters g, D, U and V . The final object properties are dependant on the deposition path and degree of infill. From a user perspective, the knowledge of the strand dimension based only on printing parameters is therefore of primary interest. A numerical model is developed to simulate the extrusion and deposition of polymer melt in FFF. This model computes the fluid dynamics coupled to heat transfer. A level-set method is implemented to implicitly track the dynamics of the melt zone in a two-phase domain. The numerical model is assessed by experimental measurements. The comparison of experimental and numerical strand shapes highlights the predictive aspect of the numerical simulation. The benefits of this model is the coupling between the polymer dynamics inside the liquefier, and the deposition behaviour upon extrusion. The coupling allows to study the influence of the liquefier geometry on the strand shape, velocity profile and pressure drop. The model highlights the predominant impact of the length of the capillary tube and diameter on the pressure drop. In particular, the pressure drop decreases when the nozzle angle increases and the length of the capillary tube decreases. An analytical model developed by Agassant et al. [START_REF] Agassant | Mise en forme des polymères: Approche thermomécanique de la plasturgie[END_REF] is assessed by the numerical model. The analytical model predicts the length of the polymer in front of the nozzle during deposition. A good agreement between the two models is found for a ratio g/D lower than 1.25. Finally the temperature behaviour is investigated, showing a predominant impact of the substrate temperature on the cooling behaviour of the first deposited strand. CHAPTER 6. GENERAL CONCLUSION AND PERSPECTIVES An extrusion velocity threshold above which extrusion fails is proposed. Simulations of the deposition of two stacked strands show an impact on the viscosity of the first deposited strand. From a user perspective, the numerical model is a real asset as a tailoring tool. This model is a preliminary tool to compute more complex phenomena, taking into account viscoelasticity and crystallisation.

Crystallisation behaviour

An experimental investigation has thus been achieved to better understand the crystallisation behaviour of semi-crystalline polymers in FFF. The study is done to assess the impact of the different printing parameters on the crystallisation behaviour. The study has focused on two polypropylenes with different thermal and crystallisation properties. Several techniques have been used such as DSC, POM, and WAXS to study the degree of crystallinity, microstructure and crystalline orientation of printed strands. Major differences of degree of crystallinity are found in the height of printed walls. The strand size and wall width also have an impact on the crystallinity due to different cooling rates. The result shows no orientation of the crystalline entities in the core of the strand for the range of printing velocities studied. Low orientation is found when performing azimuthal rotation which is attributed to the transcrystalline layers seen with POM observations. The mechanism of formation of the β modification of polypropylene in FFF is also investigated. At the heating rates encountered in FFF, the complete melting of the crystalline entities is not performed. The microstructure of the neat filament thus has a major role on the final object crystallinity. A self-seeding mechanism occurs upon crystallisation. Therefore, the β phase content increases when the cooling rate decreases. At the same time, the β content enhances with extrusion velocity.

Thermal modelling of building objects

Based on the investigation of the deposition dynamics and the crystallisation behaviour, the need of developing a numerical model on a larger scale has raised. Indeed, the numerical simulation of single strand deposition is limited by the computation time. The numerical model can only simulate short processing times without dramatically increase the computing time. Moreover, the crystallisation study has highlighted the impact of heterogeneous cooling on the crystallisation behaviour. A thermal numerical model is thus developed to simulate the process on a longer time scale. The process does not account for the fluid dynamics and thus enables to simulate the building of parts with real dimensions. The model first focuses on the thermal behaviour of an amorphous polymer. A level-set method is used to track the building front of the object. It is also used to focus re-meshing in the areas where high thermal gradient occurs. This method is fairly different to the element activation method usually implemented in these simulations [START_REF] Zhang | A parametric study of part distortions in fused deposition modelling using three-dimensional finite element analysis[END_REF][START_REF] Brenken | Development and validation of extrusion deposition additive manufacturing process simulations[END_REF][START_REF] Nath | Optimization of fused filament fabrication process parameters under uncertainty to maximize part geometry accuracy[END_REF]. The re-meshing technique enables to increase the mesh size in the areas of low temperature gradients and thus reduces the computation time. The model is assessed by experimental measurements of the cooling kinetics. Good agreement is found between the experiment and the simulation for different strand sizes. The computation of the degree of healing is done to investigate the bonding quality in an object. Most adhesion occurs during the deposition of the polymer. The crystallisation kinetics is then implemented in the thermal model. The crystallisation behaviour is computed based on a non constant cooling rate model. The latent heat of crystallisation is taken into account in the heat transfer equation. Re-meshing is also implemented to better track the crystallisation front. It is especially effective when increasing the height of the simulated object. The thermal model is used to explain the difference of degree of crystallinity highlighted by DSC measurements. The starting time of crystallisation and the crystallisation rate have an impact on the final degree of crystallinity. The impact of 6.2. PERSPECTIVES the extrusion and substrate temperature on the crystallisation behaviour is investigated. The study shows the influence of the deposition path on crystallisation kinetics when decreasing the extrusion temperature. Finally the model is used in a concrete case to determine the best parameters allowing the most homogeneous crystallisation of the wall. The result highlights the ability of the numerical model to be used as a tailoring tool before actual printing of a part. It is also used to better understand the difference of crystallinity observed experimentally. To the author knowledge, it is the first time that a thermal model of FFF have been used to study the crystallisation kinetics in a printed object.

Investigation of the distortion

Experimental and numerical investigations are done on the distortion of parts produced by FFF. The implementation of the solid mechanics coupled with the thermal modelling is done to simulate the distortion of parts during cooling. An experimental protocol is developed to perform in-situ measurements of the strain field during the printing and cooling of walls. Unlike previous studies [START_REF] Armillotta | Warpage of FDM parts: Experimental tests and analytic model[END_REF][START_REF] Brenken | Development and validation of extrusion deposition additive manufacturing process simulations[END_REF][START_REF] Nath | Optimization of fused filament fabrication process parameters under uncertainty to maximize part geometry accuracy[END_REF], DIC measurements are preferred compared to laser sensor to measure part distortion. Indeed, the use of DIC enables to obtain local displacement and strain field in every area of the object while laser sensors can only capture the global change of geometry. The experimental protocol enables to have a map of the distortion behaviour with time at specific locations of the part. The cooling rate has an impact on the strain rate and the time to reach the steady state regime. In the vertical direction, decreasing the cooling rate increases the final strain state since the shrinkage occurs on a longer time range. The experimental measurements are also used to assess the numerical model. Global and local comparisons of the displacement field are done between experimental and numerical results. The model enables to accurately predict the global displacement field in the horizontal direction. The areas with maximum displacement as well as the warping behaviour of the part is well reproduced. Discrepancies are however observed in the vertical direction as well as locally. To the author knowledge, it is the first time that local measurements of the displacement have been used to assess a numerical model in polymer FFF. The use of in-situ local measurements highlights the importance of several phenomena such as crystallisation and detachment of the part from the substrate. Several improvements of the model have thus been brought to light.

Perspectives

The numerical model of the extrusion and deposition of single strands has been developed as a preliminary step for more complex models. The implementation of the viscoelasticity and heat transfer in the liquefier can improve information produced by the model. On the other hand, the mechanical modelling of objects produced by FFF underlines several important phenomena that are not taken into account in the present model. An improvement of the accuracy of the model could extend its use on other fields such as the prediction of the mechanical properties of parts produced by FFF. The possible major improvements of the two models are listed here. Finally, the present study has highlighted key parameters to improve the process towards the printing of high functionality materials.

Thermal behaviour on single strand deposition

The present model already couples the heat transfer with the fluid dynamics. However, the polymer temperature inside the liquefier is considered equal to the extrusion temperature and the heating of the polymer is therefore not taken into account. The improvement of the model towards a full coupling of the liquefier and extrusion dynamics thus needs to account for the polymer heating inside the liquefier. The same model used in [START_REF] Pigeonneau | Heating and flow computations of an amorphous polymer in the liquefier of a material extrusion 3D printer[END_REF] can be implemented to study the fusion of the polymer from the neat filament to the extrusion. This would enables to study the fluid dynamics at the exit of the liquefier depending on the temperature profile. An assessment of the maximum extrusion velocity predicted by in Subsection 2.5.3.1 would be interesting. The full liquefier geometry is however needed to compute the heating behaviour. This would involve an increase of the computation time as the size of the computation domain increases.

Constitutive equations

Several models have studied the orientation of polymer chains during FFF when taking into account visco-elastic behaviours [START_REF] Mcilroy | Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing[END_REF][START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF][START_REF] Mcilroy | Modelling flow-enhanced crystallisation during fused filament fabrication of semi-crystalline polymer melts[END_REF][START_REF] Mcilroy | A fundamental rule: Determining the importance of flow prior to polymer crystallization[END_REF][START_REF] Xia | A numerical study of the effect of viscoelastic stresses in fused filament fabrication[END_REF]. These models compute the polymer chain orientation and disentanglement effect due to flow. These properties have an impact on the healing and crystallisation behaviours during cooling. However, the developed models do not take into account the fluid dynamics inside the liquefier. The implementation of a full CFD model taking into account the visco-elastic effects of the polymer is thus the next step of the present model. This allows the computation of polymer conformation due to fluid dynamics in the liquefier and during extrusion and deposition. The model would be based on the Roliepoly constitutive equation presented by McIlroy [START_REF] Mcilroy | A fundamental rule: Determining the importance of flow prior to polymer crystallization[END_REF]. The computation of the fluid dynamics provides a more accurate solution of the polymer chain conformation compared to previous models. It would be especially interesting for the case of semi-crystalline polymers. A coupling between the observed microstructure of single deposited strands, with the numerical model will be possible. This model will provide a better understanding of the crystallisation phenomena observed experimentally. For instance, an investigation on which parameter influences the most the microstructure between the shear rate and the cooling kinetics could be done.

On the scale of the numerical simulation of full part construction, changing the constitutive equation of the material could also be beneficial. Considering the polymer as visco-elastic with a generalized Maxwell constitutive equation could better describe the unrelaxed strain due to thermal shrinkage in the solid domain. This would imply a better characterisation of the polymer by dynamic mechanical analysis. Notably, the dependence of the polymer properties with temperature in the solid domain needs to be better characterised. In the liquid domain, a Carreau-Yasuda constitutive equation could be considered to better describe the viscosity of the polymer at low temperatures. Finally the transition from liquid to solid, depending on the cooling rates also needs to be implemented. Indeed characterising this transition through a single temperature parameter is not sufficient for polymer materials.

Characterisation of the adhesion

The thermal model developed in this study is used to predict the bonding quality based only on the temperature behaviour. Experimental measurements described in [START_REF] Coasey | Nonisothermal welding in fused filament fabrication[END_REF][START_REF] Yang | Healing of Thermoplastic Polymers at an Interface under Nonisothermal Conditions[END_REF][START_REF] Yin | Interfacial bonding during multi-material fused deposition modeling (FDM) process due to inter-molecular diffusion[END_REF] can be achieved to assess the numerical predictions. More globally, a better understanding of adhesion mechanism coupled with the thermal model can improve the bonding quality. Moreover, it would enable to account for the anisotropic properties of the printed object which could improve the shrinkage behaviour. A better description of the anisotropy would also enable predictions of the mechanical properties of the final material. The simulation of the printing of tensile specimens followed by numerical traction tests can be compared with the same experimental protocol. The comparison between the experiment and simulation could be done to assess the predictive value of the model. Finally, the experimental observation have highlighted the detachment of the object from the 6.2. PERSPECTIVES substrate during cooling. A better understanding of the interactions between the polymer and the substrate can give insight on this detachment phenomenon. A criterion can be implemented to determine when this detachment occurs in the simulation.

Change of density during crystallisation

The experimental and numerical investigation of the distortion of parts during cooling have highlighted the importance of crystallisation on the shrinkage. Indeed, important changes of density occur during crystallisation which leads to a stiff contraction. In the present model, the crystallisation is only taken into account in the heat equation. The equipments provided in this research project did not permit a full characterisation of the thermal expansion coefficient from the melt state to the solid state. The addition of a crystallisation shrinkage strain term in the thermal strain rate as proposed by Brenken et al. [START_REF] Brenken | Development and validation of extrusion deposition additive manufacturing process simulations[END_REF] can improve the shrinkage behaviour of our material.

Coupling of the two models

The coupling of the two models will enable a full description of the process from the scale of the deposition of a single strand to the construction of a full part. On a thermal point of view, the temperature field at the scale of the printed object can be used as boundary conditions in the simulation of the deposition of a single strand. The produced temperature field in the single strand can then be retrieved and used in the simulation of the scale of a printed object. This can be done by extracting the temperature field of the strand in a subdomain. A more accurate temperature behaviour of the process can then be described by this method. On the other hand, the use of the single stand deposition model can be used to extract the strand shape and temperature field. The deposited strand mesh can then be used to stack or deposit multiple strands adjacently. The coalescence between the strands could then be studied, taking into account the actual strand shape. This would also enable to study the cooling behaviour while taking into account the porosity in the wall. Avoiding the computation of fluid dynamics would save an important amount of computation time when simulating the deposition of multiple strands. However, this method cannot be used to build objects with real dimensions. Indeed, the number of elements increases with the number of strands to keep accurate strand dimensions.

High functionality materials

An improvement of the process parameters is needed to print high functionality materials. The present study points out the predominant impact of the cooling rate on the printing of a part. It influences the adhesion quality and the distortion of the part. The implementation of a printing chamber with controlled temperature which enables fast heating and cooling of the air could be coupled with the presented thermal model. The crystallisation kinetics of high functionality materials could then be better controlled. The study also shows that most adhesion occurs during the deposition of the polymer as the majority of the energy is given during this step. Applying a local heat source before this step can improve the bonding quality. Finally, the simulation of fluid dynamics in the nozzle can support the design of new liquefier geometries. This could be of particular interest when printing polymers with fillers.

1. 3 1 . 4 1 . 6

 31416 Filament deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Extrusion of the melt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Strand shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Temperature behaviour of the deposited strand . . . . . . . . . . . . . . Post deposition behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Neck growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Diffusion at welding zones . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Distortion of parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Crystallisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion and Ph.D. purposes . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Illustration of a typical extrusion-based AM process.

Figure 1 . 2 :

 12 Figure 1.2: Illustration of the melting of polymer with pressure flow removal model according to Osswald et al. [8].

Figure 1 . 4 .Figure 1 . 4 :

 1414 Figure 1.4: Visualisation of polymer chains as deformed spheres during deposition [25]. The colours indicates different locations in the strand.

Figure 1 . 5 :

 15 Figure 1.5: Failure feed velocity (V f ) as a function of temperature that can be fed into the hot end for the polymer used. open symbols: D=0.35mm-filled symbols: D=0.5 [5].

CHAPTER 1 .Figure 1 . 6 :

 116 Figure 1.6: Cross section of the strand, for various printing parameters according to Comminal et al.[4]

1. 3 .

 3 Figure1.7: Simulation of the first deposited strand shape with and without interaction of the printing head according to Bellini[START_REF] Bellini | Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design[END_REF]

CHAPTER 1 .

 1 CONTEXT AND LITERATURE REVIEW (a) Possible contact area. (b) Possible heat transfer modes.

Figure 1 . 8 :

 18 Figure 1.8: Possible contact area and heat transfer modes for a deposited strand (gray circle) depending on its location [2].

Figure 1 . 9 :

 19 Figure 1.9: Temperature profile of ABS printed at 230°C and 100mm/s. Solid lines indicates temperature from each layer. Dashed lines indicates weld temperature. L p indicates the last printed layer, L p-1 the second to last printed layer and L p-2 the third last printed layer [14].

Figure 1 . 10 :

 110 Figure 1.10: Illustration of polymer sintering [3].

CHAPTER 1 .

 1 CONTEXT AND LITERATURE REVIEW (a) Velocity vector field with shading to indicate the displacement[START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF].(b) Entanglement fraction profile[START_REF] Mcilroy | Deformation of an Amorphous Polymer during the Fused-Filament-Fabrication Method for Additive Manufacturing[END_REF].

Figure 1 . 11 :

 111 Figure 1.11: Polymer deformation after deposition.

Figure 1 . 14 :

 114 Figure 1.14: Polarized optical microscopy images of post-mortem PP for printing chamber temperature T ch of 25°C (a) and 55°C(b) [49].

CHAPTER 2 .Figure 2 . 1 :

 221 Figure 2.1: Illustration of the different numerical domains. The illustration is done at the initial state.

CHAPTER 2 .

 2 Figure 2.3: Fitting experimental values by a Carreau-Yasuda's law. T ref = 220 °C.

Figure 2 . 4 :

 24 Figure 2.4: Shift factor as a function of 1/T on semi-logarithmic scale.

2. 4 .

 4 STRAND SHAPE AND MODEL ASSESSMENT (a) U = 4 m/min, V = 2 mm/min. (b) U = 5 m/min, V = 1 mm/min.

Figure 2 . 5 :

 25 Figure 2.5: Strand cross section measured with an optical microscope.

Figure 2 . 6 : 5 :

 265 Figure 2.6: Ratio ∆W /D for different nozzle diameter, gap, and printing ratios U /V .

Figure 2 . 7 :

 27 Figure 2.7: ∆W D/g 2 as a function of U /V for two ratios g/D.

8 Figure 2 . 8 :Figure 2 . 9 :

 82829 Figure 2.8: Prediction of the strand height for different nozzle diameter and velocity. The comparison is done for different printing ratio g/D.

CHAPTER 2 .Figure 2 . 10 :

 2210 Figure 2.10: Comparison of the evolution of the strand width and height as a function of U /V for different gaps.

Figure 2 . 11 :

 211 Figure 2.11: Comparison of the global shape of the strand for 2 different ratios U /V . For all cases V = 1 m/min.

  (a) ω = 60°. (b) ω = 120°.

Figure 2 . 12 :

 212 Figure 2.12: Comparison of the pressure field for 2 nozzles with different angles ω in the converging region.

Figure 2 . 13 :

 213 Figure 2.13: Comparison of the global pressure field and velocity streamline for 2 nozzles with different angles ω in the converging region and different capillary length.

Figure 2 . 14 :

 214 Figure 2.14: Comparison of the velocity profile at one half of the converging region for 2 nozzles with different angles ω.

Figure 2 . 15 :

 215 Figure 2.15: Evolution of the viscosity at the center of the nozzle along the z axis for different nozzle angle ω. The dotted lines correspond to the starting points of the converging region.

  3 and D = 0.4 mm, respectively. Both simulations are done with the same ratio U /V = 3. This corresponds to a flow rate Q = 212 and Q = 377 mm 3 /s for D = 0.3 and D = 0.4, respectively. As expected, the pressure drop in the liquefier increases when decreasing the nozzle diameter. The total pressure drop increases from 2.3 MPa to 3.2 MPa when decreasing the diameter from D = 0.4 to D = 0.3 (a) D = 0.3 mm. (b) D = 0.4 mm.

Figure 2 . 16 :

 216 Figure 2.16: Comparison of the strand shape for different nozzle diameter D. All dimensions are normalised by the gap g.

CHAPTER 2 .

 2 FLUID DYNAMICS OF A DEPOSITED STRAND (a) g/D = 0.75. (b) g/D = 1.25. (c) g/D = 1.625.

Figure 2 . 17 :

 217 Figure 2.17: Comparison of the polymer spreading at different ratios g/D. All printings are done at D = 0.4 mm and U /V = 3 with V = 1 m/min.

  [START_REF] Boyer | Crystallization of polypropylene at high cooling rates: Microscopic and calorimetric studies[END_REF] and g/D = 1.25. The values of ∆W and ∆H are of the same order of magnitude as the numerical results. For g/D = 0.75, the numerical results find ∆W = 1.33 and ∆H = 0.3 mm and the analytical results are ∆W = 1.13 and ∆H = 0.33 mm. For g/D = 1.25, the numerical results find ∆W = 1.01 and ∆H = 0.44 mm and the analytical results are ∆W = 0.86 and ∆H = 0.44 mm. Both results are in fairly good agreements and this relation can be used for further estimations.

Figure 2 .

 2 Figure 2.19: L iso-Tg /θ g (mm) as a function of P e for the bisphenol-A polycarbonate and the acrylonitrile butadiene styrene.

  (a) g = 0.3 mm. (b) g = 0.65 mm. (c) Temperature behaviour following the z axis.

Figure 2 . 20 :

 220 Figure 2.20: Comparison of the temperature behaviour for 2 different strand shape. The size of the strand is normalised by the gap. It is therefore not representative of the actual size.

  (a) Impact of the strand height and perimeter.(b) Impact of the air and substrate temperature.

Figure 2 . 21 :

 221 Figure 2.21: Assessment of the predominant parameter between conductive and convective cooling. The comparison is done along the x axis.

  3.1 are similar. The model focuses on the deposition and thermal behaviour of successively deposited strands. Therefore only the capillary part of the nozzle is considered. The model simulates the deposition of two strands that are stacked on top of each other. The temperature and viscosity behaviour are recorded as a function of time by placing numerical sensors at specific positions in the first deposited strand. The results are presented in Figure 2.22 and 2.23.

Figure 2 . 22 :

 222 Figure 2.22: Evolution of the temperature when successively depositing two strands.

Figure 2 . 23 :

 223 Figure 2.23: Evolution of the viscosity when successively depositing two strands.

Figure 3 . 1 :

 31 Figure 3.1: Extraction areas of the samples at the bottom, top, core and surface of the printed wall.

3. 3 .

 3 COMPARISON OF THE CRYSTALLISATION OF THE TWO PRINTED PP (a) Surface. (b) Core.

Figure 3 . 2 :

 32 Figure 3.2: Microstructure of a wall printed at U =2 m/min and V =1 m/min with the PPH 3250: (a) in surface, (b) in core. The transcrystalline layers are referred as TC.

Figure 3 . 3 :

 33 Figure 3.3: Microstructure of a wall printed at U =2 m/min and V =1 m/min with the PP centaur: (a) in surface, (b) in core. The transcrystalline layers are referred as TC.

Figure 3 .

 3 Figure 3.4 provides the degree of crystallinity for four samples collected in the locations depicted in Figure3.1. Four walls have been printed with conditions summarised in Table3.3. The degree of crystallinity is larger in samples printed with the PPH 3250 than with the PP Centaur for the same printing conditions. This trend can be correlated to the size of spherulites which are larger for the PPH 3250 than for the PP Centaur. Recall that the crystallisation temperature of the PPH 3250 is larger than T c of the PP Centaur, see § 3.2.1. As both polymers are printed at the same temperature T ext = 250°C, the PPH 3250 spends less time above its crystallisation temperature during cooling. This could result in a lower degree of crystallinity, especially at the top area where both polymers are subject to the same cooling conditions. To understand this difference, measurements of the spherulites growth rate G is done by POM for the PP Centaur. The average spherulite diameter of the α phase is measured while cooling the sample at a constant rate from 210 to 50°C. An example of the shots are displayed in Figure3.5a. The growth rate is then calculated as a function of the average temperature between two measurements. The results have been compared with the growth rates of the PPH 3250 measured by Boyer et al.[START_REF] Boyer | Crystallization of polypropylene at high cooling rates: Microscopic and calorimetric studies[END_REF] in Figure3.5b. For the same range of temperature, the PPH 3250 shows a growth rate that is more than 2 decades higher than for the PP Centaur. Therefore, the polymer growth rate seems to have a predominant impact on the final degree of crystallinity compared to the crystallisation temperature.

Figure 3 . 4 :

 34 Figure3.4: Degree of crystallinity of samples taken from walls printed with the PPH 3250 and PP Centaur. "0 cm" and "1 cm" correspond to the bottom and top zones depicted in Figure3.1, respectively. "Shell" and "Core" correspond to the red and green zones, respectively. The parts printed with 4 and 6 lines are annotated "4l" and "6l", respectively.

Figure 3 . 5 :

 35 Figure 3.5: Example of the measurement of the growth rate for the PP Centaur (a) and comparison with the PPH 3250. The growth rate of the PPH 3250 is taken from Boyer et al. [75].

Figure 3 .

 3 7 depicts the 2D Debye-Scherrer patterns for the two polymers taken from samples printed at U = 2 m/min and V =1 m/min. The peaks with the highest intensities correspond to the hkl indexes 110, 040, 130, 111 and 041 of the α phase and the 300 of the β phase for the PPH 3250. For the PP centaur the peaks with highest intensities correspond to the hkl indexes 110, 040 and 111 of the α phase. The intensity of these rings are the same in all directions. This would imply that at these extrusion velocities, the shear stress is not sufficient to induce orientation of the crystalline entities. Figure 3.8 presents the 2D Debye-Scherrer patterns on samples printed with the PP Centaur. The extrusion velocity changes from 4 m/min, subfigure 3.8a to 6 m/min, subfigure 3.8b. Four other rings are noticeable for the PP Centaur, especially for the printing condition U = 6 m/min and V = 1 m/min but with much lower intensities.(a) PPH 3250. (b) PP centaur.

Figure 3 . 7 :

 37 Figure 3.7: 2D Debye-Scherrer patterns for both polypropylenes printed at U = 2 m/min and V = 1 m/min.

Figure 3 . 8 :

 38 Figure 3.8: 2D Debye-Scherrer patterns for samples printed with the PP centaur at different extrusion velocities.

Figure 3 . 9 :

 39 Figure 3.9: Identification of the crystalline peaks and orientation of the γ phase for samples printed with the PP Centaur at different extrusion velocities.

Figure 3 .

 3 Figure3.9a shows the presence of the γ phase of the PP at 2θ = 19.9°for all printing conditions. This peak corresponds to the Debye ring with the lowest intensity in Figure3.8. To ensure that the γ phase does not present orientation, azimuthal rotation at the bragg angle 2θ = 19.9°is performed. The intensity of the γ peak as a function of the azimuthal angle in transmission is presented in Figure3.9b. The peak intensity is equivalent in all directions for all printing conditions. Consequently, there is no phase orientation in transmission for any of the phases present in the PP Centaur.

3. 3 .

 3 COMPARISON OF THE CRYSTALLISATION OF THE TWO PRINTED PP (a) PPH 3250 U = 2 m/min, V = 1 m/min. (b) PP Centaur U = 2 m/min, V = 1 m/min.

Figure 3 . 10 :

 310 Figure 3.10: Intensity distribution of the α -110 and α -040 as a function of the azimuthal angle for the PPH 3250 and PP Centaur. The β -300 and γ -117 is plotted for the PPH 3250 and PP Centaur, respectively.

Figure 3 . 11 :

 311 Figure 3.11: Intensity distribution of the α -110, α -040 and γ -117 as a function of the azimuthal angle for the PP centaur printed at different extrusion velocity.

3. 4 .Figure 3 . 12 :

 4312 Figure 3.12: Comparison of the crystalline phase between a printed sample and the neat filament.

Figure 3 . 13 :

 313 Figure 3.13: Influence of the cooling rate on the latent heat of fusion of the PPH 3250 neat filament.

3. 4 .

 4 MECHANISM OF THE FORMATION OF THE β MODIFICATION OF PP IN FFF (a) Cooling from ≈ 220°C to 105°C at still air. (b) Isotherm step at 105°C after cooling at still air.

Figure 3 . 14 :

 314 Figure 3.14: Evolution of the β -300 and α -040 phases with the closest temperature history to what is encountered in FFF.

Figure 3 . 15 :

 315 Figure 3.15: Comparison of the presence of the crystalline phases for 2 heating and cooling rates. The evolution of the phase intensity with time at T = 105°C is also displayed.

Figure 3 . 16 :

 316 Figure 3.16: Evolution of the α and β phase content after 2 cycles of slow heating and fast cooling.

3. 4 .Figure 3 . 17 :

 4317 Figure 3.17: Evolution of the enthalpy of fusion of the PPH 3250 for different substrate temperature. All samples are printed at U = 2 m/min and V = 1 m/min.

Figure 3 . 18 :

 318 Figure 3.18: Microstructures for strands printed on three substrate temperatures. All strands are printed at U = 2 m/min, V = 1 m/min.

Figure 3 .Figure 3 . 19 :

 3319 Figure 3.19 depicts the enthalpy of fusion as a function of T obtained by DCS analysis. Clearly,

Figure 3 . 20 :

 320 Figure 3.20: Microstructures for strands printed with different extrusion and nozzle velocities. All strands are printed at T sub = 105°C.

Figure 4 . 1

 41 illustrates the printing experiment with the wall just before the introduction of the thermocouple on the left. The insertion of the thermocouple is depicted in the central sketch. The right picture represents the wall at the end of the experiment.

Figure 4 . 1 :

 41 Figure 4.1: Side view of the three steps of the thermocouple insertion during the wall printing composed by four filaments.

Figure 4 . 2 :

 42 Figure 4.2: Scheme of the section of a wall where the thermocouple is placed. The arrows depict the printing direction of the nozzle.

Figure 4 .

 4 Figure 4.3 depicts the temperature recording of five experiments achieved with U =4 m min -1 and V =1 m min -1 . Despite uncertainties on the accurate location of the thermocouple, the reproducibility is satisfying.

Figure 4 . 3 :

 43 Figure 4.3: Example of the average temperature evolution for a set of printing parameters.

  (a) Deposition of a sub-layer at t = nt f (b) Deposition of a sub-layer at t = (n + 1)t f

Figure 4 . 4 :

 44 Figure 4.4: Numerical printing of a wall with two steps of the printing to simulate the FFF process. The origin and the Cartesian coordinate system is depicted in sub-figure 4.4a.

4. 2 .Figure 4 . 5 :

 245 Figure 4.5: Illustration of the different meshing zones.

Figure 4 . 6 :

 46 Figure 4.6: Schematic of the domain with the two kinds of boundary conditions used in numerical computations.

Figure 4 . 7 :

 47 Figure 4.7: Flow chart of the numerical computation steps.

Figure 4 .

 4 Figure 4.8 depicts the temperature as a function of time localised on the thermocouple position for case 1. The blue solid line corresponds to the numerical results while the orange line is the temperature recorded experimentally. To help the reading of Figure 4.8, the glass transition temperature (green solid line) is also drawn.

Figure 4 . 8 :

 48 Figure 4.8: Interlayer temperature as a function of time for case 1. Comparison between numerical and experimental results. U =4.82 m/min and V =1.2 m/min

Figure 4 .

 4 Figure 4.9 gives the interlayer temperature as a function of time for cases 2 and 3. The numerical results have been obtained with the same heat transfer coefficients as case 1. The maximum temperature of each peak matches very well with the experimental measurements. As previously observed, the strands obtained numerically are more sensitive to the deposition of adjacent strands. The strand size of case 3 is smaller than case 1 explaining this increase of sensitivity. The strand size of case 2 is larger than case 1 and 3 explaining lower sensitivity. It is noteworthy that the modified thermal conductivity calculated with our method allows good prediction of the thermal behaviour. This shows that our method enables to capture the influence of porosity on cooling kinetics.

4. 3 .Figure 4 . 9 :

 349 Figure 4.9: Interlayer temperature as a function of time for cases 2 and 3. Comparison between numerical and experimental results.

Figure 4 . 10 :

 410 Figure 4.10: D h (t) determined from eq. (4.12) as a function of t with temperature taken from numerical and experimental results for (a) case 1, and (b) case 2.

and 4 .

 4 9a.

Figure 4 .

 4 10 depicts D h as a function of time for (a) case 1, and (b) case 2.

4. 3 .

 3 Figure 4.11: Temperature field ( • C) on the printing object from (a) a front view and (b) a lateral view.

Figure 4 .

 4 Figure 4.14 depicts the degree of healing over two horizontal lines. The vertical position of these two lines is 5 mm from the substrate. The first line crosses the thick wall on the right while the second line crosses the thin wall on the left of the object. These two lines are depicted in Figure 4.13. The degree of healing through the first and second line is plotted in blue and orange, respectively.

Figure 4 . 12 :

 412 Figure 4.12: Temperature field ( • C) of the object when H = 10 mm: (a) front view and (b) top view.

Figure 4 . 13 :

 413 Figure 4.13: Degree of healing of the first 10 mm of the object at the end of printing from (a) front view and (b) top view. Red arrows represent the plotted lines along which D h is plotted in Figure 4.14.

Figure 4 . 14 :

 414 Figure 4.14: Degree of healing D h through two horizontal lines of the object localised at a vertical position equal to 5 mm from the substrate.

Figure 4 . 15 :

 415 Figure 4.15: Evolution of the transformed volume fraction of crystalline entities as a function of temperature for different cooling rates.

. 31 )

 31 Figure 4.16: Evolution of the Avrami exponent as a function of the temperature.

Figure 4 . 17 :

 417 Figure 4.17: Evolution of ln ψ(T ) as a function of the temperature as red data points for the PP Centaur. The fitting of this function with a 3rd order polynomial is displayed as the plane blue curve. The plain green curve display the evolution of the temperature function for a PP model, which is the PPH 3250.

4. 4 .

 4 DEVELOPMENT OF THE MODEL FOR SEMI-CRYSTALLINE POLYMERS (a) Temperature behaviour and evolution of ξ. (b) Temperature behaviour evolution of the enthalpy of crystallisation.

Figure 4 . 18 :

 418 Figure 4.18: Comparison of the temperature behaviour when considering or not crystallisation. The evolution of the volume fraction of spherulites as well as the enthalpy of crystallisation is plotted for the case where crystallisation is taken into account.

  (a) ξ with crystallisation. (b) Temperature with crystallisation. (c) Temperature without crystallisation

Figure 4 . 19 :

 419 Figure 4.19: Comparison of the meshing in the case of a semi-crystalline and an amorphous polymer. The blue solid lines represent the meshing.

  (a) Coarse mesh without ξ re-meshing. (b) Coarse mesh with ξ re-meshing. (c) Fine mesh without ξ re-meshing.

Figure 4 . 20 :

 420 Figure 4.20: Comparison of the front of crystallisation for different meshing at the same time step.

  (a) z = 1 mm, H = 10 mm. (b) z = 5 mm, H = 10 mm. (c) z = 5 mm, H = 20 mm.

Figure 4 . 21 :

 421 Figure 4.21: Comparison of the crystallisation and thermal behaviour when considering or not re-meshing and with fine and coarse background meshing. All comparisons are done at x = 45 and y = -1.25 mm.

  (a) x = 0, y = -1.25, z = 5 mm. (b) x = 0, y = -1.25, z = 9 mm. (c) x = 45, y = -1.25, z = 5 mm.

Figure 4 . 22 :

 422 Figure 4.22: Comparison of the temperature behaviour and crystallisation kinetics as a function of time at different positions when considering ∆H f = 40 and ∆H f = 60 J/g.

4. 5 .

 5 Figure 4.23: Evolution of the C p as a function of the temperature.

  3.3.2 is achieved. Simulation of the printing of a wall fabricated at U = 2 m/min and V = 1m/min is done. Each layer is composed of 6 strands and the wall dimensions are L = 100, W = 3 and H = 10 mm. Numerical sensors(a) Z=10, t=0s. (b) Z=20, t=0s. (c) Z=10, t=167s. (d) Z=20, t=167s. (e) Z=10, t=250s. (f) Z=20, t=250s (g) Z=10, t=333s. (h) Z=20, t=333s.(i) Z=10, t=500s. (j) Z=20, t=500s

Figure 4 . 24 :

 424 Figure 4.24: Comparison of the front of crystallisation with time upon cooling for two walls of different height.

4. 5 .

 5 IMPACT OF THE PRINTING PARAMETERS ON THE CRYSTALLISATION KINETICS

Figure 4 . 25 :

 425 Figure 4.25: Front of crystallisation with time upon cooling for the wall of 20mm height. The slice of the cross section is made at the center of the wall. t = 0s corresponds to the time when the final height of the wall is reached.

  (a) y = -0.25mm, z = 5mm (b) x = 0mm, z = 9mm (c) x = 0mm, y = 1.25mm (d) Illustration of the sensors positions in the wall.

Figure 4 . 26 :

 426 Figure 4.26: Evolution of ξ(T ) and the temperature as a function of time at different locations.

  The impact of the strand dimension on the crystallisation kinetics is studied by simulating the printing of two walls with different extrusion velocities. Both walls dimensions are L = 100 mm and H = 10 mm and are composed of 4 strands in each layer resulting in different wall width. The impact of the number of strands in each layer is studied by performing two simulations at U = 2m/min and V = 1m/min with 4 and 6 strands by layers. Both walls dimensions are L = 100 mm and H = 10 mm, with W = 2 mm and W = 3 mm, respectively. The wall and strand dimensions in each case are summarised in Table4.7. The numerical sensors are placed corresponding to the locations of the DSC samples as depicted in 3.1. The comparison at the surface is done on the strand that forms the boundary surface with air in Figure 4.27b. The comparison of the core is made at the most centred strand of the wall in Figure 4.27a. Comparison between the bottom and top surfaces are done by comparing the strands closer to z = 1 mm and z = 9 mm in Figure 4.27c and 4.27d, respectively.

  (a) Core of the wall. (b) Surface of the wall (c) Bottom of the wall (d) Top of the wall

Figure 4 . 27 :

 427 Figure 4.27: Evolution of ξ(T ) and the temperature as a function of time for the printing of walls with different dimensions.

Figure 4 .

 4 Figure 4.27d shows very similar crystallization kinetics between the samples U4V1 4L and U2V1 6L. The sample U4V1 4L is maintained at a high temperature for a shorter period of time compared to the sample U2V1 6L. Two more strands are printed in each layer for U2V1 6L which explains why this sample starts cooling later. Each layer is heated for a longer period of time. This highlights that the same crystallisation time can be achieved with different printing conditions depending on the deposition path. The crystallisation rate in all cases range from 1 • 10 -2 to 1.3 • 10 -2 s -1 . The resulting degree of crystallinity is also very similar for all cases. However, the trends are different when comparing the simulated and experimental results. This highlights limitation of the model when the difference of crystallisation rate is lower than 3 • 10 -3 s -1 .

  (a) T ext = 250°C, view of the x, z plane. (b) T ext = 230°C, view of the x, z plane. (c) T ext = 200°C, view of the x, z plane. (d) Comparison of the temperature and crystallisation behaviour along the horizontal x for a strand deposited at y = -0.5, z = 5 mm at T ext = 230°C. (e) Schematic view of the time between 2 pass at the right edge of the part (f) Schematic view of the time between 2 pass at the left edge of the part

Figure 4 . 28 :

 428 Figure 4.28: Evolution of ξ(T ) for different extrusion temperature and schematic view of the deposition path.

  (a) Comparison of different extrusion temperature. (b) T ext = 230°C, with and without crystallisation.

Figure 4 . 29 :

 429 Figure 4.29: Comparison of ξ(T ) and the temperature behaviour at x = 0, y = -1.5 and z = 5 mm.

Figure 4 . 30 :

 430 Figure 4.30: Evolution of ξ(T ) for different substrate temperature.

  (a) t = 410s. (b) t = 710s. (c) t = 793s. (d) t = 840s. (e) Comparison of the temperature and crystallisation behaviour in the height of the part during the whole process.

Figure 4 . 31 :

 431 Figure 4.31: Evolution of ξ(T ) during the cooling of the part.

  ρ - → g the volumetric force density. The stress tensor is also decomposed as follows σ = s -P I = s -

5. 3 .Figure 5 . 1 :

 351 Figure 5.1: Flow chart of the global simulation process.

CHAPTER 5 .Figure 5 . 2 :

 552 Figure 5.2: Dimensions of the tensile test specimen. The thickness is ≈ 3 mm.

5. 3 .Figure 5 . 3 :

 353 Figure 5.3: Picture of a patterned plate for the calibration of the VIC ® 3D system [105].

Figure 5 . 4 :

 54 Figure 5.4: Example of DIC for a tensile test specimen.

Figure 5 . 5 :

 55 Figure 5.5: Example of a stress-strain curve of the PP Centaur: all measurements are done at 50°C.

Figure 5 . 6 :

 56 Figure 5.6: Stress-strain curve of the PP Centaur for different temperatures: the solid lines are the corresponding fit for each temperature using Equation (5.9). T (°C) σ 0 (MPa) E (MPa) m K (MPa.s m ) J (MPa) φ 50 1.2 1.61•10 2 0.6 56 5.6 15 70 1.0 7.89•101 0.6 30 3.2 20 90 0.58 3.58•10 1 0.6 15 1.3 22 100 0.21 1.94•10 1 0.6 7 0.8 23 105 0.2 6.83 0.6 5.53 0.3 25

  therefore considered as a Newtonian fluid and m = 1 in previous equation. The consistency K is determined by considering it equal to the viscosity at the Newtonian plateau. The evolution of the viscosity as a function of shear rate for different temperatures are shifted to a reference temperature. The obtained master curve is displayed in Figure 5.7a. The shift factor a T is defined by an Arrhenius law. The dependence of the viscosity with temperature is thus given by η(T ) = η 0 • a T (T ), (5.36) with a T (T ) the shift factor which depends on the temperature and η 0 the viscosity of the Newtonian plateau at a reference temperature. The activation energy is determined by fitting the evolution of a T as a function of 1/T . The linearity of the plot on a logarithmic scale displayed in Figure 5.7b, assesses the validity of the chosen shift factors. The values of the constitutive equation parameters in the fluid domain are summarised in Table 5.2. (a) Master curve. (b) a T against 1/T on a semi logarithmic scale.

Figure 5 . 7 :

 57 Figure 5.7: Determination of the constitutive equation parameters in the fluid domain for the PP Centaur. T ref = 220°C.

  (a) ε as a function of T . (b) α T as a function of T for the two experiments.

Figure 5 . 8 :

 58 Figure 5.8: Determination of the thermal expansion coefficient. The solid line is the average of the two experiments.

5. 4 .

 4 EXPERIMENTAL INVESTIGATION OF PART DISTORTION performed for 15 min after the end of the printing of the wall. This allows to study the distortion of the part upon cooling. The experimental set-up is illustrated in Figure 5.10. (a) Speckle performed after a printing stop at y = 5 mm. (b) Speckle performed after a printing stop at y = 7.3 mm. (c) Speckle performed after a printing stop at y = 10 mm.

Figure 5 . 9 :

 59 Figure 5.9: Speckle areas for the three different tests.

Figure 5 . 10 :

 510 Figure 5.10: Sketch of the experimental set-up in which the strain field is measured by DIC technique.

  (a) ε Log xx field for speckle up to z = 5 mm. (b) ε Log yy field for speckle up to z = 5 mm. (c) ε Log xx field for speckle up to z = 7.3 mm. (d) ε Log yy field for speckle up to z = 7.3 mm. (e) ε Log xx field for speckle up to y = 10 mm. (f) ε Log yy field for speckle up to y = 10 mm.

Figure 5 . 11 :

 511 Figure 5.11: Strain field in x and y directions for the three different tests. The strain field is measured up to 15 min after the end of deposition of the wall.

  (a) t = 30s. (b) t = 100s. (c) t = 300s. (d) t = 500s.

Figure 5 . 12 :

 512 Figure 5.12: Strain field in the x direction with time. The speckle is performed up to y = 10 mm, and the origin of the time is taken when the measurement begins.

  (a) y = 5 mm. (b) y = 8 mm.

Figure 5 .

 5 Figure 5.13: ϵ Log xx as a function of t for four positions along the x axis and for y =5 mm (a) and y =8 mm (b).

5. 4 .

 4 EXPERIMENTAL INVESTIGATION OF PART DISTORTION (a) ϵ Log xx at x = 5 mm. (b) ϵ Log yy at x = 30 mm.

Figure 5 .

 5 Figure 5.14: ϵ Log xx and ϵ Log yy as a function of t for three deposition heights.

Figure 5 . 15 :

 515 Figure 5.15: Comparison of the evolution of ε Log yy with the simulated temperature and transformed fraction of spherulites. The comparison is done on the same initial position x = 30 and y = 8 mm.

Figure 5 . 16 :

 516 Figure 5.16: Experimental and numerical comparison of the global displacement field in the x direction.

5. 5 .

 5 MODEL ASSESSMENT (a) Experimental -→ U y for a speckle up to y = 10mm. (b) Simulated -→ U y calculated after deposition at y = 10mm. (c) Experimental -→ U y for a speckle up to y = 7.3mm. (d) Simulated -→ U y calculated after deposition at y = 7.3mm. (e) Experimental -→ U y for a speckle up to y = 5mm.(f) Simulated -→ U y calculated after deposition at y = 5mm.

Figure 5 . 17 :

 517 Figure 5.17: Experimental and numerical comparison of the global displacement field in the y direction.

  (a) x = 95mm. (b) x = 50mm. (c) Illustration of the sensors position. (d) Illustration of the sensors position.

Figure 5 . 18 :

 518 Figure 5.18: Experimental and numerical evolution of the displacement as a function of the height. Comparison in the x direction (a) and y direction (b).

Figure 5 . 19 :

 519 Figure 5.19: Comparison of the displacement along the x direction when crystallisation is taken into account in the model or not, at x = 95 mm.

  (a) No air gap considered during cooling. (b) Air gap considered during cooling.

Figure 5 . 20 :

 520 Figure 5.20: Experimental and numerical evolution of the displacement in the x direction as a function of height, at x = 5 mm.

  and (2.15) are normalised. All spacial coordinates are normalised by the gap g between the substrate and the nozzle. The velocities are normalised by the nozzle velocity V . The time is normalised by V /g.

	CHAPTER 2. FLUID DYNAMICS OF A DEPOSITED STRAND
	Momentum Heat transfer
	ρ
	The density, conductivity, specific

Table 2 .

 2 1: Summary of all normalised variables heat and viscosity are all normalised by the polymer density ρ p , conductivity k p , specific heat C pp and viscosity at the Newtonian plateau η 0 , respectively. The temperature ranges from the air temperature T air and the extrusion temperature T ext . Therefore it is normalised as:

  .1. Only one half of the domain is considered since the problem is symmetric. The nozzle solid domain and the substrate are not included in the numerical domain. Only the polymer and air domains are considered in the computation. As depicted in Figure2.1, the nozzle geometry forms a boundary ∂Ω noz of the computation domain. Its geometry is based on the E3D-nozzle V6 with a nozzle diameter of 0.4mm, which dimensions can be found in open source. The bottom surface of the domain ∂Ω sub is considered as the substrate. The other boundaries are in contact with air.At the initial state, the interior of the nozzle is considered full of polymer. This is done by immersing the interior geometry of the tip of the nozzle in the computation domain. The immersed domain Ω pol is then taken as the initial polymer domain. It is depicted in orange in Figure2.1. The rest of the computation domain Ω air is considered as air. At the initial state, the air domain Ω air and the polymer domain Ω pol have an imposed temperature Θ = 0 and Θ = 1, respectively. Finally the nozzle boundary ∂Ω noz has an imposed temperature Θ = 1.

Table 2 .

 2

	2: Fluid mechanics and heat transfer boundary conditions in the computing domain
	depicted in Figure 2.1.

Table 2 .

 2 4: Fitting parameters of the Carreau Yasuda and Arrhenius law for the ABS.with time and see if the tests are reproducible. Shift factors are then found to overlap all the data to a reference temperature of 220°C. The behaviour of η/a T is plotted as a function of γa T . All the measurements done at different temperatures gather in a master curve. The shift of all samples to a master curve is depicted in Figure2.3. The shift factor is defined using the Arrhenius' law depicted in Equation (2.4). It is thus possible to calculate the activation energy by plotting a T as a function of 1 T . Checking the linearity of the plot on a logarithmic scale also assesses the validity of the chosen shift factors. The result is displayed in Figure2.4. The fitting parameters of the Arrhenius' law are given in Table2.4. The obtained master curve is then fitted by a Carreau-Yasuda's law given by Equation (2.1). The plot of the fitted law is also displayed in Figure2.3. The fitted parameters are summarised inTable 2.4.

	η 0	λ	a	m	E a	T ref
	Pa.s	s	-	-	kJ/mol	°C

.3.

Rheology measurements are performed to characterise the dependence of the polymer viscosity with temperature and shear rate. Viscosity measurements are done on a plate-plate rheometer ARES from TA Instruments ® in dynamic oscillations. The tests are done on a frequency range of 0.1 to 100 rad/s and a temperature range of 210 to 240°C. These tests are performed at 1% strain deformation, in the linear domain. Different samples are tested to assess how they behave 2.3. IMPLEMENTATION OF THE MODEL Figure 2.2: Flow chart of the modelling of a single strand deposition. 3.04 • 10 3 3.2 • 10 -2 0.6 0.27 115.05 220

Table 2 .

 2 6: Comparison of the inlet and extrusion flow rate for different extrusion velocities.

	Printing parameters Printing temperatures Dimensionless number Dimensionless number
							g = 0.5 mm		g = 0.65 mm
				T ext °C	230	Re	2.77 • 10 -6	Re	3.60 • 10 -6
	D	mm	0.4	T sub °C	94	Ca	1748.5	Ca	1748.5
	V m/min	1	T air °C	40	P e	84.17	P e	109.42

Table 2 .

 2

7: Numerical parameters taken for the simulations.

Table 2

 2

.7. The printing parameters are also summarised in Table 2.7. The results are shown in Figure 2.10 for both gaps.

4 Mechanism of the formation of the β modification of PP in FFF . .

  The investigation is solely experimental. It Résumé L'impact des différents paramètres d'impression sur la cristallisation lors du procédé FFF est étudié dans ce chapitre. Différentes techniques expérimentales telles que la calorimétrie différentielle à balayage (DSC), la diffusion des rayons-X (DRX) aux grands angles ainsi que la microscopie en lumière polarisée sont utilisées pour étudier le degré de cristallinité, l'orientation des phases cristallines ainsi que la microstructure du matériau imprimé. Deux polypropylènes (PP) sont étudiés afin de voir l'impact des paramètres thermiques sur la cristallisation.Les observations au microscope montrent la présence de zones trans-cristallines à l'interface entre les cordons et ainsi qu'entre les couches pour les deux PP. Les mesures de DSC mettent en lumière l'impact important de la cinétique de refroidissement sur la cristallisation. Le nombre de couches dans un mur imprimé ainsi que la taille de chaque cordon joue un rôle sur le degré final de cristallinité. Les mesures DRX montrent que le polymère ne présente pas d'orientation dans le coeur du cordon. En revanche des balayages azimutaux en DRX montrent une faible orientation des échantillons qui serait due aux zones trans-cristallines. Enfin les mesures DRX décèlent la présence de la phase β du PP dans l'un des polymères étudiés. Des mesures de DRX en température sont ainsi effectuées afin de comprendre le mécanisme de formation de cette phase en FFF. Les résultats montrent que la présence de phase β est due à une fusion incomplète des entités cristallines dans le liquéfacteur. Cela met en évidence l'importance de la microstructure initiale du filament, avant impression, sur la cristallinité finale de la pièce produite. Une diminution de la vitesse de refroidissement et une augmentation de la vitesse d'extrusion favorisent la croissance de la phase β.

	3.1. INTRODUCTION
	Chapter 3
	Impact of the printing parameters
	on the crystallisation behaviour

Contents 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Differential Scanning Calorimetry (DSC) . . . . . . . . . . . . . . . . . 3.2.4 Wide Angle X-Ray Scattering (WAXS) . . . . . . . . . . . . . . . . . . 3.2.5 Polarised Optical Microscopy (POM) . . . . . . . . . . . . . . . . . . . . 3.3 Comparison of the crystallisation of the two printed PP . . . . . . . 3.3.1 Microstructure of the two PP . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Degree of Crystallinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Crystal orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Behaviour of the crystalline phases . . . . . . . . . . . . . . . . . . . . . 3.3.4.1 Characterisation of the β content in the process . . . . . . . . . . . . . . 3.4.2 Impact of the printing parameters on the formation of β phase. . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3 .

 3 1: Position of the peaks present in the different WAXS pattern shown in this study. The peaks are attributed to their corresponding phase and (hkl) indexes.

	χ =	∆H ∆H 100	(3.1)

® . The degree of crystallinity is calculated by

Table 3 .

 3 2 times larger than the size of the spherulite for the PP Centaur and PPH 3250, respectively. The adhesion is thus promoted in walls printed with the PP Centaur. The PP Centaur has a lower fusion temperature and is softer than the PPH 3250 at ambient temperature. Since the extrusion temperature is CHAPTER 3. IMPACT OF THE PRINTING PARAMETERS ON THE CRYSTALLISATION BEHAVIOUR α spherulites Transcrystalline layer Area (µm 2 ) Width (µm) Height (µm) Core Length (µm) Surface Length (µm) 2: Comparison of the different crystalline phases' size for the PPH 3250 and PP centaur.

	PPH 3250	Core Area	2924 ± 718	62 ± 8	59 ± 8	83 ± 17	-
		Surface Area 3015 ± 452	65 ± 5	58 ± 4	73 ± 10	49 ± 7
	PP centaur	Core Area	757 ± 168	34 ± 4	27 ± 3	49 ± 16	-
		Surface Area	771 ± 137	34 ± 4	29 ± 2	56 ± 10	39 ± 10

Table 3 . 3 :

 33 Dimensions of the walls and the strands depending on the printing parameters for the PPH 3250 and PP Centaur.

	mm)

.3. Finally, a wall is printed with the PPH 3250 with U = 2 m/min, V = 1 m/min and 6 strands. After printing, samples are collected according to locations depicted in Figure 3.1. Number of strands in the width Strand's width (mm) Wall's width (mm) Wall's Height (mm) wall's length (

  2.4 are done to evaluate this assertion. A first test is conducted with the highest heating and cooling rate possible with the method. It is thus considered as the closest parameters to what is encountered in FFF. The sample is heated from ambient temperature to approximately 220°C in 30 s, which corresponds to a heating rate of approximately 360 K/min. The sample is then directly cooled in still air once it reaches the inlet temperature of 220°C. The corresponding cooling rate is approximately 20 K/min. The glass plate is set to 105°C during cooling to mimic the cooling of a deposited strand on the substrate of a 3D printer. WAXS measurements are done at different temperatures during cooling. When the polymer reaches the substrate temperature, WAXS measurements are performed at different times. Figure

Table 4 .

 4 [START_REF] Bikas | Additive manufacturing methods and modelling approaches: a critical review[END_REF] gives the thermal properties of the ABS polymer according to Costa et al.[2]. The melted polymer is deposited on a meshed substrate made of steel with thermal properties given in Table4.1.

	Properties	Notation	Unit	ABS Steel
	Density	ρ	kg/m 3	1050 7850
	Specific heat	C p	J/(kgK) 2200 477
	Thermal conductivity	k	W/(mK) 0.18 42.7

Table 4 .

 4

1: Properties of the ABS polymer according to Costa et al.

[2]

and of an AISI 4130 steel.

Table 4

 4

	.1.

Table 4 .

 4 2: Computation parameters and corresponding times for the three printing conditions.The walls are printed with our 3d printer in the working conditions provided in Table4.2. The thermal measurements are based on the experimental procedure discussed in subsection 4.2.1. The printing of the wall continues until the interlayer temperature reaches T g . Therefore, the final height depends on the printing conditions. Using the Cartesian coordinate framework represented in sub-figure4.4a, the locations of the thermocouple are x = 15, y = -0.5, z = 5 mm for case 1, x = 15, y = -0.75, z = 4.7 mm for case 2 and x = 15, y = -0.25, z = 5 mm for case 3. The printed walls height does not exceed 3 cm.The temperature of the substrate is set at T sub =94 • C. Following measurements in the surrounding air, T air is set equal to 50 • C. The temperature of the polymer fractions is set equal to T ext =200 • C in the numerical computations. Finally, the only two parameters controlling the temperature behaviour are the heat transfer coefficients h surf and h top .

Table 4 .

 4

3: Comparison of the experimental and numerical cooling rate computed for all studied cases.

Table 4 .

 4 4.

	A arr 726	s
	E a	40.6 kJ/mol
	T ref 230	• C
	Table 4.4: Parameters of the welding time given by equation (4.13) for a PC-ABS copolymer
	[36].

Table 4 .

 4 5: Values of the Avrami exponent for the different nucleation types[START_REF] Billon | Modelling of the Cooling of Semi-crystalline Polymers during their Processing[END_REF].with N 0 the initial number of potential nuclei. The growth rate G(t) is considered constant for isothermal processes. Moreover equation (4.25) can be simplified for two limiting cases. If the extended nucleation rate is purely instantaneous meaning that q is very large, or if it is purely sporadic meaning that dNe dt = Cst then the isothermal extended volume fraction simplifies as

	.27)

  Unexpected thermal phenomenon can be highlighted by the model. This should enable a better calibration of the printing parameters to get appropriate properties.CHAPTER 5. EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF PART DISTORTIONRésuméLe modèle thermique développé au chapitre précédent est amélioré afin de tenir compte de la mécanique. Le but est de pouvoir prédire la déformation de la pièce lors du refroidissement et donc l'écart de la géométrie obtenue par rapport à la géométrie souhaitée. Une loi de comportement élasto-viscoplastique est considérée pour le polymère à l'état solide. Des tests de tractions à différentes températures sont effectués afin de déterminer les paramètres de la loi de comportement. À l'état liquide, le polymère est considéré comme un fluide newtonien. Une dépendance de la viscosité à la température est implémentée par une loi d'Arrhenius.Un protocole expérimental est également développé afin d'effectuer des mesures in-situ de la déformation de la pièce, par corrélation d'images, au cours du procédé. Cela permet d'étudier localement l'évolution de la déformation lors du dépôt et du refroidissement. Le protocole expérimental est également utilisé pour valider le modèle numérique. Les mesures expérimentales ont montré que la vitesse de refroidissement avait une influence sur le taux de déformation ainsi que sur l'état final de déformation en régime stationnaire. L'augmentation de la vitesse de refroidissement accroît logiquement le taux de déformation. En revanche, dans la direction verticale, la diminution de la vitesse de refroidissement donne plus de temps au polymère pour se déformer et induit ainsi un état de déformation plus important en régime stationnaire.La confrontation entre les mesures expérimentales et numériques a montré que le modèle numérique permet une bonne prédiction du déplacement globale de la pièce dans la direction horizontale. En revanche, un écart plus important est observé selon la direction verticale. Globalement le modèle arrive bien à prédire les zones où le déplacement est le plus important ainsi que le gauchissement de la pièce. Une confrontation locale est également effectuée entre déplacement expérimentale et numérique. Les résultats révèlent que la prise en compte du changement de densité lors de la cristallisation ainsi que la prise en compte d'un critère permettant de connaître le moment où la pièce se détache du substrat, améliorerait le modèle.

	5.1. INTRODUCTION

  The constitutive equation is applied depending on the temperature field. The fluid constitutive equation is considered for every element with T ≥ T c and the solid constitutive equation is considered for T < T c . The only stress considered in the present model arises from thermal shrinkage. The thermal expansion rate is thus calculated based on the temperature field at each Two different re-meshing techniques are implemented in the simulation. The first one called HAdaptation in CimLib enables re-meshing of the domain by changing the topology of the mesh, as explained in Subsection 2.3.2. It allows to choose the appropriate mesh size depending on the location. HAdaptation is performed after deposition of each layer to re-mesh the new CHAPTER 5. EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF PART DISTORTION layer to be deposited. The second re-meshing technique in CimLib is called RAdaptation. It enables to change the nodes coordinates of the domain based on the calculated velocity field.

	The new coordinates of each nodes are calculated such that
		-→ P (x,y,z) (t i+1 ) =	-→ P (x,y,z) (t i ) + -→ u (x,y,z) ∆t i ,	(5.21)
	increment. It is defined as	εth = α T	∆T ∆t	I.	(5.20)

Table 5 .

 5 2: Constitutive equation parameters in the fluid domain, at very low shear rates.

	5.3. MECHANICAL PROPERTIES OF THE POLYMER
	η 0	m	E a	T ref
	Pa.s	-kJ/mol	°C
	1.7 • 10 3 1	43.3	200

1 . (5.35)

DIC measurements of the process indicate values of the strain rates ε lower than 1 • 10 -2 s -1 once the polymer is deposited. This will be further detailed in § 5.4.3.2. The polymer is

Table 5 .

 5

	5.5. MODEL ASSESSMENT

4: Comparison of the maximum displacement value between the experiment and the simulation in the x direction for all tests.

Table 5 . 5 :

 55 Comparison of the maximum displacement value between the experiment and the simulation in the y direction for all tests.

	Speckle height	Edge	Maximum displacement	Relative deviation
			Experiment Simulation
	(mm)		(mm)	(mm)	%
	10	Bottom	0.71	0.193	73
		Top	-0.195	-0.4835	148
	7.3	Bottom	0.68	0.199	71
		Top	-0.17	-0.36	112
	5	Bottom	0.7	0.205	46
		Top	0.015	-0.254	Not comparable

(a) U = 4 m/min, V = 1 m/min (b) U = 6 m/min, V = 1 m/min

(a) Heating at 10 K/min, cooling at ≈ 8 K/min. (b) Heating at 10 K/min, cooling at ambient air (≈ 20 K/min).

4.2. NUMERICAL FINITE ELEMENT COMPUTATION

The taken values are found to be inverted in the cited article. In the following, the actual values of degree of healing computed by the model is thus false. However, the presented results are comparative and the global trends and conclusions remain valid.

Remerciements

Chapter 4

Thermal modelling of the printing of a wall Contents Chapter 5

Experimental and numerical investigations of part distortion Contents DISTORTION by:

-→ U = - → x (t) -- → x 0 , (5.26) with -→ x (t) the position of the point at the time t. The deformation gradient F is defined as [START_REF] Rappaz | Modélisation numérique en science et génie des matériaux[END_REF]:

(5.27)

The Hencky strain tensor is then defined as:

with C the Cauchy-Green deformation tensor. Note that the Cauchy-Green tensor is symmetric and positive-definite. It is thus possible to compute its logarithm in its eigenbasis [START_REF] Gurtin | The mechanics and thermodynamics of continua[END_REF]. In the vertical y direction of a traction test, the true strain is defined as follows:

with l the length of the sample at a specific time and l 0 the initial length of the sample. An example of the resulting sample strain field in the y direction is displayed in Figure 5.4c.

Determination of the constitutive law parameters

Solid domain

As mentioned previously, the parameters of the solid constitutive equation are determined by traction tests. A tensile test machine from Erichsen is used with a 2 kN force sensor from Instron. The traction tests are performed at different temperatures to obtain the dependence of the parameters with temperature. The initial width, length and thickness of each sample are measured with a calliper before the test. The samples are maintained with pneumatic grips. The stress is measured while the samples are subject to a constant displacement velocity. The DIC cameras are set to take images every 2 s or when the force increases by 5N. The initial length of the sample is 30 mm. The displacement velocities are set to 3 • 10 -3 and 3 • 10 -2 mm/s resulting in a strain rate of 10 -4 and 10 -3 s -1 , respectively. Two different strain rates are tested to investigate their influence on the material. A minimum of three tests are done for a set of parameters to assess the reproducibility of the measurement. The evolution of the cross section during the test should be taken into account to compute the stress during the test. The evolution of the traction specimen cross section with time is written:

with w(t) the width and e(t) the thickness of the specimen as a function of time. Equation 5.29 gives the following relation for the width and thickness:

and

with w 0 the initial width of the sample, e 0 its initial thickness, ε xx and ε zz the strain in the transverse and in the thickness of the sample. The former equations gives the following relation for the evolution of the specimen cross section as a function of time:

ABSTRACT

This Ph.D. aims at the thermal and mechanical investigations of the additive manufacturing process of Fused Filament Fabrication (FFF). In this framework, the process is investigated at two different scales. Numerical models are developed to perform finite element simulations of the process. The numerical tools are supported by experimental observations and measurements. This is done to have a clear idea of the different phenomena occurring during the process as well as to assess the numerical models.

A first model focuses on the polymer fluid dynamics in the liquefier and its extrusion and deposition on a substrate. This model enables to study the influence of the nozzle geometry and printing parameters on the polymer deposition behaviour. An experimental protocol is implemented to measure the deposited strand cross section. An empirical equation is proposed to predict the strand dimensions based on the printing parameters.

A second model is developed to investigate the process on a larger scale. The first step aims to simulate the thermal behaviour of parts of several cm heights. Experimental measurements of the temperature via thermocouple are performed to assess the model. The thermal model is then adapted for the case of semi-crystalline polymers. The crystallisation kinetics is computed based on the cooling of the part. The results explain the degree of crystallinity measured experimentally. Solid mechanics is also implemented in the numerical model to account for part distortion upon cooling. An experimental protocol is developed to perform in-situ measurements of the strain during the printing of the part by digital image correlations. The combination of experimental and numerical investigations performed in this Ph.D. should provide the necessary information to better tailor the process depending on the needs of the user.

KEYWORDS

Fused filament fabrication, Thermomechanical analysis, Fluid dynamics, Numerical simulations, Polymers, Crystallisation.