of the regions, Brussels, 2020.

General Introduction

The three major energy sources nowadays are fossil fuel, natural gas and oil. These sources aren't renewable, they're quickly depleted and their prices are dramatically increasing. This is due to the accelerated economic demands and geo-political effects such as wars. The fuel prices affect directly the production costs for all the sectors. In addition, they exacerbate adverse effects on the environment. The excessive use of fossil fuels is associated with global warming and pollution due to the emission of the greenhouse gases. Thus, the European Commission presented a set of proposals to align the European Union climate and energy policies with the goal of reducing net greenhouse gas emissions by at least 55% by 2030 compared with those of 1990 [1]. In light of this, finding green and clean energy solutions became mandatory in the last few years. Hydrogen is one of the possible candidates for clean alternative fuel due to several of its advantages including [2]:

It's one of the most abundant element on earth Its combustion produces only water and heat and thus greenhouse gas emissions are eliminated It can be produced by electrolysis therefore the environmental impact related with its production is reduced One of the potential applications of hydrogen can be as vehicle motor fuel. Toyota introduced in 2014 the first commercial vehicle powered by hydrogen at the end of 2014 c.f. Figure 1 for which the total number of fuel cells used weighs 56 Kgs. With all its advantages, the main problem of hydrogen relies in its storage. It can be stored either in liquid state or as a compressed gas. The liquid hydrogen should be carried out at very low temperatures in cryogenic state since it starts to boil at -253°C [2]. This results in the need of well protected and insulated reservoirs to avoid hydrogen leakage and ensure the low temperatures required. As a compressed gas, hydrogen has a low volumetric energy density compared with fossil fuel. Therefore for the same energy output, the required hydrogen volume is bigger than that of hydrocarbons which imposes the need to bigger tanks. With an increased tank size, the material cost increases and more hydrogen is required to transport this tank along with the car, thus reducing the system efficiency.

In view of the above, reducing the reservoirs' weight is therefore a necessity in order to increase the efficiency of hydrogen powered cars. With their light weight, high specific strength and corrosion resistance, composite materials can be considered as a solution to be explored. The presence of these materials has continued to increase over the past 30 years in different sectors notably in the aeronautical and transportation industries.

Thermoset polymer composites were widely used since 1980 due to many factors including for example their better performance at elevated temperatures compared with thermoplastics. Figure 2 shows the variation of the modulus of several polymers with temperature. Phenolic polymer present the highest modulus which remains intact until high temperatures compared with the thermoplastics such as Nylon.

Figure 2 -Variation of the storage modulus with temperature for different polymers [3] Two of the major drawbacks of the thermoset resins include their long curing cycles and their inability to be recycled. In contrast, thermoplastic polymers have shorter consolidation cycles and can be potentially recycled [4]. Their processing includes in-situ consolidation and became increasingly interesting for the industrial sector during the last two decades. The principle factor is the significant productivity gains with the lack of the highly time-consuming resin curing step required in the case of thermoset polymers. Productivity is indeed a challenge for aeronautics and automobile manufacturers with the rise of the global networks and the world population. Hence, the increasing interest in thermoplastic composites mainly because they can be shaped in one-step upon cooling from the melt. Moreover, they can be recycled and reshaped because the liquid-solid transition is reversible. In addition, thermoplastic composites have an infinite shelf life which adds to the advantages over thermosets that need to be stored at low temperatures for a definite time.

Nonetheless, their processing occurs at elevated temperatures (the melting temperature of PolyEtherEtherKetone PEEK is 343°C) therefore increasing the production cost. In order to conserve the economy that can be realized by omitting the autoclave step, the aim is to present a thermoplastic material/process combination that is interesting in terms of productivity costs and meets the expectations of manufacturers and environmental authorities. It can be seen in Figure 3 that PEEK in particular presents a good candidate for thermoplastic composite structures with a good strength/price ratio. Given the above, PEEK polymer composites present potentials for application to produce hydrogen storage reservoir. [5] Thermoplastic composites reservoirs for hydrogen storage are manufactured via filament winding with insitu consolidation. This process, similarly to the tape placement, include several heating and cooling steps at elevated temperatures in order to reach the desired structure thickness. They're generally associated with high cooling rates due to the high processing temperatures and processing speeds. Moreover, thermal and crystallinity gradients are present during the process. Other inhomogeneities are associated with these processes such as the stacking sequence of the laminate, the local heating and the geometric differences between the upper and lower surfaces due to the presence of the mandrels and compaction rollers. All these factors present driving sources to the formation of residual stresses in the composite structure. These stresses can induce geometric defects through distortions caused under the bending moment resulting from the residual stresses gradient. Moreover, they can affect the performance of the structure mainly the properties governed by the resin behavior such as the transverse tensile strength [6]. Therefore, it is important to understand the stress state during and after the process. It should be noted that the residual stresses act on three levels: micro (fiber/matrix), meso (ply) and macro (structure).

In this context, Cetim, the Technical Centre for Mechanical Industry, aims to provide industrial solutions on the current development of hydrogen thermoplastic composite tanks. It inaugurated in 2014 its filament winding platform 'SPIDE-TP' used to produce thermoplastic composite reservoirs to store hydrogen. They collaborated with academic partners, the GeM (Institut de Recherche en Génie Civil et Mécanique) and LMT (Laboratoire de Mécanique et Technologie) in a common laboratory called 'COMPINNOV'. The main objective is to understand the stress state in thermoplastic composite structures during and after processing on the SPIDE-TP notably on the ply and the structure level. Therefore two thesis subjects emerged financed by the Cetim:

The research work presented in this manuscript, which deals with the residuals stresses and the material behavior after the process on the ply-level; The research work conducted by Mr. Livio DIGENNARO, who is based in ENS Paris-Saclay and studies the stress state on the structure level. This thesis takes place in the GeM laboratory and is englobed in the COMPINNOV common laboratory. The collaboration between the two research projects is based on a mutual information and knowledge sharing on different material aspects.

As previously evoked, the research work presented in this manuscript focuses on the understanding of the residual stress state and material state on the ply-level. The thesis manuscript is divided into four chapters. The first chapter offers a synthesis of the state of the art concerning carbon/PEEK composites and the filament winding and tape placement processes. It also focuses on the residual stresses: types, origins, effects in addition to the experimental and numerical methods which present in the literature to estimate the residual stresses during such manufacturing processes.

The second chapter presents the mechanical characterization techniques used to study the post-processing material behavior, in addition to the description of the method used to measure the residual strains and deformations generated by the process.

In the third chapter, the results obtained from the methods described in Chapter 2 are presented and provide the experimental characterization of the properties and behavior of the composite, in addition to the curvatures generated due to residual stresses.

Finally, the last chapter proposes a multi-physics model that incorporates the process's thermal aspect and the related crystallinity behavior in addition to the mechanical behavior of the composite. The model is validated by comparing the curvatures measured in Chapter 3 with the simulated ones. This allows the study of the development of residual stresses during and after the process. In addition to the residual stresses and strains studied, a parametric study is presented in order to understand the 1 st order parameters influencing the formation of residual strains.

1 Chapter 1: Literature review

Introduction

This chapter provides an introductory overview of the state-of-art experimental and numerical technologies related to the study of the material state and the residual stresses resulting from the filament winding and tape placement processes.

In the first part, a brief review and description of the filament winding and tape placement processes is presented. The second part focuses on the description of the composite material used in the study, its behavior and properties. The third and final part is dedicated to the description of the residual stresses and is divided into several sub-parts. The sources and types of residual stresses are first presented followed by a general overview of these stresses' effects on the composites. Next, the experimental methods used in the literature to evaluate the residual stresses and strains are summarized. The final part of this state-of-art highlights the numerical models reported in the literature for the simulation of the residual stresses either in filament winding or tape placement.

Filament winding

The high-end applications for composites were traditionally attributed to thermosets, for which the processing emerged in the beginning of the 1940s [1]- [2]. For a long time, these materials were manually manufactured which is rather a time and effort consuming task. This includes a careful lay up of unidirectional or woven plies along the desired directions and stack these plies on top of each other. In addition, the application of an external pressure and heat is mandatory for some processes. As a result and while the use of thermoset reinforced composites ensures the production of high-quality lightweight resistant structures, it is associated with high manufacturing cost. The use of thermoplastic composites eliminates the need for long curing cycles of the entire structure in an autoclave or convection oven. Therefore, these materials emerged and their use began in the 1980s for high-end applications [3].

Thermoplastic composites processing includes heating the material above the polymer's fusion temperature 𝑇 𝑚 while applying a suitable consolidation pressure, followed by the cooling down to the ambient temperature. These one-step processes with in-situ consolidation are continuously developed and improved. Their main advantages are the high production rate and energy savings compared to the processes in case of thermoset composites.

Various processes with in-situ consolidation existed traditionally for thermoplastic structures such as autoclave [4] and compression molding [5]- [6]. Their principal advantages and disadvantages are presented in Table 1.1.

In addition to the conventional fabrication processes, other techniques can be found: the automated tape placement and filament winding. The main steps are similar for both techniques and are discussed in the following.

Process Advantages Disadvantages

Autoclave  Large and complex structures can be obtained  Consistency of components produced can be achieved  Automation is difficult  Long consolidation time is necessary due to convection losses [4] Compression molding  Short processing cycles  Automation is possible  Process relate defects such as waviness and wrinkling are commonly observed [5]- [6] Table 1. 1

-Comparison of advantages and disadvantages between autoclave and compression molding for thermoplastic composites

The main advantages of the filament winding and tape placement over the autoclave and press molding are [7]:

 Fiber orientation accuracy is enhanced compared to autoclave and compression molding  Possibility of in-situ consolidation so all the process can be reduced into a single task  Constant product quality

For the automated filament winding process, there exists two different winding technologies based on the material used [8]- [9]:

 "Dry" winding, where prepreg are directly layed around the mandrel  "Wet" winding where the fibers are impregnated in a resin bath before being placed on the mandrel

The wet-winding process is usually applied to thermoset materials, however few studies discussed the possibility of processing thermoplastic composites via this type of process [10]. According to the thesis's scope, only the dry winding will be considered and discussed in the following parts.

Heating phase

During this phase, the substrate's upper surface and the incoming ply's lower surface are heated with an external source. Two important factors affect the welding quality: the processing temperature and the dwell or exposure time. The material should be heated above 𝑇 𝑚 for a sufficient time to initiate the mechanisms responsible for the welding. Low temperatures and/or short exposure time result in a low welding quality due to insufficient bonding. On the other hand, high temperatures and/or long exposure time initiate the decomposition of the polymer [11]- [12]. These two parameters are influenced by several processing conditions including the heating source, the processing temperature, the heating zone, the deposition speed and the incoming ply and heat source angle. Therefore, the choices of these conditions influence directly the bonding quality and the machine's productivity.

The melting point for semi-crystalline polymers can reach high temperatures such as 343°C for PEEK [13]- [14]. Therefore, powerful heating sources are required to supply sufficient heat flux during a short time. The heating sources used during filament winding/tape placement were extensively studied. They involve hot gas torch [11]- [15]- [16]- [17]- [18], infrared spot heaters [19]- [20]- [21], near-infrared diode laser [22]- [23]- [24]- [25]- [26]- [27] and lately LED [28].

Hot gas torches were the first to be used in automated filament winding and tape placement. The main advantage of this heating system is its low cost when compared with the other systems [11]- [29]. However, this cost becomes rapidly increasing due to the need to use inert gases such as nitrogen to prevent oxidation at elevated temperatures [11]. To increase the efficiency of this system, some solutions include preheating the tapes prior to the winding [18]- [30] or the use of a heated mandrel [31].

The infrared (IR) lamps transfer heat by radiation. They are easy to operate and cheaper than laser.

Experiments in [19] showed a better efficiency compared to hot air blower since the output heat flux is more focused on the targeted area.

Laser heating systems include the use of the CO2 laser and the near-infrared diode lasers. The main difference between the two systems is their wavelength: 10.6 µm for CO2 and 780-1064 nm for the diode.

The disadvantage with the CO2 laser is the capacity of polymers to absorb irradiations at this wavelength which may lead to their burning and oxidation [32]. On the other side, the diode laser c.f. Figure 1.1 are absorbed by the fibers and the polymer is heated by conduction therefore the burning risk may be reduced [32]. While the laser heating system is more expensive than hot gas torch and IR [11], it allows heating a more localized and controlled zone due to its high irradiation intensity [11]- [32]. In addition, the laser's low response time allows faster production rates and the possibility of controlling the temperature by changing the heat input to avoid overheating [11]. The LED-based heating system is relatively new and, to the author's knowledge, hasn't been much discussed. A numerical model was presented in [28] to predict the temperature distribution using a LED in order to improve the prediction accuracy of the temperature distribution. It was stated that, like the laser, the LED-heating system allows for fast processes due to its low response-time.

The temperature distribution between the tape and the substrate during heating is important since it determines the fusion and the degradation of the polymer. The correspondance between the heating temperature and the dwell time was reported by Cogswell for APC-2 under non-oxidizing environemnts such as nitrogen [33]. The results are presented in Table 1.2 and show that the exposure time should decrease when the exposure temperature increases to avoid the polymer's degradation. During heating, both the incoming tape and the substrate are melted with the heat source before getting in contact and initiating their consolidation.

Consolidation phase

In-situ consolidation takes place at the nip-point and is determinant for the final structure quality. In order to achieve the consolidation, two phenomenon should take place: the intimate contact and the polymer's healing [32]. Both phenomena are temperature dependent and complicated due to the successive deposition of plies associated with repeating heating and cooling.

When two surfaces get in contact without an external pressure applied, voids are formed on the interface. The formation of these voids can be prevented through the use of compaction rollers [32]. The roller's material and geometry affect the consolidation pressure applied and therefore the bonding quality. Rigid metallic rollers were widely used during automated filament winding and tape placement [16]- [13]. Other compaction systems were introduced later such as a conformable elastic roller [34]- [13] or a series of rollers for which the applied pressure can be controlled [35]. A better consolidation quality was observed when using deformable roller compared to that obtained using a rigid roller in [36] during automated tape placement. In fact, non-deformable roller leads to a shadow area prior to the nip-point [13] associated with a drop of temperature prior to the nip-point. Furthermore, the distribution of the heat between the tape and the substrate due to reflectance when using laser for example is condemned by the nip-point area geometry [37].

Intimate contact

When two surfaces are put in contact, the asperities on their rough surfaces lead to an imperfect contact between them [38]. This might result in air gaps trapped between the tape and the substrate leading eventually to low quality bonding [39]. Therefore, it is essential to define the intimate contact which is the proportion of the surfaces actually in contact. This proportion increases when the asperities are deformed under a pressure applied, hence the role of the compaction roller to apply the corresponding pressure. A number of models were developed to study the intimate contact [39, [38] & [40]. These models were later applied and compared to experimental investigations [41], [23], [13] & [42]. The intimate contact degree was mostly measured at the interfaces between the different plies based on micrographs of the studied area. [38] Dara & Loos [39 developed a model based on Weibull function to model the intimate contact and study the effect of processing parameters such as the pressure, temperature and contact time for AS4/P1700 prepreg. An intimate contact degree 𝐷 𝑖𝑐 was defined and corresponds to the ratio of the area in actual physical contact to the total interface area. However, this model was complicated and applies only for isothermal and isobaric condition, so a simplified version was proposed by Lee & Springer [38] where the asperities of the surface is represented through a geometric series of successive rectangular elements as shown in Figure 1.2.

In this model, the intimate contact degree is dependent on the width and height of the rectangular elements.

With the pressure applied, the initial length 𝑎 0 is reduced to 𝑎 and the initial width 𝑏 0 increases to 𝑏 hence the increase in the contact area. The intimate contact degree is defined in this model by Eq. 1.1.

𝐷 𝑖𝑐 = 𝑏 𝑤 0 + 𝑏 0 Eq. 1.1 𝑤 0 is the space between two consecutive rectangular elements. A more generalized expression (Eq. 1.2) for the intimate contact degree was presented in [38] and correlated between 𝐷 𝑖𝑐 , the applied pressure 𝑃 𝑎𝑝𝑝 , the viscosity of the material µ 𝑚𝑎𝑡 and the time 𝑡 c.f. Eq. 1.2

𝐷 𝑖𝑐 = 1 1 + 𝑤 0 𝑏 0 [1 + 5𝑃 𝑎𝑝𝑝 µ 𝑚𝑎𝑡 (1 + 𝑤 0 𝑏 0 ) ( 𝑎 0 𝑏 0 ) 2 𝑡] 1/2
Eq. 1.2

Mantell & Springer [43] modified the model to take into account the variation of the applied pressure and the material's viscosity with time. The modified model was applied by [40] to an AS-4 graphite fiber reinforced PEEK matrix composite. However, the average roller pressure and the isothermal condition considered in the model of Mantell & Springer in [43] are replaced by the actual pressure and temperature profiles calculated in [40]. Finally, Pitchumani & Yang [40] suggested that the surface of an AS4/PEEK composite presented a multiscale asperity structure which was observed via a profilometric scan to which they applied a Fast Fourier Transform. The model takes into account the effect of processing parameters such as the applied pressure and the material's viscosity on the intimate contact degree and the surface geometry. [41] The different models proposed were later applied by researchers to study the intimate contact during filament winding and tape placement [41]- [23]- [13]- [42]. In [41], the intimate contact degree calculated from the models of Lee & Springer [38] and Pitchumani & Yang [40] were compared to experimental measurements for AS4/PEKK composites during tape placement. The results showed lower intimate contact degree for higher processing speeds c.f. Figure 1.3. The authors attributed this to the high cooling rates experienced during the process (1000 °C/s) leading to an insufficient time above the melting temperature which prevents the flattening of asperities.

Figure 1.3 -Evolution of the intimate contact degree with different passes during tape placement for different processing velocities

Similar values for the intimate contact degree were observed in [13] for carbon-fiber/PEEK during tape placement. The low intimate contact degree was attributed to the presence of a shadowed area prior to the nip-point and the insufficient exposure time to ensure the deformation of the asperities on the surface. Similarly, the rectangular based model of Lee & Springer [38] was applied in [41] where the effect of the multiple heating and cooling during successive deposition of plies was elaborated. It was stated that with the reheating and re-melting the intimate contact degree can increase. The effect of the velocity correlated with the results in Figure 1.3: with an increase in the processing velocity, the dwell time is shortened and thus asperities wouldn't have time to deform resulting in a lower intimate contact degree.

Healing or intermolecular diffusion

Healing or intermolecular diffusion defines the creation of an adhesion between organic polymers. The adhesion is established through the inter-diffusion of polymer chains between the two polymers. It can only occur in regions where intimate contact has been established. The inter-diffusion process is described by the De Gennes's reptation theory [44] which was later reproduced by Wool [45]. The model describes the molecule's movement and the constraints to this movement due to the presence of neighboring molecules [44]- [45]- [46]. A characteristic time called the reptation or renewal time 𝑡 𝑟𝑒𝑝 was defined and corresponds to the time needed for a molecule to change its configuration. This time depends on the temperature and the polymer's molecular weight.

De Gennes's theory states that that the chain is surrounded by fixed obstacles. The chain is then free to move between the fixed obstacles but cannot cross these obstacles. The only possible chain movements then correspond to the migration of defects along the chain. De Gennes postulates that the chain threads its way through a permanent and non-deformable tube. It governs the molecular motion within the bulk to the motion of the molecular chains across the interface. Wool's theory [46] extended the De Gennes's theory to model the motion of the polymer chains across a polymer-polymer interface. Based on these two assumptions, several definitions for the degree of healing 𝐷 ℎ were presented by different researchers [47]- [43]- [48]- [49]. The definition of the healing degree given by Pitchumani & Yang [48]- [49] for nonisothermal healing condition is presented in Eq. 1.3, 𝑡 𝑤 and 𝑇 are respectively the welding time and the temperature.

𝐷 ℎ (𝑡) = [∫ 1 𝑡 𝑤 (𝑇) 𝑑𝑡 𝑡 0 ] 1/4
Eq. 1. 3 Yang & Pitchumani [48] defined a starting temperature at which healing can be initiated: glass transition for amorphous polymers and melting temperature for semi-crystalline polymers. Most of the work to predict the healing during automated tape placement for semi-crystalline composites stated that healing can occur only above the melting temperature for semi-crystalline polymers [35]- [50]- [51]. Indeed, the presence of crystalline phase during the process can limit the mobility of chains below the melting temperature [52]. This can be true assuming a complete melting of the crystals during heating. On the other hand for amorphous or low-crystallinity polymers, healing can occur even below the melting temperature due to the high mobility of amorphous chains [13]. In fact, Stokes-Griffin & Compston showed in [13] that a temperature threshold at the melting resulted in an underestimation of this healing degree and an overestimation when the threshold is defined at the glass transition. Therefore, it was stated that the influence of crystallization and melting of polymers on the healing should be taken into account when dealing with processes including fast cooling and heating rates.

Coupled Bonding

Intimate contact and healing are two coupled phenomenon. The realization of the intimate contact is mandatory for the healing to occur. In the previous sections, the different models used in the literature for the study of each phenomenon alone were presented. It is, however, important to have a coupling model taking into account both intimate contact and healing in order to define a consolidation degree which is a more global parameter than intimate contact and healing alone. For this definition, two bonding models were presented. The first one was presented in [43] where the bonding degree 𝐷 𝑏 was defined as the product of the intimate contact and healing degrees as presented in Eq. 1.4

𝐷 𝑏 = 𝐷 𝑖𝑐 * 𝐷 ℎ Eq. 1.4 The model was applied to APC-2/AS4 composites during filament winding and press molding and was later compared to experimental measurements in [53]. It assumes that intimate contact and healing occur simultaneously, which is not accurate since healing can only occur once the intimate contact has been established as already discussed. To take into account this time difference between the two phenomenon, Pitchumani & Yang [49] proposed a coupled bonding model with an incremental healing of the established intimate contact. This incremental model was later used by researchers to predict and study the bonding quality during in-situ consolidation processes [43]- [START_REF] Butler | An Analysis of Mechanisms Governing Fusion Bonding of Thermoplastic Composites[END_REF]- [51]- [13]. The choice of the bonding model parameters depends on the process. For slow processes such as resistance welding or compression molding, the long processing time allows the establishment of an intimate contact which will dominate the degree of bonding so the healing time can be neglected. However, for processes such as automated tape placement and filament winding, the processing speed is very high and the polymer is heated above its melting temperature at a rate that may exceed the one needed to initiate the intimate contact. As a consequence, authors tend to consider the healing time on the same order as the intimate contact time and therefore it cannot be neglected when studying the bonding degree [32].

Cooling phase

The last step during in-situ consolidation is the cooling phase. It can start while the roller pressure is still applied due to the thermal sink effect of the roller and continues after the pressure release until the deposition of the next ply. During the cooling phase, a semi-crystalline matrix tends to crystallize depending on the cooling rate. Furthermore, the thermal gradients and the anisotropic thermal behavior of the composite's layers lead to the development of residual stresses.

It is well known that the composite properties are defined by those of its fibers and resin. The resin's properties are directly related to the thermal history and its crystallization degree. The crystallization behavior of semi-crystalline thermoplastics is a cooling rate-dependent phenomenon and its description during processes such as tape placement and filament winding with multiple heating and cooling is mandatory to understand the changes occurring during these processes.

The correlation between the degree of crystallinity, the thermal history and the material properties was extensively studied in the literature through different models such as [START_REF] Lawrence | Prediction of Residual Stress in Continuous Fiber Semicrystalline Thermoplastic Composites: a Kinetic-Viscoelastic Approach[END_REF], [START_REF] Nejhad | Prediction of Process-Induced Stresses for In-Situ Thermoplastic Filament winding of cylinders[END_REF], [START_REF] Nicodeau | Continuous welding modeling of thermoplastic matrix composites[END_REF], [43], [53] & [START_REF] Sonmez | Modeling of Heat Transfer and Crystallization in Thermoplastic Composite Tape Placement Process[END_REF]. In addition, residual stresses' development during the cooling phase of the processes was studied by different authors such as [START_REF] Lawrence | Prediction of Residual Stress in Continuous Fiber Semicrystalline Thermoplastic Composites: a Kinetic-Viscoelastic Approach[END_REF], [START_REF] Nejhad | Prediction of Process-Induced Stresses for In-Situ Thermoplastic Filament winding of cylinders[END_REF], [START_REF] Nicodeau | Continuous welding modeling of thermoplastic matrix composites[END_REF], [START_REF] Cirino | In-situ consolidation for the thermoplastic composite ring-residual stress state[END_REF], [START_REF] Mantell | Manufacturing Process Models for Thermoplastic Composites[END_REF], [START_REF] Nejhad | Prediction of Process-Induced Stresses for In-Situ Thermoplastic Filament winding of cylinders[END_REF], [START_REF] Sonmez | Analysis of Process-Induced Residual Stresses in Tape Placement[END_REF], [27]. The different thermal, crystallization and melting models are detailed later in this chapter in addition to the ones describing the development of the residual stresses during the cooling phase of tape placement and filament winding.

While the final quality of the material is dependent on the whole process, it was established that the cooling rate is the main factor influencing the crystallinity. High cooling rates lead to lower crystallinity levels which affect the mechanical properties of the product [START_REF] Talbott | The Effects of Crystallinity on the Mechanical Properties of PEEK Polymer and Graphite Fiber Reinforced PEEK[END_REF]. The residual stresses were also shown to be directly related to the crystallinity fraction and thermomechanical properties in addition to the thermal history especially during the cooling phase of the process [START_REF] Lawrence | Prediction of Residual Stress in Continuous Fiber Semicrystalline Thermoplastic Composites: a Kinetic-Viscoelastic Approach[END_REF], [START_REF] Nejhad | Prediction of Process-Induced Stresses for In-Situ Thermoplastic Filament winding of cylinders[END_REF], [START_REF] Nicodeau | Continuous welding modeling of thermoplastic matrix composites[END_REF], [START_REF] Cirino | In-situ consolidation for the thermoplastic composite ring-residual stress state[END_REF], [START_REF] Mantell | Manufacturing Process Models for Thermoplastic Composites[END_REF], [START_REF] Nejhad | Prediction of Process-Induced Stresses for In-Situ Thermoplastic Filament winding of cylinders[END_REF], [START_REF] Sonmez | Analysis of Process-Induced Residual Stresses in Tape Placement[END_REF], [27]. Therefore, an accurate prediction of the residual stresses cannot be obtained if the thermal history during the process isn't well described and understood. To this end, the following section presents the simulated and measured thermal history during tape placement and briefs some key works on this subject.

Thermal history modelling

The objective of the thermal model is to predict the thermal history during the process through the resolution of the energy balance equation. The simplifications, the boundary conditions and the adopted solution method differ between the researchers based on the final purpose of their work. A number of heat transfer models have been proposed for either the tape placement or the filament winding of thermoplastic composites. The techniques used can be classified into categories based on three aspects:

Dimensional choice : 1-D, 2-D and 3-D Resolution method: Finite Elements, Finite Differences, Analytical Approach: Eulerian or Lagrangian. The coordinate system is supposed to be attached to the roller in the 1 st one and to the mandrel in the 2 nd one.

A number of authors focused on the resolution of a 1-D transient or steady-state thermal model to obtain the temperature variation with time through the thickness of a composite [35], [START_REF] Khan | Experimental and Simulative Description of the Thermoplastic Tape Placement Process with Online Consolidation[END_REF] and [START_REF] Weiler | Transient thermal analysis of laser-assisted thermoplastic tape placement at high process speeds by use of analytical solutions[END_REF]. Tierney & al. [35] solved the 1-D steady-state model. Khanh & al. [START_REF] Khan | Experimental and Simulative Description of the Thermoplastic Tape Placement Process with Online Consolidation[END_REF] improved the model developed in [35] to account for the variation of temperature with time during tape placement of PEEK/CF. The model was validated by comparing the simulated temperature evolution on the surface of the 13 th and 14 th ply when the 14 th ply was being placed to those measured by a thermocouple placed between the 13 th and 14 th layers of a composite laminate. Figure 1.4 presents the simulated temperature profiles during the deposition of the 15 th ply.

. 4 -Temperature profiles for a 14-ply substrate during the tape placement of the 15-th layup [START_REF] Khan | Experimental and Simulative Description of the Thermoplastic Tape Placement Process with Online Consolidation[END_REF] 21

The main observations from the results presented in Figure 1.4 are:

For the different layups, the same trend is observed: the temperature increases before the nip-point until reaching its maximum temperature. The top surface of the substrate (14 th layup) goes up to 500°C, high above the PEEK's 𝑇 𝑚 . The incoming tape temperature is higher than the substrate surface's temperature (this observation was numerically simulated and experimentally validated).

A temperature gradient occurs through the laminate thickness with a temperature decrease between the top surface (exposed to the heat source) and the bottom surface (in contact with the mandrel).

Weiler & al. [START_REF] Weiler | Transient thermal analysis of laser-assisted thermoplastic tape placement at high process speeds by use of analytical solutions[END_REF] developed a 1-D transient model to study the temperature distribution across the thickness of a PEEK/CF composite during tape placement. The study focused on the problem of heat accumulation in thin tapes and substrates, details of the model are described in the final chapter. The trend observed is however similar to the one observed in [START_REF] Khan | Experimental and Simulative Description of the Thermoplastic Tape Placement Process with Online Consolidation[END_REF] for the heating, cooling and the thermal gradients across the thickness of the composite.

More recently. Kok [14] proposed a 1-D model where the thermal problem is solved using Finite Differences (FD) for a laser assisted tape placement process. The predicted values were compared to experimental data from an infrared camera and thermocouples placed at different locations. Just as the previously cited works, a thermal gradient occurs through the laminate thickness. Finally, Kok [14] studied the effect of the roller on the temperature distribution, the results showed a drop in the nip-point temperature when a rigid roller is used compared to the use of a deformable roller. In fact, the use of a rigid roller leads to a shadowed area at the nip-point area.

The 1-D models were found to be capable of predicting the overall thermal aspect through the thickness of the composite during the process, in addition to their lower calculation time. On the other hand, these models cannot reflect the real process due to the multiple assumptions and simplifications adopted which led other researchers to adopt 2-D and 3-D models.

Nejhad & al. [START_REF] Ghasemi Nejhad | Thermal Analysis of in-situ Thermoplastic Composite Tape Laying[END_REF] solved the 2-D heat transfer problem during the tape laying analytically and numerically using FD for an APC-2 composite. The temperature profiles were predicted along the length (laying direction) and the thickness of the composite. Schäkel & al. [66] also used the FD scheme to solve the 2-D thermal problem. The results were compared to those measured by thermocouples implemented at the interfaces in Figure 1.5. Figure 1.5-(a) presents the comparison between the simulated temperature evolution and the data from a thermocouple fixed on the liner interface (named Tc1-1) during the deposition of layers 2 to 5. Other authors used the Finite Elements method to solve the 2-D thermal model in a steady-state [37], [START_REF] Sonmez | Modeling of heat transfer and crystallization in thermoplastic composite tape placement process[END_REF] or transient formulation [START_REF] Zhao | Thermal stresses in rings of thermoplastic composites made by automated fiber placement process[END_REF], [START_REF] Stokes-Griffin | A combined optical-thermal model for near-infrared laser heating of thermoplastic composites in an automated tape placement process[END_REF]. Stokes-Griffin & al. [START_REF] Stokes-Griffin | A combined optical-thermal model for near-infrared laser heating of thermoplastic composites in an automated tape placement process[END_REF] presented the results adapted from their previous 2-D FE model [34] during the tape placement of AS4/PEEK composite (𝑉 𝑓 = 0.55). the simulated results were in a good correlation with the ones measured using an infrared thermal camera.

Barasinski [START_REF] Barasinski | Modélisations du procédé de placement de fibres composites à matrice thermoplastique[END_REF] simulated the temperature distribution during tape placement by solving the thermal problem using the PGD (Proper Generalized Decomposition) method. She validated the presence of a thermal resistance between the different plies by comparing the temperature evolution at the interfaces before and after applying a final consolidation step at the end of the process. The two evolutions were shown to be different therefore suggesting the presence of a thermal contact resistance between the plies. More recently, Chinesta & al. [START_REF] Chinesta | First steps towards an advanced simulation of composites manufacturing by automated tape placement[END_REF] and Dedieu & al. [72] adopted the Proper Generalized Decomposition to solve the thermal model using constant isotropic material properties.

A more precise description of the thermal history can be obtained when the heat problem is solved in 3-D.

James & Black [START_REF] James | Thermal Analysis of Continuous Filament-Wound Composites[END_REF] predicted the steady-state temperature during the filament winding of APC-2 composites using FD. Later, Toso & al. [START_REF] Toso | Thermal Phenomena in Fiber-reinforced Thermoplastic Tape Winding Process: Computational Simulations and Experimental Validations[END_REF] studied the transient thermal model during the filament winding of thermoplastic composites. The solution was conducted in ANSYS using FE and the results were validated with experimental ones using infrared pyrometers.

The different models presented showed their ability to predict the temperature distribution during tape placement or filament winding. A general trend can be deduced from the different works regardless of the different values obtained due to the simplifications and assumptions adopted in each model:

Progressive heating until a maximum temperature as the material approaches the laser (a temperature gradient especially across the thickness was observed in the different studies with a decreasing temperature as the material is placed further away from the heated surface) Progressive cooling as the laser and the consolidation roller start to move away from the material (the temperature gradients are also observed during the cooling phase)

The substrate undergoes multiple reheating and re-cooling as a new layer is being placed with similar trends. The maximum temperature decreases with each additional layer With this complicated thermal history to which the composite is subjected during its processing, it seems crucial to well describe and understand it in order to obtain the resulting residual stresses. The resolution of the thermal equation starts with the description of the composite thermal properties which are presented later in this chapter. Moreover, in order to obtain the residual stresses and strains, the description of the composite's thermomechanical properties is necessary. To this end, a brief review of the composite thermomechanical models in addition to the crystallization and melting models is described in this chapter and finally a general presentation of the residual stresses and the related experimental and numerical key works.

Composite material system

This section serves as a brief collection and summary of the works on AS4 carbon fiber, PEEK and CF/PEEK composites to understand their behavior.

Generalities on AS4 carbon fiber

Carbon fibers are usually used in technological applications where light-weighted and resistant structures are required such as aeronautics and automobile. AS4 is a high quality based carbon fiber manufactured by Hexcel. Properties of carbon fibers include high stiffness and tensile strength, light-weight, chemical and thermal resistance. The principal properties for AS4 carbon fibers along their longitudinal direction are presented in Table 1.3 based on the producer's data sheet [START_REF]Hexcel website[END_REF].

Density (Kg/m 3 ) 1.79

Tensile modulus ( GPa)

Longitudinal 231

Radial 13

Tensile strength (MPa) 4413

Thermal expansion coefficient CTE (.10 Studies on PAN based carbon fibers showed that the behavior remains unaffected by temperature until a minimum value of 1000°C. Sauder & al. [START_REF] Sauder | The Tensile Behavior of Carbon Fibers at High Temperatures Up To 2400°C[END_REF] performed tensile tests on different carbon fibers at different temperatures. The results concern the FC2 fiber since it is PAN based so the behavior is similar to that of AS4. In fact, the Young's modulus isn't affected by the temperature until nearly 1200°C for FC2. The application of the carbon fiber in this research is for temperatures not exceeding 500°C, as a result the constant properties of carbon fibers will be adopted in the following sections. Similarly, several researchers considered constant properties for carbon fibers in different applications [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF], [START_REF] Kyriakides | On the compressive failure of fiber reinforced composites[END_REF], [START_REF] Barnes | The formation of residual stresses in laminated thermoplastic composites[END_REF], [START_REF] Lamèthe | Etude de l'adhésion de composites thermoplastiques semi-cristallins; application à la mise en oeuvre par soudure[END_REF], [START_REF] Karami | Micromechanical study of thermoelastic behavior of composites with periodic fiber waviness[END_REF] and [START_REF] Ersoy | Development of the properties of a carbon fibre reinforced thermosetting composite through cure[END_REF].

Another characteristic of carbon fibers is their anisotropic thermal and mechanical behaviors. Carbon fibers exhibit an anisotropic thermal contraction behavior. Barnes & al. measured in [START_REF] Barnes | Thermal expansion characteristics of PEEK composites[END_REF] a longitudinal CTE of -0.22• 10 -6 °C-1 and a transverse CTE of 30• 10 -6 °C-1 for AS4 carbon fibers. These values were later used by Barnes & Byerly in [START_REF] Barnes | The formation of residual stresses in laminated thermoplastic composites[END_REF]. Finally, the in-plane shear modulus for carbon fibers 𝐺 𝐹𝐿𝑇 was generally considered as a constant value and estimated around 27 MPa [START_REF] Lawrence | Prediction of Residual Stress in Continuous Fiber Semicrystalline Thermoplastic Composites: a Kinetic-Viscoelastic Approach[END_REF].

Generalities on neat resin PEEK

PEEK is a semi-crystalline thermoplastic for which the molecular structure is presented in Figure 1.6. In general, PEEK's absolute crystallinity doesn't exceed 40%. Several grades of PEEK are commercialized such as PEEK 90G, 150G and 450G which differ mainly by their melt vicosities. It is industrially and scientifically interesting due to many factors including its high chemical resistance, high impact resistance, good wear resistance and fatigue resistance. In addition, PEEK presents higher mechanical properties at ambient temperature compared to other polymers as presented in Table 1.4. The macromolecular structure of the PEEK made of benzene rings and ketone bonds is the key factor to its high resistance to thermal and chemical attacks. In fact, PEEK's thermal degradation under non-oxidizing environment doesn't occur until nearly 550°C [START_REF] Lu | Thermal stability and thermal expansion studies of PEEK and related polyimides[END_REF]. The degradation temperature decreases down to 450°C under oxidizing environments [START_REF] Chen | Analysis of crystallization kinetics of poly (ether ether ketone)[END_REF]- [START_REF] Nandan | Poly(ether ether ketone)/poly(aryl ether sulphone) blends: Thermal degradation behaviour[END_REF]. Important weight loss occurred starting at 537°C in [START_REF] Nandan | Poly(ether ether ketone)/poly(aryl ether sulphone) blends: Thermal degradation behaviour[END_REF] and a complete weight loss at 570°C under air in [START_REF] Chen | Analysis of crystallization kinetics of poly (ether ether ketone)[END_REF].

Property

PEEK PTFE PI

Tensile modulus ( GPa) 2.8 0.4 2.5

Flexural modulus (GPa) 3.7 1.5 3.2 The macromolecules are entangled and held together through weak hydrogen and Van der Waals bonds. Under the effect of heat, the weak bonds break, the polymer changes from a glassy state (hard and brittle) to a rubbery state (soft and malleable). By further increasing the temperature, the macromolecules of the crystalline phase become disorganized, the polymer becomes totally amorphous in its molten state. On cooling, the macromolecules reorganize and the weak bonds reform, the polymer returns to the glassy state. Following this, the properties of the PEEK may be highly affected by temperature and the understanding of their evolution with it may seem crucial to determine their applications and limits.

Thermal expansion coefficient

The thermal expansion behavior of a semi-crystalline polymer relies on the behavior of amorphous and semi-crystalline phases. Choy et al. [START_REF] Choy | Thermal Expansion of Poly (etherether-ketone) (PEEK)[END_REF] measured the CTE of an amorphous and a semi-crystalline PEEK 450G using thermomechanical analysis (TMA). The results are presented in Figure 1.7 and show that the CTE exhibits an important increase when passing through the glass transition temperature. This increase is more pronounced in the case of amorphous and semi-crystalline PEEK (can go up to 3 times) than in the case of crystalline PEEK. The increase of the CTE with temperature is due to the molecular relaxation of the amorphous phase leading to an increase in the specific volume. For these temperature ranges, the amorphous phase tends to relax above 𝑇 𝑔 leading to an expansion and an increase in the CTE value while the crystalline phase restrains this expansion. Therefore, an increase in the polymer crystallinity leads to a reduction in the CTE especially above the glass transition. [START_REF] Choy | Thermal Expansion of Poly (etherether-ketone) (PEEK)[END_REF] and [START_REF] Lu | Thermal stability and thermal expansion studies of PEEK and related polyimides[END_REF] The CTE for an amorphous PEEK was deduced by Lu & al. [85] from their measurements on the semicrystalline and previous ones on a totally crystallized PEEK using Eq. 1.5

𝛼 = 𝜒 𝑐 𝛼 𝑐 + 𝜒 𝑎 𝛼 𝑎

Eq. 1.5

𝛼, 𝛼 𝑐 and 𝛼 𝑎 being the thermal expansion coefficient for semi-crystalline, crystalline phase and amorphous phase respectively, 𝜒 𝑐 and 𝜒 𝑎 the volume fraction for crystalline and amorphous phase respectively. Lee & al. [START_REF] Lee | Effects of Cooling Rate on the Crystallinity and Mechanical Properties of Thermoplastic Composites[END_REF] performed tensile tests on PEEK 150P specimens of different crystallinities to study the effect of crystallinity on Young modulus. The results are presented in Figure 1.9 and show an increase in the modulus with the crystallinity until approximately 43% with a maximum value of 0.68 Msi or 4200 MPa. However, as previously established, PEEK's behavior is highly dependent on the temperature therefore the variation of its modulus with temperature was extensively studied and presented in the following.

Young's modulus

The Dynamic Mechanical Analysis 'DMA' measures the viscoelastic properties of semi-crystalline polymers such as PEEK. When the polymer is charged, its behavior has an elastic part characterized by the modulus of elasticity which designates the energy conserved and recovered during the charging cycle. The behavior of the polymer also has a viscous part characterized by the loss or viscosity modulus which is associated with the dissipated energy. Chapman &al. [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] used a micromechanical model to study the evolution of the Young's modulus for a semi-crystalline PEEK with temperature and crystallinity. The two constituents for this model are the crystalline and the amorphous phase. In his model, Chapman [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] proposed a temperature-independent crystalline phase Young's modulus of 30 GPa and Poisson's ratio of 0.3. The amorphous phase properties included a constant Poisson's ratio equal to that of the crystalline phase and a temperature-dependent Young modulus 𝐸 𝑎 as presented in 𝐸 𝑎 = 0 𝐺𝑃𝑎 T>340°C Table 1.5 -Evolution of the amorphous phase modulus function of the temperature for different temperature ranges according to [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] The results from the micromechanical model are presented in Figure 1.10. Similarly to the results from the work of Lee & al. [89] at ambient temperature, the increase in crystallinity yields an increase in the modulus. However, the maximum crystallinity in the results of Chapman is 30% so no information is given on the validity of the model for higher crystallinities. The temperature's effect can be also studied from [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF]. The modulus doesn't change with increasing temperature below the glass transition c.f. Figure 1.10. This constant value is followed by an abrupt decrease of the modulus in the glass transition zone. The shape of the decreasing curve is the same for the different crystallinities studied but no experimental validation was conducted. This model is however limited to the linear elastic behavior. The evolution of the modulus for the semi-crystalline PEEK c.f. Figure 1.11-(b) and (c) specimens is similar to the one observed by Chapman [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF]: an almost-constant value up to the glass transition temperature followed by a decrease during the glass transition temperature. In the case of the amorphous specimen, the same behavior is observed from room temperature to 𝑇 𝑔 in Figure 1.11-(a). However, the modulus value drops more abruptly during the glass transition and reaches 9 MPa while the semi-crystalline specimens' modulus drops to approximately 1900 MPa. This abrupt decrease is then followed by an increase due to the cold crystallization of the amorphous phase. The recrystallization was identified with exothermic peaks during DSC tests. In addition to the difference in the value and behavior of the storage modulus, a difference is noted in terms of the glass transition. The glass transition temperature or 𝑇 𝑔 was determined from the peak of tan δ. A difference of approximately 15°C is noted between the amorphous PEEK and the 35% crystallinity PEEK. In fact, the amorphous specimen presents a sharp and pronounced peak at around 148°C, while the semi crystalline specimens present a broader peak at around 163°C. The shift to higher glass transition temperatures with increased crystallinity fraction can be assigned to the lower amorphous mobility in presence of crystals.

Seferis [START_REF] Seferis | Polyetheretherketone (PEEK): Processing-structure and properties studies for a matrix in high performance composites[END_REF] performed also tensile tests at room temperature to evaluate the tensile modulus for PEEK at different crystallinity levels. The tensile modulus was evaluated from the linear part of the stress-strain curves after the tensile test and the variation of this modulus with crystallinity is presented in Figure 1.12.

The results are in accordance with the DMA data where an increase in the crystallinity from 9% to 33% implies an increase in the modulus from 2620 MPa or 0.38 Msi to 3378 MPa or 0.49 Msi. As previously stated, PEEK's behavior has an elastic and a viscoelastic parts. The description of these behaviors was extensively described in the literature. The main objective of this thesis is to present a model able to predict the residual stresses, and this model is highly dependent on the description of the material properties and most importantly that of PEEK since it is dependent on the temperature and therefore changes during the process. Jeronimidis &al. [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] discussed the possibility of an overestimation of the residual stresses may occur if the elastic part was solely considered. Chapman [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] and lately other authors such as Peron & al. [START_REF] Péron | Measurement and prediction of residual strains and stresses during the cooling of a glass fibre reinforced PA66 matrix composite[END_REF] discussed the possibility of taking into account an incremental constitutive law which can solve this problem. This approach is used in this study and the viscoelastic behavior won't be discussed here but readers may refer to numerous publications on the subject [START_REF] Lawrence | Prediction of Residual Stress in Continuous Fiber Semicrystalline Thermoplastic Composites: a Kinetic-Viscoelastic Approach[END_REF], [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF], [START_REF] Karimiani | Analysis of Residual Stresses in Thermoplastic Composites Manufactured by Automated Fiber Placement[END_REF].

Poisson's ratio

Coguil & Adams measured in [START_REF] Coquill | Mechanical properties of several neat polymer matrix materials and unidirectional carbon fiber-reinforced composites[END_REF] the variation with temperature of Poisson's ratio and other mechanical properties for several materials including PEEK. The specimens were provided by ICI chemicals but no information was provided on their grade. The tests were performed at three temperatures below the glass transition of PEEK: 23, 82 and 121°C and the results are presented in Poisson's ratio increases from 0.41 at 21°C to 0.44 at 121°C which presents a difference of 7%. Ogale & McCullough [95] conducted tensile tests on PEEK 45G specimens. They reported Poisson's ratio values for semi-crystalline PEEK with a 33% crystallinity. The results showed an approximate constant value that varied from 0.378 at 23°C to 0.379 at 130°C and 0.377 at 145°C. However, the authors considered a constant Poisson's ratio value of 0.378 for their work for simplification. This value was later used by Unger & Hansen in [START_REF] Unger | The Effect of Thermal Processing on Residual Strain Development in Unidirectional Graphite Fibre Reinforced PEEK[END_REF]. A constant value of 0.3 for the Poisson's ratio for PEEK was reported by Lawrence & Seferis in [START_REF] Lawrence | Material Response of a Semicrystalline Thermoplastic Polymer and Composite in Relation to Process Cooling History[END_REF] but no explication on the source of this value was given, as is the case for Chapman &al. in [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF]. In a recent study, Poisson's ratio for 3-D printed specimens made of PEEK 151G was evaluated at room temperature from static tensile tests. The tests were performed on amorphous and semi-crystalline materials without mentioning the crystallinity. The values reported for the Poisson ratio were similar for both materials on an average of 0.38. The difference between the values obtained from these studies can be attributed to the testing conditions or different grades of PEEK.

Shear modulus

Considering PEEK as an isotropic material, the shear modulus can be estimated from Eq. 1.6

𝐺 𝑚 = 𝐸 𝑚 2(1 + 𝜈 𝑚 )
Eq. 1.6 Based on Eq. 1.6, the shear modulus can be estimated from Young modulus and Poisson's coefficient. As it was shown in previous sections, PEEK's elastic modulus is highly dependent on temperature while Poisson's ratio can be considered as constant. As a consequence, the shear modulus follows a similar trend to that of Young's modulus. This dependence was observed by Borgna [START_REF] Borgna | Etude des proprietes de composites a matrice thermoplastique thermostable au-dela de leur temperature de transition vitreuse[END_REF] for a semi-crystalline PEEK 150G as shown in APC-2 prepreg are characterized by their high fracture toughness, excellent solvent resistance and low moisture absorption [33]. Moreover, they can be stored at ambient temperature and have an indefinite shelf life [START_REF]Hexcel website[END_REF]. The thermo-physical properties are presented in Table 1.6 at ambient temperature. The anisotropic thermal conductivity is related to the fiber's anisotropic behavior. These properties may be used as inputs to the thermal models presented in Section 1.

Thermal expansion coefficient

As discussed in Section 1.3.2.1, the PEEK resin's isotropic CTE exhibits a dependence on the crystallinity and the temperature especially above the glass transition. Moreover, the carbon fiber's CTE is anisotropic. As a result, PEEK/AS4 composite exhibits an anisotropic behavior which might be dependent on the temperature and the crystallinity. The longitudinal behavior (along the fiber axis) is governed by the fiber while PEEK controls the transverse behavior.

Jeronimidis & Parkyn [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF] measured the longitudinal and transverse thermal expansion coefficient 𝛼 𝐿 and 𝛼 𝑇 respectively of APC-2 composites using high temperature strain gauges. The measurements were conducted up to 240°C. 𝛼 𝐿 seems to be temperature independent with a value of 0.6• 10 -6 °C-1 . On the other hand, 𝛼 𝑇 remains approximately constant equal to 29.1• 10 -6 °C-1 between room temperature and 143C°C then undergoes an increase up to 75• 10 -6 °C-1 when the material passes through its glass transition estimated around 143°C. The values measured by Jeronimidis & Parkyn [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF] are similar to the ones reported by Chapman & al. [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] and originally presented by Cogswell in [START_REF] Cogswell | The processing Science of Thermoplastic Structural Composites[END_REF] from room temperature until 343°C, which corresponds to the PEEK's fusion temperature. No details were given on the measurements of the CTEs, and the authors reported a varying longitudinal and transverse CTE when passing through the glass transition.

In a series of three publications, Barnes et al. focused on the thermal expansion behavior of PEEK composites [START_REF] Barnes | Thermal Expansion Behaviour of Thermoplastic Composite Materials[END_REF], [START_REF] Barnes | Thermal expansion characteristics of PEEK composites[END_REF]. In the first publication [START_REF] Barnes | Thermal Expansion Behaviour of Thermoplastic Composite Materials[END_REF], the authors conducted measurements of the CTE for PEEK/AS4 on temperatures ranging from -180°C to +120°C. The measurements confirmed a low longitudinal CTE as stated by Jeronimidis [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF] and Cogswell [START_REF] Cogswell | The processing Science of Thermoplastic Structural Composites[END_REF]. The results for 𝛼 𝑇 are presented in Figure 1.15. Similarly to previous results presented by Jeronimidis [99], Cogswell [START_REF] Cogswell | The processing Science of Thermoplastic Structural Composites[END_REF] and Barnes & al.

[101], [START_REF] Barnes | Thermal expansion characteristics of PEEK composites[END_REF], 𝛼 𝑇 remains approximately constant with a mean value of 30• 10 -6 °C-1 until 𝑇 𝑔 around 141°C followed by an increase to 80• 10 -6 °C-1 . However, an additional increase of the transverse CTE was reported around the melting onset temperature approximated to 320°C. The decreasing value of the CTE starting approximately 340 °C in Figure 1.15 was attributed to slipping in the experimental procedure and therefore results above 330°C were not considered. Unger & Hansen [START_REF] Unger | The Effect of Thermal Processing on Residual Strain Development in Unidirectional Graphite Fibre Reinforced PEEK[END_REF] also measured 𝛼 𝐿 and 𝛼 𝑇 of APC-2 composite using strain gauges during cooling. Below 𝑇 𝑔 , the average value for the transverse CTE was approximated to 34.3• 10 -6 °C-1 and 66• 10 -6 °C-1 above 𝑇 𝑔 .

Cogswell &al.

[100]

Jeronimidis & al.

[99]

Barnes & al. [START_REF] Barnes | Thermal Expansion Behaviour of Thermoplastic Composite Materials[END_REF], [START_REF] Barnes | Thermal expansion characteristics of PEEK composites[END_REF] Unger & al. [START_REF] Unger | The Effect of Thermal Processing on Residual Strain Development in Unidirectional Graphite Fibre Reinforced PEEK[END_REF] 𝜶 𝑳 (• 10

Longitudinal and transverse modulus

The longitudinal modulus 𝐸 𝐿 of a unidirectional laminate can be estimated by a simple law of mixture c.f. Eq. 1.7.

𝐸 𝐿 = 𝑉 𝑓 𝐸 𝑓𝐿 + (1 -𝑉 𝑓 )𝐸 𝑚 Eq. 1.7
It was stated in previous sections the high stiffness of the carbon fiber which can be around 231 GPa and remains constant even at high temperatures. PEEK's modulus drops upon heating when reaching 𝑇 𝑔 . With a fiber's volume fraction 𝑉 𝑓 around 59-60%, the APC-2 composite presents a high longitudinal modulus even at elevated temperatures. The dependence of 𝐸 𝐿 on temperature was measured by Wang & Sun in [START_REF] Wang | Experimental Characterization of Constitutive Models for PEEK Thermoplastic Composite at Heating Stage during Forming[END_REF] for a 15-ply APC-2 pre-consolidated laminate with 𝑉 𝑓 =61%. The authors reported a decrease from 130 GPa at 23°C to 124 GPa at 315°C which corresponds to a decrease of less than 5%. Later, Wang & Sun [START_REF] Wang | Thermoelastic Behavior of PEEK Thermoplastic Composite during Cooling from Forming Temperatures[END_REF] and Grogan [START_REF] Grogan | Damage and permeability in linerless composite cryogenic tanks[END_REF] used a piece-wise linear interpolation to present the evolution of the longitudinal modulus of APC-2 function of temperature. A constant longitudinal modulus for a press-molded APC-2 of 130 GPa up to 300°C was reported by Jeronimidis & Parkyn [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF]. Several other researchers considered a constant value of the longitudinal modulus for unidirectional APC-2 such as [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF], [START_REF] Shih | On-line consolidation of thermoplastic composites[END_REF] and [START_REF] Chinesta | First steps towards an advanced simulation of composites manufacturing by automated tape placement[END_REF]. No study on the dependence of the longitudinal modulus of a unidirectional APC-2 on the crystallinity was found, this is in fact logical since the fibers pilot the composite behavior along this direction.

The transverse behavior, on the other side, is mainly controlled by the resin for which the behavior is highly dependent on both the temperature and crystallinity. Press-molded semi crystalline APC-2's Young's modulus was measured using a Dynamic mechanical thermal analysis (DMTA) in [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF]. No information was given on the fiber content, however the transverse modulus shows a small decrease until the glass transition temperature followed by an abrupt decrease. This modulus trend is similar to that of the semi crystalline PEEK as presented in section 1.3.2.2. Similar results were obtained by Chapman & al. [77] . The authors in [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] used micromechanics to model the thermomechanical properties of the semi-crystalline PEEK manufactured by compression molding using temperature-dependent amorphous phase properties and constant crystalline properties as presented in section 1.3.2.2.

Unger & Hansen [START_REF] Unger | The Effect of Cooling Rate and Annealing on Residual Stress Development in Graphite Fibre Reinforced PEEK Laminates[END_REF] measured the dependence of the transverse modulus on temperature and crystallinity. They studied the behavior of APC-2 in its amorphous, nominally cooled and annealed (at 225°C) from the amorphous cases using DMTA. The results are presented in Barnes & Byerly [START_REF] Barnes | The formation of residual stresses in laminated thermoplastic composites[END_REF] also used DMTA to measure the variation of the transverse modulus with temperature for several composites including the APC-2. The values were normalized according to the modulus obtained with a tensile test and were compared with a previous study [START_REF] Mcguinness | Characterisation of thermoplastic composite melts in rhombus-shear: the picture-frame experiment[END_REF]. Similar trend as [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF], [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] and [START_REF] Unger | The Effect of Cooling Rate and Annealing on Residual Stress Development in Graphite Fibre Reinforced PEEK Laminates[END_REF] for the transverse modulus of a semi-crystalline APC-2 was obtained.

In addition to experimental measurements, the inverse law of mixture c.f. Eq. 1.8 showed its capacity to predict the transverse modulus of several unidirectional materials [START_REF] Herakovich | Mechanics of fibrous composites[END_REF].

𝐸 𝑇 = 𝐸 𝑚 𝐸 𝑓 (1 -𝑉 𝑓 )𝐸 𝑓 + 𝑉 𝑓 𝐸 𝑚
Eq. 1.8

In-plane Poisson's coefficient

The studies on the dependence of Poisson's coefficient of APC-2 on temperature and crystallinity are much less numerous than the ones in the case of Young modulus or thermal expansion coefficient.

It was rather assumed a constant in-plane Poisson coefficient 𝜈 𝐿𝑇 of 0.3 for APC-2 in numerous studies such as [START_REF] Tzeng | Thermal residual stress analysis for in situ and post-consolidated composite rings[END_REF], [START_REF] Sonmez | Analysis of process-induced residual stresses in tape placement[END_REF], [START_REF] Lü | Effects of tape tension on residual stress in thermoplastic composite filament winding[END_REF], [START_REF] Grogan | Damage and permeability in linerless composite cryogenic tanks[END_REF], [START_REF] Borgna | Etude des proprietes de composites a matrice thermoplastique thermostable au-dela de leur temperature de transition vitreuse[END_REF]. Shih [START_REF] Shih | On-line consolidation of thermoplastic composites[END_REF]in his thesis reported a constant value of 0.28 previously reported by Mantel & al. in [53] for which the authors didn't precise the source of this value. strain gauges. The authors report a value within 4% of that published by the manufacturer which is equal to 0.3. Kyriakides & al. [START_REF] Kyriakides | On the compressive failure of fiber reinforced composites[END_REF] measured the same coefficient for an APC-2 with a 60% volume fraction. Uematsu & al. [112] measured the variation of 𝜈 𝐿𝑇 with temperature using the same technique in [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF] and [START_REF] Kyriakides | On the compressive failure of fiber reinforced composites[END_REF]. The results are presented in The results showed that the Poisson's ratio for APC-2 remains relatively constant with temperature. The values measured by Uematsu are slightly higher than those previously measured which is due to the higher fiber volume fraction (62%) in the case of Uematsu & al. [112] compared to previous studies (59 and 60%).

Karimiani [START_REF] Karimiani | Analysis of Residual Stresses in Thermoplastic Composites Manufactured by Automated Fiber Placement[END_REF] performed calculations in Abaqus to estimate the variation of properties of a PEEK/Carbonfiber with a volume fraction crystallinity of 59%. The calculated value was 0.339, which is a bit higher than the previously reported value for Poisson's coefficient. It should be however noted that no experimental validation was conducted in the sake of the validity of this value.

In-plane shear modulus

The in-plane shear modulus for a unidirectional fiber reinforced polymer can be calculated using Eq. 1.9

𝐺 𝐿𝑇 = 𝐺 𝑚 𝐺 𝑓 (1 + 𝑉 𝑓 ) + 𝐺 𝑚 (1 -𝑉 𝑓 )
𝐺 𝑓 (1 -𝑉 𝑓 ) + 𝐺 𝑚 (1 + 𝑉 𝑓 )

Eq. 1.9

Jeronimidis & al. [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF] measured 𝐺 𝐿𝑇 below the material's glass transition and the reported value was 5.1 GPa and considered constant over this temperature range. Barnes & al. [79] presented the variation of the in-plane shear modulus for PEEK/AS4 with temperature from a previous study [START_REF] Sun | Experimental Characterization of Constitutive Models for PEEK Thermoplastic Composite at Heating Stage during Forming[END_REF]. Uematsu & al. [112] measured the elastic modulus 𝐸 0 , 𝐸 90 and 𝐸 45 for unidirectional PEEK/AS4 along the direction 0°, 90° and 45° respectively. Uematsu & al. [START_REF] Uematsu | Delamination behavior of a carbon-fiber reinforced thermoplastic polymer at high temperatures[END_REF] used Eq. 1.10 to obtain the in-plane shear modulus.

1 𝐺 𝐿𝑇 = 4 𝐸 45 -( 1 𝐸 0 + 1 𝐸 90 - 2𝜈 𝐿𝑇 𝐸 0 )
Eq. 1.10

The trend followed by the in-plane shear modulus with temperature observed by Barnes & al. [79] and Uematsu & al. [112] is similar to the one observed for the in-plane transverse elastic modulus. The same trend is observed by Borgna [START_REF] Borgna | Etude des proprietes de composites a matrice thermoplastique thermostable au-dela de leur temperature de transition vitreuse[END_REF]. Borgna measured at room temperature a modulus of approximately 6.9 GPa which is higher than 5.2 GPa, the one reported by the manufacturer [START_REF]Hexcel website[END_REF]. No explanation was given on the cause of this difference, however it may be attributed to the difference in the measurement techniques, since uniaxial tensile tests were considered in [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF], [START_REF] Barnes | The formation of residual stresses in laminated thermoplastic composites[END_REF], [START_REF] Uematsu | Delamination behavior of a carbon-fiber reinforced thermoplastic polymer at high temperatures[END_REF] and [START_REF]Hexcel website[END_REF] while the properties were measured by rectangular torsion using a rheometer in [START_REF] Borgna | Etude des proprietes de composites a matrice thermoplastique thermostable au-dela de leur temperature de transition vitreuse[END_REF].

In this section, the APC-2 properties and their dependence on temperature and crystallinity were presented. Most of the works presented were related to compression molded specimen. The composite properties discussed in this part after tape placement/filament winding and the effect of the process parameters especially the mandrel temperature wasn't highly discussed in the literature.

Fusion and crystallization of PEEK and PEEK/AS4 composite

In this section, a general overview on literature key-works on the morphology of PEEK, melting and crystallization of PEEK models and the effect of the presence of carbon fibers will be presented.

Morphology of PEEK

Microscopic observations showed that PEEK presents a spherulites crystal morphological structure with spherulites size between 1 and 10 µm [START_REF] Dawson | X-ray data for poly(aryl ether ketones)[END_REF], [START_REF] Lovinger | Electron-microscopic investigation of the morphology of a meltcrystallized polyaryletherketone[END_REF], [START_REF] Bassett | On crystallization phenomena in PEEK[END_REF], [START_REF] Gardner | Structure, crystallization and morphology of poly (aryl ether ketone ketone)[END_REF] & [START_REF] Lu | Synthesis, morphology, and melting behavior of poly(ether ether ketone) of different molecular weights[END_REF]. The size of the spherulites is highly dependent on the material's thermal history.

Fusion mechanisms

Blundell & Osborn [START_REF] Blundell | The morphology of poly(aryl-ether-ether-ketone)[END_REF] performed DSC scans on a series of PEEK samples crystallized isothermally at different temperatures as shown in Figure 1.17. The DSC scans showed a double melting behavior for semicrystalline specimens (curves (B) to (F)). For the different curves, the upper endotherm occurs around 335°C as was also reported by Cebe & Hong [START_REF] Cebe | Crystallization behaviour of poly(ether-ether-ketone)[END_REF]. On the other side, the lower endotherm shifts to a lower temperature as the crystallization temperature decreases. The authors in [START_REF] Blundell | The morphology of poly(aryl-ether-ether-ketone)[END_REF] associated the lower endotherm to the melting of crystals present prior to the DSC scan. On the other side, the high-temperature scan correspond to the melting of crystals formed due to the simultaneous melting and crystallization during the heating scan.

Figure 1.17 -DSC heating scans for (A) amorphous PEEK -(B),(C),)(D),(E) and (F) semi-crystalline

PEEK crystallized isothermally at 320°C, 310°C,270°C,230 °C and 200°C respectively [START_REF] Blundell | The morphology of poly(aryl-ether-ether-ketone)[END_REF] The dual melting endotherm was later observed by several authors including Cebe & Hong [START_REF] Cebe | Crystallization behaviour of poly(ether-ether-ketone)[END_REF] and Lee & Porter [START_REF] Lee | Double-melting behavior of poly(ether ether ketone)[END_REF]. While Lee & Porter [START_REF] Lee | Double-melting behavior of poly(ether ether ketone)[END_REF] adopted the same explanation for the dual melting behavior of PEEK as Blundell & Osborn [START_REF] Blundell | The morphology of poly(aryl-ether-ether-ketone)[END_REF], Cebe & Hong [START_REF] Cebe | Crystallization behaviour of poly(ether-ether-ketone)[END_REF] attributed the low-temperature endotherm to the fusion of unstable crystals existing between the lamellae.

Another characteristic of the polymer is the thermodynamic melting point of a hypothetical crystal of infinite thickness 𝑇 𝑚 0 [START_REF] Dawson | X-ray data for poly(aryl ether ketones)[END_REF]. It was determined in the literature using the Hoffman-Weeks relation [START_REF] Hoffman | Melting process and the equilibrium melting temperature of polychlorotrifluoroethylene[END_REF] and estimated between 385°C and 395°C for PEEK [START_REF] Blundell | The morphology of poly(aryl-ether-ether-ketone)[END_REF], [38]. If a polymer is heated up to a temperature not sufficiently higher than its thermodynamic fusion temperature 𝑇 𝑚 0 , some nuclei may survive the melting conditions and increase the crystallization rate [START_REF] Jonas | Thermal stability and crystallization of poly(aryl ether ether ketone)[END_REF] & [START_REF] Choe | Nonisothermal crystallization kinetics of poly(etheretherketone) (PEEK)[END_REF]. It's therefore essential to define the processing temperature and the time spent at this temperature. For APC-2 prepreg, the effect of molding temperature and time on the microstructure and performance of the material was studied in [START_REF] Vu-Khanh | Processing-Structure-Property Relations in PEEK/Carbon Composites Made from Comingled Fabric and Prepreg[END_REF]. An optimum temperature window between 400 and 450°C was defined to ensure a complete suppression of any residual nuclei. The numerical models to simulate the fusion of PEEK or PEEK composites is however rarely studied in the literature. To our knowledge, the only model was proposed by Maffezolli & al. [START_REF] Maffezzoli | Welding of PEEK/carbon fiber composite laminates[END_REF] and is discussed in the last chapter.

Crystallization of PEEK and PEEK/AS4 composites

The crystallization of a polymer is a two stages process: primary and secondary. The primary crystallization is also divided in two steps: nucleation and growth.

 Nucleation can be defined as a spatial organization of molecules into a nucleus with the ability to grow if it's stable.

 Growth is the second phase, it includes the packing of molecules into a three-dimensional order. Many factors such as temperature play a key role into slowing the disordering processing, thus allowing the growth of three dimensional crystallites by adding more chains to the stable nucleus formed in the nucleation phase.

The heterogeneous nuclei produced during the primary crystallization can fill the inter-crystallite voids during the secondary crystallization.

On the macroscopic scale, the crystallization kinetics models deal with the progressive shift of the melt phase into crystallites. This transformation relies on the description of the relative or absolute degree of crystallinity function of time. The material is subjected to non-isothermal conditions during processing on the SPIDE-TP, therefore the models describing the non-isothermal crystallization are the main interest. However, they are deduced from theories adapted to isothermal conditions.

Experimental techniques include differential scanning calorimetry (DSC), infrared spectroscopy, optical microscopy, wide and small angle X-ray scattering and electron microscopy [START_REF] Blundell | Variations in the crystal lattice of PEEK and related para-substituted aromatic polymers: 1. Thermal expansion[END_REF], [START_REF] Velisaris | Crystallization kinetics of polyetheretherketone (peek) matrices[END_REF], [START_REF] Harris | A Study of the crystallisation kinetics in PEEK and PEEK composites[END_REF], [START_REF] Doumeng | A comparative study of the crystallinity of polyetheretherketone by using density, DSC, XRD, and Raman spectroscopy techniques[END_REF].

PEEK's isothermal crystallization is studied using the Avrami's theory for which the general equation is Eq. 1.11:

𝜒 𝑣𝑐 (𝑡) = 𝜒 ∞ [1 -𝑒 -𝐾 𝐴𝑣 𝑡 𝑛 𝐴𝑣 ]
Eq. 1.11

𝜒 𝑣𝑐 , 𝜒 ∞ , 𝐾 𝐴𝑣 and 𝑛 𝐴𝑣 are the volume fraction of crystallized material at the time t, the equilibrium volume fraction of crystallinity, the Avrami kinetic coefficient and the Avrami exponent respectively. The Avrami kinetic coefficient is function of the pressure and temperature and describes the ability of a polymer to crystallize. Several previous studies showed that the plot of Avrami's equation describe a dual crystallization model: the primary and secondary crystallization processes. The primary crystallization involves the threedimensional growth of spherulites with an Avrami exponent is evaluated between 2.5 and 3 [START_REF] Velisaris | Crystallization kinetics of polyetheretherketone (peek) matrices[END_REF], [START_REF] Cebe | Crystallization behaviour of poly(ether-ether-ketone)[END_REF].

The second process is generally associated with an Avrami exponent of 1.5 and corresponds to secondary crystallization [START_REF] Cebe | Crystallization behaviour of poly(ether-ether-ketone)[END_REF].

Since the Avrami model ignores the presence of amorphous regions trapped between the crystallite lamellae, Tobin model was developed to model the isothermal crystallization of PEEK [START_REF] Tobin | Theory of phase transition kinetics with growth site impingement. III. Mixed heterogeneou/homogeneous nucleation and nonintegral exponents of the time[END_REF]. This model takes into account both homogeneous and heterogeneous nucleation.

Ozawa's theory [START_REF] Ozawa | Kinetics of non-isothermal crystallization[END_REF] is a an extended Avrami model to study the crystallization of PEEK under a constant cooling rate [START_REF] Addonizio | Study of the non-isothermal crystallization of poly(ethylene oxide)/poly(methyl methacrylate) blends[END_REF] c.f. Eq. 1.12.

𝜒 𝑣𝑐 = 𝜒 ∞ [1 -𝑒 𝑘 𝑂𝑧𝑎𝑤𝑎 (𝑇) 𝑎 𝑛 ]
Eq. 1.12 𝑘 𝑂𝑧𝑎𝑤𝑎 (𝑇) is the cooling rate function and n is a constant. Ozawa's model was proved capable of predicting the non-isothermal crystallinities for polymers such as poly (ethylene terephthalate) or PET [START_REF] Ozawa | Kinetics of non-isothermal crystallization[END_REF]. This was not the case for PEEK, where in fact, Ozawa's model plotted in the logarithmic form of Eq. 1.12 resulted in a series of curved lines with different slopes [START_REF] Cebe | Crystallization behaviour of poly(ether-ether-ketone)[END_REF] as shown in Two observations were made from these results, the first one is the effect of the secondary crystallization not taken into consideration in the Ozawa's model. The second one is regarding to the changing slopes, which states that the coefficient n is not constant with temperature which is not the consideration in Ozawa's model. Lee & Springer [38] adopted the Ozawa's model to predict the crystallinity of PEEK 150P and PEEK/AS4 composites during manufacturing. The authors suggested a validation of the crystallization model used, however they only compared the average crystallinity over the ply thickness to DSC tests and no experimental validation was conducted to characterize the through-thickness crystallization.

Velisaris & Seferis [START_REF] Velisaris | Crystallization kinetics of polyetheretherketone (peek) matrices[END_REF] proposed a model based on a dual Avrami expression to take into account the two mechanisms but different parameters should be considered since they observed a shift from the first to second mechanism in the presence of fibers.

Other works conducted observations on smaller scale to study the effect of the presence of carbon fiber. Blundell & al. [134] performed wide angle X-ray scattering " WAXS" observations on APC-2. The results showed an influence of the carbon fibers on the growth of spherulites which were observed for cooling rates higher than 10°C/min. They suggested that crystallization of PEEK in the presence of carbon fibers could be thought as a competition between the growth of PEEK's spherulites and a trans-crystalline growth emerging from the fibers' surface. And, therefore, the Avrami's model cannot hold with a constant value of n if both spherulites and transcrystallinity develop at the same time.

Application of the melting and crystallization models of PEEK/AS4 composites during tape placement and filament winding

Mantel & Springer [START_REF] Mantell | Thermoplastic composites for aerospace[END_REF] studied the crystallization behavior of APC-2 during tape placement using Lee & springer's model [38] and that of Velisaris & Seferis [START_REF] Velisaris | Crystallization kinetics of polyetheretherketone (peek) matrices[END_REF]. Nejhad & al. [56] also used the model developed by Velisaris & Seferis [START_REF] Velisaris | Crystallization kinetics of polyetheretherketone (peek) matrices[END_REF] to study the evolution of crystallinity during filament winding of APC-2 prepreg. The results showed a through-the-thickness crystallinity gradient which was manifested by a properties gradient. However, no experimental validation was conducted to validate the crystallinity model applied. Tierney & Gillespie [START_REF] Tierney | Crystallization kinetics behavior of PEEK based composites exposed to high heating and cooling rates[END_REF] implemented the same model considered by Mantel & Springer with the same parameters in [START_REF] Mantell | Thermoplastic composites for aerospace[END_REF] for a single ply heated with a gas torch.

Choe & Lee [START_REF] Choe | Nonisothermal crystallization kinetics of poly(etheretherketone) (PEEK)[END_REF] proposed a non-isothermal crystallization model where the crystallization of PEEK is presented as a combination of two nucleation-growth mechanisms, an assumption backed by previous researchers [START_REF] Keith | Crystallization of polymers from the melt and the structure of bulk semicrystalline polymers[END_REF]. The expression is based on Tobin's theory of phase transition kinetics and given by Eq. 1.13:

𝛼̇(𝑡) = 𝜅 1 exp (- 3𝐸 𝑑 𝑅𝑇 ) 𝑒𝑥𝑝 (- 3𝛹 1 𝑇 𝑚 0 𝑇(𝑇 𝑚 0 -𝑇) ) 𝑡 2 [1 -𝛼(𝑡)] 2 + 𝜅 2 𝑒𝑥𝑝 (- 3𝐸 𝑑 𝑅𝑇 ) 𝑒𝑥𝑝 (- (3𝛹 1 + 𝛹 2 )𝑇 𝑚 0 𝑇(𝑇 𝑚 0 -𝑇) ) * [1 -𝛼(𝑡)] 2 ∫ (𝑡 -𝑤) 2 [1 -𝛼(𝑤)]𝑑𝑤 𝑡 0
Eq. 1.13 𝛼̇(𝑡), 𝑇 𝑚 0 , 𝐸 𝑑 , 𝛹 1 , 𝛹 2 are the rate of variation of relative crystallinity at time t, the equilibrium melting temperature, the activation energy of diffusion of crystallization segments across the phase boundary, the constant related to the free energy of formation of a critical nucleus, the constant related to the free energy of formation of a growth embryo, κ1 and κ2 are kinematic parameters. The values for the different parameters were obtained through DSC measurements.

This model was used by Sonmez & Hahn in [START_REF] Sonmez | Modeling of heat transfer and crystallization in thermoplastic composite tape placement process[END_REF] during the cooling phase of an automated tape placement process to simulate the crystallization of PEEK/AS4 composite. The authors made the assumption that the crystallization kinetics applied to a fiber reinforced matrix are the same ones applied to a neat matrix as was shown in [38]. They were also the only ones to incorporate the model of Maffezolli & al. [START_REF] Maffezzoli | Welding of PEEK/carbon fiber composite laminates[END_REF] to calculate the residual degree of crystallinity during tape placement of APC-2 composite.

Nakamura also proposed a non-isothermal crystallization model from a generalized Avrami equation presented in Eq. 1.14.

𝜒 𝑣𝑐 = 𝜒 ∞ [1 -𝑒 -(∫ 𝐾(𝑇)𝑑𝑡 𝑡 0 ) 𝑛 ]
Eq. 1.14

A differential form of Eq. 1.14 was presented by Patel & Spruiell in [START_REF] Patel | Crystallization kinetics during polymer processing-Analysis of available approaches for process modeling[END_REF] to simulate the non-isothermal crystallization of Nylon-6. The authors found that Nakamura's integral and differential forms overestimate the crystallinity. They attributed this overestimation to a lack of consideration of an induction time for nucleation. To our knowledge, Nakamura's model applied on PEEK or PEEK/AS4 composites has been investigated only by Nicodeau [START_REF] Nicodeau | Continuous welding modeling of thermoplastic matrix composites[END_REF] in her thesis where a crystallization kinetics law was proposed with modifications on parameters. More recently, Gordnian [START_REF] Gordnian | A Rate-Type Crystallization Kinetics Model for Process Modelling of Carbon Fibre PEEK Matrix Composites[END_REF] proposed an approach to model the crystallization of PEEK/AS4 composites. An empirical model was introduced based on multiple DSC measurements which were used to fit the model proposed and determine the model parameters. The model took into account an induction time which was considered as a limitation when using the Nakamura's model as stated previously. The model was able to predict the non-isothermal crystallization of PEEK/AS4 as shown in Figure 1.19.

Finally, the crystallization of a polymer is associated with a crystallization shrinkage which adds to the thermal deformation during the tape placement and filament winding. However, this part showed that the research presented studied on the evolution of crystallinity during the process and didn't focus on implementing the crystallization shrinkage to calculate its effect on residual stresses during tape placement or filament winding. In this work, the crystallization shrinkage for PEEK is presented in the last chapter of this manuscript.

Residual stresses in thermoplastic composites

Residual stresses are locked-in stresses that remain in the structure after the processing and are associated with structural properties and the processing techniques. Several factors induce the formation of these stresses and include for example the differences in thermo-mechanical properties between the different component of the composites and thermal gradients. These gradients are accentuated in processes with localized heating such as tape placement and filament winding. In addition, the imposed geometric constraints such as the presence of a mandrel or the pre-tension applied to the plies may induce the formation of residual stresses. Their presence influences the performance of the part and may cause distortions and dimensional changes. Counteracting these undesired geometrical defects requires the use of different methods that can easily become time consuming. Residual stresses can also cause delamination, cracks and reduction of mechanical properties of the material. As a result, the structure can fail earlier than expected. Thus, to ensure a reliable structure, residual stresses should be accounted for during the design process.

The first step is understanding their causes and origins. These causes may be pronounced in some processes more than others. For instance, laser assisted tape laying and filament winding are associated with local heating which is not the case for other processes such as compression molding. Therefore thermal gradients may develop in the thickness direction leading to crystallinity and thermo-mechanical properties gradients which are driving sources to the formation of residual stresses.

Based on their origin, residual stresses can be studied on three different levels: microscopic, mesoscopic and global level. Parlevliet & al. presented three complete reviews [START_REF] Parlevliet | Residual stresses in thermoplastic composites-A study of the literature-Part I: Formation of residual stresses[END_REF], [START_REF] Parlevliet | Residual stresses in thermoplastic composites -a study of the literature. Part II: Experimental techniques[END_REF] and [START_REF] Parlevliet | Residual stresses in thermoplastic compositesa study of the literature . Part III : Effects of thermal residual stresses[END_REF] on the origins of residual stresses, their characterization and their effect on the behavior of composites. While it is insignificant to recite the information presented in the three mentioned references, it's important for the sake of the thesis to recall briefly the origins of the residual stresses. Thus the following sections present a brief description of these origins especially those related to the laser assisted tape placement and filament winding.

Formation of residual stresses in thermoplastic composites during the process 1.4.1.1 Formation of residual stresses on the micro-scale

On the micro or constituent level, residual stresses arise mainly due to thermal and crystallization effects. The governing parameter is the mismatch in the CTE between the fiber and the matrix. Studies on the carbon fiber showed highly anisotropic thermal expansion behavior as previously discussed in Section 1.3.1. This ultimately leads to different behavior between the radial and longitudinal directions.

Moreover, the heterogeneity between the PEEK and the carbon fiber induces different shrinkages, ultimately leading to residual stresses. Upon cooling from the melt, the thermoplastic matrix shrinks higher than the fiber due to their higher CTE in addition to crystallization shrinkage associated with the formation of crystals in semi-crystalline PEEK. This results in residual strains in the fiber and the matrix. If the fiber and the matrix are bonded, then compressive residual stresses develop in the longitudinal direction of the fiber which are balanced by tensile residual stresses in the matrix as shown in The residual stresses level are highly influenced by the difference between the service temperature and the one at which residual stresses start building-up known as the stress-free temperature 'SFT'. The SFT depends on the polymer morphology whether it's a total amorphous or semi-crystalline polymer. The SFT for amorphous polymers was defined as its glass transition temperature by several authors [START_REF] Barnes | The formation of residual stresses in laminated thermoplastic composites[END_REF], [START_REF] Peeters | Thermally-Induced Shapes of Unsymmetric Laminates[END_REF], [START_REF] Cowley | The measurement and prediction of residual stresses in carbon-fibre/polymer composites[END_REF].

In the case of the semi-crystalline polymer, the SFT may be located at any temperature in the crystallization range due to the load-bearing capacity of the crystalline phase [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF], [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF].

Formation of residual stresses on the meso-scale

On the meso-scale, thermoplastic composite plies exhibit anisotropic behavior due to the fibers architecture thus leading to different CTEs along the material principal directions. As an example, in a [0/90] laminate made of two plies of the same material, the plies oriented at 0° have a lower CTE than those oriented at 90°C along the x-direction c.f. Table 1.7. Therefore, during cooling c.f. Figure 1.21, the 0° oriented plies impose a tensile stress on the 90° oriented plies and are in their turn imposed to a compressive stress state. Another driving source for the formation of residual stresses at the ply level is the sucessive laying up of the plies. This is characteristic to the tape placement and filament winding processes and is included in the complex thermal history resulting from these processes. It was introduced in several studies dealing with residual stresses during tape placement and filament winding [START_REF] Nejhad | Prediction of Process-Induced Stresses for In-Situ Thermoplastic Filament winding of cylinders[END_REF], [27]. The deposition of a heated ply on a cooler substrate generates additional gradients through the structure's thickness eventually modifiying the residual stresses and strains states. Finally, thermal gradients may be present through the thickness of one ply due to the complex thermal history evoked during automated tape placement and filament winding which is associated with local heating thus thermal gradients through the composite thickness.

Formation of residual stresses on the global scale

On a global level or the level of the manufactured structure, the driving sources for residual stresses formation include high cooling rates gradients through the thickness of the part. These through-thickness thermal gradients are pronounced during automated tape placement and filament winding more than other processes due to the local heating.

Additional sources include geometrical constraints due to the mandrel and tension applied to the incoming plies [START_REF] Cirino | In-situ consolidation for the thermoplastic composite ring-residual stress state[END_REF], [START_REF] Eduljee | Elastic response of postand in situ consolidated laminated cylinders[END_REF]. The residual stresses arise due to the difference in the thermal expansion coefficients and stiffness between the material and the mandrel.

As discussed, residual stresses arise at different levels and due to several factors that may be linked, complicated and numerous making their distinction a rather difficult task. However, and for the following parts of this thesis, the focus will be made on residual stresses generated at the mesoscale or the ply-scale while incorporating the effects at the micro-level such as the thermomechanical properties of the constituents.

Effect of the residual stresses on composites

The presence of residual stresses, even prior to the application of external loads, induces defects in the composite structure related to the following: the quality of the manufactured part by inducing defects such as fiber waviness which may degrade the compressive strength of the composite, the structure's mechanical properties the dimensional accuracies.

Fiber waviness as shown in Figure 1.22 is one of the defects related to the quality of the structure and caused by the presence of residual stresses. It's in fact the deviation of fibers from a mean direction as shown in Figure 1.22. Kugler & Moon [START_REF] Kugler | Identification of the Most Significant Processing Parameters on the Development of Fiber Waviness in Thin Laminates[END_REF] concluded in their study that the driving parameters are the mismatch in thermo-mechanical properties between the composites and the mandrel and the cooling rates. A decrease of the compressive strength was observed in previous works which increased with the fiber waviness [START_REF] Adams | Effects of layer waviness on the compression fatigue performance of thermoplastic composite laminates[END_REF] & [START_REF] Joyce | Compression strength reduction in composites with in-plane fiber waviness[END_REF]. Another defect related to the final structure quality is micro cracks or transverse cracking (perpendicular to the fiber direction) c.f. Figure 1.23. They occur primarily when the tensile residual stresses exceed the matrix's yield strength. The cracks are usually initiated in the matrix along the fiber/matrix interface [START_REF] Gentz | In-plane shear testing of medium and high modulus woven graphite fiber reinforced/polyimide composites[END_REF], [START_REF] Gentz | In-plane shear testing of medium and high modulus woven graphite fiber reinforced/polyimide composites[END_REF]. Micro cracks reduce the component's flexural modulus and can initiate delamination. Inter laminar debonding or delamination occurs due to the presence of micro cracks and a discontinuous stress state between different plies [START_REF] Parlevliet | Residual stresses in thermoplastic compositesa study of the literature . Part III : Effects of thermal residual stresses[END_REF]. Delamination may eventually lead to loss of stiffness and strength and to the failure of the structure [START_REF] Unger | A Method to Predict the Effect of Thermal Residual Stresses on the Free-Edge Delamination Behavior of Fibre Reinforced Composite Laminates[END_REF]. In addition to quality related defects, the presence of residual stresses influences the dimensional accuracy of the composite structure. From these defects, the laminates curvature and warpage shown in Figure 1. 24 were extensively mentioned in the literature [START_REF] Wang | Thermoelastic Behavior of PEEK Thermoplastic Composite during Cooling from Forming Temperatures[END_REF], [START_REF] Unger | A Method to Predict the Effect of Thermal Residual Stresses on the Free-Edge Delamination Behavior of Fibre Reinforced Composite Laminates[END_REF], [START_REF] Cowley | The measurement and prediction of residual stresses in carbon-fibre/polymer composites[END_REF], [START_REF] Bapanapalli | A linear finite element model to predict processing-induced distortion in FRP laminates[END_REF]. The warpage of the laminates is one of the most challenging problem for the fabrication of composite structures. It's a direct result of the anisotropic thermal gradients during the fabrication [START_REF] Tseng | Prediction of Shrinkage and Warpage of Fiber Reinforced Thermoset Composite Parts[END_REF], the stacking sequence and the interaction between the mandrel and the surface in contact with the mandrel. This enhances the through-thickness thermal gradients due to the thermal sink effect of the mandrel in addition to unsymmetrical boundary conditions between the bottom (in contact with the mandrel) and the upper surface. Additionally, stress gradients through the material thickness during tape placement and filament winding caused by the successive deposition of plies may eventually lead to warpage. The influence of the presence of residual stresses extends to the mechanical behavior of these structures.

Cowley & Beaumont [START_REF] Cowley | The measurement and prediction of residual stresses in carbon-fibre/polymer composites[END_REF] studied the effect of the residuals stresses on several cross-ply laminates. The first ply failure was adopted and it includes the experimental determination of the applied tensile load at which the 1 st transverse cracking occurs in the 90° oriented plies. The results showed that the transverse residual stresses decrease significantly the transverse tensile strength of the plies from 61.4 MPa to 28.6 MPa in the case of carbon fiber reinforced poly (ether sulfone) matrix.

The flexural strength is also affected by with a reduction between 37% and 45% of the transverse flexural strength for glass fiber reinforced polypropylene [START_REF] Guillén | The Influence of Cooling Rate on the Fracture Properties of a Thermoplastic-Based Fibre-Metal Laminate[END_REF].

A number of models were developed to study the influence of residual stresses on the fracture toughness such as [START_REF] Blundell | Crystalline morphology of the matrix of PEEK-carbon fiber aromatic polymer composites. I. Assessment of crystallinity[END_REF], [START_REF] Unger | A Method to Predict the Effect of Thermal Residual Stresses on the Free-Edge Delamination Behavior of Fibre Reinforced Composite Laminates[END_REF]. Nairn & al. [156] studied the effect of residual stresses on the mode I energy release rate for double cantilever beam (DCB) specimens and deduced that an error of around 76% can be made when calculating the fracture toughness while neglecting the residual stresses.

Experimental techniques to characterize and measure the residual stresses

A detailed description of the different experimental techniques used to measure the residual stresses can be found in [START_REF] Parlevliet | Residual stresses in thermoplastic composites -a study of the literature. Part II: Experimental techniques[END_REF]. The experimental methods can be either destructive or non-destructive. They are mainly based on the measurement of residual strains. Then, the residual stresses can be obtained from the strains through the elastic coefficients of the involved materials.

The non-destructive techniques are rather expensive and sometimes complicated to install. From these methods, one may cite the Moiré interferometry [START_REF] Kang | Application of Moire Interferometry to the Analysis of Residual Stresses in Metal-Matrix Composites[END_REF] & [START_REF] Schulz | Determination of residual stress and thermal history for IM7/977-2 composite laminates[END_REF] and more recently the fiber Bragg grating sensors [START_REF] Yadav | In-line residual strain monitoring for thermoplastic automated tape layup using fiber Bragg grating sensors[END_REF]. The Moiré interferometry method consists on the projection of interference pattern which acts a reference at the beginning of the measurements (during cooling) and then following its deformation on the surface based on the reflection of the light. The residual stresses can be deduced based on these deformations from the classical laminate theory. In the same manner, and using the laminate theory to measure the residual stresses, the curvature measurement c.f. Figure 1.25 is another non-destructive technique extensively used [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF], [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF], [START_REF] Cowley | The measurement and prediction of residual stresses in carbon-fibre/polymer composites[END_REF], [START_REF] Warnet | On the effect of residual stresses on the transverse cracking in cross-ply carbonpolyetherimide laminates[END_REF], [START_REF] Sglavo | Procedure for residual stress profile determination by curvature measurements[END_REF], [START_REF] Péron | Measurement and prediction of residual strains and stresses during the cooling of a glass fibre reinforced PA66 matrix composite[END_REF]. This technique is relatively simple, covers a wide range of materials and can be combined with other techniques to give stress profile such as the layer removal technique. On the other side, it's limited to simple shapes. This technique will be presented and used later in this work since it's widely used so it can be trusted in addition to its previously listed advantages.

Figure 1.25 -Measurement of curvature for unsymmetrical laminate [START_REF] Cowley | The measurement and prediction of residual stresses in carbon-fibre/polymer composites[END_REF] Another number of studies used experimental techniques based on embedded sensors such as strain gauges [START_REF] Unger | The Effect of Thermal Processing on Residual Strain Development in Unidirectional Graphite Fibre Reinforced PEEK[END_REF], [START_REF] Wisnom | Mechanisms generating residual stresses and distortion during manufacture of polymer-matrix composite structures[END_REF], fiber optic sensors [START_REF] Okabe | Effect of thermal residual stress on the reflection spectrum from fiber Bragg grating sensors embedded in CFRP laminates[END_REF], [START_REF] Chehura | Strain development in curing epoxy resin and glass fibre/epoxy composites monitored by fibre Bragg grating sensors in birefringent optical fibre[END_REF] and [START_REF] Yadav | In-line residual strain monitoring for thermoplastic automated tape layup using fiber Bragg grating sensors[END_REF] were utilized to measure the residual strains. The main advantage of these techniques is the high resolution of measured strain. However, the sensor used may be sensitive to high temperatures Destructive techniques were also used in several studies to assess the levels of residual stresses. They include first ply failure [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF], [START_REF] Cowley | The measurement and prediction of residual stresses in carbon-fibre/polymer composites[END_REF], layer removal [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF], [START_REF] Parlevliet | Residual stresses in thermoplastic composites -a study of the literature. Part II: Experimental techniques[END_REF] and blind hole drilling [START_REF] Cowley | The measurement and prediction of residual stresses in carbon-fibre/polymer composites[END_REF] & [START_REF] Parlevliet | Residual stresses in thermoplastic composites -a study of the literature. Part II: Experimental techniques[END_REF]. The first ply failure technique is one of the most used and said to be the most precise [START_REF] Parlevliet | Residual stresses in thermoplastic composites -a study of the literature. Part II: Experimental techniques[END_REF] but is however limited to cross-ply laminates. During the first-ply failure, tensile load is applied on a cross-ply laminate and the value corresponding to the first ply failure is recorded. Classical laminate theory can be used to determine the stress corresponding to this recorded load. The difference between this measured stress and the transverse tensile strength of the unidirectional laminate gives insight into the level of transverse tensile residual stress. The layer removal technique is said to be inaccurate since the curvature produced is sensible to the errors that could be performed during the abrasion of successive layers [START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF]. In addition, the abrasion technique prevents any characterization of the removed material which becomes a waste. However, Chapman & al.

[77] used a modified form of the layer removal called the process simulated laminate, during which Kapton release films are placed between the plies at the desired locations. Therefore, the layers can be easily removed after the process and the removed material can still be used for characterization. The blind hole drilling consists in bonding strain gauges to the surface of the composite material in order to measure the induced strain change caused by the formation of the hole. It's adapted to most of the materials and the strain gauges can be replaced by Moiré interferometry. However, the calibration is rather difficult since the stresses aren't uniform through the hole depth. To this end, an enhanced method called the incremental hole drilling method was developed [START_REF] Sicot | Influence of experimental parameters on determination of residual stress using the incremental hole-drilling method[END_REF] & [START_REF] Akbari | Characterization of Residual Stresses in a Thin-Walled Filament Wound Carbon/Epoxy Ring Using Incremental Hole Drilling Method[END_REF]. It is similar to the traditional hole drilling method but in this method, the hole is repeatedly pierced in an incremental way.

A more detailed description of these techniques can be found in [START_REF] Parlevliet | Residual stresses in thermoplastic composites -a study of the literature. Part II: Experimental techniques[END_REF]. With each technique having its drawbacks and advantages, an ultimate experimental technique doesn't really exist but it's a rather more suitable choice depending on the application and the available techniques. The methods used in the case of automated tape placement and filament winding are, to the author knowledge, the fiber Bragg grating sensors [START_REF] Yadav | In-line residual strain monitoring for thermoplastic automated tape layup using fiber Bragg grating sensors[END_REF], curvature measurement [START_REF] Qureshi | In situ consolidation of thermoplastic prepreg tape using automated tape placement technology: Potential and possibilities[END_REF] and Digital Image Correlation [START_REF] Karimiani | Analysis of Residual Stresses in Thermoplastic Composites Manufactured by Automated Fiber Placement[END_REF]. Qureshi & al. [167] used the curvature measurement to obtain the residual strains in AS4/PEEK laminates manufactured with automated tape laying. An overall residual strain was determined for three tool temperatures: ambient, 65°C and 150°C. The residual strain was equal to 0.24, 0.22 and 0.09% for the mandrel temperatures of ambient, 65°C and 150°C respectively. Karimiani [START_REF] Karimiani | Analysis of Residual Stresses in Thermoplastic Composites Manufactured by Automated Fiber Placement[END_REF] used the Digital Image Correlation to measure the residual strains on the surface of a composite laminate during the automated tape placement between the time after laying the ply down and its cooling. This method allows the determination of the surface residual strain only. Yadav & al. [START_REF] Yadav | In-line residual strain monitoring for thermoplastic automated tape layup using fiber Bragg grating sensors[END_REF] used embedded fiber Bragg grating sensors to monitor the variation of strains during an automated tape laying process for a carbon fiber polyamide composite. The sensors were glued on the 2 nd layer of a 15-ply UD laminate and the strains were measured on the surface of the 2 nd ply during the placement of layer 3 to 15. The conversion from residual strains to stresses occurred through classical mechanical calculation and the longitudinal residual stress at the surface ply was presented function of the tool temperature. The results are presented in and show that the tool temperature affect the surface stress by generating a compressive stress of -21 MPa for a tool at 30°C. As the tool temperature increases, therefore decreasing the cooling rates, the stress magnitude decreases (in absolute value) and may be equal to zero for a certain tool temperature, around 115°C in this case.

Modelling the residual stresses during automated tape laying

This part will cover the key-models developed to predict the residual stresses formations during different processes of thermoplastic composites. It's however possible to find reviews on residual stresses in composites such as [27].

Chapman & al. [START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] presented a model to predict the macroscopic in-plane residual stress state. The model developed is based on the classical laminate theory (CLT) with an incremental aspect to account for the variation of the mechanical properties. A crystallization kinetics model based on Velisaris & Seferis [START_REF] Velisaris | Crystallization kinetics of polyetheretherketone (peek) matrices[END_REF] was implemented to evaluate through-thickness crystallinity gradients and thus thermo-mechanical properties gradients. The composite was characterized using a first order viscoelastic model. The SFT was function of the thermal history equal to the temperature where the elapsed time at a certain material point is equal to the relaxation time. The elapsed time for a certain material point is equal, according to Chapman, to the time spent for this point to reach its temperature from the melting point. The model showed its capability to predict the residual stresses for compression molding. For a cooling rate of 35°C/s a tensile residual stress of 20 MPa was estimated which presents 25% of the material tensile strength of 79 MPa c.f. Figure 1.26. However, this work wasn't applied to tape placement or filament winding and was experimentally verified through the results of layer removal of compression molded APC-2 unidirectional laminates.

Cirino & Pipes [START_REF] Cirino | In-situ consolidation for the thermoplastic composite ring-residual stress state[END_REF] predicted the stress state in PEEK/AS4 composite rings after filament winding via an analytical model based on linear elastic theory with the assumptions of plane stress for a cylindrically orthotropic ring subjected to a uniform internal stress and temperature range. The final stress state was obtained by superimposing the contribution of the thermal loads due to in-situ consolidation and mechanical loads due to winding tension and the pressure applied by the mandrel. The authors investigated the effect of winding tension and mandrel properties on the formation of residual stresses. The results showed that the application of a constant winding tension leads to a compressive stress state in the radial direction and a tensile stress state in the circumferential one. It was also shown that the stress state can be modified according to the winding tension profile. For the radial direction, a constant and an increasing winding tension results in compressive stress state in the composite ring. The stress-state can be tailored to become tensile with a decreasing winding tension as shown in Figure 1.27. The drawbacks of this work were considering constant composite material properties. In addition, a constant temperature change was adopted through the thickness and therefore no temperature gradient was introduced in this work. Finally, no experimental validation was performed. Mantell & Springer [START_REF] Mantell | Manufacturing Process Models for Thermoplastic Composites[END_REF] developed a model to simulate the residual stresses for PEEK/AS4 thermoplastic composites during tape laying and filament winding. The model incorporated non uniform temperature distribution through the thickness of the laminate. It accounted for the matrix's crystallinity from Lee & Springer's [38] model and the variation of the matrix's viscosity with temperature. The stress-strain submodel in [START_REF] Mantell | Manufacturing Process Models for Thermoplastic Composites[END_REF] is based on the classical laminate theory for the case of tape placement. The results of this model were presented in a complementary paper [53] but only concerned the temperature and intimate contact models. No results were presented concerning the crystallinity gradients nor the residual stresses and strains.

Nejhad & al. [START_REF] Nejhad | Prediction of Process-Induced Stresses for In-Situ Thermoplastic Filament winding of cylinders[END_REF] presented a 3-D thermoelastic model for filament winding of PEEK/AS4 composites. The strains and stresses due to thermal loadings and crystallization shrinkage are calculated via an incremental method. However, the computational time was excessive for simple cases which made it even harder to perform the calculations for more complicated cases. Moreover, the numerical results weren't compared to experimental values to assess the validity of the model. Sonmez &al. [START_REF] Sonmez | Analysis of process-induced residual stresses in tape placement[END_REF] performed a 2-D finite element model to predict the residual stresses through the thickness of a cross-ply PEEK/carbon-fiber composite. A state of plane strain was assumed during the modelling, the model was assumed to be quasi-static with the frame being placed on the deposition head.

The experimental validation of the numerical model was performed on specimens fabricated using press molding. The comparison between the experimental and numerical residual stresses verified the ability of the numerical model to predict the residual stress as presented in Figure 1.28. The experimental values were obtained using the layer removal technique. Sonmez & al. [START_REF] Sonmez | Process optimization of tape placement for thermoplastic composites[END_REF] later imported the model developed in [START_REF] Sonmez | Analysis of process-induced residual stresses in tape placement[END_REF] and optimized the model by refining the mesh. In the same manner as [START_REF] Sonmez | Analysis of process-induced residual stresses in tape placement[END_REF], an initial residual stress state is applied to the ply under deposition however two conditions are applied. The first one in the case of a substrate's temperature exceeding the SFT, the residual stresses in this case are supposed to be relaxed and the initial stresses are equal to zero. The second case is the case where the substrate temperature doesn't exceed SFT. In both of the publications, the transverse tensile residual stresses are in the order of 150 MPa for the 90° oriented plies which exceed the tensile strength of the material around 75 MPa, the material is therefore expected to fail prematurely.

Chinesta & al. [START_REF] Chinesta | First steps towards an advanced simulation of composites manufacturing by automated tape placement[END_REF] developed a two-dimensional model to calculate the residual stresses during the tape placement of PEEK/Carbon-Fiber based on finite element. The behavior of the material is considered thermoelastic with temperature independent thermo-mechanical properties. The residual stresses were calculated during the deposition of one ply on a stress-free 1-ply substrate. Dedieu & al. [27] performed also a two-dimensional thermoelastic model to calculate the residual stresses using finite element. The winding of the plies is realized through iterative calculation. The model was however formulated for a temperature difference between the processing and ambient temperature. The processing temperature was set below the glass transition temperature of PEEK.

More recently, Peron & al. [START_REF] Péron | Measurement and prediction of residual strains and stresses during the cooling of a glass fibre reinforced PA66 matrix composite[END_REF] introduced a model to estimate the residual stresses which is based on the modified form of the CLT proposed by Hyer [START_REF] Hyer | Calculation of the room-temperature shapes of unsymmetric laminates[END_REF]. Hyer [START_REF] Hyer | Calculation of the room-temperature shapes of unsymmetric laminates[END_REF] proposed a model based on the classical laminate theory but with some modifications. These modifications were introduced to predict the observed cylindrical shapes of thin unsymmetrical laminates [START_REF] Hyer | Some Observations on the Cured Shape of Thin Unsymmetric Laminates[END_REF] which couldn't been predicted with the classical laminate theory. They included non-linear relations between the strains and deformations. The model developed by Hyer is however limited to the study of cross-ply laminates subjected only to thermal loading with constant mechanical properties during the whole cooling process with a uniform temperature through the thickness of the laminate. This model was later reproduced by Peron & al. [START_REF] Péron | Measurement and prediction of residual strains and stresses during the cooling of a glass fibre reinforced PA66 matrix composite[END_REF] with several enhancements to the model where a proper thermal history, crystallization gradients and temperaturedependent thermo-mechanical properties were considered.

Conclusion on the literature review

In this chapter, the brief introductions on the material properties and behavior, the process and the residual stresses showed that a large literature already exists on these different topics. Concerning the tape placement process, it's usually associated with high thermal and crystallinity gradients in addition to geometrical constraints leading to the formation of residual stresses in the final part. Several studies dealt with the experimental characterization of the thermal gradients which lead ultimately to crystallinity gradients. Some works tried to simulate the presence of these crystallinity gradients. However, the experimental validation wasn't carried out in the mentioned studies which is the first gap identified. In addition, the transverse behavior of APC-2 plies was extensively studied but the majority of the cited articles were concerned with the effect of temperature and crystallinity on the transverse modulus value and behavior. This leads to second gap identified where the transverse behavior wasn't directly linked to the effect of the process and the possibly generated crystallinity gradients on this behavior.

In light of this and in order to solve these two gaps, it is proposed in the next chapters to study the behavior of the laminates after the process using DMA and DSC. A combined analysis of the obtained results can give insight into the effect of the process on the material behavior mainly controlled by the resin and possible presence of crystallinity gradients.

While the number of studies concerned with the residual stresses in thermoplastic composites are numerous, it seems that a third gap can be identified regarding simulating the residual stresses. The two models that can be selected as the most complete ones are those of Chapman and Sonmez where in-plane and instantaneous residual stresses are discussed. However, the predicted values were only compared to the results of press-molding due to the lack of data on tape placement.

From this, the objective of this work can be introduced: predicting the residual stresses using a simple method which can incorporate the different aspects of the process influencing the residual stresses and validate this method thanks to data obtained from tape placement process.

The numerical model is based on the modified laminate theory proposed by Peron & al. and based on the works of Chapman and Hyer for press molding. This method is adapted in this work to the tape placement and incorporates a thermal, crystallization and melting models which were shown to highly influence the residual stresses formation.

The experimental validation is performed through the curvature measurement which was shown to be a simple method and proved its ability to simulate the residual stresses. In addition to the curvature measurement, the post-process quality of the material may give insight into the real scenario during the process which couldn't been online controlled. These experimental investigations require methods presented in the next chapter.

Chapter 2: Materials and Experimental Methods

Introduction

The objective of this chapter is to present the studied material and the experimental techniques used to study the behavior of the composite material fabricated using the SPIDE-TP. In the first part, a description of the materials used is presented. Next, the SPIDE-TP machine used to fabricate the different laminates studied in this work is described. The methods and devices used for the characterization of the material are then presented along with the corresponding standards. The part devoted to the description of the experimental techniques ends with a description of the method used for the curvatures measurements. While the residual stresses aren't directly characterized in this part, the curvature measurement introduces the access to the residual stresses by validating a numerical model through which the level of the stresses can be calculated.

In order to better understand the whole work scheme, an overview of the general methodology is presented at first.

Overview of the general methodology

The work during this thesis consisted of several tasks related to the characterization of the material and the determination of the residual stresses in laminates fabricated on the filament winding machine SPIDE-TP. 1 presents an overview of the general methodology followed during this work. The laminates processed using the SPIDE-TP are divided into two parts according to their experimental destination. The first group is used for the experimental characterization of the material after the process. The objective of the experimental characterization is to understand the effect of the process parameters on the thermomechanical behavior in addition to quality related parameters such as the fiber content and the presence or the absence of defects. The second batch of the fabricated specimens is destined to the estimation of residual stresses. The determination of the residual stresses is however a combination between an experimental and a numerical part. Experimentally, the laminates' curvatures at room temperature and their variation with temperature are measured. However, the access to the residual stresses from these curvatures occurs through a numerical model that will be detailed in another chapter. In this chapter, a description of the different techniques used for both material characterization and curvature measurement is presented.

Presentation of the studied materials

The composite material selected for this study is the PEEK/AS4 from Tencate Cetex® commercially known by APC-2 (Aromatic Polymer Composite). It is provided in the form of pre-impregnated tapes with a width around 6.35 mm, a thickness around 0.150 mm and a volumetric fiber volume fraction around 59% according to the manufacturer. No information was provided on the uncertainties concerning the dimensions described, in addition no access to the raw prepreg was possible during this thesis work so no measurements of these uncertainties was possible.

The APC-2 is a high-performance composite for which a detailed description based on the literature was presented in the previous chapter. The matrix is the thermoplastic PEEK with a grade 150G. The reinforcements are AS4 carbon fibers embedded in the matrix to form the unidirectional pre-impregnated ply used in this work

The unidirectional plies supplied by Tencate Cetex® were then processed on the SPIDE-TP, a filament winding machine based in the Cetim in Nantes. A detailed description of the SPIDE-TP is presented in the following section.

Processing

This part of the chapter concerns the production of the specimens used in this work to study the residual stresses. For the sake of clarity, this part is divided into two sub-parts: in the first one, the machine is described in general with its different parts and the general procedure to fabricate thermoplastic composite parts and devices on this machine. In the second part, the specimens fabricated on this machine and the processing parameters used for their fabrication are presented.

Presentation of the SPIDE-TP machine

As discussed in the introduction, this thesis concerns the SPIDE-TP which is an automated filament winding machine designed to produce structures with a revolution-symmetry around their axis such as reservoirs and pipes as shown in A general description of the winding procedure on the SPIDE-TP is presented here, the main parts of the SPIDE-TP mentioned here can be seen in Figure 2.3. Prepreg tapes are placed on a coil feeding a deposition head. On the deposition head, a series of rollers exert tension on the incoming ply in order to ensure a suitable and controlled deposition of each ply on the mandrel or on the substrate, i.e. the already deposited plies. A laser source mounted on the deposition head heats the incoming ply and the substrate up to a predetermined temperature higher than the melting temperature of the resin which is around 343°C for the semi-crystalline PEEK [1], [2], [3], [4], [5]. The laser source attached to the SPIDE-TP generates an electromagnetic radiation with a power varying between 100 and 13000 Watts. To obtain a specific power, the user can select a combination between four available wavelengths: 808, 920, 980 and 1015 𝑛𝑚. PEEK, like other thermoplastics, cannot absorb wavelengths between 400 and 1600 𝑛𝑚 [6] and thus acts as a transparent material when subjected to the laser source [6]. The carbon fibers, on the other side, are capable of absorbing these wavelengths and therefore heat the PEEK by conduction [7].

A silicone roller also mounted on the deposition head ensures the application of a compaction load on the tape and substrate in contact. The laser heat and the compaction applied by the silicone roller lead to the welding of the structure. The plies are winded on a mandrel mounted on two supports. To accomplish the winding and obtain a mandrel-shaped structure, the mandrel rotates around its axis while the deposition head moves horizontally and vertically to cover each one of the different surfaces of a mandrel. Several mandrels are available and can be mounted on their support, they're generally made of steel and can be either heated or unheated. While the heated mandrel remain a more expensive solution compared to the conventional unheated mandrel, it may promote better crystallinities and therefore better properties since it lowers thermal gradients though the thickness of the fabricated part [8].

This process is temperature-controlled, therefore the laser power and the optics angle are adjusted to maintain a certain control temperature. To this end, a thermal camera is mounted on the deposition head next to the laser source c.f.

The specimens processed on the SPIDE-TP

For this work, twelve different laminates were elaborated on the SPIDE-TP, six using a mandrel at ambient temperature and six using a heated mandrel at 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 =180°C. An obround shaped mandrel was used during the processing of the different laminates c.f. This procedure was adopted in order to produce different plates simultaneously and reduce the processing time since the machine was highly occupied during the thesis. The laminates were either unidirectional 'UD' or cross-ply laminates [90 𝑛 /0 𝑛 ]. The obround-shaped specimen was cut into different laminates based on their thicknesses and lay-up sequences using a MAC ALISTER MTC750L cutter with a diamond blade in the presence of water to avoid the overheating of the composite material. As a result, six laminates for which the dimensions are presented in Table 2.1 were obtained. The difference in the surfaces' dimensions are due to problems during processing such as the non-adhesion between the plies, in addition to possible losses in materials during the cutting process.

Table 2.1 -List of the fabricated laminates with their lay-up sequences and surface dimensions

The mean values of the thicknesses presented in Table 2.1 slightly diverge from the theoretical ones which can be obtained by multiplying 0.15 mm (the theoretical thickness of an individual ply) by the number of plies in each laminate. The reasons for this divergence could be the squeeze flow effect i.e. the flow along the thickness direction due to the application of the compaction roller compensated by an expansion along the width direction. Another reason includes the fact that the 0.15 mm thickness provided by Tencate is a mean value that could cause differences during the process. In addition, it can be due to the overlapping of two adjacent plies in order to ensure their welding. This overlapping leads to an increase in the measured thickness whenever the measurement occurs at the overlapping zone as represented in Figure 2.6.

Figure 2.6 -Schematic representation of the effect of overlapping two adjacent tapes on the thickness:

The principal parameters to consider during processing on the SPIDE-TP include the deposition speed, nippoint temperature, orientation angle of the laser, roller pressure and mandrel temperature [9], [10], [11] & [12]. The processing parameters selected are listed in Table 2.2 and correspond to the mandrels. The laser power isn't presented with the processing parameters since it's not user selected but rather automatically adjusted via the SPIDE-TP to maintain the nip-point temperature selected at 410°C. The orientation of the laser optics can be varied and is dependent on the user's choice. The substrate is thicker than the incoming ply therefore requires longer time to be heated until melting. In addition, the ply is thermally isolated due to the presence of the silicone roller. Therefore, the laser angle should be well selected to ensure a suitable distribution of the laser power between the substrate and the incoming ply. In the case of the specimens presented in Table 2.1, the orientation of the laser was selected as automatic as presented in Table 2.2. This choice allows the machine to vary slightly the laser's orientation to conserve the selected nip-point temperature during the process.

Processing Parameter Value

Deposition speed 𝑣 𝑑𝑒𝑝 (𝑚/min) 7

Optics Angle Automatic

T nip-point (°C) 410 °C

Table 2.2-Processing parameters selected on the SPIDE-TP to fabricate the specimens

Material characterization

In the previous chapter, a general overview of previous studies on the influence of the thermal history on the APC-2 behavior and properties was presented with different transverse behavior depending on the material crystallinity. The latter is highly dependent on the thermal history of the material. Moreover, the residual stresses are directly influenced by the thermomechanical properties of the material. It is essential to understand and study the thermomechanical behavior of the composites elaborated on the SPIDE-TP in order to move to the next step which is the study of the residual stresses. In this part, the techniques and specimens used to study the behavior of the material are presented.

Dynamic mechanical analysis 'DMA'

The dynamic mechanical analysis, known as DMA, is a widely used technique to study the viscoelastic behavior of the material. This technique allows the measurement of the material's properties specifically its modulus as a function of temperature, frequency and/or time. During the measurements, a small deformation is applied to a sample in a cyclic fashion. This makes it possible to obtain the resulting force. Knowing the stress mode applied and the geometry of the sample, it is then possible to determine the different characteristics such as the complex modulus which is the objective of this part.

DMA was used to study the storage modulus 𝐸 ′ indicating the material's ability to store and return energy.

The second parameter is the damping factor tan 𝛿 also called the loss factor defined as the ratio of the loss modulus to the storage modulus.

DMA tests include amplitude, frequency, humidity and time sweep measurements in addition to the temperature ramp. For this study, only the temperature ramp was used and will be briefly described in this part. However, a detailed description of each of the DMA testing method can be found in references such as [13], [14]. The temperature ramp includes applying sinusoidal load while varying the temperature under constant frequency. This type of measurement gives insight not only on the mechanical properties but also on the transition regions of the specimen such as the glass transition.

Equipment, specimens and loadings applied

For this work, a METTLER TOLEDO DMA 1 shown in Figure 2.7 was used to perform the measurements. The objectives of these measurements include the characterization of the evolution of the storage modulus 𝐸 ′ and the damping factor tan 𝛿 of the composite specimens. These two evolutions give insight on the effect of the process conditions and related thermal histories on the composite behavior. From these measurements, the glass transition temperature 𝑇 𝑔 can be studied and the effect of the process on this temperature can be established. In addition, the measurements obtained from these tests allow a first approximation of whether or not the high cooling rates and gradients during the SPIDE-TP affect the crystallinity of the material by affecting its storage modulus. The information on the crystallinity of the material is determinant to understand the input required for the numerical model in the next steps such as the modulus and its evolution during the process. The storage modulus and damping factor were measured under the three-point bending test as shown in The tests were conducted on unidirectional specimens taken from the plates described in Table 2.1. From each unidirectional plate, three rectangular specimens were cut using a MAC ALISTER MTC750L cutter equipped with a diamond blade with rim of 180 mm. They were then polished using several glass sandpapers to ensure a constant section with parallel sides. The measurements concern the transverse direction only since it was shown in the previous chapter that the longitudinal one isn't highly affected by the temperature. Therefore the width dimension 𝑏 of the specimens in Figure 2.9 is parallel while h designates the specimen thickness normal to the fiber direction. at least five times lower than the device's supports which is around 1000 N/mm for the three-point bending device should be higher than the device's membrane stiffness indicated to be around 2.5 N/mm. With these two conditions verified, the supports won't be deformed under the applied load and the stiffness measured will be that of the specimen not the membrane. Since the thickness ℎ is imposed from the processing on the SPIDE-TP, the width 𝑏 and the free sample length 𝐿 0 were chosen to respect the two conditions described earlier in this paragraph regarding the specimen stiffness.

𝑻 𝒎𝒂𝒏𝒅𝒓𝒆𝒍 =25°C

𝑻 𝒎𝒂𝒏𝒅𝒓𝒆𝒍 =180°C A total of 18 specimens for which the dimensions are shown in Table 2. 3 were tested in DMA. The differences between the dimensions of the specimens in the case of the cold mandrel and the ones using the mandrel at 180°C are due to the initial modulus chosen to set these dimensions.

These specimens were subjected to a heating from 25 to 250°C at a rate of 2°C/min and then maintained at 250°C for 10 minutes before starting to cool down. The temperature of the specimen is continuously measured using a thermocouple attached to the DMA testing head as shown in Figure 2.8. The method usually recommended by the European norm ISO 6721-11 is usually through the inflection point in the elastic modulus however another standard [17] recommends the use of the loss factor peak which is also used in this work as previously extensively used in the literature for PEEK and PEEK composites [18], [19], [20].

Determining the glass transition temperature from DMA tests

Testing the transverse modulus at ambient temperature

Following the three-point bending tests on DMA on the specimens produced using the cold mandrel, it was found rather suitable to test the unidirectional specimens cut from the same plate on a standard mechanical test such as the static three-point bending machine at ambient temperature to compare with the one measured by the DMA tests. This judgment was taken considering the results of the DMA tests that will be presented in the following chapter, in addition to recommendations from previous works on the same material such as Barnes & Byerly [21]. In fact, DMA provides a qualitative presentation of the dependence of the modulus on the temperature. However, the value of the provided modulus can be the subject of errors due to several criteria including:

Uncertainties on the sample dimensions DMA stiffness which can be close to that of the specimen and can therefore cause erroneous values

The three-point bending tests were carried along the transverse direction to measure the transverse modulus similarly to the DMA tests on an INSTRON® 3369 machine for which the maximum capacity is 10 KN c.f. The specimens are 6 and 9 plies UD rectangular bars. The 3 plies laminates couldn't be tested since their thickness in the order of 0.52±0.04 mm doesn't comply with the recommendations in the European standard ISO 14125 [22]. The dimensions for the tested specimens are presented in Table 2. 4 and match the standards of the ISO 14125. The specimens were cut using the MAC ALISTER MTC750L cutter and polished using glass sandpapers to ensure a homogeneous section across their length.

Table 2.4 -Dimensions of the specimens for the three-point tests on the Instron 3369 machine

The control and the acquisition during the tests were done through the Bluehill® software connected to the Instron® machine. The software also provides at the end of each test a load-displacement curve. To calculate the flexural modulus, the deflections 𝑠 ′ and 𝑠 ′′ corresponding to the flexural strain 𝜀 𝑓𝑙𝑒𝑥𝑢𝑟𝑎𝑙 ′ = 0.05 % and 𝜀 𝑓𝑙𝑒𝑥𝑢𝑟𝑎𝑙 ′′ = 0.25% respectively are measured according to the standard ISO 14125 using Eq. 1.10.

𝑠 =

𝜀 𝑓𝑙𝑒𝑥𝑢𝑟𝑎𝑙 * 𝐿 0 4𝑏ℎ 3 ( 𝛥𝐹 𝛥𝑠 ) Eq. 2.3

𝛥𝐹 and 𝛥𝑠 are the difference of respectively the load and mid-point displacement between the loads and displacements corresponding to the flexural strains of 0.05% and 0.25% according to the ISO 14125 [22].

Differential scanning calorimetry 'DSC'

The differential scanning calorimetry also known as DSC is a widely known and used technique to study the heat effects associated with phase transitions and chemical reactions as function of temperature. The main idea behind a DSC measurement is to record, for the same temperature, the difference in heat flow between the studied sample and a reference. These measurements detect the occurrence of phase transitions whether they're endothermic such as the fusion or exothermic such as crystallization. The glass transition can also be characterized even if it's not a thermodynamic transition. In fact, it's associated with a change in the thermal capacity of the specimen which can be observed through the differential calorimetric measurement.

The DSC used in this work is a Q200 Differential Scanning Calorimeter by TA Instruments presented in Figure 2.13. This DSC machine is part of the heat flux design machines. It's made of a single furnace in which the specimen in its sealed pan and the empty reference sealed pan are placed together and subjected to a heat flux. The DTA or Differential Thermal Analysis modulus measures the temperature difference between the sample and the reference. Four specimens were selected from each presented in Table 2.1. In addition, three specimens taken from the APC-2 prepreg tapes before being processed on the SPIDE-TP were tested in order to study the behavior of the material before the process. The specimens weighed between 5 and 14 mg, and were placed each in a sealed aluminum pan. Each specimen presents the whole thickness of the laminate from which it was extracted. From the first thermal cycle, the heating at constant rate allows the determination of the glass transition temperature, the melting temperature and the melting enthalpy. It can also detect the presence of a cold crystallization effect. Cold crystallization is the organization of amorphous chains into crystals under the heating energy once the polymer's temperature reaches its glass transition. This effect is highly possible in the case of the specimens processed on the SPIDE-TP due to the high cooling rates that could have prevented the total crystallization of the semi-crystalline PEEK resin. The crystallization temperature can be detected during cooling. The second thermal cycle is used to compare values with the first one, especially for the cold crystallization that should be normally absent in the second cycle.

While the glass transition, melting and crystallization temperatures can be directly obtained from the modules included in the DSC machine, the crystallinity degree is calculated from the thermogram obtained at the end of each DSC measurement c.f. Figure 2.15. In fact, and as previously mentioned, thermoplastic polymers can crystallize either during cooling from the melt (hot crystallization) or during heating from the solid vitreous state (cold crystallization). Therefore, the crystallinity fraction 𝜒 𝑐 (%) is calculated from the heating scan part of the thermogram by using Eq. 2.4 [23].

𝜒 𝑐 = 𝛥𝐻 𝑚 -𝛥𝐻 𝑐𝑐 𝐶 𝑚 𝛥𝐻 𝑚 100 x 100 Eq. 2.4

𝛥𝐻 𝑚 and 𝛥𝐻 𝑐𝑐 are the melting enthalpy (J/g) and the cold crystallization enthalpy (J/g) obtained from the DSC heating scan. 𝛥𝐻 𝑚 100 is the theoretical melting enthalpy of an ideal fully crystallized PEEK estimated to be 130 J/g [23] and 𝐶 𝑚 is the PEEK mass fraction. The cold crystallization enthalpy may not exist whenever the material tested has already reached its maximum crystallinity and in this case 𝛥𝐻 𝑐𝑐 is equal to zero.

The enthalpies corresponding to the melting and cold crystallization correspond to the area under the peaks in

Fiber content and microscopic observation

The material's supplier indicate a volumetric fiber fraction of 59%. This is a mean value and is subjected to local variations. Those variations and inhomogeneity can be the results of several material or process related parameters. In fact, the pre-impregnated plies can present defects related to the fiber volume fraction present initially in the supplied tapes. The process can also affect the volume fraction crystallinity and induce defects such as delamination between consecutive plies, voids and resin rich areas. These defects influence the behavior of the material under service conditions and may allow us to understand other observations or results obtained from a different experimental technique such as DMA or DSC.

As a result, the choice was made to observe the microstructure of specimens elaborated on the SPIDE-TP.

To this end, plastic molds c.f. The specimens are rectangular shaped as shown in Figure 2.16-(b) with a length L between 1 and 2 cm and a width of 1.5 to 2.5 cm. The dimensions of the specimens aren't subjected to high precision requirements but should however range between the values stated earlier in order to fit in the plastic molds. The specimens were then observed using a KEYENCE® VHX-7000 digital microscope.

Curvature measurement

The determination of the residual stresses from the experimental techniques correlated with numerical or analytical models was widely discussed in the literature and a brief summary of these techniques was presented in the previous chapter. From these techniques, there exist some based on the in-plane and out-of-plane deformations such as the interferometry based methods and the curvature measurement methods. The curvature measurement methods use optical techniques to assess the deformation measured such as a travelling telescope [21], [24], conventional cameras [25] and the Digital Image Correlation (DIC) [26].

In this work, the access to the residual stresses level in the laminates processed on the SPIDE-TP occurs through the curvature measurement technique. In order to understand the whole technique, a description of the causes and the establishment of the curvature is presented first. For instance, and as previously discussed, the residual stresses build-up occurs during the cooling phase of the process due to the multiscale anisotropic behavior: between the fiber and the matrix and between plies at different orientations or different properties.

For a unidirectional APC-2 tape, the behavior along the fiber direction is governed by the carbon fiber with a high modulus and a low thermal expansion coefficient. On the other side, the transverse behavior is governed by the PEEK thermoplastic matrix with a lower modulus and a higher CTE compared to the carbon fiber. As a consequence, the contraction during cooling along the transverse direction is higher than the one occurring along the longitudinal direction. Therefore, the deposition of successive plies during tape placement and filament winding induces the formation of residual stresses when the adjacent plies are of different orientation or present different properties due to temperature or crystallization difference. The residual stresses gradients through the laminate thickness induce a bending moment which leads to the formation of a curvature when the specimen is separated from the mandrel. If a symmetrical APC-2 crossply [(90/0) 2 ] laminate is considered as shown in Figure 2.17-(a), the residual stress state along the local principal axis is the same for each ply even if the plies are deposited at the same time such as in the press molding (no successive deposition is considered). However, the bending moment along the x and y-axis is null due to the counterbalancing effect of the symmetry. This doesn't apply to a non-symmetric laminate made of the same material presented in Figure 2.17-(b). In fact, due to unbalanced stress state macroscopically, these laminated tend to curve under the effect of the bending moment which is not null in this case. In the case of a unidirectional laminate with successive deposition using localized heating, thermal gradients may occur leading eventually to a stress gradient through the composite thickness and therefore a curvature may occur.

Setting up the curvature measurement system and taking the photos

A number of specimens with a length of 150 mm and a width of 15 mm were cut from the laminates presented in Table 2.1 using a MAC ALISTER MTC750L cutter with a diamond blade. They were then subjected to a finishing polish using grit glass sandpapers to get accurate dimensions and remove any burrs or defects present at their edges that may alter the measurements. The number of specimens selected from each laminate are presented in Table 2.5. This number corresponds to the specimens fabricated with the mandrel at ambient temperature. The same number is considered for the specimens fabricated with the heated mandrel at 180°C.

Laminate sequence Number of specimens

[90 3 /0 3 ] Unsymmetrical 6 [([90 3 /0 3 ]) 2 ] 3 [90 6 ] Unidirectional 3 [([90 3 /0 3 ]) 2 /90 3 ] Symmetrical 3

Table 2.5 -Number of specimens for each laminate sequence used to perform the curvature measurements

The objective of this section is to measure the curvatures of the specimens at room temperature and their variations with temperature. The final objective is the comparison between the measured curvatures and the numerical ones calculated using a numerical method presented in Chapter 4. Once the experimental and numerical results correlate, the numerical model is validated and the residual stresses calculated through it can be considered as representatives of the real residual stress state. The measured curvatures can also be considered as input to the numerical model based on the modified laminate theory and estimate the residual stresses level from the laminate stiffness.

To this end, a curvature measurement system was mounted and presented in Four heating and cooling cycles were applied with the following maximum temperatures: 120, 150, 180 and 200°C. Other higher temperatures could've been exploited, however due to time constraints the measurements were limited to the mentioned temperatures. The steps of the general procedure for this set of specimens are as follows:

1) Six specimens named figuratively to understand from 𝐴 1 to 𝐴 6 c.f. Figure 2.19 are placed in the oven using the fold back clips while making sure that no contact exists between the different specimens. 2) In the oven, a graduated steel rule is placed next to the specimens as shown in Figure 2.18 in order to set a scale during the measurement of the curvatures.

3) The digital camera is placed on its tripod on a solid table at the same height level as the Binder from the ground. The distance between the camera and the oven was approximately 270 cm. This distance was chosen after several trials along with changing the magnification of the camera lens in order to have a good compromise between avoiding any astigmatism related problems and having a complete image of all six specimens. 4) Once the measurement assembly is mounted, a first set of photos are taken of the specimens and the graduated rule at room temperature. 5) Next, a glass window is mounted behind the door of the Binder and in front of the specimens. In fact, the climate chamber has a solid opaque door and isn't equipped with a glass door to avoid any heat losses when the photos are taken. Therefore, the glass window minimizes the heat losses when the photos are taken. Once the glass window is mounted, another photo of the specimens and the graduated rule is taken before removing the graduated rule.

Figure 2.19 -Illustration of the different specimen through the different heating and cooling cycles in the climate chamber

Once all these steps are complete, cycle 1 in Figure 2.19 starts with the first heating from 30°C up to 120°C at 1°C/min. Photos are taken every 10°C after the specimens spend 10 mins at the required temperature to ensure a homogenized temperature distribution across their volume. After reaching the 120°C temperature, the cooling begins and similarly to the heating phase, photos are taken every 10 degrees. However, the cooling rate cannot be controlled in the same manner as the heating one because the climate chamber isn't equipped with an appropriate cooling system but rather with a sole ventilator. The first maximum temperature chosen as 120°C is smaller than the glass transition temperature of the PEEK typically known around 143°C [27], [28], [29].

During cycle 2, 𝐴 1 and 𝐴 2 are maintained in the Binder while the remaining four are removed as shown in Figure 2.19. 𝐴 1 and 𝐴 2 are subjected again to the different steps 1,2,4 and 5 while the camera's position was maintained. However, for this heating phase, the maximum temperature is 150°C which falls above the glass transition temperature. The specimens are heated at a rate of 1°C/min and photos are taken each 20°C until 120°C. After taking the corresponding photos at 120°C, the specimens are directly heated until 150°C. A first set of photos was taken after 10 minutes. Next, photos were taken every 10 minutes until 30 minutes have passed. The curvature was checked during this 10 minutes intervals and it was approximately the same after 20 minutes and 30 minutes. Therefore, a final set of photos was taken before starting the cooling phase. The specimens were maintained for 30 minutes on this temperature in order to check for any changes of the specimens' curvatures during the isothermal run.

Cycle 3 includes the specimens named 𝐴 1 to 𝐴 4 which are heated up to 180°C following the same procedure as cycle 2. This set of measurements allow the measurement of the evolution of curvature with temperature to check if any changes occur during the isothermal heating on 180°C due to an incomplete cold crystallization at 150°C or due to relaxation of stresses. In addition, comparing the curvatures of the couples allow the conclusion on whether the thermal path affects or not these curvatures i.e. whether any differences are observed if the specimens are heated to 150°C then to 180°C or directly to 180°C.

At last, the two remaining specimens 𝐴 5 and 𝐴 6 are placed back with the four specimens heated until 180°C and the same process is repeated by heating up to 200°C and then cooling down (cycle 4 in Figure 2.

19).

The same procedure is adopted for the remaining laminates listed in Table 2.1. However, , the number of specimens is reduced to three and all three specimens in each case are tested up to 120°C, 150°C and 180°C in the same manner as the procedure described for the [(90 3 /0 3 )] laminates. This decision was based on the measurements results of [(90 3 /0 3 )] laminates.

Extraction of the curvatures' evolutions

The variation of the curvature of the specimens with temperature was monitored, as previously said, using a Nikon D600 camera. The photos are taken at different temperature steps. For each set of specimens heated and cooled simultaneously in the oven, the post processing of the photos occurs simultaneously. This is due to the presence of the graduated steel ruler at the beginning of the heating phase which serves to set the scale for the measurement. This scale is attributed for the different photos of the different specimens at different temperature steps.

The photos are post-processed on the ImageJ software. The first step is to import an Image Sequence which is the set of successive photos taken at different time steps starting with the one containing the steel ruler at ambient temperature. Next, the scale is set based on the graduation of the steel ruler and applied for the different photos. Once the scale is set, different points were selected on the edge of each specimens and their coordinates (𝑥 𝑖 , 𝑦 𝑖 ) were automatically obtained by the ImageJ c.f. The curvature 𝜿 (1/m) for each specimen is equal to the inverse of the radius of the circle as presented in Eq. 2.7.

𝜿= 1 𝑟 𝑐

Eq. 2.7

Conclusion

In this chapter, the material selected for this study which is pre-impregnated PEEK/AS4 plies with a volumetric fiber fraction of 59% is presented. The material was processed on the SPIDE-TP to produce unidirectional and cross-ply laminates listed in Table 2.1.

The material is subjected to different characterization techniques such as DMA, DSC, static three-point bending tests and microscopic observations to understand its state after the process. After the characterization of the material, the curvature measurement method is presented.

In the following of the chapter, a brief review of the techniques is presented, they're applied on specimen elaborated using different processing conditions to study the effect of process parameters such as the mandrel temperature:

1. DMA is adopted to study the transverse modulus in addition to the glass transition temperature. The value of the modulus is adjusted through a standard three-point bending test. 2. Microscopic observations were realized to study the fiber content and detect the presence of defects. 3. DSC technique is also employed to determine the different characterizing temperatures such as the glass transition, crystallization and melting temperatures in addition to the crystallinity fraction after the process. 4. The variation of the curvature with temperature for the different specimens was realized through the method explained in Section 2.7. The following chapter is dedicated to present the results obtained from these experimental techniques.

3 Chapter 3: The effect of the process on the material behavior and its curvatures: experimental results

Introduction

As discussed in the first chapter, laser assisted tape placement and filament winding calls for a complex thermal history and induces the formation of residual stresses in the final product. In addition, it was particularly stressed in the literature review that the thermomechanical properties of the PEEK/AS4 composites are highly dependent on the crystallinity and temperature. Therefore it seems essential to study the properties of the laminates produced on the SPIDE-TP in order to:

Verify their suitability for a certain application Understand their dependence on process parameters Moreover, the main objective of this thesis is to evaluate the residual stresses during tape placement or filament winding on SPIDE-TP. The literature review presented in Section 1.4.1 emphasized the relation between the process parameters (mainly the thermal history) and the residual stresses. Hence, the evaluation of the residual stresses implies an adequate description of the process and its parameters. However, a live access to the parameters during processing is a difficult task. An alternative post-process study of the produced laminates' properties may give insight into the process via a backward analysis.

The above discussed points are complex and call for several techniques which were presented in the previous chapter. The present chapter focuses on the presentation and the analysis of the results and is divided into two main parts:

Cross-sectional micrographs

This section aims to present the results regarding the post-processing quality in terms of fiber content, consolidation quality and presence of defects in the processed laminates. As it was shown in Chapter 1, the quality of the final part is directly related to the thermal load during the process in addition to the effects of the mandrel and the roller. This section is divided into a qualitative and quantitative description of the laminate morphology. While specimen from different laminates were prepared according to the protocol described in Section 2.6, the microscopic observations of the 15-plies symmetrical laminate [(90 3 /0 3 ) 2 /90 3 ] are only presented here due to difficulties concerning the polishing of thinner laminates that tended to flow in the resin.

Qualitative analysis of cross-sectional micrographs

Cross-sectional micrographs of the 15-plies symmetric [(90 3 /0 3 ) 2 /90 3 ] processed with the cold and the heated mandrel were analyzed to study the influence of the process on the final quality. The main three aspects studied were the fiber content, the morphology of the interface between the plies and the presence of defects. A closer look to the interfaces between the 90° and 0° oriented plies in case of the cold and heated mandrel can be seen in Figure 3.2. It shows that in both cases inhomogeneities along the laminate thickness can be detected. In addition to the interlaminar voids, resin rich areas can also be seen such as

Quantitative analysis of cross-sectional micrographs

Following the qualitative analysis of the cross-sections of the different laminates, this section provides the quantitative analysis part. It aims into identifying the fiber content in the selected laminates. To this purpose, two exemplary photos are presented in Regarding the fiber volume fraction, the values measured correspond to 50±5% and 54±6% in the case of the cold and the heated mandrel respectively. These values are obtained based on the measurements taken from approximately 20 images for each specimen and using the microscope program. These values are less than the one indicated by the manufacturer for the prepreg corresponding to 59% but don't present a high deviation from this value. Moreover, the value given by the manufacturer is a mean value and can present heterogeneities across the prepreg tape used. This may also explain the difference between the measured value and the one indicated by the manufacturer.

The results presented in this section don't seem to highlight much differences between the cold and heated mandrel. Therefore, it seems crucial, for future study, to enlarge the specimens' number and geometries in order to understand thoroughly the effect of the mandrel temperature on the processed laminate geometry.

Differential scanning calorimetry analysis

DSC tests have been carried out on specimens extracted from plates manufactured using the cold and heated mandrels according to the testing protocol described in Paragraph 2.

1 st Heating

The behavior of the material after the process can be studied during this heating run. The main difference between the unheated and heated mandrel can be detected during the 1 st heating. An exothermic peak (at 173°C for the specimen presented in Figure 3.5-(a)) is clearly present in the case of the cold mandrel. In contrast, this exothermic peak is either absent or too small to be detected in the case of the heated mandrel as seen in Figure 3.

5-(b).

The occurrence of the cold crystallization in the case of the cold mandrel suggests the presence of regions for which the crystallization hasn't been completed during the process. The effect of the mandrel temperature and consequently the thermal history on the crystallinity of the material can be established from Figure 3.5 as follows: a homogeneous thermal distribution and slow cooling rates (obtained using the heated mandrel) reduces the presence of amorphous regions compared to an inhomogeneous thermal distribution and fast cooling (in the case of an unheated mandrel).

An endothermic peak characteristic of the melting of the crystalline phase within the composite is detected at higher temperatures (at 345°C in

Cooling

The exothermic peak during the cooling run (blue curve) corresponds to the melt crystallization of the macromolecular structure of the matrix. No important differences between the specimens manufactured with the two mandrels can be mentioned since the cooling occurs from the melt after erasing the previous thermal history of the material. Once the specimen reaches the ambient temperature, it's subjected to a 2 nd heating (pink curve).

2 nd Heating

Once the specimen reaches the ambient temperature, it's subjected to a 2 nd heating (pink curve). In the case of the cold mandrel and in contrast to the 1 st heating, no exothermic peak is detected upon heating the specimen a second time. It can be concluded that the specimen was able to fully crystallize during its cooling from the melt in the DSC. The difference in terms of the cold crystallization between the 1 st and the 2 nd heating suggests the presence of partially crystallized regions in the material after processing on the SPIDE-TP (1 st heating run) that can crystallize upon slow cooling from the melt (2 nd heating run).

Glass transition

The glass transition is a second order phase transition associated with a change in the material's behavior from glassy to rubbery upon heating and vice versa upon cooling. A modification of thermomechanical properties is associated to this behavioral change. Therefore, the glass transition is considered as a parameter of high interest when studying amorphous and semi-crystalline materials. It can be determined via several ways including DSC and DMA. In this part, the results from DSC scans are presented.

A comparison of the measured 𝑇 𝑔 before and after cooling from the melt is presented in In the case of the cold mandrel, the shift of 𝑇 𝑔 by 8°C between the 1 st and 2 nd heating is in accordance with a previous work on pure PEEK 380G [4]. The DSC tests performed by Atkinson [4] revealed a 𝑇 𝑔 of 145°C and 152°C for the amorphous and semi-crystalline specimen respectively. In addition, Doumeng [8] evaluated in her thesis the dependence of the glass transition temperature and the crystallinity of PEEK 450G on the annealing temperature and the annealing time. The author showed that the material is able to recrystallize partially when the annealing time increases and that 𝑇 𝑔 increases with the annealing time and crystallinity c.f. Figure 3.7. This result also correlated with the observations in Table 3.1. In fact, increasing the crystallinity fraction ( through slow melt crystallization) increases 𝑇 𝑔 due to the lower amorphous phase mobility [4].

(a) (b)

Fusion

The endothermic peak in the heating ramp of the DSC scan corresponds to the fusion of the crystalline phase of the PEEK resin. The fusion temperatures presented in Table 3.1 are close to 343°C, the melting point of the PEEK resin [9].

The measured temperatures in Table 3.1 are close, however a decrease of the melting point by few degrees is observed between the 1 st and 2 nd heating for the specimen with the cold mandrel. The difference is not very important (less than 2 degrees) but it follows the trend pointed out in Doumeng's thesis [8] on the dependence of the melting temperature on the annealing time and the crystallinity. The results from [6] presented in Figure 3.8 show that increasing the PEEK's crystallinity via cold crystallization decreases the fusion temperature by around 20 degrees. The shift in the case of the PEEK/AS4 in Table 3.1 is around 2 degrees rather than the 20 degrees in [6]. The comparison in [6] occurs between an amorphous (crystallinity=0) and a semi-crystalline, while the comparison in Table 3.1 occurs between two semi-crystalline with different crystallinities.

The DSC scans in Figure 3.5 show different melting endotherm shapes between the 1 st and the 2 nd heating in the case of 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 = 𝑇 𝑎𝑚𝑏𝑖𝑒𝑛𝑡 and 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 = 180°𝐶. The endotherm during the 1 st heating ramp presents a single peak and is wider than the second one. On the other side, two peaks are detected during the 2 nd heating: a minor and a major c.f. Figure 3.9 (red circles). This behavior was observed for the different specimens. The double melting behavior of PEEK was widely discussed in the literature and a summary of the previous publications on this subject was presented in the previous chapter. The first peak (highlighted with the dashed red circle in Figure 3.9) corresponds to the fusion of thin lamellae formed during isothermal crystallization. These lamellae may recrystallize during the heating and form more stable thicker lamellae.

The second peak (highlighted by the continuous red circle in Figure 3.9) corresponds to the fusion of thick lamellae which were already present in the specimen in addition to the ones formed during the recrystallization of the thin lamellae [10], [11], [12], [13], [14] & [15]. The double melting peaks in Figure 3.9 appear after a crystallization of the specimen form the melt (380°C) during the first cooling. This is in accordance with the results of Alvaredo & al. [16] on PEEK 90G reinforced with graphene nano-platelets composites presented in Concluding on the melting behavior according to the DSC scans:

The fusion temperature 𝑇 𝑚 doesn't seem highly influenced by the mandrel temperature (a maximum difference of less than 2 °C is recorded between the melting temperatures in the two cases) Similarly, the melting behavior isn't affected by the mandrel temperature ( one melting peak is observed for the specimens during the 1 st heating in case of cold or heated mandrel) Crystallizing from the melt (380°C) induces a change in the melting behavior (two endothermic peaks) in the case of the cold and heated mandrels. Based on the literature, a morphological change is induced in a semi-crystalline PEEK when subjected to an annealing resulting in the double melting peak behavior.

Crystallization Temperatures

An exothermic peak is detected during the 1 st heating of the specimens manufactured with the cold mandrel.

It indicates the occurrence of a cold crystallization around 𝑇 𝑐𝑐 =172.5°C. No exothermic peak is detected for the specimens manufactured with the heated mandrel. The mean value measured for 𝑇 𝑐𝑐 correlates with previously reported values in the literature for PEEK resin and PEEK composites [6], [17]. Risteska & al. [17] found 𝑇 𝑐𝑐 of 173°C for a Tencate® PEEK 150G reinforced with HTS45 carbon fibers and processed using laser assisted tape placement. The cold crystallization occurs in the amorphous region of a polymer. According to Figure 3.5(a), the fusion endotherm is bigger than the exothermic peak of the cold crystallization. This suggests that a part of the crystals melted at high temperatures were already present after the process and didn't all result from the cold crystallization. Therefore, it can be concluded that the specimens processed using a cold mandrel didn't reach their maximum crystallinity after the process. Regarding the crystallization occurring at high temperatures, the measured temperatures for the specimen are close and no significant differences can be mentioned.

The cold crystallization temperature is considered as a key parameter for the post-processing treatment. This is due to the fact that post-processing treatments such as the annealing in the objective of enhancing the manufactured laminates crystallinity fraction are usually carried out at temperatures close to the cold crystallization temperature. However, the crystallization temperature at high temperatures of the prepreg prior to the process is important since it affects the evolution of the crystallization rate during the process and the formation of the residual stresses [18].

Crystallinity fraction

The crystallinity fractions are calculated using the method in Section 2.5.3 for the specimen manufactured using the cold and hot mandrels are presented in The crystallinity resulting from the process is estimated during the 1 st heating. In case of the cold mandrel, the maximum crystallinity in The difference of crystallinity fraction between the cold and heated mandrel is directly related to their different thermal histories. The main factors included are the cooling rate (its value and the presence of gradients) and the processing temperature. Heating with a laser which is a focused heat source at high deposition speeds is associated with high cooling rates and thermal gradients due to the local heating. These cooling rates can range between -3500°C/s and -380°C/s such as in [19] for a placement speed of 0.1 m/s.

The crystallinity fractions measured after the process in Figure 3.12 , notably in the case of the cold mandrel, are lower than the values reported in the literature for autoclaved PEEK/AS4 composites such as the 40% in [20]. As previously said, an important factor influencing the crystallinity fraction is the cooling rate. This dependency was previously studied by several authors [21], [22] & [23]. The values reported in the literature are presented in Figure 3.13. The results are scattered due to either the type of resin and fiber in the composite or to the different techniques used to measure the crystallinity such as Wide Angle X-ray Diffraction (WAXS) [21] and DSC in [22] and [23]. Regardless of the scattering in the values, the different results show a similar trend of a decreasing crystallinity with an increasing cooling rate. The cooling rates during tape placement or filament winding with in-situ consolidation are generally elevated. According to the results from Figure 3.13, the crystallinities corresponding to these high cooling rates range from 25% to 0%, therefore the results obtained in this study in Figure 3.12 correlate with the values in the literature. The use of a heated mandrel leads to a more homogeneous temperature distribution across the manufactured composite and slower cooling rates. Therefore, it seems logical to obtain higher crystallinities in the case of the heated mandrel. In case of the heated mandrel, the increase between the 1 st and 2 nd heating is less pronounced compared to the cold mandrel case. Nevertheless, the cold crystallization peak to which is assigned the crystallinity increase in the case of the cold mandrel isn't explicitly present during the 1 st heating scan for the heated mandrel. The use of the heated mandrel may induce a more homogeneous temperature distribution and a slower cooling rate compared to the cool mandrel specimen. However, a thermal gradient is inevitably present due to the laser heat source, therefore a non-isothermal crystallization takes place even with a heated mandrel. This non-isothermal crystallization can lead to a crystallinity gradient through the specimen thickness. The crystallinity increase after the slow cooling from the melt in the case of the heated mandrel may be explained by either one or both of the following:

Small amorphous regions existed after processing using the heated mandrel which were crystallized during the slow cooling from the melt in the DSC The slower and more homogeneous cooling in the DSC allowed a suitable growth of the crystals therefore occupying a bigger volume which resulted in higher crystallinity fraction.

Finally, the crystallinity fraction calculated from DSC scan can be considered as a mean value since it represents the crystallinity of the whole specimen. In fact, the exothermic cold crystallization in the case of the cold mandrel indicates one of the two following suggestions:

The specimen presents crystalline regions through which amorphous regions are distributed across the whole specimen The presence of amorphous layers between semi-crystalline layers ( with relative crystallinity that could go up to 1)

The presence of amorphous regions in the polymer can be detected through experimental techniques or via validated numerical models. The experimental techniques include for example progressive cutting through the specimen thickness and measuring the removed parts crystallinities or through tomographic methods [24]. However, due to time related limitations, the experimental method to study the presence of crystallinity gradients through the thickness was not performed. It will however be mentioned as one of the perspectives and the presence of this gradient is investigated numerically in the next chapter. The different specimens show a similar trend: an increase in the load with increasing the displacement until the failure. The failure behavior is however different between the specimen. In the case of the unheated mandrel, a sudden drop in the load is obvious for the 6 and 9 plies specimen. In the case of the heated mandrel, there seems to be a divergence in the behavior at the break between the specimens. These differences can be the result of defects present prior to the tests such as resin rich areas, internal cracks, voids, etc. The flexural moduli calculated from these tests are presented in Figure 3.16. In case of the cold mandrel, the modulus is 6.5±0.32 GPa and 6.1±0.6 GPa for the 6 and 9 plies unidirectional specimen respectively c.f. Figure 3.16-(a). These values increase for the heated mandrel to 7.7±0.12 GPa and 7.8±0.27 GPa for the 6 and 9 plies specimen respectively. The differences between the moduli in the case of each mandrel may be due measurement inaccuracies and/or defects present in the specimen prior to the measurement. Therefore, the results show little dependency of the measured modulus on the thickness i.e. the number of layers (the comparison is made between 6 and 9 plies). The dependence of the modulus on the mandrel temperature is more important with an increase of 20% and 28% for the 6 and 9 plies specimen respectively between the cold and heated mandrel.

Mechanical behavior

99 The modulus increase in the heated mandrel case is directly related to the crystallinity increase. In fact, the specimen manufactured with the heated mandrel present higher levels of crystallinity compared to the ones manufactured using the cold mandrel based on the DSC measurements c.f. Figure 3.12. For the 6-plies specimen, the modulus increases from 6.45 GPa for the cold mandrel to 7.76 GPa for the heated mandrel which corresponds also to an increase of crystallinity from 21.3% to 27%. The modulus passes from 6.1 GPa to 7.84 GPa between the cold and heated mandrel in the case of the 9-plies specimen with an increase from 23 to 28% in the crystallinity.

As discussed in the previous chapter, the values of the modulus from the static three-point bending tests were used to calibrate the results of the DMA tests presented in the following section.

Thermomechanical behavior of unidirectional laminates: Results of the DMA tests

The variation of the transverse storage modulus 𝐸 22 and the damping factor 𝑡𝑎𝑛 𝛿 with temperature are exploited through the DMA tests. As discussed in the previous chapter, two consecutive heating are applied on the DMA tests. The first one allows to study the effect of the process parameters (mainly the mandrel temperature) on the material behavior while the second one translates the effect of the annealing at 250°C on this behavior.

Figure 3.17-(a)

shows the evolution of 𝐸 22 for the unidirectional specimen manufactured using the cold mandrel during the first heating and the second heating on the DMA test. In the case of the heated mandrel, the differences in the behavior during the first and second cooling are negligible, therefore the curves in Figure 3.17-(b) represent the evolution of 𝐸 22 during the first heating only. The specimen extracted from the same plate behave similarly so only one specimen for each case is presented in Figure 3.17. 3.2 lists the maximum standard error and the corresponding temperature at which this maximum error occurs between the specimens for the different laminate sequences studied in case of the unheated and heated mandrel. In the case of the cold mandrel, the maximum standard error can be identified at ambient temperature for the three laminate sequences and at 171°C and 165°C for the 6 and 9 plies UD respectively. The maximum value is 15% and can be due to dimensional differences between the different specimen. However, the errors are considered acceptable and the specimens with the same laminate sequence present the same evolution of their modulus with the temperature. The specimen manufactured with the unheated mandrel present lower storage modulus compared to the ones manufactured using the heated mandrel between the ambient temperature and temperatures close to 200°C. For instance, at 30°C, the DMA measured for the [90 3 ] specimen a transverse modulus of 6.25 GPa in the case of the cold mandrel compared to 7.82 GPa in the case of the heated mandrel. At 120°C, prior to the glass transition the transverse modulus for the [90 6 ] specimens is 5.75 GPa and 7.39 GPa for the cold and heated mandrel respectively.

However, at elevated temperature (above 200°C), the moduli of the different specimen seem to be similar. The same observation can be made from the comparison between the values during the first and second heating in the case of the heated mandrel. This is the result of the higher crystallinities in the case of the heated mandrel presented in the previous section.

Below the glass transition, the storage modulus is almost constant in the glassy state (below 𝑇 𝑔 ). For example, 𝐸 22 for the 6 plies UD specimen manufactured using the unheated mandrel remains close to 6.1 GPa below 105 °C ( 𝑇 𝑔 -28°𝐶).

Above the glass transition temperature, the composite behaves similarly to the phase below the glass transition. The modulus is almost constant and remains close to 2 GPa above 190°C ( 𝑇 𝑔 + 60°𝐶) until 250°C for the 6 plies UD specimen. Moreover, the storage modulus in the glassy state is 3 times higher than the one in the rubbery state. The same observation can be made for the remaining specimen.

During the glass transition phase, the main differences between the specimens arise. The specimen manufactured using the unheated mandrel present an abrupt and sudden decrease of their modulus during the glass transition phase as depicted in Figure 3.17-(a) during the first heating. The modulus reaches 0.5 GPa, 1 GPa and 1.1 GPa for the 3-plies, 6-plies and 9-plies UD specimen respectively. Once these values are reached, the modulus re-increases with temperature and reaches values around 2 GPa. The modulus increase after the sudden drop occurs within the same temperature range as the exothermal cold crystallization recorded with the DSC tests in the previous part. The increase in the modulus can therefore be attributed to the increase in the crystallinity during the cold crystallization. This behavior was observed for amorphous APC-2 after quenching in [25], [26] & [27].

In the case of the heated mandrel and the annealing, i.e. after cooling from 250°C, the specimen behave as an APC-2 composite with a semi-crystalline resin. The modulus of the material with a semi-crystalline resin decreases gradually during the glass transition. After this decrease and until 250°C, 𝐸 22 remains close to 2 GPa which corresponds to 1/3 of the modulus below the glass transition. This behavior is typically observed in literature for APC-2 with a semi-crystalline resin [25], [26], [27].

In the case of the cold mandrel, the specimen behaves similarly to an amorphous material during the glass transition, however the results of the DSC tests on specimen extracted from the same plates don't suggest a complete amorphous material (the crystallinities were observed between 19 and 21 % c.f. Figure 3.12). In light of this, a crystallinity gradient may be present through the thickness of the material that can induce this decrease during the glass transition of the material. This gradient can be the direct result of the thermal history during processing with the unheated mandrel associated with local heating and fast cooling rates.

On the other side, the use of a heated mandrel results in a more homogeneous temperature distribution along the thickness therefore omitting the gradient of crystallinity through the thickness.

In addition to the transverse modulus, the glass transition temperature 𝑇 𝑔 can be exploited from the variation of the damping factor 𝑡𝑎𝑛 𝛿 with temperature as discussed in the previous chapter. The glass transition temperatures correspond to the peak of 𝑡𝑎𝑛 𝛿 and are labelled in

Curvature measurements

This part concerns the measurement of curvatures as described in Section 2.7. The curvatures of the specimen at ambient temperature and their variation with temperature during different thermal cycles are presented. The importance of the results presented in the following is twofold:

1. Understand the effect of the mandrel temperature on the material behavior and correlate with the results obtained previously in this chapter concerning the experimental characterization of the specimen after the process. 2. Establish the set of experimental data that will be used in the next chapter to validate the numerical model and access to the residual stresses.

Evolution of the curvature of unsymmetrical cross-ply [𝟗𝟎 𝟑 /𝟎 𝟑 ] specimens manufactured with the cold mandrel

Maximum temperature: 120°C

The 1 st set of measurements was conducted on a series of [90 3 /0 3 ] unsymmetrical cross-ply laminates processed using the cold mandrel. The results for the first heating up to 120°C and the corresponding cooling are presented in Figure 3.20. The maximum standard deviation in Figure 3.20 is 0.088 𝑚 -1 and is obtained at ambient temperature. This shows a good reproducibility of the measurements over the whole temperature range. The initial curvature is 2.6±0.14 𝑚 -1 which corresponds to the specimens' mean curvature before any heat applied. The curvature smoothly decreases upon heating to 120°C and increases during the subsequent cooling. Slight differences exist between the curvatures during heating and cooling and can be attributed to measurement errors. The maximum difference is equal to 0.26 𝑚 -1 at ambient temperature which can be accepted in light of the measurements' errors and possible presence of defects in the specimens. Similarly to the trend observed in Figure 3.20, the curvature of the specimen decreases upon heating. A smooth decrease with a fairly linear shape can be detected until 120°C with a slope of -0.024𝑚 -1 . °𝐶-1 as it can be seen on the left of Figure 3.21. The curvature measured at 150°C (before the isothermal run for 30 min) shifts away from this linear increase (c.f. Figure 3.21 on the left) and is higher than the theoretical one (red cross) if the linear trend was to be followed. In addition, an increase in the specimens' curvatures from 0.89 𝑚 -1 to 1.03 𝑚 -1 (an increase of around 16.4%) occurs during the isothermal run for 30 mins at 150°C. In the case of the specimen subjected to the heating to 120 and 180°C, the behavior is similar to the one reported in Figure 3.21. For the first set of specimen previously subjected to the annealing for 30 minutes at 150°C, the increase in the curvature during the isothermal run at 180°C is around 3%. This increase reaches a value around 17% in the case of the specimen annealed only at 180°C. Nevertheless, the values of the curvatures at the end of the isothermal run at 180°C are close with a slightly higher value for the ones previously heated at 150°C (0.69 𝑚 -1 compared to 0.67 𝑚 -1 ). Figure 3.23-(a) and (b) present the linear fittings for both sets of specimens during heating and cooling from 180°C respectively. In both cases, the experimental curvature deviates from the linear prediction during heating with a clear offset as it can be seen in Figure 3.23-(a). In fact, for both cases, if the linear evolution was followed, the theoretical curvatures would've been negative at 180°C.

The results in Figure 3.22 show a difference of 0.17 𝑚 -1 between the curvatures of the two sets of specimen. This value corresponds to the difference between the mean curvatures of the specimens subjected to an annealing at 150°C then to 180°C and the ones subjected to the annealing only at 180°C, the former being higher.

Maximum temperature: 200°C

The variation of the specimen curvature with temperature during the heating and cooling phases of the annealing at 200°C are presented in than the glass transition temperature 𝑇 𝑔 of the material measured between 133°C (DMA c.f. Figure 3.18) and 140°C (DSC c.f. Figure 3.6). Below the glass transition, the only parameter influencing the curvature is the thermal deformations caused by the temperature change. For thermoplastics, these deformations are reversible during heating and cooling from the same temperatures. Slight differences exist between the curvatures during heating and cooling for the same temperature and can be attributed to measurement errors.

This behavior changes when the specimens are heated to 150, 180 and 200°C where an offset between the curvatures during heating and cooling is present as it can be seen in Cebe & Hong [29] showed that a cold crystallization may begin slightly above the glass transition and can continue up to 270°C. This cold crystallization is accompanied by a crystallization shrinkage which plays a major part in the variation of the curvature during this isothermal step. In addition to the crystallization shrinkage, the increase in the material's crystallinity is associated with a variation of its properties notably the Young modulus and the thermal expansion coefficient. By comparing the slopes of the linear fittings in Figure 3.21and Figure 3.23, one cannot directly conclude on the evolution of properties from the slopes because the values obtained during heating and cooling for the different cases are very similar. However, the DMA results for the cold mandrel case in Figure 3.17 showed an increase in the modulus once the material is heated above its glass transition temperature which also correlates with the occurrence of a cold crystallization detected in Figure 3.10.

The results concerning the curvature evolution recall previously reported results by Barnes & Byerly in [30] even though the maximum temperature in their case is 310°C and thus can induce different behavior. They measured the variation of the curvatures of unsymmetrical cross-ply PEEK/AS4 and PEEK/IM8 composites function of temperature during heating and cooling from 310°C. The results obtained are presented in Figure 3. 25 where an offset exists between the curvatures during heating and cooling. It was to an increase in the crystallinity when the material is heated at high temperatures but no information was further given and no DSC tests were presented to show the original crystallinity of the material.

Evolution of the curvature for the unidirectional specimen: cold mandrel

The presence of a curvature when studying unsymmetrical cross-ply laminates is logical due to the anisotropic behavior between the plies at different stacking orientations. This anisotropy is, on the other hand, irrelevant when dealing with unidirectional (UD) laminates and therefore this type of laminates is expected to be flat, theoretically. This flatness may be possible in the case of thermoforming or compression molding assuming a symmetrical cooling and that there exists no geometric defects such as local fiber misalignments or fiber volume fraction gradients. When dealing with processes including local heating and high thermal gradients during the process such as the case of filament winding or tape placement on the SPIDE-TP, the thermal history induces the presence of properties gradients through the laminate thickness leading to strains gradients. In addition, the presence of a mandrel at the bottom surface of laminate adds to the heterogeneities. The result is therefore a residual stresses distribution inducing a bending moment leading to a curvature of the specimens. This is the case of the UD laminates presented in this part for which a curvature was observed at ambient temperature directly after the cutting process. The same annealing cycles adopted for the cross-ply [90 3 /0 3 ] presented previously were applied on three 6-plies unidirectional laminates obtained under the same processing conditions. The evolution of the mean curvature for the three UD laminates during heating and cooling from 120°C is presented in Figure 3.26 compared to that of the unsymmetrical [90 3 /0 3 ] laminates for the same heatingcooling cycle. The mean curvature for the unidirectional laminates at room temperature is equal to 0.38 𝑚 -1 which corresponds to approximately 15% of that for the cross-ply laminates. The difference between the two values presents the effect of the stacking sequence which is more pronounced in the case of the crossply laminates. On the other hand, the UD specimen curvature manifests the effect of the thermal history which evokes local heating and fast cooling rates leading to high thermal gradients across the thickness of the laminate. Other effects may participate to the formation of this curvature including for example geometric defects such as small angle deviations between the plies in the UD.

On another note, the evolution of the mean curvature for the UD laminates is quite similar to the previous observation on the cross-ply laminates: a decrease during heating and an increase during cooling. No offset between the curvatures during heating and cooling is observed as in the case of the cross-ply laminates for a heating at a maximum temperature of 120°C.

The same specimen were subjected to annealing at 150°C and the evolution of the curvatures during this annealing is presented in Figure 3.27. The results are compared to the cross-ply laminates curvatures for the same thermal cycle. The difference between the curvatures at ambient temperature at the beginning and the end of the annealing at 150°C is 0.58 𝑚 -1 for the UD laminates and 0.52 𝑚 -1 for the cross-ply laminates. The behavior of the UD laminates for the remaining two annealing cycles at 180°C and 200°C is similar to that for the cross-ply laminates with an insignificant change in the residual curvatures at the end of both cycles compared to that at the end of 150°C annealing cycle. The use of a heated mandrel logically leads to a more homogenized temperature distribution across the laminate volume resulting in a reduced residual stress state compared to the cold mandrel. The strip specimen curvature in the first case is expected to be lower. Moreover, the mandrel's temperature is 180°C which is higher than the glass transition temperature. This results in a material constantly above its glass transition temperature, the crystallization is therefore always possible.

At room temperature, the [90 3 /0 3 ] strips present a mean curvature of 2.28 𝑚 -1 which corresponds to 88% of the curvature of the same laminates in the case of the un-heated mandrel. This ratio reduces to less than 1% for the unidirectional strips.

The [90 3 /0 3 ] strips mean curvature in Figure 3.28 follows the same trend as the previously discussed curvatures with a decrease upon heating and an increase during cooling. The offset between the curvatures during heating and cooling from 120°C is negligible and may be due to measurements error. The curvatures of the unidirectional laminates remain negligible regardless of the temperature. The increase in the curvature during the isothermal run at 150°C is negligible for the [90 3 /0 3 ] strips as shown in Figure 3.29, which is in accordance with the DSC results where no cold crystallization is measured that can lead to a crystallization shrinkage and a modification in the materials' thermomechanical properties. However and despite this, Figure 3.29 shows that the curvatures during cooling from 150°C are higher than the ones measured during heating. The increase is however around 12% which is lower than the one observed for the unsymmetrical laminates (around 20%) manufactured with the cold mandrel. In the previously discussed results, the difference between the cooling and the heating scans was attributed solely to the crystallization effects, i.e. the crystallization shrinkage and the variation of the properties. In this case, the discussion is different. For instance, the DSC scans didn't show a cold crystallization taking place in the case of the heated mandrel. However, the crystallinity increases during the 2 nd heating run, therefore suggesting the presence of amorphous regions which could've crystallized during the 1 st scan but the crystallization enthalpy was too small to be detected by the DSC scan. As a consequence, the effect of the crystallization on the curvature increase cannot be omitted. In addition to the crystallization effect on the curvatures, the curvatures increase can be also linked to the stress relaxation due to the viscoelastic nature of the PEEK.

Evolution of the[(𝟗𝟎 𝟑 /𝟎 𝟑 ) 𝟐 ] specimen curvature: cold mandrel

The [(90 3 /0 3 ) 2 ] specimen were subjected to the same annealing cycles as the [90 3 /0 3 ] presented in Section 3.5.1. The different specimen were processed under the same conditions and using the same mandrel in order to evaluate the effect of thickness on the curvature. The results for the first heating and cooling from 120°C are presented in Figure 3.30 for [90 3 /0 3 ] and [(90 3 /0 3 ) 2 ] specimen. The mean curvature of the [(90 3 /0 3 ) 2 ] specimen at room temperature before the heating phase is equal to 0.44 𝑚 -1 compared to 2.58 𝑚 -1 for the thinner [90 3 /0 3 ] specimen. The difference reveals the effect of geometric conditions such as the thickness and thus the plies number on the laminate curvature. It can be concluded from the results that increasing the number of layers leads to a decrease in the curvature. This is in agreement with the work of Jun & Hong [31] and Davis [32]. It was shown in both publications that the room-temperature shape of unsymmetrical laminates decrease with increasing the number of layers marked by 'n' in c.f. The evolution of the curvatures with temperature for the symmetric [(90 3 /0 3 ) 2 /90 3 ] laminates for both mandrels (heated and unheated) is presented in Figure 3.33 for the first heating and cooling from 120°C.

As it can be seen, the strips with the unheated mandrel present a mean curvature at ambient temperature which is negligible compared to that in the case of unsymmetrical [90 3 /0 3 ] in Figure 3.20 but the causes are worth considering. As previously discussed, the causes of the formation of residual stresses and thus the appearance of a curvature are numerous, the main factors being the thermal history and the stacking sequence of the laminate. If a symmetric laminate is considered, this laminate should theoretically be flat with a null curvature. This is the direct result of the counterbalancing effects of balancing plies at each side of the midplane. In a real case scenario such as the one presented in Figure 3.33, process related parameters such as the presence of a mandrel at the lower surface of the laminate induces strain gradients through the laminate thickness. In addition, process related defects such as small deviations from the theoretical orientations or an inhomogeneous fiber content across the laminate can result in the formation of a curvature. These defects can explain for example the curvatures in the case of the specimen with 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 = 180°𝐶.

Other reasons can generate the curvature which are more pronounced in the case of an unheated mandrel and can explain the difference between the curvatures of the two sets of specimen in Figure 3.33. In fact, the thermal history when processing with an un-heated mandrel leads to thermal gradients due to local heating. These thermal gradients lead to anisotropic behaviors between adjacent plies due to their different thermo-mechanical properties and thus the formation of residual stresses which were not balanced with the symmetric stacking sequence. The curvature observed in Figure 3.33 is the manifestation of these stresses.

Conclusion

The present chapter is divided into two main parts: the first one focuses on the characterization of specimen with different thicknesses and stacking sequences manufactured with a cold and a heated mandrel. The main objective of this part is to study the influence of the process on the material properties and morphology. The second part focuses on the measurement of curvatures after the process and during different thermal cycles.

The main results from the first part can be resumed as follow:

The crystallinity fraction resulting from processing on the SPIDE-TP with a cold mandrel is estimated between 20 and 22%. The use of a heated mandrel results in a crystallinity fraction between 28 and 30%. This can be directly related to the slower cooling in the case of the heated mandrel and the fact that the material is kept above its glass transition temperature. The heating ramp during the DSC scans highlights the presence of an exothermic peak characteristic of the cold crystallization in the case of the cold mandrel. The melting enthalpy is however higher than the cold crystallization enthalpy. It can be concluded that processing using a cold mandrel results in a partially crystallized polymer which can present a homogeneous partial crystallinity or completely amorphous regions and completely crystalline regions. The crystallinity of this polymer may be increased under thermal treatments such as the annealing.

A lower glass transition temperature (140.8°C) is obtained after processing on the SPIDE-TP with a cold mandrel, while the 𝑇 𝑔 in case of the heated mandrel increases to 148.18°C. The glass transition temperature can be increased for the cold mandrel processed specimens via post-process thermal treatments. In this work, the thermal treatment applied was a slow cooling from the melt. The shift in 𝑇 𝑔 can be attributed to a higher crystallinity fraction.

The melting behavior was shown to be unaffected by the mandrel temperature. A single melting peak was observed in both cases (cold and heated mandrel). After crystallizing from the melt, a double melting peak was observed. The double melting peaks were attributed to two lamellae species: thick and thin. The thick lamellae are present after processing regardless of the mandrel temperature. The thinner lamellae on the other side result from melt crystallization upon slow cooling from the melt. The transverse modulus governed by the polymer's behavior showed high dependency on the processing conditions (mandrel temperature). Processing with a cold mandrel results in a lower transverse modulus at ambient temperature compared to the processing with a hot mandrel. This is in direct relation with the crystallinity which follows the same dependency on the mandrel temperature.

The DMA tests on specimens manufactured using the cold mandrel showed a behavior close to that of amorphous PEEK with a dramatic decrease of the modulus during the glass transition. A small increase of the modulus was measured upon heating above the glass transition which coincided with the cold crystallization depicted in the DSC scans. The specimen processed with the hot mandrel present a typical behavior of a semi-crystalline polymer.

On the other side, the main results from the curvature measurements are:

In the case of the cold mandrel, the comparison between the curvatures obtained at ambient temperature prior to any thermal load applied may give several insights summarized in For the different specimen tested, heating the specimens up to 120°C (below their glass transition) doesn't induce any change in the curvatures upon cooling. No change in the curvature is recorded during the isothermal run at 120°C and no offset between the curvatures at the heating and cooling phases was measured. This observation is valid regardless of the mandrel temperature.

For the same number of plies, a cross-ply stacking sequence will directly lead to higher curvatures compared to unidirectional laminates (Figure 3 When an annealing at a temperature above the glass transition temperature is applied to specimen processed using the cold mandrel, an increase in the curvature is observed during the isothermal run ( The curvatures during the cooling from the different annealing cycles for the specimen manufactured using the cold mandrel present an offset from the ones during the heating phase. This offset increases when the annealing temperature increases from 150 to 180°C (Figure 3.24). This increase is however reduced for an annealing temperature of 200°C compared to 180°C (Figure 3.24). In order to understand the major driving factors behind this increase, one shall compare the curvatures in different cases: o From the annealing at 150°C, the effect of the cold crystallization is directly observed with a crystallization shrinkage leading to higher curvatures in addition to modified thermomechanical properties during cooling compared to the heating phase due to an enhanced crystallization. o If the specimen annealed successively at 150 and 180°C are considered, the results in Figure 3. 22 show that the curvature increase during the isothermal ramp at 180°C is negligible compared to the one measured at the 150°C ramp. The straightforward conclusion is that the cold crystallization may be completed during the annealing at 150°C and the resulting curvatures during cooling from 180°C should not diverge from the ones measuring during cooling from 150°C. This is however contradicted in Figure 3.22 with an additional offset of the curvatures between the cooling from 150°C and 180°C. Therefore, the cold crystallization related effect cannot be the sole responsible for the offset and a stress relaxation may contribute to this offset. o The stress relaxation effect is again manifested in the results of Figure 3.29 where an offset between the cooling and heating curvatures is recorded even though no increase in the curvature was measured during the isothermal run at 150°C. The DSC curves validated the absence of a cold crystallization in the case of specimens fabricated using the heated mandrel.

The different points stressed above show that processing using a cold mandrel leads to lower crystallinities and properties compared to processing using a heated mandrel.

From the curvature measurement, it seems clear that residual stresses do result from processing on the SPIDE-TP. The main effects addressed are the thermal history during processing and the stacking sequence. It's clear that the thermal history influences highly the residual stresses which is manifested via the curvatures of the unidirectional laminates. Moreover, we can assume from the DMA tests the presence of an amorphous region. The presence of the amorphous region should be considered when predicting the residual stresses since they lead to a lower stress free temperature and therefore lower the residual stresses.

The prediction of the thermal history and the crystallinity variation through the thickness of the laminate takes place in the next chapter. They'll serve as inputs, alongside the material's properties, to estimate the curvatures of the specimens.

Chapter 4: Numerical Study

Introduction

The evaluation of the residual stresses and strains states in a thermoplastic composite laminate during and after the process is based on the determination of the mechanical (elastic, viscoelastic), thermal (temperature) and crystallization (variation of the degree of crystallinity and therefore of matrix based properties) histories. The expression of the variation of the system energy function of these three variables gives an insight into the mechanical constitutive behavior law, the heat equation and the crystallization/melting evolution.

It consists into three major steps:

Study of the heat problem to determine the evolution of the temperature during the process (heating and cooling).

Examine the crystallization and melting during the process which is directly related to the thermal history through a thermo-physical coupling by adequate models.

Once the temperature distribution and the crystallization/melting behavior are described, the evolution of the material properties function of the temperature and the crystallinity can be estimated. In addition, the determination of the thermal and crystallization related deformations can be realized through the constitutive law in order to finally estimate the residual stresses.

In this chapter, the three major steps are discussed and the models used in each step is described. Therefore, the chapter begins with a description of two thermal models, the first one based on the Poper Generalized Description or "PGD" and the second one is an analytical model. The results of both models are presented, however due to time limitations, the results of the PGD based model only are retained for the next parts.

The thermal history obtained serves as input to the crystallization and melting models which are also analytical models. While the thermal model feeds the crystallization and melting models, the crystallization enthalpy is neglected and doesn't serve as input to the thermal model.

Next, the modified laminate theory which is the approach on which the work is based to estimate the residual deformations and stresses is described. The mechanical behavior model is based on a criteria that defines the beginning of the residual stresses formation which is the Stress Free Temperature (SFT). The numerical study is carried out in Matlab to simulate the processing of two 6-plies laminate (cross-ply [90 3 /0 3 ] and UD) using the cold mandrel. The predicted curvatures are compared to the measured ones in the previous chapter to validate the model. Once the model is validated, the residual stresses estimated are presented and discussed. Finally, a parametric study is realized to understand the effect of some parameters on the formation of the residual stresses during the process. These parameters include the choice of the SFT and the simplification of the through-thickness material behavior description by reducing the elements number.

This allows a better understanding of the first-order parameters influencing the formation of the residual stresses during processing on the SPIDE-TP in the final objective of obtaining a model simple enough to limit the calculation times yet including all the main parameters leading to the formation of the residual stresses.

Heat transfer and temperature distribution during the process

In Section 1.2.4, a review of the models simulating the thermal transfer during tape deposition were presented. Two thermal models were adopted during this work, the 1 st one is based on the work of Barasinski [1] & Perez [2] while the 2 nd one is based on the work of Weiler & al. [3]. The choice was initially to develop a suitable model to simulate the processing on the SPIDE-TP, however due to time limitations, the decision was to switch to available tools which led to the choice of the first model. With the need to validate the thermal history obtained from the 1 st model in order to proceed to the following parts and the technical difficulties related with this validation, the 2 nd analytical model was chosen. It's based on simple equations and its results can be compared with those obtained from the 1 st model. The thermal properties serving as input to the models are first presented followed by a brief description of these models and their results.

Material thermal properties

The composite material is naturally anisotropic and inhomogeneous. However the numerical thermal model considers an equivalent homogeneous material with effective properties that can be anisotropic. The resolution of the thermal problem requires the knowledge of the following effective composite properties: density ρ (Kg/m³), specific heat 𝑪 𝒑 (J/ (Kg. °C)), longitudinal and transverse thermal conductivities 𝑲 𝑳 and 𝑲 𝑻 (W/m. °C). The model developed by Barasinski [1] considered constant thermal properties. In this work, the thermal conductivities are constant while the density and the specific heat are considered as temperature-dependent properties.

Thermal conductivity

AS4 carbon fiber is anisotropic with a longitudinal thermal conductivity 𝐾 𝐿 (along their axis) higher than the lateral one 𝐾 𝑇 (along their radius) while PEEK is isotropic material. Cogswell [3] reported the dependence of 𝐾 𝐿 and 𝐾 𝑇 on temperature for APC2 c.f. Figure 4.1. The variation of the thermal conductivity with temperature in both directions is negligible, therefore they'll be considered constant throughout this work. 𝜌, 𝜌 𝑓 and 𝜌 𝑚 are the densities of the composite, fiber and resin respectively. 𝜌 𝑓 may be considered temperature independent [4] equal to 1790 𝐾𝑔. 𝑚 -3 . PEEK's density on the other side was evaluated as a function of temperature by Curliss & al. [5]. By applying the law of mixture for a 𝑉 𝑓 = 0.59 with the values 123 from [4] & [5], a linear dependency of the composite density on the temperature can be considered c.f.

Density

Specific heat

Lelouet [4] measured in her thesis the variation of the specific heat with temperature using DSC for PEEK and PEEK/AS4. The results are presented in Figure 4.3 alongside the results of two previous studied on PEEK from Cogswell [7] and Lamèthe [8]. The offset between the values were attributed to the different grades of PEEK. The graphs in the right side of Figure 4.3 show the evolution of the composite specific heat 𝐶 𝑝 with temperature obtained using the law of mixture [6] in Eq. 4.2 for 𝑉 𝑓 = 0.59. 𝐶 𝑝𝑓 and 𝐶 𝑝𝑚 correspond to the specific heat of fiber and matrix respectively.

𝐶 𝑝 = 𝑉 𝑓 𝐶 𝑝𝑓 + (1 -𝑉 𝑓 )𝐶 𝑝𝑚 Eq. 4.2

𝐶 𝑝𝑚 varies linearly with temperature c.f. Figure 4.3 while 𝐶 𝑝𝑓 = 1143 𝐽. 𝐾𝑔 -1 . °𝐶-1 and is temperature independent [4].

Based on the following, Table 4.1 presents a briefing on the thermo-physical properties of PEEK-AS4 used.  The heat transfer is only studied through the thickness direction. According to [12], the laser spot is usually larger than the tape and the welding zone which causes an overlapping on the left and right side of this zone therefore justifying the fact that the width direction is neglected in this model.

 During the process, heat exchange occurs within the composite and between the composite and its surrounding. The flow from the studied composite to its surrounding such as the air, the roller and the mold is more important when the composite is heated. A thermal resistance and a temperature difference are associated with each of these two heat flow processes. Weiler [3] studied the resulting values of the associated thermal resistances. From these values, it was concluded that during tape placement, heat will flow internally rather than with the surrounding.

 During cooling, the dominant mode is internal conduction  No thermal resistance exists between the tape and the substrate after contact  The interlayer resistance is neglected in this study. This assumption relies on a comparison of a series of thermal resistances and it was concluded that the thermal resistance between the composite and its surrounding is much higher than the thermal resistance between the plies.

As a result of the different assumptions and simplifications implied, a 1-D equation Eq. 4.4 of the heat problem was suggested by the authors rather than the complex 3-D problem.

𝜌𝐶 𝑝 𝑣 𝑑𝑒𝑝 𝜕𝑇 𝜕𝑡 = 𝐾 𝑧 𝜕 2 𝑇 𝜕𝑧 2
Eq. 4.4

𝜌, 𝐶 𝑝 and 𝐾 𝑧 are the composite effective density, specific heat and transverse thermal conductivity respectively while 𝑣 𝑑𝑒𝑝 is the placement speed. The initial condition applied suggests that the initial temperature of the material 𝑇 𝑖 is equal to that of the surroundings 𝑇 0 . In addition, the following boundary conditions apply for the surface in contact with the laser i.e. z=0 Eq. 4.5 and the back surface i.e. z=d Eq. The resolution of the nonhomogeneous heat problem in Eq. 4.4 requires the separation of the solution into two phases: the heating and the cooling.

For the heating, the analytical solution depends on the body behavior in terms of heat accumulation. Indeed, the authors in [12] defined two bodies : finite and semi-infinite. Finite bodies are those where heat reaches the backside while it doesn't in the case of semi-infinite bodies. The distinction can be made using the Fourier number 𝐹𝑜 defined using Eq. 4.7 𝑒 -(𝑛𝜋) 2 𝐹 0 ) Eq. 4.8

In case of semi-infinite slabs with Fo < 0.2, which is the case of the substrate when its thickness increases, another analytical solution was proposed to estimate the temperature distribution presented by Eq. 4.9

𝑇(𝑧, 𝑡) = 𝑇 0 + 2𝑞 0 ′′ 𝐾 √𝛼 √ 𝑡 𝑝 𝑖𝑒𝑟𝑓𝑐 ( 𝑧 √ 4𝛼𝑡 𝑝 )
Eq. 4.9

For the cooling phase, the analytical equation in Eq. 4.10 was obtained assuming that the dominant mode of transfer during cooling is the internal conduction: In addition, Weiler & al. [12] modified Eq. 4.10 so that a specific surface temperature 𝑇 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 is chosen according to which the flux required and the temperature distribution through the thickness are estimated. The resulting equation for the heat flux is presented in Eq. 4.11. This is indeed an important feature of this model because it simulates the control of the processing mode on the SPIDE-TP which is temperature-controlled: a nip-point temperature is chosen and the laser power is modified to maintain the specified temperature

𝑞 0 ′′ = (𝑇 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 -𝑇 0 )𝐾 𝐿 (𝐹 0 + 1 3 -( 2 𝜋 2 ) . ∑ 1 𝑛 2 ∞ 𝑛=1
. 𝑒 -(𝑛𝜋) 2 𝐹 0 ) Eq. 4.11

The results from the simplified analytical model presented in [12] were compared to those of two finite elements numerical models one is complex and the other one is simplified for the same materials and same processing conditions to obtain the through-thickness temperature. For more information on the two numerical models, the reader may refer to the thesis of Weiler in [14]. Figure 4.6 presents the comparison between the simulated temperature values. The temperatures obtained using the analytical model on the tape surface was for all the cases higher than the numerically predicted ones. A difference of 10°C with the simplified numerical model and 38°C with the complex numerical model occurs after 0.4ms of heating for the tape. These differences decrease to1°C and 12°C for the substrate. The differences in the cooling phase are reduced as presented in Figure 4.6. The authors attributed the differences to the averaged thermal properties in case of the analytical model while a temperature-dependency of the properties was assumed in the complex numerical model. Moreover, the boundary condition regarding the roller at the backside of the tape affected its thermal behavior. Regardless of these differences, the authors judged the analytical model a good approximation of the complex model.

Predicted thermal history

The results of the two thermal models discussed are presented for the 6-plies laminate and since the results are presented in a 1-D form, they can be applied in the case of both the cross-ply [90 3 /0 3 ] and the UD.

Temperature distribution according to the PGD resolved model 4.2.3.1.1 2-D static temperature distribution

As evoked in the previous paragraph, the thermal model gives the temperature distribution across the length and thickness of the laminate during the deposition of the last ply. The deposition of the first single ply is not considered because the configuration is not the same and the substrate is just the steel mandrel.

In the 2-D form of the model, the results for n=2,3,4,5 and 6 plies are presented in A) and as it moves from the left to the right beneath the laser (Point B) and its successive locations when it moves away from the nippoint (from Point C to Point E). In order to obtain the time vector 𝑡 𝑖 for the different time steps ''i'', Eq. 4.12 is used.

𝑡 𝑖 = 𝑥 1 -𝑥 𝑚𝑜𝑑,𝑖 𝑣 𝑑𝑒𝑝 Eq. 4.12 𝑥 1 is the x-position at the right of the domain (value of 1 m and an index i equal to 1501), 𝑥 𝑚𝑜𝑑,𝑖 is the position vector along x-axis from the right to the left ( ''i'' starts from 1 until 1501 at a step of 1). In this manner, Point A corresponds to t=0, point B corresponds to t=0.63s, etc... By applying this to the different x-axis points for each node, the evolution of the temperature with time for the node is obtained in the form presented in In fact, a time gap equal to 0.001 s was considered between the depositions of two consecutive layers. In reality, the time is much higher. However the hypothetical value added isn't supposed to affect the results presented afterwards since a purely elastic behavior of the material is chosen below T g . Moreover, the crystallization and the melting occur at temperatures above the glass transition, therefore the cooling rate below this temperature is insignificant to the results. Since the deposition of a single ply couldn't been simulated, it's considered that during its placement, the temperatures at the different nodes of Ply 1 (Node 1 to Node 11 in For information on the time steps and the convergence study, the reader may refer to Appendix A. The results presented in Figure 4.11 show that the different plies behave similarly during the process with a time delay according to their deposition. The maximum differences temperature differences are related to the temperature during the second peak for the different plies. The absolute values of these differences are reported in Table 4.4. The highest differences occur between Ply 1 and the rest of the plies and is attributed to the presence of the mandrel in contact with this ply therefore imposing a different boundary condition than the contact between two adjacent plies.

Table 4.4 -Maximum differences between the temperatures of the plies during their second peak

In the following, the temperature evolution for only the 1 st ply is detailed and a similar trend applies for the remaining plies. As the ply approaches the laser, the temperature through its thickness increases and reaches its maximum when it's directly subjected to the laser c.f. Once ply 1 the deposition of ply 1 on the mandrel is completed, ply 2 starts to be placed on top of it. The evolution of the contact temperatures between the substrate and the tape is presented in Figure 4.14.

The substrate and the incoming ply surfaces which are in contact don't have the same temperature directly. This is due to the presence of a thermal contact resistance between the plies considered in the model.

Figure 4.14 -Evolution of the substrate and the incoming tape contact surfaces temperatures during the deposition of Ply 2 (incoming ply) on top of Ply 1(substrate). (a): High cooling rates zone and (b) : Slower cooling rates

As the process continues and more plies are being placed, the temperature through the thickness of Ply 1 changes with trend similar to Figure 4.13 and Figure 4.14 but with different maximum temperatures. At the end of the process, it can be seen from Figure 4.11-(a) that the temperature of the nodes through the thickness of Ply don't reach the glass transition temperature while it did during the deposition of the remaining plies. Therefore, it can be concluded that during the deposition of the sixth ply on top of a 5 ply substrate, 4 out of 5 layers are heated above the glass transition while the remaining one which is located at the bottom of the substrate isn't heated above 𝑇 𝑔 .

Cooling rates

During the process, the material undergoes significant change in the cooling rate which doesn't stay constant throughout its cooling. The cooling rate can be averaged into two stages: the high cooling rate stage (up to 2200°C/s) followed by a slower cooling stage (ranging between 400°C/s and 20°C/s) for the tape at the contact surface with the substrate. In addition, cooling rates gradients exist through the thickness of the substrate and the incoming ply thicknesses as it can be seen in Cooling rates gradients are also observed during the whole process and the difference through the thickness of the substrate becomes bigger as its thickness increases. For instance, it can be seen in Figure 4. 16 on the right that a maximum difference around -2000°C/s exists between the top of the substrate (Node 1 of Ply 2) and its bottom (Node 11 of Ply 1). This difference is reduced as the consolidation occurs but can still be as high as -400°C/s (at the 14 th second of the process) between the top and the bottom.

Temperature distribution for a 6-plies laminate according to the analytical model

During the process, the substrate thickness is defined with two values for each ply deposition: before and after the consolidation. Once deposited, the incoming ply becomes the substrate and cools down after the laser heating. During the placement of the remaining plies, the heat flux required to heat the incoming ply is always defined for a thickness of 0.15 mm during 𝑡 𝑝 . In the case of the substrate, the heat flux is defined during 𝑡 𝑝 for a thickness corresponding to 0.15*(n) with n being the number of the pre-deposited plies. The irradiation time 𝑡 𝑝 can be estimated from the placement speed 𝑣 𝑑𝑒𝑝 and the irradiated length 𝑥 𝑖𝑟𝑟𝑎𝑑𝑖𝑎𝑡𝑒𝑑 using Eq. 4.13. The irradiated length can be obtained from the SPIDE-TP laser precisions and is equal to 46 mm while the placement speed corresponds to 0.15 m/s.

𝑡 𝑝 = 𝑥 𝑖𝑟𝑟𝑎𝑑𝑖𝑎𝑡𝑒𝑑 𝑣 𝑑𝑒𝑝

Eq. 4.13

The irradiation time is estimated around 0.31s which is very close to the one obtained from the previously discussed model based on the PGD which was equal to 0.37 s (delay between the beginning of the heating and the highest peak during the process obtained from the different results in The results presented show temperature profiles which are qualitatively close to the ones predicted using the PGD solved model. The different plies exhibit the same history delayed in time according to their deposition except for Node 6 in Ply 1 for example for which the cooling behavior is different than the rest of the nodes. Moreover, some differences are noted between the maximum temperatures reached: for example the third peak of Ply 1 reaches a maximum temperature of 313°C while the third peak of Ply 2 corresponds to a maximum temperature of 340°C.

In the next part of this chapter, the results from the PGD model will be used rather than the ones obtained from the analytical model. Several factors motivate this choice. First, the preliminary results from the analytical model are promising, however due to the multiple assumptions and simplifications implied, some results need better improvement and understanding such as the behavior of Node 6 of Ply 1 in Figure 4.18-(a). Moreover, a very fast cooling rate that can go up to 1610*10 6 °C/s is predicted for the 1 st cooling time step when the material suddenly decreases from 410°C to 250°C. These results need to be better investigated, but due to time limitations they couldn't been better studied. The results from the PGD model are better correlating with the literature results. Finally, the PGD model used in this work was extensively studied and compared to experimental results in the works of Barasinski and Perez.

While the results of the PGD model are considered in the remaining parts of this chapter, one of the drawbacks of this model is the incability to reach to solution algorithm developed. This establishes a motivation for a future study to asses the simplifactions thatneed to be changed in the analytical model in order to solve the two previously mentioned problems encountered with its results.

Temperature evolution during the curvature measurements in the climate chamber

The two previously presented models simulate the thermal history during the manufacturing of the investigated laminates. The same laminates were later subjected to a heating and cooling to different maximum temperatures in a climate chamber in order to monitor their curvatures. The simulation of these thermal cycles is however simpler than the process and include a simple linear increase in temperature from room temperature after demolding until the maximum temperature considered followed by a subsequent linear cooling at it can be seen in Figure 4. 19 presenting the heating to 120°C and the subsequent cooling. The different points presented along the curve in Figure 4.19 correspond to the temperatures at which the curvatures were calculated during the heating and cooling. The same profile is considered when the laminates are heated up to different temperatures. In the analytical model, the temperature profile is considered directly uniform through the thickness of the laminate since at each time step, the laminate was conserved for around 10 minutes ( in the experimental part) at the considered temperature to ensure a homogeneous distribution of the temperature across its volume.

Validation of the predicted thermal history model

The validation of the predicted thermal history should've occurred through experimental measurements by using thermocouples embedded in the laminate during its fabrication. However, due to technical difficulties, the access to the machine during the work of the thesis was not always possible, therefore the experimental validation of the thermal numerical model wasn't possible during the project time. However, the predicted thermal histories are similar to previously reported ones on similar processes as discussed in the literature review in Section 1.2.4. The previously reported thermal histories during tape placement of PEEK/Carbon fiber composites present thermal profiles similar to the one predicted using the PGD model as it was shown in the first chapter. The comparison of the values obtained from the PGD model and previously cited works in Section 1.2.4 such as [17], [18], [19] and [20] will evidently show different temperature values. This is in fact directly related to different process conditions and materials. It should be however noted that the PGD model was previously validated in the works of Barasinski [1] & Perez [2]. Based on the uncertainties implied with the analytical model and the need to further improve it by checking the asusmtpions considered, the choice was to work with the results of the PGD resolution.

Modelling the melting and crystallization during the process

The thermal history during cooling has major implications on the material properties and crystallinities. For instance, a slower cooling rate during the bonding phase induces longer residence time which enhances the crystals fusion. Between the crystallization and the glass transition temperatures, a slower cooling rate directly leads to higher crystallinity levels. 16 show a cooling rate gradient through the thickness of the substrate (from 2200°C/s to 1627°C/s) and the incoming ply (from 1670°C/s to 1420°C/s). This gradient may lead to gradients in residual crystal nuclei who skipped the total melting and lead to eventual crystallinity gradients through the thickness.

The description of the crystallization kinetics is important to accurately predict the residual stresses. The literature review in Section 1.3.2 and 1.3.3 showed the high dependence of the composite matrix dominated properties such as the transverse modulus and CTE on the crystallinity. The variation of this crystallinity during the process directly induces the variation of these properties. Moreover, the crystallization during the cooling from the melt is associated with a shrinkage which, similarly to the thermal deformation, influences the residual stresses Some studies on the residual stresses in fiber reinforced thermoplastic composites neglected the effect of crystallization on the stresses [21], [22] & [23]. Other authors modelled the crystallization kinetics during the process to evaluate the crystallization related heat source in the energy balance equation and the variation of the properties with the crystallinity fraction [24] in addition to crystallization shrinkage [25], [26]. The SFT in the cited studies was either based on the peak crystallization temperature or through the relaxation time when the behavior of the resin was defined as viscoelastic. This definition is however straightforward and doesn't account for the presence of a crystallinity gradient through the thickness of a composite which may require the definition of several SFTs depending on the crystallinity of each layer.

While the crystallization takes place during the cooling phase, it's preceded by the melting of the crystals when the polymer is heated above its fusion temperature. However, with the high heating rates and the thermal gradients during the automated tape placement, some nuclei may escape the melting and influence the final crystallinity fraction.

Moreover, with the PEEK's successive melting and crystallization during the continuous deposition, it's critical to incorporate a model capable of predicting the melting and the crystallization during the process which is the objective of this section.

Presentation of the models and conditions applied

Choe & Lee [27] proposed a non-isothermal crystallization model where the crystallization of PEEK is presented as a combination of two nucleation-growth mechanisms [28]. The expression derived is based on Tobin's theory of phase transition kinetics and given by Eq. 4.14:

𝛼̇(𝑡) = 𝑑𝛼(𝑡) 𝑑𝑡 = 𝜅 1 exp (- 3𝐸 𝑑 𝑅𝑇 ) 𝑒𝑥𝑝 (- 3𝛹 1 𝑇 𝑚 0 𝑇(𝑇 𝑚 0 -𝑇) ) 𝑡 2 [1 -𝛼(𝑡)] 2 + 𝜅 2 𝑒𝑥𝑝 (- 3𝐸 𝑑 𝑅𝑇 ) 𝑒𝑥𝑝 (- (3𝛹 1 + 𝛹 2 )𝑇 𝑚 0 𝑇(𝑇 𝑚 0 -𝑇) ) * [1 -𝛼(𝑡)] 2 ∫ (𝑡 -𝑤) 2 [1 -𝛼(𝑤)]𝑑𝑤 𝑡 0
Eq. 4.14 𝑇 𝑚 0 , 𝐸 𝑑 , 𝛹 1 and 𝛹 2 are respectively the equilibrium melting temperature, the activation energy of diffusion of crystallization segments across the phase boundary, the constant related to the free energy of formation of a critical nucleus and the constant related to the free energy of formation of a growth embryo. κ1 and κ2 are kinematic parameters. 𝛼̇(𝑡) presents the rate of variation of the relative crystallinity 𝛼(𝑡) = 𝑋 𝑡 (𝑡)/𝑋 ∞ (𝑡) at time t with 𝑋 𝑡 (𝑡) and 𝑋 ∞ (𝑡) are the volume fraction crystallinity and the ultimate volume fraction crystallinity at time t. The values for the different parameters were obtained by Choe & Lee [27] through DSC measurements. The crystallization model is applied during heating and cooling between 𝑇 𝑚 (342°𝐶) and 𝑇 𝑔 (143°𝐶).

In this study, the melting kinetics are based on a Maffezolli model [29]. It was initially applied to the resistance welding process and later used by several authors during the tape placement [30], [31] and [32]. It's based on integrating the rate of heat absorbed during the melting process. From the integrating equation, the crystallinity fraction X t at time t can be obtained using Eq. 4.15 applied whenever the material is heated above its T m (342°C). t in and X t (t in ) correspond to the time and the crystallinity fraction obtained when the material reaches its T m during heating. K A can be calculated using Eq. 4.16

𝐾 𝐴 = 𝐾 0 exp (- 𝐸 𝑎 𝑅𝑇 ) Eq. 4.16
𝐾 0 , 𝐸 𝑎 , and 𝑅 present the pre-exponential factor equal to 5.05*10^31, the activation energy equal to 397 KJ/mol.K and the universal gas constant.

The crystallization and melting models are successively applied to the plies placed on the mandrel as they're continuously heated, cooled and reheated during the process according to The temperature profiles presented in the previous part and obtained from the PGD solved model (Figure 4.11) are used as input to the melting and crystallization models. The tapes are supposed having an initial crystallinity equal to 0.34.

Predicted crystallinity evolution during the process

As in the case of the temperature history, the behavior of the 1 st ply's nodes is described and it can be applied to the remaining plies. The temperature history and the corresponding melting behavior for nodes 1 to 11 of Ply 1 during its deposition are presented in Figure 4.21-(a). The maximum temperature is 421°C for Node 1 directly exposed to the laser source with a difference of 22°C with Node 11 located at the other side from the laser with a temperature around 399°C. While all the nodes are heated above the melting temperature, the melting model predicts a gradient in the residual crystals. For instance, Node 11 has a residual crystallinity of 26% compared to a negligible (less than 3%) crystallinity for Node 1. The melting behavior is related to the temperature and the time spent at this temperature: the higher the temperature and time spent, the higher the melted crystals fraction. For the convergence on the time steps number, the reader may refer to Appendix A.

Figure 4.21 -Evolution of the temperature and crystallinity fraction for the different nodes of Ply 1 during its deposition. Continuous lines (-) present the temperature while dashed lines (-o) present the crystallinity

The melting of crystals is followed by their crystallization during the cooling from 320°C which is presented in During the deposition of Ply 3, the maximum temperature reached through the thickness of Ply 1 is 248°C (at Node 11). This temperature implies that no re-melting occurs through the thickness of Ply 1. It can however induces a re-crystallization since the temperatures of the different nodes range between the glass transition and the crystallization temperatures with a very negligible increase compared to the crystallinity fraction once the deposition of the 1 st ply is achieved.

It can be concluded that the model predicts that the crystallinity fraction through the thickness of a ply 'n' is mainly established during its deposition. Small changes may be induced due to the re-melting during the deposition of the next ply 'n+1' and this effect disappears once the ply 'n+2' starts to be placed. The crystallinities may be slightly increased during the deposition of the remaining plies since the ply can be reheated above its glass transition temperature. However, the contribution of the crystallization during cooling to the final crystallinity is negligible and this is directly related to the high cooling rates.

Finally, the crystallinity fractions through the thicknesses of the different plies are presented in The model predicts the presence of a crystallinity gradient through each ply thickness and through the substrate thickness. A layer exists between each two consecutive plies presenting a lower crystallinity than the rest of the ply's thickness (<5%). The crystallinity fraction is in general maximum at Node 11 for each ply. This results from the melting during the deposition of the ply.

While an experimental validation of the through-thickness crystallinity fraction couldn't been carried out during the scope of the thesis, one may compare the average crystallinity overall the laminate thickness with that measured using DSC. The average crystallinity through the thickness of the different nodes is 19.7% which is close to 23.5% measured for the 6-plies laminates using DSC. The difference is 16% which can be accepted within the scope of the errors from the model and the measurements using DSC.

The importance of the predicted crystallinity gradients evolution during the process is twofold:

The gradient of crystallinities through the thickness leads to a gradient in properties which may induce residual stresses even in the case of unidirectional laminates. This may contribute to the understanding of the non-negligible curvatures for UD specimen measured in the previous chapter.

If the predicted curvatures were to be similar to those measured, this can be an indirect method to validate the thermal model.

A gradient in the crystallinity as observed in this part leads naturally to different SFTs considered to estimate the residual stresses. In fact, a high crystallinity fraction resin indicates an SFT closer to the crystallization temperatures, on the other hand an amorphous resin as predicted between the layers starts to lock-down stresses during cooling from the glass transition temperature.

In the next and final part of this chapter, the predicted thermal histories in addition to the predicted crystallinities are implemented in the mechanical model to estimate the residual stresses. A part will be dedicated to study the influence of the SFT on the residual stresses.

Prediction of the residual stresses and strains during tape placement on the SPIDE-TP

Modified laminate theory

The classical laminate theory known as "CLT" evolved in the 1960s and presents a set of equations to describe the behavior of laminated composites and predict the residual stresses present in these composites.

Several assumptions are considered in the CLT to solve different problems [33]:

1) The laminate is formed of perfectly bonded layers 2) The material's behavior is linearly elastic 3) Each layer is a homogeneous medium with effective properties and can be isotropic, orthotropic or transversely isotropic 4) Plane stress condition apply for the laminate 5) The displacements 𝑢 (along x-direction), 𝑣(along y-direction), and 𝑤 (along z-direction)are small compared to the thickness 6) Kirchhoff hypothesis for deformation is assumed valid and therefore according to this hypothesis:

(i) -the normal lines to the mid-plane of the laminate remain straight and normal to the deformed mid-plane and (ii)the length of the normal lines to the mid-plane remain constant This theory applies for thin plates, and therefore the thickness is small compared to the other dimensions. Consequently, the first terms of the displacement u and v are retained while the w-displacement is constant resulting in Eq. 4.17-Eq. 1.10 𝑢 = 𝑢 0 (𝑥, 𝑦) -𝑧

𝜕𝑤(𝑥, 𝑦) 𝜕𝑥

Eq. 4.17 The plane stress state condition in each layer implies that the transverse shear components 𝜀 13 =𝜀 31 and 𝜀 23 = 𝜀 32 are negligible. The Kirchhoff condition on constant length of the normal to the mid-plane requires that the transverse normal strain 𝜀 33 = 𝜀 𝑧 is null and the out-of-plane displacement 𝑤 is only function of 𝑥 and 𝑦 coordinates. Adding these statements to the small deformations assumption and the definition of the displacements yield the following in-plane strain-displacement relations Eq. 4.20 to Eq. 4.22 to obtain the total strains at any point with a location 𝑧 from the mid-plane along the z-axis: In these equations, 𝑢 and 𝑣 represent the in-plane displacement, 𝜀 𝑥𝑥 , 𝜀 𝑦𝑦 and 𝛾 𝑥𝑦 are the in-plane strains and finally 𝜅 𝑥 , 𝜅 𝑦 and 𝜅 𝑥𝑦 represent the mid-plane curvatures. Eq. 4.20 to Eq. 4.22 can be written in the matrix form as follows:

𝜀 11 = 𝜀 𝑥𝑥 = 𝜕𝑢 𝜕𝑥 = 𝜕𝑢 0 𝜕𝑥 -𝑧 𝜕 2 𝑤 𝜕𝑥 2 =
{𝜀} = {𝜀 0 } + 𝑧{𝜅} Eq. 4.23
Considering a laminate made of n laminae as shown in Figure 4.24, the stresses {𝜎} 𝑘 at a location 𝑧 in the kth layer of a laminate can be obtained by combining Eq. 4.23 and Hooke's law which is applicable due to the plane stress assumption. These stresses are expressed in the form of Eq. 4.24

{𝜎} 𝑘 = [𝑄 ̅ ] 𝑘 {𝜀 0 } + [𝑄 ̅ ] 𝑘 𝑧{𝜅}
Eq. 4.24

[𝑄 ̅ ] 𝑘 is the transformed reduced stiffness of the kth layer in which the stresses are calculated. The expressions for the transformed reduced stiffness matrix can be found in Appendix C. Eq. 4.24 allows the determination of the stresses at a specific point of coordinate z from the mid-plane. The classical laminate theory defines for the laminate, the in-plane forces per unit length {𝑁 𝑥 , 𝑁 𝑦 , 𝑁 𝑥𝑦 } or {𝑁} corresponding to the integral of the planar stresses defined in Eq. 4.24 over the thickness of the laminate. The in-plane forces per unit length tensor can be obtained by a summation of the integrals over the thickness of each individual ply. The moments per unit length {𝑀 𝑥 , 𝑀 𝑦 , 𝑀 𝑥𝑦 } or {𝑀} are also defined by the CLT as the integral over the laminate thickness of the force times the moment arm. The details for these calculation can be found in [33].

In conclusion the laminate constitutive equation can be written as Eq. 4.25

{ 𝑁 𝑀 } = [ 𝐴 𝐵 𝐵 𝐷 ] { 𝜀 0 𝜅 } Eq. 4.25
[𝐴], [𝐵] and [𝐷] represent the in-plane stiffness, the bending-stretching coupling and the bending stiffness respectively. They can be calculated using Eq. 4.26-Eq. 4.28.

[𝐴] = ∑[𝑄 ̅ ] 𝑘 (𝑧 𝑘 -𝑧 𝑘-1 )

𝑁 𝑘=1

Eq. 4.26

[𝐵] = 1 2 ∑[𝑄 ̅ ] 𝑘 (𝑧 𝑘 2 -𝑧 𝑘-1 2 ) 𝑁 𝑘=1
Eq. 4.27

𝜀 𝑥𝑥 = 𝜕𝑢 0 𝜕𝑥 + 1 2 ( 𝜕𝑤 𝜕𝑥 ) 2 -𝑧 𝜕 2 𝑤 𝜕𝑥 2
Eq. 4.32

𝜀 𝑦𝑦 = 𝜕𝑣 0 𝜕𝑦 + 1 2 ( 𝜕𝑤 𝜕𝑦 ) 2 -𝑧 𝜕 2 𝑤 𝜕𝑦 2
Eq. 4.33

𝜀 𝑥𝑦 = 1 2 ( 𝜕𝑢 0 𝜕𝑦 + 𝜕𝑣 0 𝜕𝑥 + ( 𝜕𝑤 𝜕𝑥 ) ( 𝜕𝑤 𝜕𝑦 )) -𝑧 𝜕 2 𝑤 𝜕𝑥𝜕𝑦 , Eq. 4.34
The strain-displacement relations defined in Eq. 4.29-Eq. 4.32 were used to formulate the problem of finding the deformed shape of the laminate by examining the total potential energy 𝑊 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 . Since the mechanical work is ignored by Hyer in his model, the total potential energy is defined as the integral of the strain energy 𝑊 𝑠𝑡𝑟𝑎𝑖𝑛 over the volume 𝑉

𝑊 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 = ∫ 𝑊 𝑠𝑡𝑟𝑎𝑖𝑛 𝑑𝑉 𝑉 Eq. 4.35
The strain energy accounts, in Hyer formulation, for only the elastic and thermal deformations. By using components from the reduced stiffness matrix of the laminate 𝑄 𝑖𝑗 ̅̅̅̅ and the thermal expansion coefficients 𝛼 𝑘𝑙 , Eq. 4.35 can be written as Eq. 4.36 while the condition of plane stress applies

𝑊 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 = 1 2 ∫ 𝜎 𝑖𝑗 𝜀 𝑖𝑗 𝑑𝑉 𝑉 Eq. 4.36
With 𝜎 𝑖𝑗 = 𝑄 𝑖𝑗 ̅̅̅̅ (𝜀 𝑘𝑙 -𝛼 𝑘𝑙 ∆𝑇)

Eq. 4.37

∆T is presented in Hyer formulation as the difference in material temperature during the cooling from the processing temperature. Hyer postulated the form in Eq. 4.42 for w in order to take into account the saddlelike shape and the two cylindrical shapes that he observed for the laminates. The integration then allows to determine the forms of the mid-plane displacements u 0 and v 0 as presented in Eq. 4.38-Eq. 4.39

𝑤(𝑥, 𝑦) = 1 2 (𝑎𝑥 2 + 𝑏𝑦 2 ) Eq. 4.38 𝑢 0 (𝑥, 𝑦) = 𝑐𝑥 - 𝑎 2 𝑥 3 6 - 𝑎𝑏𝑥𝑦 2 4
Eq. 4.39

𝑣 0 (𝑥, 𝑦) = 𝑑𝑦 - 𝑏 2 𝑦 3 6 - 𝑎𝑏𝑥 2 𝑦 4 Eq. 4.40
𝑎, 𝑏, 𝑐 and 𝑑 are constants representing generalized coordinates that need to be determined, with 𝑎 and 𝑏 being the negative of the curvatures in the x and y direction respectively while 𝑐 presents the negative of the twist curvature. The form attributed to the out-of-plane deflection 𝑤(𝑥, 𝑦) correlated well with the observations on the possibilities for the deformed shape of the laminate as observed in [34] which include a saddle-like shape (𝑎 = -𝑏) and the two cylindrical (𝑎 = 0, 𝑏⧧0) and (𝑎⧧0, 𝑏 = 0). In order to find the different constants 𝑎, 𝑏, 𝑐 and 𝑑, the first derivative of the total potential energy with respect to these constants should be equal to zero as shown in Eq. 4.41. The set of equations which derives from Eq. 4.41 lead to three possible solutions with only one of them being stable. In order to find the stable solution, a stability criterion was used by Hyer and includes the second derivative of the total potential energy to be positive.

𝛿𝑊

In a second paper on the subject, Hyer [36] modified the polynomial forms associated with 𝑢 0 and 𝑣 0 presented in Eq. 4.39 and Eq. 4.40 respectively. These modifications were conducted on the mid-plane displacements only since the functional form adopted for the out-of-plane displacement correlated well with the experimental data and could be easily measured. The in-plane displacements are however not always evident for the observer. The main motivation behind the modifications was to allow a more spatial variation of the strain. The functional forms adopted in [36] allowed the in-plane elongation strains to vary in the space function both x and y in contrast to the previous work where 𝑢 0 and 𝑣 0 are function of only y and x respectively. The functional forms for the mid-plane strains are presented in Eq. 4.42-Eq. 4.43

𝜀 𝑥 0 = 𝑎 1 + 𝑎 2 𝑥 2 - 𝑎𝑏𝑦 2 4
Eq. 4.42

𝜀 𝑦 0 = 𝑏 1 + 𝑏 2 𝑦 2 - 𝑎𝑏𝑥 2 4
Eq. 4.43

𝑎 1 , 𝑎 2 , 𝑏 1 and 𝑏 2 in Eq. 4.42 and Eq. 4.43 are constants to be determined. These functional forms of the inplane elongation mid-plane strains correspond to cross-ply laminates for which the in-plane shear strain 𝛾 𝑥𝑦 0 is negligible compared the elongation strains. After determining these functional forms, the procedure for the energy minimization is similar as presented before which was developed in [35].

The approach presented by Hyer constitutes a baseline for other models modelling the behavior of unsymmetrical laminates. For instance, Jun & Hong [37] used the same resolution process as Hyer but modified the displacement expressions for 𝑢 0 and 𝑣 0 as presented in Eq. 4.44-Eq. 4.45 while conserving the same expression for the out-of-plane displacement 𝑤 as presented in Eq. 4.38. These modifications were implemented to study the effect of the residual in-plane shear strain for cross-ply laminates which is generally pronounced near the edges and was neglected in the study of Hyer.

𝑢 0 (𝑥, 𝑦) = 𝑥(𝑎 1 - 𝑎 2 6 𝑥 2 + 𝑎 3 𝑦 2 )
Eq. 4.44

𝑣 0 (𝑥, 𝑦) = 𝑦(𝑏 1 - 𝑏 2 6 𝑦 2 + 𝑏 3 𝑥 2 )
Eq. 4.45 This part served as an introductory to the model proposed by Hyer and the modifications implemented to this model in order to have an overview of the theory used. The algorithm used, based on the equations presented in this part, to solve the problem are presented later in this chapter. It should be finally mentioned that the classical and modified laminate theory are used for originally flat laminates which is a principal drawbacks of this model. However, it is implemented in this work since the aim is to investigate the residual stresses on the ply-level. From this point, one can link with the thesis of L. Degennaro (mentioned in the introductory part of this manuscript) which deals with the residual stresses at the structure level.

Input and assumptions considered with the model

This part presents the inputs to the model mainly the constituent properties and the equations used to evaluate the unidirectional prepreg properties. Second, the incremental aspect of the model is presented where the calculation scheme of the stresses and strains are presented. Finally, the SFT is defined in addition to other assumptions.

Thermo-mechanical properties of the composite

The thermomechanical properties of the carbon fiber AS4, the resin PEEK and the composite PEEK/AS4 were presented in Sections 1.3.1, 1.3.2 and 1.3.3. The identification of these properties is crucial to correctly estimate the residual stresses using the modified laminate theory presented earlier. These properties include  the in-plane longitudinal and transverse thermal expansion coefficients 𝛼 𝐿 and 𝛼 𝑇 respectively  the in-plane longitudinal and transverse elastic moduli 𝐸 𝐿 and 𝐸 𝑇 respectively  the in-plane shear modulus and Poisson's coefficient 𝐺 𝐿𝑇 and 𝜐 𝐿𝑇  the in-plane longitudinal and transverse crystallization shrinkage coefficients 𝛽 𝐿 and 𝛽 𝑇 respectively

The UD composite effective properties are calculated using the equations presented in Appendix B and in [38] for a fiber volume fraction 𝑉 𝑓 of 59% as described by the manufacturer data sheet. A brief recall of these equations can be found in Appendix B. In this work, it was assumed that no voids exist in the unidirectional prepreg.

These properties depend on those of the carbon fiber and the PEEK resin. The results reported in the literature and presented in Section 1.3.3 show that the fiber properties aren't highly sensitive to the temperature. Therefore, the properties of the fibers are considered temperature-independent as shown in Table 4.5.

Property Value Source

𝑬 𝑭𝑳 (GPa) 231 [39] 𝑬 𝑭𝑻 (GPa) 15 [7] 𝝊 𝑭𝑳𝑻 0.26 [40] 𝜶 𝑳𝒇 (°𝑪 -𝟏 ) -9 *10 -7

[41]

𝜶 𝑻𝒇 (°𝑪 -𝟏 ) 7.2*10 -6

[41]

𝑮 𝑭𝑳𝑻 (GPa) 27.58 [41] Table 4.5 -Carbon fiber properties input to the thermo-mechanical model predicted fast cooling rated predicted, the crystallization of the material takes place during its cooling. This crystallization is associated with a shrinkage due to the crystals densification and should be taken into account in the composite constitutive equation as presented in [26]. The fiber doesn't undergo any crystallization therefore its coefficient is equal to zero. PEEK resin's crystallization coefficient can be estimated from PvT measurements using the same method presented in [38]. The PvT measurements are those conducted by Pignon in his thesis [44] presented in Since the crystallization occurs during the cooling phase i.e. after the roller has passed, the values selected to determine the crystallization shrinkage are those corresponding to the 30 MPa pressure (the lowest one) to be as close as possible to the atmospheric pressure. As previously said, the volumetric crystallization shrinkage coefficient is estimated using the same method in [38] and the volumetric shrinkage results are presented in

Incremental aspect of the model

The linear-thermo-elastic law presented in Eq. 4.37 used to describe the behavior of the material was replaced by an incremental form in previous studies to evaluate the residual stresses [25] and [48]. The incremental constitutive law takes into account the variation of the material properties with temperature and crystallinity during the process. The literature review in Section 1.3.3 and DMA tests in Section 3.4.2 showed that this variation is not negligible in the process temperature range (from ambient up to temperatures above the melting point). The crystallization related shrinkage can also be taken into account through the incremental constitutive law. An adequate modelling of the variation of these properties contributes to a better estimation of the residual stresses level. The increment of stress ∆σ ij can be calculated for the increments of the deformations using Eq. 4.48

∆𝜎

Stress-free-temperature 'SFT'

The stress free-temperature 𝑇 𝑆𝐹 is defined as the temperature residual stresses start to relax upon heating and start to appear upon cooling. For the PEEK and PEEK composites, SFT is highly dependent on the crystallinity and thus on the cooling rate. In the literature, the PEEK's SFT was generally defined either as the glass transition temperature for amorphous resin or a crystallization temperature ranging between 289 and 320°C for semi-crystalline matrix.

Jeronimidis & al. [21] heated different APC-2 laminates in an enclosed space until becoming flat. They suggested that the residual stresses are completely removed at the temperature corresponding to the flat state of the specimens. The stress-free-temperature reported by Jeronimidis & Parkyn in [21] was 310°C for press-molded laminates with a cooling rate of 3°C/min.

Chapman & al. [25] incorporated a viscoelastic constitutive model to describe a transition from viscous to thermoelastic behavior of the properties in the matrix dominated direction. The authors in [25] defined a transition temperature 𝑻 * at which the material's behavior changes from viscous to thermoelastic upon cooling. This temperature was defined as the SFT in [25] because it was assumed that viscous material relax instantaneously the stresses which can build up during the thermoelastic regime. The estimation of the transition temperature was based on the comparison between an elapsed time and the relaxation time. The former was estimated from the thermal model while DMA tests allowed the definition of the latter. The transition temperature was defined as the point where elapsed time is equal to relaxation time. For a certain point in the laminate, if the elapsed time is higher than the relaxation time, the authors assumed that the stresses are relieved. An example of the transition temperature estimated in [25] for APC-2 for a surface cooling rate of 35°C is presented in Sonmez & al. [23] compared their results for an SFT equal to 280°C as determined by Unger & Hansen [49] and 310°C according to Jeronimidis & al. [21]. The authors claimed that their results didn't show much difference for the two different SFTs, so they considered an SFT of 280°C in their work.

Peron & al. [48] considered an SFT equal to the peak crystallization temperature of a glass fiber reinforced polyamide 66 similarly to [50] and [21].

Based on the previous, the following assumption is considered in this work: whenever the material is heated above the SFT, the stresses are considered to be completely and instantly relaxed. During cooling, the residual stresses don't start to be locked-in the composite until its SFT is reached. The value of the SFT is however dependent on the experimental observations in Sections 3.4.2 and 3.5. In fact, the DSC tests showed a composite with a semi-crystalline PEEK resin for which the crystallinity goes up to 21% after processing using the unheated mandrel. While this observation suggests a semi-crystalline material for which the SFT is closer to the crystallization temperature, the transverse modulus behavior obtained through the DMA tests contradicts this suggestion. The observed behavior is closer to an amorphous PEEK with a brutal decrease of the modulus until a very low value during the glass transition. The conclusion from the two observations (DSC and DMA) is that a possible crystallinity gradient exists through the composite thickness. Unfortunately, due to the lack of time and equipment this crystallinity gradient couldn't been experimentally observed, however it may be simulated if present with the coupled crystallization-melting-thermal model. The semi-crystalline elements present an SFT equal to their crystallization temperature while the amorphous ones present an SFT equal to the glass transition temperature. While two SFTs should be defined based on the material's crystallinity, the presence of amorphous layers may induce the relaxation of any stresses locked-in the semi-crystalline elements. This simplification is adopted in the following and therefore  The SFT of the composite corresponds to 𝑇 𝑔 =143°C.

Additional assumptions

The described model is applied to a 6 plies unidirectional and cross-ply unsymmetrical [903/03] laminate. The laminate has a length of 150 mm, a width of 15 mm and a thickness of 0.9 mm in order to present the strips for which the curvatures were measured in Section 3.5. The small deformation (classical laminate theory) can be used for long narrow strips with the condition that 𝜅 𝑦 =0 [51].

The geometry and the through-the-thickness nodes are predefined for the simulation. The nodes belonging to a ply which hasn't been placed yet have their temperatures equal to zero and their local reduced stiffness matrix [𝑄 ̅ ] is considered equal to zero.

Finally, during the process, the laminate is attached to the mandrel therefore out-of-plane deformations cannot occur i.e. 𝑤(𝑥, 𝑦) in Eq. 4.35 is set equal to zero until the end of the process. Once the process ends, the laminate detaches from the mandrel, out-of-plane deformation 𝑤 occurs and the bending moment generates the specimen curvature.

General algorithm solution and equations used

This part serves as briefing to the equations implemented and a general form of the algorithm used to finally obtain the curvature and the residual stresses.

1) The fiber and the matrix properties presented in Section 4.3.2.1 are used to calculate the unidirectional ply properties for a range of temperatures and crystallinities using the equations in Appendix B.

2) The temperature distribution for each time step and for each node is obtained using the thermal model in Section 4.2.3.1.

3) The temperature distribution obtained is used with the crystallization and the melting model to obtain the crystallinity evolution presented in Section 4.2.4.2.

Once these three points are established, their results can be used in the mechanical model using the theory proposed by Hyer. The steps presented in the following take place during each time step. Once all those steps are realized, they are reused in the next time step until the end of the process 4) The temperature and crystallinity through the nodes are defined at the beginning of the step and the stiffness matrices are calculated using the equations in Appendix C with the corresponding properties. 5) The displacements u 0 , v 0 and w are established function of the parameters using Eq. 4.44, Eq. 4. 45 and Eq. 4.38 respectively. The parameters for each time step correspond to their value in the previous step (value known) in addition to an increment (unknown value) for which the value is determined at the end of the time step 6) The strains increments are then calculated using the displacements expressions, the thermal and crystallinity contribution 7) The stresses' expression can then be established from the strains and the stiffness matrices 8) Once the strains and stresses are defined function of the unknown parameters, the energy equation is presented and the stability criterions take place. This allows the determination of the unknown parameters presented in the displacements expressions. These unknown parameters calculated at each step present an increment and are then added to the previous value to obtain the total deformation, stresses and strains.

Results and discussions

During processing on the SPIDE-TP, the laminate is subjected to a thermal history which includes multiple heating and cooling in addition to thermal and crystallinity gradients through its thickness as previously shown. This leads to the presence of properties gradients through the laminate thickness even in the case of symmetrical laminates. This effect is aggravated when dealing with an unsymmetrical layup which induces additional heterogeneities through the laminate thickness. The consequence of these heterogeneities is a non-uniform residual stresses distribution through the thickness which causes a bending moment once the laminate is separated from the mandrel leading to shape distortions such as curvatures. The complete coupled thermal-physical-mechanical model simulated the thermal history, crystallinity/melting and mechanical behavior during the process and the following heating and cooling phases in the climate chamber. It predicts the curvature of the specimen after its separation from the mandrel. The validation of the model occurs upon the comparison between the predicted curvatures and the ones experimentally measured presented in Section 3.5. It was previously stated that due to the PGD thermal model limitations, the numerical study will be only limited to 6 plies laminates i.e. the unsymmetrical cross-ply [90 3 /0 3 ] and the 6-plies unidirectional laminate in the case of the cold mandrel.

Curvatures and residual stresses for the cross-ply unsymmetrical [𝟗𝟎 𝟑 /𝟎 𝟑 ]

Stress state

In this part, the distribution and evolution of the mesoscopic residual stresses (at the ply level) through the thickness of the different plies during the process (while the curvature is null), after the demolding (when the curvature occurs after separation from the mandrel) and after the cooling from 120°C are presented. The mesoscopic stresses can be estimated at the ply local coordinates or at the global coordinate axis. In order to understand the final stress state after the process, the development of these stresses during the process is crucial. To this end, the in-plane residual stresses components in the global coordinates 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 are presented at different times of the process.

 Placement of Ply 1

According to the thermal model results presented in Figure 4.12, the placement of Ply 1 occurs between 0 and 6.63s. The temperature through Ply 1 thickness remains higher than 𝑇 𝑔 c.f. During the deposition of Ply 1, a stress-free state is assumed until 𝑇 𝑔 is reached. Once 𝑇 𝑔 is reached, the material passes into a glassy state and the transverse modulus of the material suddenly increases with a through-the-thickness gradient due to the crystallinity gradient (c.f. Since the SFT is set equal to 𝑇 𝑔 , the strain due to crystallization is neglected. Hence if Eq.4.44 is reformulated in the form of Eq. 4.46, it shows that the stress increment is directly proportional to the stiffness matrix and the difference (∆ε 0 -∆ε 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 ).

∆𝜎 𝑖𝑗 = 𝐶 𝑖𝑗𝑘𝑙 (∆ε 0 -∆ε 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 )

Eq. 4.50

Along the x-axis which corresponds to the transverse direction of Ply 1, the different elements tend to thermally shrink as it can be seen in Figure 4.28-(c) upon cooling leading to an initial compressive stress state along the thickness. However, due to the temperature gradient in addition to the crystallinity gradient through the thickness, the elements shrink asymmetrically. This ultimately results in a thermal stress gradient through the ply thickness. The upper elements which are stiffer due to their higher crystallinity restrain the shrinkage of the lower elements, therefore the shrinkage of each element is restrained by its neighboring stiffer element. This ultimately leads to a tensile stress increment through the ply's thickness.

As the cooling process continues, the residual stress profile develops through the thickness with compressive stresses on the upper elements and tensile stresses on the lower ones. The stress gradients obtained through the laminate thickness is directly related to the crystallinity and thermal gradients through the thickness. where only Ply 1 reached its SFT while Ply 2's temperature was higher, therefore Ply 2 couldn't sustain any residual strains or stresses yet. The stress increment is dependent on the thermal strain and the reduced matrix stiffness, therefore the complex distributions presented in The evolution of the in-plane residual stresses 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 are presented in Since the temperature for the remaining plies are higher than those for Ply 1, the final state of the residual stresses and strains for the remaining plies (Ply 2 to Ply 6) correspond to the ones obtained during the placement of Ply 6 in the case of the 6 plies laminate simulated in this study.

Based on this, the stress evolution during the placement of Ply 3 and 4 are similar to the ones obtained during the placement of Ply 2 and will be erased during the deposition of the next plies, therefore the evolution of the residual stresses are presented in the following during the deposition of Ply 5 and 6.

 Placement of Ply 5 If the x-axis is considered, it corresponds to the transverse direction for the 90°oriented plies and the longitudinal one for the 0° oriented plies. During the cooling phase, the substrate cooling is inhomogeneous and Ply 1 which is contact with the mandrel reaches the glass transition temperature first followed by the upper plies. Along the x-axis, the 90° oriented plies tend to shrink while the 0° oriented plies tend to slightly elongate. The 0° oriented plies being stiffer than the 90° oriented ones, they block their shrinkage. Therefore, a tensile stress increment is imposed on the 90° oriented plies while the 0° oriented plies are subjected to a compressive stress state c.f. Finally, the placement of Ply 6 occurs and the laminate made of the 5 previously laid plies are heated along with the incoming Ply 6. As it was previously mentioned and according to the thermal model, the different plies except Ply 1 are heated above their glass transition temperature. This ultimately leads to a complete relaxation of the stresses during their heating above the glass transition temperature. In case of Ply 1 which is not heated above the glass transition temperature, only a part of the residual stresses resulting from the During the process, the specimen's curvatures are null since the laminates are attached to the mandrel. Once separated, the laminate presents, under the bending moment effect, a curvature at 30°C which was measured equal to 2.58 𝑚 -1 compared to a numerically predicted value of 2.72 𝑚 -1 which is a difference of 8%. This curvature decreases to upon heating to 1.05 𝑚 -1 (experimental) vs 1.04 𝑚 -1 (numerical) at 120°C. While the model predicts a linear decrease of the curvature, the experimentally measured curvatures present a nonlinear evolution. This leads to differences between the two sets of values with a maximum offset equal to 0.5 𝑚 -1 occurring at 60°C. This difference can be attributed to two factors:

 Fiber misalignments occurring in the real scenario which is ignored in the model where the fibers are considered perfectly parallel  In the model, the evolution of the thermo-mechanical properties such as the resin modulus and the CTE were considered constant below the glass transition. However, DMA tests presented in Section 3.4.2 showed that the transverse modulus slightly decreases non-linearly below the glass transition temperature. In addition, the thermal expansion coefficient of the resin was considered in the model constant below the glass transition temperature. However, the literature review in Section 1.3.3.2 showed that the thermal expansion coefficient slightly varies between the ambient and the glass transition temperature. Combining an adequate evolution of both the modulus and the CTE below the glass transition may enhance the numerical results and lead to a non-linear evolution of the curvature with temperature.

During cooling from 120°C to the ambient temperature, the model predicts a linear increase in the strips curvatures following the same values predicted during heating. Moreover, as in the case of the experimental curvatures, no change in the crystallinity or the properties was predicted when the material is heated to 120°C. The differences between the predicted and the experimentally measured curvatures can be attributed to the multiple assumptions and simplifications considered in the model. From these factors, one may cite:

 The geometrical effect of the pre-tension applied to the pre-impregnated plies prior to their deposition and the mechanical load of the roller which were ignored in the model  The simplified thermo-mechanical properties evolution considered  Defects such as fibers misalignments or voids which weren't considered in the model Before demolding along the x-axis corresponding to the transverse one for the 90° oriented plies (with the negative z) and the longitudinal direction for the 0° oriented plies (positive z), the 90° oriented plies are submitted to a tensile stress with a maximum value of 25.2 MPa located at the lower element of Ply 1 i.e. the one in direct contact with the mandrel. On the other hand, the 0° oriented plies are submitted to a compressive stress with a minimum value of -48 MPa located in the lower element of Ply 4 which is also in direct contact with the 90° plies. This compressive stress increases up to -6 MPa for the upper element of Ply 6. The stresses obtained at the end of the process (before demolding) depend on the process. In fact, the temperature evolution presented in Figure 4. 39 show that the two elements of Ply 1 are heated above their glass transition temperature during the placements of Ply 1 till Ply 5. The model considers an immediate relaxation of stresses whenever the material is heated above its glass transition. Therefore, for the first ply placed, the residual stresses at the end of the process correspond to the ones developed during the placement of Ply 5 and Ply 6. Consequently, the residual stresses for the remaining plies result from the placement of Ply 6 only. Along the x-axis, the deformation of the 90° oriented plies i.e. Ply 1 and Ply 3 (which tend to shrink upon cooling) is higher than that for the 0° oriented plies i.e. Ply 4 and Ply 6 (which tend to slightly elongate upon cooling) due to the more pronounced effect of the resin with a higher CTE compared to the fiber. With the 0° oriented plies being stiffer, the shrinkage of the 90° oriented plies is restricted. This ultimately lead to a tensile stress applied on the 90° plies and a compressive stress applied on the 0° oriented plies. The evolution of 𝜎 𝑥𝑥 for the upper and lower elements of Ply 1, 3, 4 and 6 during cooling after Ply 6 was placed is presented in Figure 4.43. The thermal model in fact shows that 𝑇 𝑔 is reached consecutively from Ply 1 to Ply 6. Between t= 34.4s till t=35.3s which correspond to the activation of the cooling from 𝑇 𝑔 for Ply 1, 2 and 3 which are all oriented at 90°, the increase of the residual stresses remains negligible with a maximum tensile stress of 2 MPa in the lower element of Ply 2. The stresses are however inhomogeneous for the plies due to the temperature and crystallinity gradient. An important increase of the stresses is however noticed once Ply 4, oriented at 0°, is placed on top of Ply 3. This result is predictable since the major driving source for the residual stresses is the discrepancy in the thermal behavior between the plies at different orientations. A similar analysis can be made along the y-axis where the lower 3 plies (oriented at 90°) restrain the shrinkage of the upper three plies ultimately leading to a compressive stress in the lower 3 plies and a tensile stress in the 3 upper plies.

The level of the predicted residual stresses are close to those reported using the 11-elements/ply scheme. In fact, along the x-axis the maximum tensile stress is equal to 28.8 MPa in the case of the 11-eleemts/ply compared to 25.2 MPa for the 2-elements/ply case. Along the same direction, the maximum compressive stress in the 11-elemts/ply case is equal to -15.2 MPa compared to -48 MPa in the second case. The difference in the maximum compressive value and the locations of the maximum and minimum residual stresses are different can be attributed to the crystallinity gradients.

After demolding: Once the laminate is separated from the mandrel, the uneven distribution of residual stresses through the laminate thickness generates a bending moment and the specimen curvature occurs leading to a redistribution of the stresses through the laminate thickness. The 90° oriented plies remain under tension with a minimum value of 2.4 MPa at the lower element of Ply 1 and a maximum value of 33.1 MPa in the upper element of Ply 3. On the other side, Ply 4 and the lower element of Ply 5 are under compression with a minimum value of -160 MPa at the lower element of Ply 4. The upper element of Ply 5 and Ply 6 become subjected to a tensile stress with a maximum value of 74.5 MPa for the upper element of Ply 6.

After cooling from 120°C:

The curvatures predicted by the model presented in Figure 4. 41 show that no offset exists between the values during cooling and heating since no change is induced in the material (120°C is lower than the glass transition temperature and therefore no changes are expected to occur). This results, after cooling to ambient temperature from 120°C, to the same stress state as the one predicted for the same temperature after demolding.

From the results presented earlier, the transverse stress calculated in the local coordinates of the different plies can be calculates. The results are presented in Along their transverse direction, the different plies are subjected to a tensile stress since their shrinkage is restricted by their surrounding stiffer plies. By applying the maximum stress criterion approach, the maximum stress in the laminate is equal to 25.2 MPa (before demolding) and 33.1 MPa (after demolding) occurring respectively in the lower element of Ply 1 and the upper element of Ply 3 oriented at 90°. The transverse tensile strength of the unidirectional APC-2 composite found in the literature [39] corresponds to 78 MPa and is taking here as the critical transverse strength value 𝜎 22,𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 above which transverse cracks start to be formed. Even though the maximum predicted transverse stresses are of 25.2 and 33.1 MPa, they are almost equal to half of 𝜎 22,𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 , therefore the laminate is already charged to around half of its transverse strength. This again proves the necessity to take into account the presence of residual stresses after the process in order to either reduce them or define the structure usage limits in order to prevent transverse cracking from occurring.

Evolution of the curvature when the material is heated to 150°C using the 2 nodes per ply resolution scheme

Using the same process and therefore with the same stress state predicted previously, the evolution of the curvature and the stresses during the heating to 150°C and subsequent cooling was studied. The main difference with the previous case is the possible occurrence of the cold crystallization during heating above the glass transition temperature. Figure 4.45 presents the predicted crystallinity using the models described earlier during the heating to 150°C in the oven and the subsequent cooling. The results show the occurrence of the cold crystallization upon heating the material above its glass transition where the crystallinity across the laminate thickness increases to the maximum value set to 34%. During heating, an offset is recorded with a maximum difference of 0.6 𝑚 -1 at ambient temperature. It was already stated in Section 3.5.1 that a difference between the curvatures during heating to 120°C and 150°C exists. In contrast to the results in the case of the maximum temperature of 120°C where no offset was predicted between the cooling and heating evolutions, the model predicts in the case of the maximum temperature of 150°C, an offset between the cooling and the heating. This offset is due to the increased crystallinity predicted during heating above the glass transition temperature and presented in Figure 4.45. This crystallinity is accompanied by a crystallization shrinkage which was calculated in the case of PEEK resin and presented in Figure 4.26 in addition to the modification of the composite properties if the crystallization increases. The curvature predicted during cooling seems to be able to predict the measured one. However, for the different temperatures presented, the predicted curvatures are slightly lower than those measured. The maximum difference occurs at 0.175 𝑚 -1 at 100°C. The differences during cooling between the numerically and experimentally measured curvatures may be attributed to relaxation of stresses which wasn't taken into account in the model.

Curvatures and residual stresses for the 6-plies unidirectional laminate

The previous section was dedicated to the study of the residual stresses obtained in the case of a 6-plies cross-ply laminate [90 3 /0 3 ]. The results showed that the thermal history induced crystallinity gradients through the plies' thickness in addition to thermal gradients. These inhomogeneities led to the presence of residual stresses gradients even after the placement of only unidirectional plies such as the results presented in Based on the results presented in Figure 4.47, the curvature predicted at T=30°C for the unidirectional laminate is equal to 0.37𝑚 -1 compared to an experimentally measured value of 0.38 𝑚 -1 with a standard error of 0.06𝑚 -1 . The predicted curvature is therefore close to that numerically predicted at ambient temperature. However, the evolution of the experimentally measured curvatures are different than the predicted one. In fact, the experimentally measured curvature decreases from 0.38 𝑚 -1 at T=30°C to 0.22 𝑚 -1 at 120°C which presents a decrease of approximately 42%. On the other hand, the model predicts a small decrease of the curvature of around 0.8%. The differences between the two sets of curvatures may have several reasons such as the presence of fiber misalignments which weren't accounted for in the numerical model and other defects.

Regardless of the differences between the numerically predicted and experimentally measured curvatures at elevated temperatures during cooling and heating, a good correlation is noted at ambient temperature which corresponds to the curvature after demolding. This result shows the effect of the thermal history on the residual strains and stresses. Thus, the coupled model and despite the simplifications and assumptions adopted is able to predict the effect of the thermal history. The residual stresses occurring through the laminate thickness at the end of the process i.e. before demolding and after demolding can consequently be predicted using the presented model. The results regarding the in-plane residual stresses components are presented in The results in Figure 4. 48 show a residual stress gradient through the laminate thickness for both directions.

Along the x-axis, the different plies tend to shrink since it's their transverse direction. The thermal model presented in the previous sections of this chapter showed that a thermal gradient is present throughout the laminate thickness inducing an inhomogeneous thermal behavior.

Before demolding Along the x-axis, a stress gradient is present where the lower layer (i.e. the one in contact with the mandrel) is subjected to a compressive stress of -4.1 MPa followed by a tensile stress of 0.6 MPa in the next layer. In fact for the six lower layers (from z=0.45 to z=0), the residual stresses differences between the plies is more important than the remaining layers (from z=0 to z=0.45). The maximum tensile stress occurs at the surface element with a value of 2.2 MPa. The gradients are also observed along the y direction.

After demolding Once the laminate separates from the mandrel, the bending moment leads to the occurrence of the curvature just as in the case of the [90 3 /0 3 ] laminate and therefore causes a modification in the residual stresses distribution along the 'x' and 'y' direction. Along the 'x' direction, the maximum tensile stress occurs at the surface layer and is equal to 3.7 MPa while the maximum compressive stress is equal to -5.9 MPa at the lowest layer. It was previously stated that the Stress-Free-Temperature is highly dependent on the polymer's crystallinity. At a first glimpse, the crystallinity gradient obtained using the coupled thermo-physical model through the laminate thickness imposes the presence of different SFTs based on each ply's crystallinity. The choice in this work was to choose one SFT corresponding to the glass transition temperature of the PEEK's resin. The choice of this SFT was previously discussed and is related to the presence of an amorphous layer between the semi-crystalline laminates according to the crystallization model. The presence of this amorphous layer with its very low modulus allows the relaxation of the stresses accumulated within the semi-crystalline elements. This assumption showed that the model is capable of predicting the curvatures of the specimen in the previous parts.

Case A SFT=143°C

Case B SFT=200°C

Case C SFT=320°C

Table 4.7 -The Stress-free-temperature (SFT) selected for the parametric study on the influence of the SFT on the predicted curvatures and residual stresses According to Figure 4.49, the higher the SFT considered, the higher the curvatures predicted. In fact, the major source of the residual stresses as previously discussed is the thermal residual stresses which are linked with the temperature difference between the SFT and the room temperature. An increase in the SFT increases the resulting residual stresses. Moreover, an increased SFT may allow the occurrence of residual stresses through the different plies earlier. In fact, for an SFT=143°C, it was previously shown that the different plies ( Ply 2 to Ply 6) present a final residual stress state resulting from the deposition of Ply 6 only. This is due to their heating above their SFT throughout the process which led a relaxation of the stresses. When the SFT is higher, for example 200°C or 320°C, other plies such as Ply 2 present a final residual stress originating from the deposition of Ply 5 and Ply 6.

It seems clear that the model presented is highly affected by the SFT considered. For the case presented in this study, the crystallinity gradient leading to amorphous layers between semi-crystalline layers allowed the consideration of a single SFT which is equal to the glass transition temperature. This is in fact possible because the amorphous layers, as presented in Section 3.4.2 through the measurement of the DMA, present a very low transverse modulus when heated above the glass transition temperature. This eliminates the constraints on the semi-crystalline polymers to deform freely. However, this special crystallinity gradient may not be the case for different processing techniques. Therefore, for future works, it may be interesting to evaluate the validation of the model with different SFTs based on the material temperature and the viscoelastic behavior of the resin such as the case of Chapman [25].

4.3.3.3.2

The effect of a simplified thermal history on the curvatures and the residual stresses: considering a simultaneous deposition of all the plies rather than the successive deposition

In the previous section, the reduction of the number of elements from 11 to 2 in each ply with one of them being amorphous and the other one semi-crystalline showed its capability to predict the specimens curvatures. It allowed the reduction of the simulation time from around 192 hours to 4 hours. Another simplification that can be considered is a simultaneous deposition of all the plies which reduces the number of time step by 6 and thus going from 3000 to 500 time steps for the process. In fact, several previously published works such as [25] and [52] on the residual stresses during tape placement or filament winding compared their numerically predicted values with the experimental values obtained from press molding i.e. using a simultaneous simulation due to the lack of experimental values coming from tape placement or filament winding.

In order to evaluate the influence of this simplification on the curvatures and the residual stresses at the end of the process, a simulation was conducted where the 6 plies of the cross-ply [90 3 /0 3 ] are processed simultaneously while adopting the simplification of the elements numbers in each ply from 11 to 2 similarly to the previous section. The thermal history is the same for the different plies and is equal to that of Ply 1 during its placement presented in Along the x-axis, the simultaneous deposition of all the plies leads to a stress profile similar in shape to the one predicted using the successive deposition. The 90° oriented plies (negative z) are subjected to tensile residual stresses while the 0° oriented plies are subjected to a compressive stress state. However, the residual stresses in the first case are lower than the ones in the second case. The maximum tensile residual stresses is located at the upper element of Ply 3 (oriented at 90°) for both simulations and is equal to 8.5 MPa for the case of simultaneous deposition compared to 29 MPa in the case of successive deposition. On the other hand, the minimum compressive stress is equal to -8.3 MPa for the simultaneous compared to -59 MPa for the successive deposition. This corresponds to an underestimation of the maximum stresses by 70% and 86% for the tensile and compressive cases respectively.

In the previous section, it was concluded that for all plies except Ply 1, the residual stresses at the end of the process result from the ones accumulated during the placement of Ply 6 only. In the case of Ply 1, the residual stresses result from those accumulated during the placement of Ply 5 and Ply 6. Based on this, a simulation where the plies are considered placed simultaneously shouldn't lead to different results than the one with the successive placement. The difference however occurs and it's evaluated as an underestimation by 40% for the curvatures, 70% for the maximum tensile stress and 85% for the maximum compressive stress. The difference is related directly to the thermal history. In fact, during the successive deposition, the thermal history showed that during the cooling phase after Ply 6 has been placed, a thermal gradient is induced in the laminate thickness as it can be seen in On the other hand, the stress increment is highly dependent on the thermal strain which is in its turn dependent on the temperature difference between two successive plies. A successive deposition induces major temperature difference ΔT gradient through the laminate thickness as cooling occurs as it can be seen in The results presented here show the effect of the thermal history on the residual stresses' prediction. In fact, the thermal history includes the effects of the thermal gradients leading to thermal stress increments gradients. Moreover, it includes the effect of the successive deposition of the plies which lead to different behavior through the laminate thickness with the plies closer to the mandrel cool before the upper plies and thus solidify before them. These effects were shown in this part for the case of a six-ply laminate and may be more pronounced in case of a thicker laminate which should be verified in future works.

Conclusion

In this chapter, a model based on the modified laminate theory was used to estimate the internal stresses generated during the processing of a thermoplastic laminate composite manufactured using the SPIDE-TP.

In a first step, the temperature evolution during the process was simulated using two previous models from the literature for the deposition of a 6-plies laminate. The first one is based on the PGD from the works of Barasinski & al. [1]. The second model is an analytical model where the equations proposed by Weiler [12] were used and modified to take into account the successive deposition of the plies during processing on the SPIDE-TP. The results predicted by the two models were similar, however those of the PGD based model were selected for the next parts of this chapter mainly because the cooling rate predicted using the analytical model were very high and not representative of any previously reported value. The predicted temperature profiles showed an increase in the temperature for the incoming ply above the resin's melting temperature 𝑇 𝑚 . In addition, thermal gradients occur through the thickness of the laminate during its processing. In fact, the laser heats the incoming ply and the mandrel surface above the melting temperature of the PEEK resin to ensure their melting in order to ensure their welding together. The laser heat propagates through the laminate thickness by conduction and the different plies of the substrate are heated. Their temperatures are however dependent on the substrate thickness.

The next step was to couple the thermal model with a melting/crystallization model to study the evolution of the crystallinity during the process. The thermal gradients predicted using the thermal model led to crystallinity gradients through the laminate thickness and through each ply's thickness. These gradients suggested the presence of an amorphous layer between each consecutive plies and an increase in the crystallinity through the material's thickness. This effect wasn't experimentally verified due to time limitations, however the observations from the DMA and DSC results in the previous chapter may suggest that this observation may be correct.

Finally, the coupled thermal-physical-mechanical predictive model was used to simulate the development of residual stresses induced by the processing on the SPIDE-TP. The presence of the amorphous layer between the semi-crystalline layers of the laminate could possibly lead to a relaxation of any residual strains during the process due to the low value of the amorphous phase above the glass transition temperature. Therefore, the model used an SFT=143°C to define the temperature at which the residual strains and stresses start to be locked-in the laminate. The predicted curvatures during heating and cooling up to 120°C were compared to those experimentally measured in the previous chapter. The comparison suggested a good estimation of the measured curvatures in the case of the heating to 120°C using the proposed model and the assumptions and simplifications considered. Based on this, the evolution of the stresses and strains during the process was determined from the proposed model which led to the determination of the residual stresses during and after the process.

The time required to run a simulation for 6 plies was evaluated around 192 hours, it was therefore suggested to reduce the ply number through the thickness. To this end, the numbers of elements was reduced from 11 to 2 for each ply with one of them being amorphous and the other one semi-crystalline. The curvatures predicted using this approach were similar to those in the case of the crystallinity gradient through the thickness. This reduced the simulation time to 4 hours. As previously discussed, this analysis and those assumptions were applied to the case with the special crystallinity gradient obtained under the evoked processing conditions which led to the prediction of amorphous layers between the different plies. However, the crystallinity through the laminate thickness may be completely different and thus the SFT should be defined function of temperature during the process in a more complex manner.

A residual stress gradient was predicted in the case of the unsymmetrical cross-ply [90 3 /0 3 ] laminate with a maximum tensile stress of 25.2 MPa at its lower surface. This value is higher than the third of the PEEK/AS4 transverse tensile strength equal to 78 MPa. It's therefore critical and should be taken into account during the design.

The curvatures and the residual stresses were predicted in the case of a 6-plies UD laminate using the same approach. The curvature predicted at room temperature was shown similar to that experimentally measured. The predicted evolution was however not similar to the experimentally measured one. The differences were attributed to possible defects that weren't taken into account in the model. However, since the initial curvature after the process was similar, it was judged that the model was able to reproduce the residual stresses and strains for 6-plies UD laminate. This showed the effect of the thermal history on the residual stresses and strains.

Based on the two-elements per ply approach, the curvatures were predicted in the case of heating and cooling from 150°C. The model predicted a cold crystallization occurring upon heating above the glass transition which led to a crystallization shrinkage and a modification of the composite properties and allowed the simulation of the offset between the curvatures during heating and cooling from 150°C which was experimentally observed.

Based on the two-elements per ply approach, the effect of the complex thermal history was studied where a simultaneous deposition of all the plies was adopted which, if validated, could further reduce the calculation time. However, this approach led to an underestimation of the curvatures and the residual stresses by 40 and 70% respectively. This underestimation was directly related to the thermal history which includes a successive deposition of the plies which leads to thermal gradients through the thickness leading to higher residual stresses compared to the simultaneous deposition of all the plies.

Moreover, the effect of the SFT choice was studied. The results showed that the higher the SFT considered, the higher the residual stresses are and an adequate SFT equal to the glass transition is sufficient to predict the residual stresses. Finally, a two-SFT simulation was conducted and the initial results were presented and should be inspected in future works.

To conclude, the residual stresses and strains generated after processing on the SPIDE-TP can be predicted using a coupled thermal-physical-mechanical model based on the modified laminate theory. The major driving sources that should be taken into account in order to obtain accurate results with the minimum time are the thermal history which includes the thermal gradients and the stacking sequence. The choice of the stacking sequence is important. Indeed UD stacks generate curvatures lower than those generated in the case of an unsymmetrical cross-ply laminate with the same plies numbers. This shows the effect of the thermal history while the curvatures in the case of unsymmetrical laminate combine the effects of both the thermal history and the stacking sequence.

General Conclusion and Perspectives

The work presented in this manuscript focused on the tape placement of composite materials with a thermoplastic matrix. This process has the particularity of presenting high cooling rates in addition to thermal gradients due to the local heating with the laser heat source. Therefore, the final part properties may be different than those obtained with conventional processes such as compression molding. Moreover, high levels of residual stresses may be present through the laminate geometry due to the processing mechanisms.

In light of this, this work aimed to apply and combine experimental and numerical methods to characterize the initial state of semi-crystalline thermoplastic polymers and to assess the level of residual stresses which may, affect the structure properties and behavior.

The bibliographical chapter has introduced the essential concepts for understanding the material properties and behavior, the process and the residual stresses' origins and causes. While numerous studies were listed in the first chapter dealing with the different topics of concern, this bibliographic study showed that gaps can be identified. For instance, a complete study of the first state of the APC-2 composite including the material behavior and its residual strains and curvatures is required to link the process parameters with the final result. Moreover, the models simulating the residual stresses and strains during tape placement or filament winding validated their models using experimental results from different manufacturing processes data such as press molding or didn't incorporate essential phenomenon occurring such as the presence of crystallization gradients.

In this perspective, the originality of this study was to simultaneously apply numerical models and experimental techniques to characterize the residual strains and stresses of thermoplastic composites processed via laser assisted tape placement. In addition, experimental techniques were used to study the material behavior and link it with the process conditions and mechanisms.

In the second chapter and in light of the given, various experimental devices, listed in Table , were developed and/or used during this thesis to evaluate the impact of the process on the material behavior and residual stresses. The materials unto which these techniques were applied were processed on a tape placement machine using two mandrels at different temperatures (ambient and 180°C). This also allowed to assess the effect of the mandrel temperature on the material. The effect of the mandrel temperature on the laminate properties tested in Chapter 2 wasn't highly discussed in the literature. In fact, the first chapter highlighted the results from literature concerning the effect of the mandrel temperature on the crystallinity and the effect of the crystallinity on the properties.

The third chapter was devoted to present and discuss the results obtained using the experimental techniques presented in Chapter 2. It was divided into two parts: the first one focuses on the characterization of specimen with different stacking sequences manufactured with a cold and a heated mandrel. The second part focuses on the measurement of curvatures after the process and during different thermal cycles.

The results showed that the use of heated mandrel leads ultimately to better results in terms of a higher crystallinity fraction and mechanical properties. The crystallinity fraction resulting from processing using a mandrel at ambient temperature is estimated between 20 and 22% and go up to 30% in case of a heated mandrel. The lower crystallinity fraction in case of the cold mandrel was linked with the occurrence of a cold crystallization during DSC tests. From this observation, one may propose the presence of amorphous fraction resulting from processing using a cold mandrel. This directly results from the elevated cooling rates gradients taking place during the process. Moreover, the differences between the specimens manufactured with the mandrels were also observed in terms of the glass transition temperature especially. A lower glass transition temperature of 140.8°C is obtained using the cold mandrel compared to the value of 148.18°C obtained using the heated mandrel. The direct result of the crystallinity differences is the different transverse modulus since this property is dominated by the matrix. The DMA results normalized with the three-point bending tests showed high dependency of the transverse modulus value on the processing conditions (mandrel temperature). At ambient temperature, the measured transverse modulus for the cold mandrel case was 6.45 GPa for the 6-plies specimen and 6.1 GPa for the 9-plies specimen compared to 7.76 GPa and 7.84 GPa respectively in case of the heated mandrel. The differences of the modulus between the specimens of different thicknesses were linked to possible heterogeneities through the laminate geometry. The differences between the two mandrels didn't only include the value differences, the DMA tests showed that the behaviors of the laminates are different at elevated temperatures especially during the glass transition phase.

The DMA tests on specimens manufactured using the cold mandrel showed a behavior close to that of amorphous PEEK with a dramatic decrease of the modulus during the glass transition. A small increase of the modulus was measured upon heating above the glass transition which coincided with the cold crystallization depicted in the DSC scans. The specimen processed with the hot mandrel present a typical behavior of a semi-crystalline polymer.

On the other side, the curvature measurements also showed the differences between the mandrel temperatures for the different laminates studied where the curvatures were lower in case of the heated mandrel compared to the similar stacking sequences manufactured suing the cold mandrel. This result addresses the thermal history effect on the residual strains and stresses. The use of a heated mandrel generates a more homogenous temperature distribution compared to the cold mandrel which limits the thermal gradients through the laminate thickness during the process. This ultimately reduces the residual strains and stresses generated from the process.

The effect of the stacking sequence was also addressed where, for the same number of plies, a cross-ply stacking sequence will directly lead to higher curvatures compared to unidirectional laminates. At room temperature and prior to any annealing, the curvature of the [90 3 /0 3 ] laminate is approximately 6.6 times the curvature of 6-plies unidirectional laminates processed under the same conditions. This result directly emphasizes the effect of the stacking sequence on the residual stress during the processing. The cold crystallization observed with the DSC tests in case of the cold mandrel was again manifested when an annealing at a temperature above the thermoplastic transition temperature is applied to specimen processed using the cold mandrel. In fact, an increase in the curvature was observed during the isothermal and is the direct result of the shrinkage that accompanies the cold crystallization highlighted via exothermic peaks during the heating run of DSC tests. The curvatures during the cooling from the different annealing cycles for the specimen manufactured using the cold mandrel presents an offset from the ones during the heating phase. This offset increases when the annealing temperature increases from 150 to 180°C. This increase is however reduced for an annealing temperature of 200°C compared to 180°C. From the annealing at 150°C, the effect of the cold crystallization is directly observed with a crystallization shrinkage leading to higher curvatures in addition to modified thermo-mechanical properties during cooling compared to the heating phase due to an enhanced crystallization. When a specimen is annealed successively at 150 and 180°C, the results showed that the curvature increase during the isothermal ramp at 180°C is negligible (3%) compared to the one measured at the 150°C ramp (17%) for the case of unsymmetrical cross-ply laminate. The straightforward conclusion is that the cold crystallization may be completed during the annealing at 150°C and the resulting curvatures during cooling from 180°C should not diverge from the ones measuring during cooling from 150°C. This was however with the occurrence of an additional offset of the curvatures between the cooling from 150°C and 180°C. Therefore, the cold crystallization related effect cannot be the sole responsible for the offset and a stress relaxation may contribute to this offset.

The different points stressed above show that processing using a cold mandrel leads to lower crystallinities and properties compared to processing using a heated mandrel. The curvature results proved the presence of residual strains and therefore stresses resulting from the process. The main factors addressed are the thermal history during processing and the stacking sequence. It's clear that the thermal history influences highly the residual stresses which is manifested via the curvatures of the unidirectional laminates.

The prediction of the thermal history and the crystallinity variation through the thickness of the laminate takes place in the fourth and final chapter. They serve as inputs, alongside the material's properties, to estimate the curvatures of the specimens. Once the numerically estimated curvatures correlate with the measured ones, the model is validated and the residual stresses can be estimated.

First, a thermal model was adapted from previous works by Barasinski in her thesis to simulate the thermal history during the process. Due to limitations on the model, the thermal model was only applied for the cold mandrel case. The results showed the presence of thermal gradients through the thickness of the laminate during its processing. In fact, the laser heats the incoming ply and the mandrel surface above the melting temperature of the PEEK resin to ensure their melting in order to ensure their welding together. The laser heat propagates through the laminate thickness by conduction and the different plies of the substrate are heated. Their temperatures are however dependent on the substrate thickness. Cooling rate gradients were simulated through the thickness of the substrate (from 2200°C/s to 1627°C/s) and the incoming ply (from 1670°C/s to 1420°C/s). These gradient may lead to gradients in residual crystal nuclei that skipped the total melting and lead to eventual crystallinity gradients through the thickness. Their high values also result in low crystallinity fractions such as those obtained from the DSC tests.

The next step was to couple the thermal model with a melting/crystallization model to study the evolution of the crystallinity during the process. The thermal gradients predicted using the thermal model led to crystallinity gradients through the laminate thickness and through each ply's thickness. These gradients suggested the presence of an amorphous layer between each two consecutive plies and an increase in the crystallinity through the material's thickness.

Finally, the coupled thermal-physical-mechanical predictive model was used to simulate the development of residual stresses induced by the processing on the SPIDE-TP. The presence of the amorphous layer between the semi-crystalline layers of the laminate could possibly lead to a relaxation of any residual strains during the process until the amorphous layers solidify i.e. starting the glass transition temperature during cooling. Therefore, the model used an SFT=143°C to define the temperature at which the residual strains and stresses start to be locked-in the laminate. The predicted curvatures during heating and cooling up to 120°C were compared to those experimentally measured in the previous chapter. The comparison suggested a good estimation of the measured curvatures using the proposed model and the assumptions and simplifications considered. Based on this, the evolution of the stresses and strains during the process was determined from the proposed model which led to the determination of the residual stresses after the process.

The results presented show residual stresses gradients through the laminate thickness and high level of residual stresses especially along the 90° oriented plies. These plies are mostly subjected to tension along their transverse direction with a residual stress equal to 28.1 MPa which almost equal to half of the material ultimate transverse strength (78 MPa).

The time required to run a simulation for 6 plies was evaluated around 192 hours, it was therefore suggested to reduce the ply number through the thickness. The curvatures predicted were similar to those in the case of the crystallinity gradient through the thickness. This reduced the simulation time to 3 hours. This result suggest that one of the key factor to be considered during the simulation of the residual stresses at the ply level is the presence of an amorphous zone and a semi-crystalline zone for a ply. The consideration of a detailed crystallinity gradient through the laminate thickness isn't therefore a necessity to model the residual stresses at the ply level.

While the thermal model couldn't been experimentally verified, it showed good correlation with previously reported thermal profiles on similar processes. Moreover, the curvatures predicted for the UD laminates showed a good correlation with the experimentally reported ones at ambient temperature which further validates the model used.

The effect of the complex thermal history was studied where a simultaneous deposition of all the plies was adopted which, if validated, could further reduce the calculation time. However, this approach led to an underestimation of the curvatures and the residual stresses by 40 and 70% respectively. This underestimation was directly related to the thermal history which includes a successive deposition of the plies which leads to thermal gradients through the thickness leading to higher residual stresses compared to the simultaneous deposition of all the plies.

Moreover, the effect of the SFT choice was studied. The results showed that the higher the SFT considered, the higher the residual stresses are and an adequate SFT equal to the glass transition is sufficient to predict the residual stresses.

Perspectives

If this study has made it possible to contribute to a better understanding of the residual stresses and strains during tape placement of semi-crystalline composites, it also open up new research perspectives. In fact, a coupled thermal-physical-mechanical model has been established to model the residual stresses and strains during and after tape placement. The model was applied to 6-plies laminates only and the temperature and the crystallinity profiles weren't experimentally validated. The thermal model based on the PGD was chosen to deal with however this model is hard to understand since no access was granted to the code and the maximum number of plies simulated was 6. An analytical model was presented based on previous work and showed promising results. Some difficulties were however present when dealing with the results obtained. This model can be regarded closely in a future work in order to correctly understand the errors and limitations for it to be implemented.

Simultaneously, the temperature profile through the laminate thickness needs to be experimentally measured in order to validate any thermal model used. This work opens up to a following experimental campaign regarding the measurement of the crystallinity gradient through the laminate thickness when it's processed on the SPIDE-TP.

Once the corresponding thermal model coupled with the crystallization and fusion models are validated, another window can be established dealing with a sensitive study on the effect of the process parameters on the crystallinity and therefore properties of the manufactured laminate.

On the mechanical model, the properties variations were simple and an elastic behavior was attributed to the material which is not completely representative of the material complex behavior. This also led to small differences between the model proposed and the measured curvatures. Hence, an adequate description of the properties especially regarding the viscoelastic behavior of PEEK need to be quantified and linked to the equations presented in this study so that their influence can be understood.

Moreover, a simple definition of the SFT was used in this work. While this approach seemed capable of predicting the residual stresses and strains for the laminates used in this study, it may be suitable to study the effect of the presence of several SFTs according to the material crystallinity and temperature which can be done by adopting a viscoelastic behavior for the resin.

The original thrust of this project was related to residual stresses and strains occurring in thermoplastic composite laminates at the ply level. The residual stresses are also manifested on the micro-level i.e. between the fiber and the resin. It may be interesting to incorporate the thermal history and the crystallinity gradients to study the residual stresses at this level in future works.

𝑁 𝐿1 =9006 total steps which yields 1501 steps for each of the six plies placed, 𝑁 𝐿2 =4500 steps, 𝑁 𝐿3 =3000 steps, 𝑁 𝐿4 =1500 steps and finally 𝑁 𝐿5 =600 steps. The results presented in Figure 6.1 correspond to the bottom of the 1 st , 2 nd and 3 rd ply placed. It should be however mentioned that the same analysis applies for the rest of the nodes but those nodes are chosen for the sake of clarification.

It can be observed that the difference between the results of the simulations with 𝑁 𝐿1 , 𝑁 𝐿2 , 𝑁 𝐿3 and 𝑁 𝐿4 steps is very small that it cannot be detected in the figures presented. However, the use of 600 steps underestimates the temperature mainly that of the bottom for Layer 1. The choice is therefore to use either 3000 or 1500 steps. In order to correctly choose the steps number, the convergence study was also conducted using the coupled thermal-melting-crystallization model. The results presented in Figure 6.2 correspond to the crystallinity fraction obtained at all the nodes of the 6 plies at the last step of the simulation when using the different steps number 9000, 4500,3000 and 1500. The crystallinity fractions predicted at the end of the processes when using 9000, 4500 and 3000 steps are close and the differences are negligible. However, the simulation using 1500 steps shows clear offsets with the original results (for Ply 1 and Ply 6 especially). It's therefore concluded that 3000 time steps are the optimal choice. Avec tous ses avantages, le principal problème de l'hydrogène est son stockage. Les principaux modes de stockage de l'hydrogène sont soit à l'état liquide (cryogénique), soit sous forme de gaz Les composites polymères thermodurcissables ont été largement utilisés depuis 1980 en raison de nombreux facteurs, notamment les meilleures performances à des températures élevées par rapport aux thermoplastiques. La Figure 2 montre la variation du module de plusieurs polymères avec la température. Le polymère phénolique présente le module le plus élevé qui reste intact jusqu'à des températures élevées par rapport aux thermoplastiques tels que le nylon. Deux des inconvénients majeurs des résines thermodurcissables sont leurs longs cycles de durcissement et leur incapacité à être recyclées. En revanche, les polymères thermoplastiques ont des cycles de consolidation plus courts et peuvent être potentiellement recyclés [4]. De plus, leur fabrication peut inclure une consolidation in situ. Ce type de procédés est devenu de plus en plus intéressant pour le secteur industriel au cours des deux dernières décennies. Le principal facteur est les gains de productivité importants avec l'absence de l'étape de durcissement de la résine très chronophage requise dans le cas des polymères thermodurcissables. La productivité est en effet un enjeu pour les constructeurs aéronautiques et automobiles avec l'essor des réseaux mondiaux et de la population mondiale. D'où l'intérêt croissant pour les composites thermoplastiques principalement parce qu'ils peuvent être façonnés en une seule étape lors du refroidissement de la masse fondue. De plus, ils peuvent être recyclés et remodelés car la transition liquide-solide est réversible. Enfin, les composites thermoplastiques ont une durée de conservation infinie, ce qui ajoute aux avantages par rapport aux thermodurcissables qui doivent être stockés à basse température pendant un temps défini.

A. Appendix D: Résumé en français

Néanmoins, leur traitement se produit à des températures élevées (la température de fusion du PolyEtherEtherKetone PEEK est de 343°C) augmentant ainsi le coût de production. Afin de conserver l'économie qui peut être réalisée en supprimant l'étape d'autoclavage, l'objectif est de présenter une combinaison matière thermoplastique/procédé intéressante en termes de coûts de productivité et répondant aux attentes des industriels et des autorités environnementales. On peut voir sur la Figure 3 que le PEEK en particulier présente un bon candidat pour les structures composites thermoplastiques avec un bon rapport résistance/prix. Compte tenu de ce qui précède, les composites polymères PEEK présentent des potentiels d'application pour produire un réservoir de stockage d'hydrogène. Enfin, le dernier chapitre propose un modèle multi-physique qui intègre l'aspect thermique du procédé et le comportement de cristallinité associé en plus du comportement mécanique du composite. Le modèle est validé en comparant les courbures mesurées au chapitre 3 avec celles simulées. Cela permet d'étudier l'évolution des contraintes résiduelles pendant et après le procédé. En plus des contraintes et déformations résiduelles étudiées, une étude paramétrique est présentée afin de comprendre les paramètres du 1er ordre influençant la formation des déformations résiduelles.

Enfin, une conclusion générale sur les résultats majeurs de l'étude est présentée ainsi que des pistes de travaux futurs. En résumé rapide, les essais DSC, DMA et les essais de flexion statiques ont montré une différence de comportement entre les échantillons obtenus par les deux mandrins froid et chaud. Les différences montrées dans les travaux reportés dans les paragraphes précédents concernent le taux de cristallinité, les températures de transition de phase lors de changement de température notamment la température de transition vitreuse, les valeurs du module transverse et son comportement durant la transition vitreuse. Les résultats essentiels pouvant être retenus sont :

Résultats et conclusion

 L'utilisation des deux mandrins induit des matériaux avec une résine semi-cristalline cependant les stratifiés dans le cas du mandrin froid présentent une fraction de cristallinité inférieure à celle obtenue dans le cas du mandrin chaud  Cette fraction de cristallinité plus inférieur conduit à l'obtention d'un module transverse inférieur dans le cas du mandrin froid comparé à celui du mandrin chaud  En observant leur comportement en température, les échantillons se comportent différemment autour de la transition vitreuse. Le mandrin froid conduit à un matériau semicristallin qui se comporte comme un matériau amorphe autour de la transition vitreuse ce qui suggère la possibilité de la présence d'un gradient de cristallinité à travers l'épaisseur.

Ces résultats permettent d'avoir une vue sur le comportement du matériau après le procédé et sur l'effet de la température du mandrin sur ce comportement. Ils ne sont pas directement liés aux contraintes et déformations résiduelles après le procédé, mais malgré ça, leur exploitation peut contribuer à la compréhension du déroulement du procédé. Cette compréhension influencera éventuellement les conditions considérées dans le modèle de prédiction des contraintes vue la difficulté de contrôler en ligne le procédé.

De l'autre côté, les contraintes résiduelles pouvaient être déterminées via la méthode de mesure des courbures. Selon le chapitre bibliographique, cette méthode est simple et a pu montrer son efficacité [15], [16], [17], [18], [19], [20] et en plus, elle a été déjà utilisée dans le laboratoire GeM. Dans la suite, les résultats principaux des mesures de la variation des courbures. Dans le cas du mandrin froid, la comparaison entre les courbures obtenues à température ambiante avant l'application de charges thermiques peut donner plusieurs indications résumées dans la Figure A.9.

La comparaison entre les stratifiés [90 3 /0 3 ] et ((90 3 /0 3 ) 2 (flèche verte) montre l'effet de l'épaisseur c'est-à-dire le nombre de couches : l'épaisseur (c'est-à-dire le nombre de couches) diminue la courbure pour les mêmes conditions de traitement. L'évaluation des états de contraintes et de déformations résiduelles dans un stratifié composite thermoplastique pendant et après le procédé repose sur la détermination des propriétés mécaniques (élastiques, viscoélastiques), histoire thermique (température) et de cristallisation (variation du degré de cristallinité). L'expression de la variation de la fonction d'énergie du système de ces trois variables donne un aperçu sur la loi de comportement mécanique, de l'équation de la chaleur et de l'évolution de la cristallisation/fusion. Dans la suite, les modèles utilisés pour simuler chaque variante sont décrits.

Dans un premier temps, l'évolution de la température au cours du procédé a été simulée à l'aide de deux modèles issus de la littérature pour le dépôt d'un stratifié de 6 plis. Le premier basé sur la PGD est issu des travaux de Barasinski. Le deuxième modèle est un modèle analytique où les équations proposées par Weiler [21] ont été reprises et modifiées pour tenir compte du dépôt successif des plis lors du traitement sur le SPIDE-TP. Les résultats prédits par les deux modèles étaient similaires, cependant ceux du modèle basé sur le PGD ont été sélectionnés pour les parties suivantes principalement parce que la vitesse de refroidissement prédite à l'aide du modèle analytique était très élevée et non représentative de toute valeur rapportée précédemment. Les profils de température prédits ont montré une augmentation de la température du pli en cours de dépôt au-dessus de la température de fusion de la résine cf. Dans le cas des stratifiés asymétriques tel que les [90 3 /0 3 ], une courbure est présentée et a été mesurée auparavant. Compte tenu de ça, la théorie du stratifié apparaît comme un bon candidat pour traduire le comportement du matériau lors du refroidissement. Hyer a modifié la théorie classique des stratifiés (CLT) [24] afin de reproduire le comportement des stratifiés composites à plis croisés asymétriques soumis à des variations de température. Sa théorie repose sur l'hypothèse du champ de déplacement afin de reproduire les formes observées expérimentalement des stratifiés composites. Dans ce but, le déplacement hors-plan 'w' est exprimé selon Eq. 1.10 avec a et b des constantes représentant les courbures selon la direction 'x'et 'y' respectivement 𝑤(𝑥, 𝑦) = 1 2 (𝑎𝑥 2 + 𝑏𝑦 2 ) Eq. A.1

En utilisant plusieurs hypothèses géométriques données dans [24], les champs de déplacement dans le plan approchés u 0 et v 0 sont donnés par Eq. A.2et Eq. 1.10 respectivement.

𝑢 0 (𝑥, 𝑦) = 𝑐𝑥 -𝑎 2 Enfin, le modèle couplé thermique-physique-mécanique a été utilisé pour simuler l'évolution des contraintes résiduelles induites par le traitement sur le SPIDE-TP. Le modèle de calcul des contraintes et déformations résiduelles se base sur théorie modifiée des stratifiés introduite par Hyer [24]. La présence de la couche amorphe entre les couches semi-cristallines du stratifié pourrait éventuellement conduire à un relâchement d'éventuelles contraintes résiduelles au cours du procédé du fait de la faible valeur de la phase amorphe au-dessus de la température de transition vitreuse. Les courbures et les contraintes résiduelles ont été prédites dans le cas d'un stratifié UD 6 plis en utilisant la même approche. La courbure prédite à température ambiante s'est avérée similaire à celle mesurée expérimentalement. L'évolution prédite n'était cependant pas similaire à celle mesurée expérimentalement. Les différences ont été attribuées à d'éventuels défauts qui n'ont pas été pris en compte dans le modèle. Cependant, étant donné que la courbure initiale après le processus était similaire, il a été jugé que le modèle était capable de reproduire les contraintes et déformations résiduelles pour le stratifié UD à 6 plis. Ceci a montré l'effet de l'histoire thermique sur les contraintes et déformations résiduelles.

Sur la base de l'approche à deux éléments par pli, les courbures ont été prédites dans le cas d'un chauffage et d'un refroidissement à partir de 150°C. Le modèle a prédit une cristallisation à froid se produisant lors d'un chauffage au-dessus de la transition vitreuse qui a conduit à un retrait de cristallisation et à une modification des propriétés du composite et a permis de simuler le décalage entre les courbures lors du chauffage et du refroidissement à partir de 150°C qui a été observé expérimentalement.

Sur la base de l'approche à deux éléments par pli, l'effet de l'histoire thermique complexe a été étudié où un dépôt simultané de tous les plis a été adopté qui, s'il était validé, pourrait encore réduire le temps de calcul. Cependant, cette approche a conduit à une sous-estimation des courbures et des contraintes résiduelles de 40 et 70 % respectivement. Cette sous-estimation est directement liée à l'histoire thermique qui comprend un dépôt successif des plis qui conduit à des gradients thermiques à travers l'épaisseur conduisant à des contraintes résiduelles plus élevées par rapport au dépôt simultané de tous les plis.

De plus, l'effet du choix SFT a été étudié. Les résultats ont montré que plus la SFT considérée est élevé, plus les contraintes résiduelles sont élevées et un SFT adéquat égal à la transition vitreuse est suffisant pour prédire les contraintes résiduelles. Enfin, une simulation à deux SFT a été menée et les premiers résultats ont été présentés et devraient être inspectés dans les travaux futurs.

Perspectives

Si cette étude a permis de contribuer à une meilleure compréhension des contraintes et déformations résiduelles lors de la mise en place de rubans de composites semi-cristallins, elle ouvre également de nouvelles perspectives de recherche. En fait, un modèle couplé thermiquephysique-mécanique a été établi pour modéliser les contraintes et déformations résiduelles pendant et après le placement de la bande. Le modèle a été appliqué aux stratifiés 6 plis uniquement et les profils de température et de cristallinité n'ont pas été validés expérimentalement. Le modèle thermique basé sur le PGD a été choisi pour traiter mais ce modèle est difficile à comprendre car aucun accès n'a été accordé au code et le nombre maximum de plis simulés était de 6. Un modèle analytique a été présenté basé sur des travaux antérieurs et a montré des résultats prometteurs. Certaines difficultés étaient cependant présentes dans le traitement des résultats obtenus. Ce modèle pourra être examiné de près dans un futur travail afin de bien comprendre les erreurs et les limites de sa mise en oeuvre.

Simultanément, le profil de température à travers l'épaisseur du stratifié doit être mesuré expérimentalement afin de valider tout modèle thermique utilisé. Ce travail débouche sur une campagne expérimentale suivante concernant la mesure du gradient de cristallinité à travers l'épaisseur du stratifié lors de son traitement sur le SPIDE-TP.

Une fois le modèle thermique correspondant couplé aux modèles de cristallisation et de fusion validé, une autre fenêtre peut être établie portant sur une étude sensible de l'effet des paramètres du procédé sur la cristallinité et donc les propriétés du stratifié fabriqué.

Sur le modèle mécanique, les variations de propriétés étaient simples et un comportement élastique a été attribué au matériau qui n'est pas tout à fait représentatif du comportement complexe du matériau. Cela a également conduit à de petites différences entre le modèle proposé et les courbures mesurées. Par conséquent, une description adéquate des propriétés, en particulier concernant le comportement viscoélastique du PEEK, doit être quantifiée et liée aux équations présentées dans cette étude afin que leur influence puisse être comprise.

De plus, une définition simple de la SFT a été utilisée dans ce travail. Alors que cette approche semblait capable de prédire les contraintes et déformations résiduelles pour les stratifiés utilisés dans cette étude, il peut être approprié d'étudier l'effet de la présence de plusieurs SFT en fonction de la cristallinité et de la température du matériau, ce qui peut être fait en adoptant un comportement viscoélastique pour la résine.

L'objectif initial de ce projet était lié aux contraintes et déformations résiduelles se produisant dans les stratifiés composites thermoplastiques au niveau des plis. Les contraintes résiduelles se manifestent également au niveau micro, c'est-à-dire entre la fibre et la résine. Il peut être intéressant d'intégrer l'histoire thermique et les gradients de cristallinité pour étudier les contraintes résiduelles à ce niveau dans les travaux futurs.

Figure 1 -

 1 Figure 1 -2015 Toyota Mirai[2]

Figure 3 -

 3 Figure 3 -Tensile strength and corresponding prices for the most used polymers[5]

Figure 1 . 1 -

 11 Figure 1.1 -Photograph of the laser system used with the filament winding machine, SPIDE-TP in Cetim

Figure 1 . 2 -

 12 Figure 1.2 -Rectangular elements representing the surface's asperities[38]

Figure 1 . 5 -

 15 (b) presents the comparison between the simulated contact temperature evolution between different layers and the data from the corresponding thermocouples. The temperature profiles presented in Figure1.5 follow the same trend previously described. In addition, the presence of the thermal gradient can be observed and is qualitatively similar to the results in Figure1.4.

Figure 1 . 5 -

 15 Figure 1.5 -Comparison between the numerical and experimental temperature values for (a)-the lower surface of the 1 st layer ( contact between the liner and the 1 st layer) and (b)-contact temperatures between different layers[START_REF] Schäkel | Temperature analysis for the laser-assisted tape winding process of multi-layered composite pipes[END_REF]

Figure 1 . 6 -

 16 Figure 1.6 -Molecular structure of PEEK

Figure 1 . 7 -

 17 Figure 1.7 -Temperature dependence of the thermal expansion coefficient of the PEEK for different crystallinities[START_REF] Choy | Thermal Expansion of Poly (etherether-ketone) (PEEK)[END_REF]

  Lu et al. [85] used TMA to determine the linear CTE of semi-crystalline PEEK 450G with a 0.35 volumetric crystallinity fraction. The results showed again a dependence of the CTE on the temperature. They are presented in Figure 1.8 alongside those from Choy & al. [88] for a crystallinity fraction of 0.34 for comparison. It can be seen that the evolution of the linear CTE is similar for the two studies between 60°C and 180°C. On the other side, the values at room temperature are different and may be due to measurements protocols in each case. At elevated temperatures, Choy & al. limited their measurements to 200°C while Lu & al. measured the linear CTE until 250°C.

Figure 1 . 8 -

 18 Figure 1.8 -Evolution of the linear CTE with temperature for PEEK 450G from[START_REF] Choy | Thermal Expansion of Poly (etherether-ketone) (PEEK)[END_REF] and[START_REF] Lu | Thermal stability and thermal expansion studies of PEEK and related polyimides[END_REF]

Figure 1 . 9 -

 19 Figure 1.9 -Tensile modulus of PEEK 150P function of degree of crystallinity[START_REF] Lee | Effects of Cooling Rate on the Crystallinity and Mechanical Properties of Thermoplastic Composites[END_REF]

Figure 1 . 10 -

 110 Figure 1.10 -Predicted Young modulus for PEEK based on micromechanics model[START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] Bas performed in[START_REF] Bas | Reinforcement effect and molecular motions in semicrystalline PEEK films: Mechanical and physical modelings. I[END_REF] an experimental study on the influence of both crystallinity and temperature on the elastic or storage modulus E' and loss tangent tan δ of PEEK. No indication was given on the PEEK's grade. The Dynamic Mechanical Analysis 'DMA' experiment was performed on as-received amorphous specimens and other specimens subjected to annealing at different temperatures. DSC measurements were carried to identify the specimens' crystallinities. The variation of log E' and tan δ for amorphous PEEK, semi-crystalline PEEK with a volumetric crystallinity fraction of 23% and 35% are presented in Figure 1.11-(a), (b) and (c) respectively.

Figure 1 . 11 -

 111 Figure 1.11 -Variation of the storage modulus and loss tangent for (a) amorphous PEEK, (b) PEEK with 23%volumetric crystallinity and (c) 35% volumetric crystallinity[START_REF] Bas | Reinforcement effect and molecular motions in semicrystalline PEEK films: Mechanical and physical modelings. I[END_REF]

Figure 1 . 12 -

 112 Figure 1.12 -Variation of the tensile modulus for PEEK with different crystallinities according to tensiletests[START_REF] Seferis | Polyetheretherketone (PEEK): Processing-structure and properties studies for a matrix in high performance composites[END_REF]

Figure 1

 1

Figure 1 . 13 -

 113 Figure 1.13 -Evolution of the Poisson's ratio with temperature measured during static tests in[START_REF] Coquill | Mechanical properties of several neat polymer matrix materials and unidirectional carbon fiber-reinforced composites[END_REF]

Figure 1 .

 1 14-(a).Talbot & al.[START_REF] Talbott | The Effects of Crystallinity on the Mechanical Properties of PEEK Polymer and Graphite Fiber Reinforced PEEK[END_REF] measured the variation of the shear modulus of a PEEK 150P as a function of the volume fraction crystallinity of the PEEK resulting from different cooling rates. The results are presented in Figure1.14-(b), they show an increase in the shear modulus with increasing crystallinity which is in accordance with the trends observed for the Young's modulus.

Figure 1 .

 1 Figure 1.14 -(a) Variation of the shear modulus for PEEK with temperature, values reproduced from [98] and (b)-Dependence of the shear modulus for PEEK on crystallinity, values are reproduced from [62]

Figure 1 . 15 -

 115 Figure 1.15 -Variation of the transverse CTE of PEEK/AS4 with temperature[START_REF] Barnes | Thermal Expansion Behaviour of Thermoplastic Composite Materials[END_REF]

Figure 1 . 16 .

 116 The amorphous specimen behaves similarly to the amorphous PEEK presented in Figure1.11-(a). The behavior of the nominally cooled and annealed specimens is similar to that of the semi-crystalline PEEK as shown in section 1.3.2.2 and in[START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF] and[START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF]. As the temperature increases, the semi-crystalline modulus's value drops down from 8.5 GPa (at ambient temperature) to approximately 1.2 GPa at 300 °C corresponding to almost 15% of the modulus at ambient temperature as shown in Figure1.16.

Figure 1 . 16 -

 116 Figure 1.16 -Evolution of the transverse modulus with temperature for amorphous, nominallycooled and annealed form amorphous of APC-2 measured using DMTA from[START_REF] Unger | The Effect of Cooling Rate and Annealing on Residual Stress Development in Graphite Fibre Reinforced PEEK Laminates[END_REF]

Figure 1 . 18 .

 118 If Ozawa's model was adapted to correctly predict the behavior of PEEK's non-isothermal crystallization, the results in Figure 1.18 would've been straight parallel lines.

Figure 1 . 18 -

 118 Figure 1.18 -Ozawa's logarithmic equation plot function of different cooling rates[START_REF] Cebe | Crystallization behaviour of poly(ether-ether-ketone)[END_REF]

Figure 1 . 19 -

 119 Figure 1.19 -Comparison between predicted using Gordnian model and experimental results for non-isothermalcrystallization of PEEK/AS4 composite[START_REF] Gordnian | A Rate-Type Crystallization Kinetics Model for Process Modelling of Carbon Fibre PEEK Matrix Composites[END_REF]

Figure 1

 1

Figure 1 . 20 -

 120 Figure 1.20 -Effect of cooling on residual stress state in a matrix surrounding a fiber when bonding occurs[START_REF] Barnes | Thermal expansion characteristics of PEEK composites[END_REF]

Figure 1 . 21 -

 121 Figure 1.21 -Residual strains and stresses at the ply level[START_REF] Parlevliet | Residual stresses in thermoplastic composites-A study of the literature-Part I: Formation of residual stresses[END_REF]

Figure 1 . 22 -

 122 Figure 1.22 -Image illustrating the fiber waviness in T300/P1700 laminate[START_REF] Kugler | Identification of the Most Significant Processing Parameters on the Development of Fiber Waviness in Thin Laminates[END_REF]

Figure 1 .

 1 Figure 1.23 -SEM image of transverse micro cracks in carbon fiber reinforced thermoplastics[START_REF] Gentz | In-plane shear testing of medium and high modulus woven graphite fiber reinforced/polyimide composites[END_REF]

Figure 1 . 24 -

 124 Figure 1.24 -Warpage of composite plate[START_REF] Bapanapalli | A linear finite element model to predict processing-induced distortion in FRP laminates[END_REF]

Figure 1 . 26 -

 126 Figure 1.26 -Residual stresses distribution through the thickness for a 40-ply APC-2 laminate predicted viaChapman's model[START_REF] Chapman | Prediction of Process-Induced Residual Stresses in Thermoplastic Composites[END_REF] and compared to experimental results

Figure 1 . 27 -

 127 Figure 1.27 -Effect of varying the winding tension profile on the radial stress in filament winded compositerings[START_REF] Cirino | In-situ consolidation for the thermoplastic composite ring-residual stress state[END_REF]

Figure 1 . 28 -

 128 Figure 1.28 -Comparison between simulated and experimentally measured transverse normal residual stressdistribution within a 40 ply APC-2 unidirectional laminate[START_REF] Sonmez | Analysis of process-induced residual stresses in tape placement[END_REF]

Figure 2 . 1 -

 21 Figure 2.1 -Overview of the general methodology

Figure 2 .

 2 Figure 2.1 presents an overview of the general methodology followed during this work. The laminates processed using the SPIDE-TP are divided into two parts according to their experimental destination. The first group is used for the experimental characterization of the material after the process. The objective of the experimental characterization is to understand the effect of the process parameters on the thermomechanical behavior in addition to quality related parameters such as the fiber content and the presence or the absence of defects. The second batch of the fabricated specimens is destined to the estimation of residual stresses. The determination of the residual stresses is however a combination between an experimental and a numerical part. Experimentally, the laminates' curvatures at room temperature and their variation with

Figure 2

 2

Figure 2 . 2 -

 22 Figure 2.2 -Photos of thermoplastic structures fabricated on the SPIDE-TP: (a) Pipe and (b) Reservoir

Figure 2 . 3 -

 23 Figure 2.3 -Photo of the SPIDE-TP machine in Cetim, Nantes-France

Figure 2 . 4 .

 24 During the process and in a continuous manner, this camera captures thermal images which are transferred to a specific software controlling the SPIDE-TP. A processing window allows the user to follow the continuous control during which an automatic variation of the laser power occurs in order to maintain a constant value of the nip-point temperature 𝑇 𝑛𝑖𝑝-𝑝𝑜𝑖𝑛𝑡 . The processing window presented in Figure2.4-(b)shows three concerned temperature zones: the tape, nip-point and substrate temperatures. The nip-point temperature is considered as the maximum temperature in the processing window (b) generally located around the mid-plane between the incoming ply and the substrate. Once the fabrication is complete, either a direct demolding of the fabricated piece from the mandrel occurs, or a posttreatment such as annealing is applied before demolding.

Figure 2 . 4 -

 24 Figure 2.4 -(a) Photo of the thermal camera and the laser mounted on the deposition head of the SPIDE-TP; (b) Typical processing window of the thermal control during the deposition of a ply on a substrate on the SPIDE-TP

Figure 2

 2

Figure 2 . 5 -

 25 Figure 2.5 -(a) Schematic presentation of the obround-shaped laminate fabricated on the SPIDE-TP; (b) and (c) Photo and schematic presentation of respectively the upper and lower surfaces of the laminate in (a)

Figure 2 . 7 -

 27 Figure 2.7 -DMA 1 METTLER TOLEDO

Figure 2 . 8

 28 Figure 2.8 at a frequency of 1 Hz. This testing mode is suitable for stiff materials i.e. in the range of high modulus such as thermoplastic and thermoset resins and composites as well as metals and ceramics. A maximum deflection of 10 μm was imposed.

Figure 2 . 8 -

 28 Figure 2.8 -The three point bending test on the DMA 1 METTLER TOLEDO machine

Figure 2 . 9 - 1 𝐾

 291 Figure 2.9 -Dimensions of the specimen tested in 3-point bending DMA test to obtain the transverse modulus

Figure 2 . 10 -

 210 Figure 2.10 -Points selected from DMA obtained curves to determine the glass transition temperature [15]The glass transition temperature 𝑇 𝑔 indicates the change of the amorphous phase from a brittle glassy state to a more viscous rubbery state. It can be measured via several experimental techniques including the DMA tests. Three methods exist to determine the glass transition temperature according to the European standard ISO 6721-11[16]:

Figure 2 .

 2 11-(a).

Figure 2 .

 2 Figure 2.11 -(a) Photo of a specimen during a three-point bending test on the Instron 3369 machine (b) -Schematic presentation of the three point test and the specimens' dimensions

Figure 2 . 12 -

 212 Figure 2.12 -Determining the flexural modulus from the load-displacement curve obtained through the 3-point bending test

Figure 2 . 13 -

 213 Figure 2.13 -Photo of the DSC Q200 from TA Instruments used to perform the DSC tests During the DSC tests, the specimens were heated from room temperature at 25°C until 380°C at a heating rate of 10 K/min. The specimen are maintained for 15 mins at 380°C to erase their thermal history. Finally, they're cooled from 380°C until 25°C at -2 K/min. The sample thermal cycle is applied a second time on the samples in order to test the effect of the slow cooling from the melt on the different characteristics studied. A schematic presentation of the thermal cycle applied to the specimens is presented in Figure 2.14. The time ′𝑡 0 ′ in Figure 2.14 indicate the time between two thermal cycles. This time is variable and depends on the conditions of the testing. It's however irrelevant to the test results.

Figure 2 . 14 -

 214 Figure 2.14 -Thermal load applied to the specimens tested under DSC

Figure 2 .

 2 15 which correspond to the integral of the heat flow function of time using Eq. 1.10

Figure 2 . 15 -

 215 Figure 2.15 -Illustration of the thermogram obtained during the heating scan of a DSC test

Figure 2 .

 2 16-(a)were used to place specimens c.f. Figure2.16-(b) which were selected from different locations in the plates presented in Table2.1. Next, a resin flow is poured on the specimens placed in the mold in order to obtain specimens casted in resin blocks c.f. Figure2.16-(c).The resin mixed with the hardener poured on the specimens requires three to four hours to cure at room temperature. Once cured, the solid block of resin containing the specimens can be removed from the plastic molds, At this point, the resin blocks shown in Figure2.16-(d) can be polished on a Struers® TegraForce-5 polisher using several glass sandpapers starting from big grit size of P320 down to the small grit size of P4000.

Figure 2 .

 2 Figure 2.16 -(a) Plastic mold in which the specimen destined for microscopic observation is placed, (b) -Example of specimens to be placed in the plastic mold, (c) -two specimens coated in resin inside the plastic mold and (d) specimens in the resin block after resin curing

Figure 2 . 17 -

 217 Figure 2.17 -Schematic presentation of -(a) Symmetrical laminate and (b) Unsymmetrical laminate

Figure 2 . 18 .

 218 The specimens were placed by series of six in a drying and heating oven FED 115 from Binder®. They were held on a metallic bar in the Binder using fold back clips as it can be seen in Figure2.18, with their lateral side section facing a Nikon D600 digital camera with a Nikkor 18-105 mm f/3.5-5.6G ED VR lens (Nikon,Inc., Tokyo, Japan) under ambient room light. The fold back clips are used to avoid the impact of gravity and friction on the curvature evolution that could have a high impact if the samples were just put on a horizontal plate. This allows the determination of the strips' curvatures by analyzing the images taken by the camera. The first set of measurements was conducted on the [(90 3 /0 3 )] laminates in the case of 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 = 25°𝐶.

Figure 2 . 18 -

 218 Figure 2.18 -Curvature measurement experimental setup

Figure 2

 2

Figure 2 . 20 - 2 ( 2 (

 22022 Figure 2.20 -Extracting the points coordinates in ImageJ in order to obtain the corresponding curvatureThe coordinates of the three different points (𝑥 1 , 𝑦 1 ), (𝑥 2 , 𝑦 2 ) and(𝑥 3 , 𝑦 3 ) for the different specimens are then introduced in a Matlab code to solve the system of equations Eq. 2.6 and obtain 𝑥 𝑐 , 𝑦 𝑐 and 𝑟 𝑐 the abscissa, ordinate and radius of the center of the circle passing through these three points.

A

  . Characterization of the material after the process: this part deals with the results from experimental techniques used to evaluate the laminates' properties, crystallinity fraction and different transition temperatures. It includes several sub-parts: Microscopic observations of sections from different produced laminates Differential Scanning Calorimetry (DSC) analysis Static Three-Point flexural tests Dynamic mechanical analysis (DMA) tests B. Measurement of the curvatures of different specimens produced on the SPIDE-TP: As discussed previously, the curvatures are measured at room temperature after the process. Their variation with temperature during heating and cooling for different thermal cycles were also monitored. The results presented in this chapter correspond to the following specimens manufactured with an unheated (at room temperature) and a heated (at 180°C) mandrel. Unsymmetrical [90 3 /0 3 ] specimens 6-plies unidirectional specimens Symmetrical [([90 3 /0 3 ]) 2 /90 3 ] specimens

Figure 3 . 1 -Figure 3 . 1 -

 3131 Figure 3.1 -Cross sections of the [(90 3 /0 3 ) 2 /90 3 ] specimen manufactures using (a) Cold mandrel & (b) Heated mandrel Figure 3.1-(a) & (b) present an example of the analyzed cross-section in case of the cold and heated mandrel respectively. In both cases, defects are present and include principally voids located between the 90° and 0° (red arrows) oriented plies owing to lack of consolidation affecting the inter-ply intimate contact.

Figure 3 . 2 -

 32 (c) for the cold mandrel and Figure 3.2-(d) for the heated mandrel. The resin rich areas are indicative of good consolidation quality. The presence of interlaminar voids between plies of different orientations can be attributed to different factors such as insufficient roller pressure, insufficient heating power, overheating leading to eventual deconsolidation once the roller pressure is released or time to melt the resin crystals and allow their migration between adjacent plies.

Figure 3 . 2 -

 32 Figure 3.2 -Cross-sections of the interfaces between the 90° and 0° oriented plies in the symmetric [(90 3 /0 3 ) 2 /90 3 ] laminate in the case of left column :(a) & (c) -Cold mandrel and right column : (b) & (d) -Heated mandrelWhile the voids between the 90° and 0° oriented plies are large and can be easily detected, those between the plies of the same orientation are much less apparent. The most of voids are located between the plies of different orientations. In fact, in Figure3.2-(c) & (d), it seems difficult to identify the interfaces between the three plies forming the 90° oriented plies, therefore indicating a good consolidation quality. This was also observed in several previous studies which stated a good consolidation quality when the interfaces between plies couldn't been identified[1],[2]

Figure 3 . 3 -

 33 (a) & (b) for the cold and the heated mandrel respectively. The fiber content was analyzed at a numerical zoom of 500 times on the microscope. The different constituents are identified with the fibers shown in dark grey, the matrix in light grey and defects related to the polishing protocol and/or voids in dark circles.

Figure 3 . 3 -

 33 Figure 3.3 -Samples of micrographs for symmetrical 15-plies laminates manufactured with (a)-Cold mandrel and (b)-Heated mandrel in order to study the fiber content

5 . 3 .

 53 In this part, the results from the different DSC tests are presented and discussed to quantify: The glass transition temperature 𝑇 𝑔 The fusion temperature 𝑇 𝑚 The cold crystallization temperature 𝑇 𝑐𝑐 The crystallinity fraction for the different specimens In order to better understand the results afterwards, a typical DSC curve is presented in Figure 3.4 where the different phase transitions of a polymer can be detected.

Figure 3 . 4 -

 34 Figure 3.4 -Heating and cooling scans during DSC tests showing the different transitions of a polymer[3] Thermoplastic polymers such as PEEK can exist, according to their temperature, in three states: glassy, rubbery and melted. The change in the temperature induces a change in the polymer's state. DSC scans can be used to detect the transition between the polymer's different states through a change in the thermogram curve.In reference to Figure3.4, during the heating scan, the glass transition is detected first through a shift in the baseline which translates the difference in the specimen's thermal capacity before and after the transition. It occurs in the amorphous fraction of the polymer and is characterized by the breaking of the Van Der Waals bonds by the application of a certain amount of heat. The crystallites regions are maintained during the glass transition due to their strong covalent bonds. The breakage of these bonds occurs at a higher temperature defined by the melting temperature 𝑇 𝑚 . Heating the material beyond its melting temperature leads eventually to its oxidation and thermal degradation. An exothermic peak is presented in Figure3.4 during the heating scan after the glass transition. It corresponds to the cold crystallization which takes place during heating between 𝑇 𝑔 and 𝑇 𝑚 . Another exothermic peak is recorded during the cooling phase close to the melting temperature. The melt crystallization generally results in large size crystals compared to the cold crystallization during which a slower molecular motion is induced.

Figure 3 . 5 -Figure 3 . 5 -

 3535 Figure 3.5 -DSC thermogram obtained for the specimens with (a) -𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 = 𝑇 𝑎𝑚𝑏𝑖𝑒𝑛𝑡 and (b) -𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 = 180°𝐶

Figure 3 . 5 -

 35 (a) and 344°C in Figure 3.5-(b)). It translates the breaking of covalent bonds and the melting of the crystals.

Figure 3 . 6 .

 36 The specimens manufactured using the cold mandrel present a 𝑇 𝑔 of 140.8°C during the 1 st heating which increases to 148°C during the 2 nd heating. The value during the 2 nd heating is close to the one measured for the heated mandrel case which is 148.19°C and doesn't vary much during the 2 nd heating run. The values presented here are in accordance with the literature for the 𝑇 𝑔 of PEEK/AS4 and pure PEEK in the order of 140.4-152°C[4],[5],[6] &[7].

Figure 3 . 6 -

 36 Figure 3.6 -The glass transition temperatures measured from DSC scans during the 1st and 2nd heating of specimen manufactured with the cold and unheated mandrels

Figure 3 . 7 -

 37 Figure 3.7 -Evolution of the crystallinity (a) and the glass transition temperature (b) with the annealing time at 156°C [8] In conclusion, the use of an unheated mandrel leads to a lower 𝑇 𝑔 (140.8°C) compared to the heated mandrel (148.19°C). A thermal treatment such as cooling slowly from the melt (380°C) increases the 𝑇 𝑔 (148.67°C).

Figure 3 . 8 -

 38 Figure 3.8 -Evolution of the crystallinity (a) and the fusion temperature (b) with the annealing time at 156°C [8]

Figure 3 . 9 -

 39 Figure 3.9 -Melting behavior during the 1st and 2nd heating on the DSC for the specimen manufactured with the cold mandrel

Figure 3 . 10 -

 310 Figure 3.10 -DSC thermogram for PEEK reinforced with graphene composites with different weight fractions during (a)-Crystallization from the melt (b) -Heating after the crystallization from the melt and (c) -heating from the glassy state[16]

Figure 3 . 10 .

 310 The double melting behavior is highlighted in Figure 3.10-(b) after a crystallization from the melt. The specimen heated from the glassy state don't show a double melting behavior c.f. Figure 3.10-(c).

Figure 3 . 11 -

 311 Figure 3.11 -Cold crystallization exothermic peak detected on the DSC thermogram for specimen manufactured with the cold mandrel

Figure 3 . 95 Figure 3 . 12 -

 395312 Figure 3.12 -Crystallinity fraction calculated from the DSC tests during the first and second heating for the specimen manufactured using (a) -Unheated mandrel & (b) -Heated mandrel

Figure 3 .

 3 12-(a) is 23.5% calculated for the 6-plies UD laminate while the lowest value is 20.5 % corresponding to the thickest 15-plies symmetric laminates. On the other hand, the maximum and minimum values in the case of the heated mandrel in Figure 3.12-(b) are respectively 28.7 and 30.2 % corresponding to the 9-plies UD and [90 3 /0 3 ] laminates. In both cases, the different specimens present close crystallinities.

Figure 3 . 13 -

 313 Figure 3.13 -Crystallinity fraction for PEEK/AS4 composites function of cooling rate from previous worksAn increase in the crystallinity fraction is measured during the 2 nd heating cycle compared to the 1 st one in Figure3.12-(a) for the cold mandrel case. The histograms presented in Figure3.12 are superimposed and presented Figure3.14 to highlight the increase. Figure3.14 presents the crystallinity fractions for the different specimen in the case of the cold mandrel calculated from the 2 nd heating cycle. The grey part corresponds to the crystallinity obtained after the process (calculated from the 1 st heating) while the blue part indicates the increase due to the annealing at 380°C. The effect of the annealing at 380°C followed by the slow cooling isn't negligible and can go up to an increase of 14.4% ( the blue part of the symmetric [(90 3 /0 3 ) 2 /90 3 ] laminate).

Figure 3 . 14 -

 314 Figure 3.14 -Crystallinity fraction calculated from the 2nd heating of specimen manufactured with the cold mandrel (Grey bars: crystallinity fraction resulting from the process, Blue bars: Increase due to the annealing at 380°C)

Static 3 - 3 . 4 . 1 Figure 3 .

 33413 Figure 3.15 presents the results of the static three-point bending tests of the UD specimens manufactured using the cold mandrel with 6 plies (Figure 3.15-(a)) and 9 plies (Figure 3.15-(b)). The results in case of the heated mandrel are presented for the 6 plies in Figure 3.15-(c) and 9 plies in Figure 3.15-(d). The different specimens show a similar trend: an increase in the load with increasing the displacement until the failure. The failure behavior is however different between the specimen. In the case of the unheated mandrel, a sudden drop in the load is obvious for the 6 and 9 plies specimen. In the case of the heated mandrel, there seems to be a divergence in the behavior at the break between the specimens. These differences can be the result of defects present prior to the tests such as resin rich areas, internal cracks, voids, etc.

Figure 3 .

 3 Figure 3.15 -Load-displacement curves for different samples in three-point bending tests for specimen manufactured using unheated mandrel with (a) -6 plies and (b) -9 plies respectively, and specimen manufactured using heated mandrel with (c) -6 plies and (d) -9 plies respectively

Figure 3 . 16 -

 316 Figure 3.16 -Transverse modulus estimated from three point bending tests on unidirectional specimen manufactured with the cold and the heated mandrel in (a) and its variation with the crystallinity in (b)

Figure 3 . 17 -

 317 Figure 3.17 -Variation of the storage modulus with temperature from DMA tests for the specimen manufactured with (a) -unheated mandrel during the first and second heating on the DMA & (b) -with the heated mandrel during the first heating on the DMA Table3.2 lists the maximum standard error and the corresponding temperature at which this maximum error occurs between the specimens for the different laminate sequences studied in case of the unheated and heated mandrel. In the case of the cold mandrel, the maximum standard error can be identified at ambient temperature for the three laminate sequences and at 171°C and 165°C for the 6 and 9 plies UD respectively. The maximum value is 15% and can be due to dimensional differences between the different specimen. However, the errors are considered acceptable and the specimens with the same laminate sequence present the same evolution of their modulus with the temperature.

The comparison of the behavior during the first heating in Figure 3 . 17 -

 317 (a) and Figure 3.17-(b) translates the effect of the mandrel temperature while the comparison between the behavior during the first and second heating in Figure 3.17-(a) addresses the effect of the annealing.

Figure 3 .

 3 18-(a)for the 3-plies, (b) for the 6-plies and (c) for the 9-plies UD specimen manufactured using the unheated mandrel.102The glass transition temperatures in the case of the unheated mandrel range between 133 and 136 °C according to the DMA tests. It increases to 152°C during the second heating.

Figure 3 . 18 -

 318 Figure 3.18 -Loss factor 𝑡𝑎𝑛 𝛿 function of temperature measured during DMA tests on specimen manufactured with unheated mandrel with (a) -3 plies, (b) -6 plies & (c) -9 plies during the first and the second heating Another observation can be made regarding the evolution of 𝑡𝑎𝑛 𝛿 with temperature during the 1 st heating run for the specimens in case of the cold mandrel. During the glass transition, a 1 st peak is detected, from which the glass transition temperatures have been deduced. This peak is associated with the breaking of the Van Der Waal bonds in the amorphous part of the resin. It's however succeeded by another one, smaller in intensity which can be clearly detected for [90 3 ] and the [90 6 ] laminates in Figure 3.18-(a) & (b). This peak is associated with the effect of the recrystallization which takes place during the cold crystallization. The same observation with double peaks was also reported by Cortes & al. [28] when plotting the loss modulus 𝐺 ′ vs. temperature for amorphous PEKK. The authors attributed the second peak to the effect of recrystallization. The values of 𝑇 𝑔 during the second heating is close to the ones measured for the specimen manufactured with the heated mandrel which are presented in Figure 3.19. The variation of the glass

Figure 3 . 19 -

 319 Figure 3.19 -Loss factor 𝑡𝑎𝑛 𝛿 function of temperature measured during DMA tests on specimen manufactured with heated mandrel with (a) -3 plies, (b) -6 plies & (c) -9 plies

Figure 3 . 20 -

 320 Figure 3.20 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] laminates fabricated using the un-heated mandrel during heating and cooling rom 120°C

Figure 3 . 21 -

 321 Figure 3.21 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] laminates fabricated using the un-heated mandrel during heating and cooling from 150°C

Figure 3 . 22 -

 322 Figure 3.22 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] laminates fabricated using the un-heated mandrel during heating and cooling from 180°CNext, two specimen previously subjected to the first heating and cooling from 120°C were added to the other two heated until 150°C. Therefore, a total of four specimens were placed in the oven and were heated simultaneously until 180°C. These specimens are divided into two sets: the first one gathers the two specimens subjected to consecutive heating up to 120, 150 and 180°C while the second set gathers the two specimens subjected to heating at 120 and 180°C only. The evolution of the curvatures with temperature for the two sets of specimens is presented in Figure3.22. The curvature of the specimen previously heated to 120 and 150°C (denoted by Heating-150-180°C) is equal to 3.7 𝑚 -1 compared to 2.8 𝑚 -1 for the ones previously heated to 120°C only. The evolution of the curvature with temperature follows the same trend as in the two previous cases with a decrease upon heating and an increase upon cooling.

Figure 3 . 23 -

 323 Figure 3.23 -Linear fittings for the evolution of the curvatures in case of the unsymmetrical [90 3 /0 3 ] laminates fabricated using the un-heated mandrel during heating (a) and cooling (b) from 180°C

Figure 3 . 24 .

 324 At the end of the cooling from 200°C, the residual curvatures for the specimens heated successively to 150, 180 and 200°C and the ones heated to 180°C then 200°C are close with slight differences.

Figure 3 . 24 -

 324 Figure 3.24 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] laminates fabricated using the un-heated mandrel during heating and cooling from 200°C

Figure 3 . 21 ,Figure 3 .22 and Figure 3 . 24 .

 3213324 According to the DMA and DSC scans, 𝑇 𝑔 was measured between 133°C and 140°C. The DSC scans in the case of the specimen manufactured with the cold mandrel show a cold crystallization occurring upon heating the specimen above their glass transition temperature c.f. Figure3.11.

Figure 3 . 25 -

 325 Figure 3.25 -Variation of the dimensionless curvature, i.e. the curvature divided by the thickness, function of temperature for unsymmetrical cross-ply (a) -PEEK/AS4 composites and (b) -PEEK/IM8 composites[30]

Figure 3 . 26 -

 326 Figure 3.26 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] vs. unidirectional 6-plies laminates fabricated using the un-heated mandrel during heating and cooling from 120°C

Figure 3 . 27 -Figure 3

 3273 Figure 3.27 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] vs. unidirectional 6-plies laminates fabricated using the un-heated mandrel during heating and cooling from 150°CAccording to Figure3.27, the UD laminates behave in the same manner as the cross-ply laminates whenever they're heated above their 𝑇 𝑔 . The curvature decreases during heating until reaching 150°C. During the isothermal run for 30 min at 150°C, the curvature increases from 0.18 to 0.21 𝑚 -1 which corresponds to an increase around 17%. Upon cooling, the curvature increases and an offset from the recorded curvatures during the heating phase is observed similarly to the cross-ply laminates in

3. 5 . 3

 53 Curvature evolution with temperature of the [𝟗𝟎 𝟑 /𝟎 𝟑 ] and unidirectional specimen: 𝑻 𝒎𝒂𝒏𝒅𝒓𝒆𝒍 = 𝟏𝟖𝟎°𝑪The evolution of the curvatures for the [90 3 /0 3 ] and the 6-plies UD strips cut from laminates manufactured using the heated mandrel at 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 =180°C during heating and cooling from 120°C is presented in Figure3.28.

Figure 3 . 28 -

 328 Figure 3.28 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] vs. unidirectional 6-plies laminates manufactured using the heated mandrel at 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 =180°C during heating and cooling from 120°C

Figure 3 . 29 -

 329 Figure 3.29 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] vs. unidirectional 6-plies laminates fabricated using the heated mandrel at 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 =180°C during heating and cooling from 150°C

Figure 3 . 30 -

 330 Figure 3.30 -Evolution of the curvature with temperature for unsymmetrical [90 3 /0 3 ] vs. [(90 3 /0 3 ) 2 ]laminates fabricated using the un-heated mandrel during heating and cooling from 120°C

Figure 3

 3

Figure 3 .

 3 Figure 3.31 -(a -) Effect of the number of layers on the principal curvatures of a [+45 𝑛 /-45 𝑛 ] 𝑇 laminates [31] & (b) -Effect of the thickness (number of layers) on the radius of curvature of unsymmetrical laminates made with A: 2 plies, B: 4 plies and C: 8 plies [32]

Figure 3 .

 3 Figure 3.32 presents the evolution with temperature of the curvature for the [(90 3 /0 3 ) 2 ] laminates annealed at 150°C. The trend is similar to that of the [90 3 /0 3 ] : Increase of the curvature during the isothermal run at 150°C ( 15% increase for 16.4% in the case of [90 3 /0 3 ] laminates) Offset between the curvatures during cooling and heating

Figure 3 . 32 -

 332 Figure 3.32 -Evolution of the curvature with temperature for unsymmetrical [(90 3 /0 3 ) 2 ] laminates fabricated using the un-heated mandrel during heating and cooling from 150°C

Figure 3 . 33 -

 333 Figure 3.33 -Evolution of the curvature with temperature for symmetric [(90 3 /0 3 ) 2 /90 3 ] laminates manufactured using the mandrels at ambient temperature and at 𝑇 𝑚𝑎𝑛𝑑𝑟𝑒𝑙 =180°C during heating and cooling from 120°C

Figure 3

 3

Figure 3 . 34 -Figure 3 . 26 ,Figure 3 . 26 & 3 . 34 )Figure 3 . 28 &

 334326326334328 Figure 3.34 -Comparison of the measured curvature for the different specimen after the cutting prior to any thermal load o The comparison between the [90 3 /0 3 ] and the ((90 3 /0 3 ) 2 laminates (green arrow) shows the effect of thickness i.e. the number of layers: Increasing the thickness (i.e. the number of layers) decreases the curvature for the same processing conditions. o The comparison between the UD and the [90 3 /0 3 ] specimen curvatures may directly give an insight on the thermal history effect on the residual stresses and the curvatures. It's clear

. 26 ,Figure 3 . 27 ,Figure 3 . 28 &

 26327328 Figure 3.29) which translates into a higher residual stresses state. At room temperature and prior to any annealing, the curvature of the [90 3 /0 3 ] laminate is approximately 6.6 times the curvature of 6-plies unidirectional laminates processed under the same conditions.

Figure 3 . 21 ,Figure 3 . 22 ,

 321322 Figure 3.21, Figure 3.22, Figure 3.24, Figure 3.32,). This increase is the direct result of the shrinkage that accompanies the cold crystallization highlighted via exothermic peaks during the heating run of DSC tests.The curvatures during the cooling from the different annealing cycles for the specimen manufactured using the cold mandrel present an offset from the ones during the heating phase. This offset increases when the annealing temperature increases from 150 to 180°C (Figure3.24). This increase is however reduced for an annealing temperature of 200°C compared to 180°C (Figure3.24). In order to understand the major driving factors behind this increase, one shall compare the curvatures in different cases:o From the annealing at 150°C, the effect of the cold crystallization is directly observed with a crystallization shrinkage leading to higher curvatures in addition to modified thermomechanical properties during cooling compared to the heating phase due to an enhanced crystallization. o If the specimen annealed successively at 150 and 180°C are considered, the results in Figure3.22 show that the curvature increase during the isothermal ramp at 180°C is negligible compared to the one measured at the 150°C ramp. The straightforward

Figure 4 . 1 -

 41 Figure 4.1 -Variation of longitudinal and transverse thermal conductivities of unidirectional APC2 with temperature [3] (-) Transverse;(--) Longitudinal Moreover, Lelouet [4] measured 𝐾 𝑇 of PEEK/AS4 composite supplied with 𝑉 𝑓 = 0.62 using hot plates. She measured a 𝐾 𝑇 value of 0.43 W.𝑚 -1 .𝐶 °-1 at 50°C. The longitudinal thermal conductivity 𝐾 𝐿 is estimated equal to 4.6 W.𝑚 -1 .𝐶 °-1 by Lelouet [4]. This value was taken from the results in Figure 4.1 from the measurements of Cogswell [3].

Figure 4 . 2 -

 42 Figure 4.2 -Evolution of 𝜌 𝑚 , 𝜌 𝑓 and 𝜌 with temperature from [5], [4] & [6]The theoretical density of a fiber reinforced polymer can be estimated from the law of mixture according to Agarwal & al.[6] using Eq. 4.1

Figure 4

 4

Figure 4 . 3 -

 43 Figure 4.3 -Evolution of the specific heat function of temperature from[7], [8] & [4] . On the left: for PEEK, on the right : for PEEK/AS4 obtained in [4] using Eq. 4.2

  presents the heat flux into the body. The insulation condition at the back-side Eq. 4.6 accounts for the heat accumulation considered by the authors in[3]. A schematic presentation of the coordinate system is presented in Figure4.5.

Figure 4 . 5 -

 45 Figure 4.5 -Schematic presentation of the simplified analytical model boundary conditions and coordinates system

Figure 4 . 6 -

 46 Figure 4.6 -Comparison between the analytical and numerical surface temperatures from[12]

Figure 4 . 8 -

 48 Figure 4.8 -Example of an extraction of the Temperature controls data for the tape, substrate and contact zone during processing on the SPIDE-TP

Figure 4 . 9 -

 49 (a), (b), (c), (d) &(e) respectively. The process parameters are constant for the different simulations and the difference was the number of plies. The whole domain has a length of 1 m and a thickness of 0.15*n mm with 'n' equal to the ply number in each simulation. It allows the determination of the temperature at each point during the deposition of the upper ply. Each ply is divided into 11 nodes through its thickness. The nodes are numbered from 1 to 11 from the bottom to the top i.e. the 1 st one (Node 1) is the closet to the mandrel. In the model, the nip-point is located at x=0.92m.

Figure 4 . 9 -plies 4 . 2 . 3 . 1 . 2

 4942312 Figure 4.9 -Temperature distribution across the length (x-direction) and thickness(y-direction) of a laminate during the deposition of the upper ply for (a) -2 plies, (b) -3 plies, (c) -4 plies, (d)-5 plies and (e) -6 plies 4.2.3.1.2 Switch from 2-D static to 1-D temperature evolution with time In order to obtain an adequate representation of the process, a switch from the 2-D configuration i.e. the results presented in Figure 4.9 to a 1-D time-dependent configuration is required. Let's consider for example Node 5 in Figure 4.9-(a). Different points are selected to explain the determination of the temperature

Figure 4 .

 4 10-(b).

Figure 4 .

 4 Figure 4.10 -(a) Example of the Different locations of a node during the processing on the SPIDE-TP in the 2D model and (b) -corresponding temperature evolution with time for the same time 4.2.3.1.3 Results and discussion By applying this procedure to all the nodes in Figure 4.9, the temperature evolution with time of the different nodes is obtained and presented in Figure 4.11. A temperature equal to 0 in the different figures indicate that the ply hasn't been deposited yet. The sudden drop of temperature to 25°C at the end of each deposition is considered in the thermal model to reduce the simulation time (presented by // in Figure 4.10).In fact, a time gap equal to 0.001 s was considered between the depositions of two consecutive layers. In reality, the time is much higher. However the hypothetical value added isn't supposed to affect the results presented afterwards since a purely elastic behavior of the material is chosen below T g . Moreover, the crystallization and the melting occur at temperatures above the glass transition, therefore the cooling rate below this temperature is insignificant to the results. Since the deposition of a single ply couldn't been simulated, it's considered that during its placement, the temperatures at the different nodes of Ply 1 (Node 1 to Node 11 in Figure 4.10-(a)) are equal to those of Ply 2 during its deposition (Node 12 to Node 22 in Figure 4.10-(a)). For the sake of clarity, only the evolution for Node 1 (bottom), node 6 (middle) and node 11 (top) for each ply are presented in Figure 4.11. The temperatures for the remaining nodes follow the same trend and range between the values of the extreme nodes (Node 1 and Node 11).

Figure 4 .

 4 10-(a)) are equal to those of Ply 2 during its deposition (Node 12 to Node 22 in Figure 4.10-(a)). For the sake of clarity, only the evolution for Node 1 (bottom), node 6 (middle) and node 11 (top) for each ply are presented in Figure 4.11. The temperatures for the remaining nodes follow the same trend and range between the values of the extreme nodes (Node 1 and Node 11).

Figure 4 . 11 -

 411 Figure 4.11 -Evolution of the temperature with time during the manufacturing of 6 plies laminate for the nodes located at the bottom, the midplane and the surface of (a)-Ply 1, (b)-Ply 2, (c)-Ply 3, (d)-Ply 4, (e)-Ply 5 & (f)-Ply 6 using the PGD solved model

Figure 4 .

 4 12-(a).The different nodes of the ply are heated above the PEEK's documented fusion temperature 𝑇 𝑓𝑢𝑠𝑖𝑜𝑛 = 343°𝐶. A temperature gradient is observed during the heating phase with a difference of 25°C between Node 1 and 11 c.f.

Figure 4 .

 4 12-(a).This difference during heating may affect the fusion behavior of the composite and induce a crystallinity gradient through the ply thickness. As the material flows away from the laser, it starts to cool down and the nodes temperatures gradually decrease. The trends observed for the different nodes are similar to previously reported data on the tape placement of PEEK/AS4 composites[15] [16].

Figure 4 .

 4 Figure 4.12 -(a) Evolution of the nodes temperature with time during the placement of Ply 1 and (c) -Zoom on the temperature gradient during the 1 st peak

Figure 4 .

 4 13-(a) presents a schematic description. The top surface of Ply 1 and the bottom surface of Ply 2 are directly exposed to the laser beam. The nodes temperatures of both plies exceed the PEEK's melting temperature c.f.Figure 4.13-(b). The maximum temperatures reached by the tape and the substrate at the interfaces i.e. Node 1 for Ply 2 and Node 11 for Ply 1 are 424 °C and 410°C respectively. The values obtained are close to those of the thermal camera data presented in Figure4.8 with a small overestimation in the case of the incoming tape. The substrate maximum temperature is reached after the contact between the two surfaces as it can be seen in Figure4.13-(b) which is evident since the incoming tape is at a higher temperature therefore increasing that of the substrate through conduction when they're in contact.

Figure 4 .

 4 Figure 4.13 -(a): Schematic presentation of the nodes through the thickness of Ply 1 and 2 when Ply 2 is being placed on top of Ply 1 and (b): Zoom on the temperature evolution for the nodes of Ply 1 (continuous lines) and Ply 2 (dashed lines) during the 2 nd peak i.e. when the 2 nd ply is being placed on top of Ply 1

Figure 4 .

 4 15 for example during the placement of Ply 2 on top of Ply 1.

Figure 4 . 15 -

 415 Figure 4.15 -Evolution of the cooling rate with time during the deposition of Ply 2 on top of Ply 1 through the thickness of the considered plies with a zoom on the zone with the highest cooling rates The same trends described for Ply 1 are repeated for the subsequent layers deposition with slightly different values. For instance, the temperature profiles for Node 1 and Node 11 corresponding to the substrate layers (Ply 1 and 2) during the placement of Ply 3 are presented in Figure 4.16.

Figure 4 . 16 -

 416 Figure 4.16 -Evolution of the temperatures (on the left) and cooling rates (on the right) with time for Node 1 and Node 11 corresponding to the substrate layers (Ply 1 and Ply 2) during the deposition of Ply 3

Figure 4 . 12 -

 412 Figure 4.16. The magnitude of the surface heat flux 𝑞 0 ′′ required to achieve a surface temperature of 410°C after 0.31 s for the different ply number is presented in Figure 4.17.

Figure 4 . 17 -

 417 Figure 4.17 -Heat flux estimated to reach a 𝑇 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 = 410°𝐶 for different thicknesses according to the analytical modelJust as in the previous model, the total length considered is 1m and the total time is equal to 1/0.15 = 6.67 𝑠. The material therefore cools down from 𝑡 𝑝 until 6.67 s. The thermal history predicted using the adapted analytical model presented in Section 4.2.2.2 is presented in Figure4.18 for the different plies. Just as in the case of the previous model, three nodes are selected for the presentation for the sake of clarity.

Figure 4 . 18 -

 418 Figure 4.18 -Evolution of the temperature with time during the manufacturing of 6 plies laminate for the nodes located at the bottom, the midplane and the surface of (a)-Ply 1, (b)-Ply 2, (c)-Ply 3, (d)-Ply 3, (e)-Ply 5 & (f)-Ply 6 using the analytical model

Figure 4 . 19 -

 419 Figure 4.19 -Evolution of the temperature during heating to 120°C and subsequent cooling in the climate chamber where the curvatures measurements occurred

Figure 4 . 15 &

 415 Figure 4.

  𝑋 𝑡 (𝑡) = 𝑋 𝑡 (𝑡 𝑖𝑛 ) (1 -0.5 ∫ 𝐾 𝐴 𝑑𝑡

Figure 4 . 20 .Figure 4 . 20 -

 420420 Figure 4.20 -Application conditions for the crystallization and melting models during the process

Figure 4 .

 4 21-(b).The increase in the crystallinity Ply 1 is negligible and this directly results from the high cooling rates presented in the previous section.

Figure 4 . 22 -

 422 Figure 4.22 -Evolution of the temperature and crystallinity fraction for the different nodes of Ply 1 during the heating (a) and cooling (b) phases when Ply 2 is being placed. Continuous lines (-) present the temperature while dashed lines (-o) present the crystallinity On the other hand, during the deposition of Ply 2 on top of Ply 1, the different nodes of Ply 1 are reheated above PEEK's 𝑇 𝑚 but at lower temperatures compared to the ones presented in Figure 4.21. The melting model predicts a re-melting of the different nodes. It can be seen in Figure 4.22-(a) on the left that the crystallinities of nodes 8 to 11 (11 being the closest to the laser) decrease more importantly than nodes 1 to 7. For example, the crystallinity of Node 11 decreases from 27 % at the end of the placement of Ply 1 to 24% during the reheating of Ply 1 when Ply 2 is deposited. The crystallinity fractions of different nodes reincrease during their cooling (Figure 4.22-(b)) however this increase remains negligible due to the high cooling rates (a maximum increase of 0.05% at Node 11).

Figure 4 . 23 .

 423 The results show a similar behavior for all the plies. The differences in terms of the crystallinity values are due to The different number of reheating and the re-cooling to which each ply is subjected  The maximum temperature differences between the plies presented in Table4.4 .

Figure 4 . 23 -

 423 Figure 4.23 -Predicted crystallinity fractions through the thickness of the six plies at the end of the process

Figure 4 . 24 -

 424 Figure 4.24 -Schematic presentation of a laminate made of n laminae

Figure 4 .

 4 25 along with the glass transition and crystallization zones.

Figure 4 . 25 -

 425 Figure 4.25 -PvT measurements on PEEK 150G presented in [44] for three different pressures (30 MPa in black, 60 MPa in red and 90 MPa in green)

Figure 4 . 26 .

 426 The volumetric shrinkage coefficient 𝛃Vol of PEEK is equal to 4.46% according to Figure4.26, with PEEK being an isotropic material, its linear crystallization coefficient 𝛃m is equal to 1/3 of 𝛃Vol.

Figure 4 . 26 -

 426 Figure 4.26 -Variation of the total𝜀 𝑡𝑜𝑡𝑎𝑙 , thermal 𝜀 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 and crystallization 𝜀 𝑐𝑟𝑦𝑠𝑡𝑎𝑙𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 volumetric shrinkages for a 150G PEEK submitted to 30 MPa

Figure 4 . 27 .

 427 The authors estimated a transition temperature in the range of 160°C.

Figure 4 . 27 -

 427 Figure 4.27 -Determination of the SFT by comparing of the elapsed and relaxation time in the surface and the center of a 40 ply UD APC-2 laminate processed via compression molding [25]

Figure 4 .

 4 12 until t=4.1s. The stresses remain consequently null after the heating until the SFT of the material is reached. The 11 elements of Ply 1 are cooled below the glass transition temperature simultaneously at 4.1 s after the beginning of the tape placement, therefore they start to carry residual stresses simultaneously. Between 4.1 s and 6.63 s, 184 time steps exist. For the sake of clarity, the evolution of the in-plane residual stresses 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 are presented in Figure4.28-(a) and (b) respectively for only the lower element 1 (Bottom 1) and the upper element 11 (Upper 1) of Ply 1 from t=4.3s to t=6.63s. The evolution of the thermal strain increments along with the temperatures for the same elements during the same time scale is presented in Figure 4.28-(c) and (d). The residual stresses for the remaining elements located between those of the presented elements based on their location .

Figure 4 . 28 -

 428 Figure 4.28 -Evolution of the in-plane stresses 𝜎 𝑥𝑥 (a), 𝜎 𝑦𝑦 (b) and the strain increments 𝑑𝜀 𝑥𝑥 (𝑐) 𝑎𝑛𝑑 𝑑𝜀 𝑦𝑦 (𝑑) with the temperature for the lower surface of Ply 1 (Bottom1) i.e. the one in contact with the mandrel and its upper one (Upper 1) with time during the cooling from 𝑇 𝑔 to ambient temperature after the deposition of Ply 1

Figure 4 .

 4 29 green rectangle).

Figure 4 . 29 -

 429 Figure 4.29 -Evolution with time of the transverse modulus 𝐸 22 for the unidirectional Ply 1 through its thickness

Figure 4 .

 4 30 presents the stress profile along x-axis and y-axis at the end of the placement of Ply 1. The model predicts a maximum tensile stress along x-axis at element 1 (the lower surface) equal to 𝜎 𝑥𝑥,𝑚𝑎𝑥 = 0.3 MPa.

Figure 4 . 30 - 1 

 4301 Figure 4.30 -In plane residual stresses components 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 through the laminate thickness after the placement of Ply 1  Placement of Ply 2 The deposition of Ply 2 on top of Ply 1 leads, according to the thermal model, to a complete melting of Ply 1 and Ply 2 c.f.Figure 4.13. Therefore, the stress state evoked by Ply 1 during its placement and presented in Figure 4.28 and Figure 4.30 is erased when Ply 2 and Ply 1 are heated. The thermal model results presented in Figure 4.13 showed that the substrate made of Ply 1 reaches 𝑇 𝑔 before the incoming Ply 2. As a consequence, Ply 1 starts to accumulate residual stresses slightly before Ply 2. The initial stresses states 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 through the laminate thickness at time t=10.84s i.e. once both plies' temperature becomes lower

Figure 4 . 13 .

 413 Therefore, the stress state evoked by Ply 1 during its placement and presented in Figure 4.28 and Figure 4.30 is erased when Ply 2 and Ply 1 are heated. The thermal model results presented in Figure 4.13 showed that the substrate made of Ply 1 reaches 𝑇 𝑔 before the incoming Ply 2. As a consequence, Ply 1 starts to accumulate residual stresses slightly before Ply 2. The initial stresses states 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 through the laminate thickness at time t=10.84s i.e. once both plies' temperature becomes lower than 𝑇 𝑔 is presented in Figure 4.31-(a) & respectively.

Figure 4 . 31 -

 431 Figure 4.31 -In plane residual stresses components 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 through the laminate thickness at time t=10.84 s after the placement of Ply 2 on top of Ply 1

Figure 4 .

 4 32-(a)shows the transverse modulus trough the thickness of Ply 1 and Ply 2 at t=10.84s. The modulus gradient translates the crystallinity gradient with an amorphous layer at the bottom of Ply 2 i.e. between Ply 1 and 2.

Figure 4 .

 4 32-(b)shows the strain increment at t=10.8s

Figure 4 .

 4 32 explain the stress gradient through the thickness in Figure 4.31. The strain increment along the y-direction is similar to that observed in Figure 4.28-(d) with both plies tending to elongate.

Figure 4 .

 4 Figure 4.32 -(a) The transverse modulus through the thickness of Ply 1 and 2 at time t=10.84s when the two plies' temperatures are below the glass transition temperature and (b) -the strain increment Δεₓₓ t time t=10.8s (black curve) and t=10.84s (blue curve) through the thickness of Ply 1 and 2

Figure 4 .

 4 28-(a) & (b) respectively for only the lower elements (Bottom 1 and Bottom 2) and the upper elements (Upper 1 and Upper 2) of Ply 1 and Ply 2 during their cooling after the placement of Ply 2.

Figure 4 . 33 -

 433 Figure 4.33 -Evolution of the in-plane residuals stresses 𝜎 𝑥𝑥 (a) and 𝜎 𝑦𝑦 (b) for the lower surface of Ply 1 and Ply 2 (Bottom 1) i.e. the one in contact with the mandrel and their upper ones (Upper 1 and Upper 2) with time during their cooling after the deposition of Ply 2 on top of Ply 1 The thermal model showed that the temperature through the thickness of Ply 1 increases above the glass transition with the deposition of Ply 3, 4 and 5. In fact, Figure 4.12-(a) shows that the lower element of Ply 1 which corresponds to the lowest temperature across the substrate temperature is above T g during the deposition of Ply 5. This consequently leads to an erase of the residual stresses and strains which accumulated through the thickness of Ply 1 during its deposition until the deposition of Ply 4, just as presented in Figure 4.33 where it was considered that no residual stresses are present above the glass transition. On the other hand, Figure 4.11-(a) shows that the upper surface of Ply 1 is lower than T g during the placement of Ply 6. Therefore, the final residual stresses and strains through the thickness of Ply 1 depends on the ones accounted for during the placement of Ply 5 and Ply 6.

Figure 4 . 34 -

 434 Figure 4.34 -Evolution of the in-plane residuals stresses 𝜎 𝑥𝑥 (a) and 𝜎 𝑦𝑦 (b) for the lower surfaces of Ply 1, 2, 3, 4 and 5(Bottom) and their upper ones (Upper) with time during their cooling after the deposition of Ply 5

Figure 4 .

 4 34-(a) and (b). The final stress state through the 5 plies laminate thickness at the end of the cooling before Ply 6 starts to be placed is presented in Figure 4.35.

Figure 4 . 35 - 5 

 4355 Figure 4.35 -In plane residual stresses components 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 through the laminate thickness at time t=33.3 s after the placement of Ply 5  Placement of Ply 6

Figure 4 . 37 -

 437 Figure 4.37 -Comparison between the experimentally measured and numerically predicted curvatures of the [90 3 /0 3 ] strip specimen manufactured using the cold mandrel during heating and cooling from 120°C

Figure 4 . 41 -

 441 Figure 4.41 -In plane residual stresses components 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 through the [90 3 /0 3 ] laminate thickness simulated using two elements for each ply and obtained at T=30°C before and after demolding and after cooling from 120°C

Figure 4 .

 4 42-(a) & (b) presents the change in the strains increment 𝑑𝜀 𝑥𝑥 and 𝑑𝜀 𝑦𝑦 for the upper and lower elements of Ply 1, 3, 4 and 6 during their cooling after Ply 6 was placed.

Figure 4 . 42 -

 442 Figure 4.42 -Evolution with time of the incremental in-plane strains 𝑑𝜀 𝑥𝑥 (a) and 𝑑𝜀 𝑦𝑦 (b) for the lower and upper surfaces of Ply 1, 3,4 and 6 during their cooling after the deposition of Ply 6

Figure 4 . 43 -

 443 Figure 4.43 -Evolution with time of the in-plane stresses 𝜎 𝑥𝑥 (a) and 𝜎 𝑦𝑦 (b) for the lower and upper surfaces of Ply 1, 3,4 and 6 during their cooling after the deposition of Ply 6

Figure 4 . 44 .

 444 Verifying this stress component is crucial because it lies in the direction controlled by the matrix and can easily become critical.

Figure 4 . 44 -

 444 Figure 4.44 -In plane local transverse residual stresses components 𝜎 22 through the [90 3 /0 3 ] laminate thickness simulated using two elements for each ply and obtained at T=30°C before and after demolding and after cooling from 120°C

Figure 4 . 45 -

 445 Figure 4.45 -Evolution with time of the crystallinity fraction for the lower and upper surfaces of Ply 1, 3,4 and 6 during their heating and cooling in the oven up to 150°C after the process The resulting curvatures predicted during heating and cooling from 150°C are presented in Figure 4.46.During heating, an offset is recorded with a maximum difference of 0.6 𝑚 -1 at ambient temperature. It was already stated in Section 3.5.1 that a difference between the curvatures during heating to 120°C and 150°C exists. In contrast to the results in the case of the maximum temperature of 120°C where no offset was predicted between the cooling and heating evolutions, the model predicts in the case of the maximum temperature of 150°C, an offset between the cooling and the heating. This offset is due to the increased crystallinity predicted during heating above the glass transition temperature and presented in Figure4.45. This crystallinity is accompanied by a crystallization shrinkage which was calculated in the case of PEEK resin and presented in Figure4.26 in addition to the modification of the composite properties if the crystallization increases. The curvature predicted during cooling seems to be able to predict the measured one. However, for the different temperatures presented, the predicted curvatures are slightly lower than those measured. The maximum difference occurs at 0.175 𝑚 -1 at 100°C. The differences during cooling between the numerically and experimentally measured curvatures may be attributed to relaxation of stresses which wasn't taken into account in the model.

Figure 4 . 46 -

 446 Figure 4.46 -Comparison between the experimentally measured and numerically predicted curvatures of the [90 3 /0 3 ] strip specimen manufactured using the cold mandrel during heating and cooling from 150°C

Figure 4 .30 and Figure 4 . 31 .

 4431 With the absence of theoretical differences in the ply orientations, the major source behind these stresses is the thermal history. The results presented in Section 3.5.2 for the UD laminate in case of the cold mandrel proved the presence of a curvature at ambient temperature. The model presented in this chapter was used to simulate the curvature obtained for the UD laminate. The same thermal history and assumptions as the previously discussed case of the cross-ply laminate are used. The simulation takes place using the 2-elements/ply which proved its capability of predicting the curvatures in case of the crossply laminate. Based on this, the predicted curvature for the 6-plies UD laminate[90 6 ] are presented in Figure4.47 and compared to the experimentally measured curvatures during heating to 120°C and the subsequent cooling.

Figure 4 . 47 -

 447 Figure 4.47 -Comparison between the experimentally measured and numerically predicted curvatures of the UD [90 6 ] strip specimen manufactured using the cold mandrel during heating and cooling from 120°C

Figure 4

 4

Figure 4 . 48 -

 448 Figure 4.48 -In plane residual stresses components 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 through the [90 3 /0 3 ] laminate thickness simulated using two elements for each ply and obtained at T=30°C before and after demolding and after cooling from 120°C

4. 3 . 3 . 3 residual stresses 4 . 3 . 3 . 3 . 1

 33343331 Parametric study: Effect of varying parameters on the curvatures and the The effect of the Stress-Free-Temperature on the curvatures and the residual stresses: Application to the 6-plies cross-ply laminate manufactured with the cold mandrel

Figure 4 . 49 -

 449 Figure 4.49 -Comparison between the numerically predicted curvatures of the [90 3 /0 3 ]strip specimen manufactured using the cold mandrel during heating and cooling from 120°C for the different SFT selected In this paragraph, the effect of the choice of the SFT is presented where different SFTs were chosen using the same method as Paragraph 4.3.3.1.1. The values chosen to study the effect of the SFTs are presented in Table 4.7. The corresponding curvatures at the end of the process are presented in Figure 4.49.

Figure 4 .

 4 39 between 0s and 6.6s. The predicted curvatures during heating and cooling from 120°C are presented in Figure4.50 compared to those predicted in the previous section i.e. with the successive deposition of the 6 plies.

Figure 4 . 50 -

 450 Figure 4.50 -Comparison between numerically predicted curvatures of the [90 3 /0 3 ] strip specimen manufactured using the cold mandrel during heating and cooling from 120°C for the case of simultaneous deposition of all the plie and the successive deposition

Figure 4 . 51 -

 451 Figure 4.51 -In plane residual stresses components 𝜎 𝑥𝑥 (a) and 𝜎 𝑦𝑦 (b) through the [90 3 /0 3 ] laminate thickness simulated with the successive and simulated depositions and obtained at T=30°C before demolding

Figure 4 . 52 -

 452 (a). The plies through the laminate thickness reach their SFT successively which simulates a successive deposition of the plies, which is not the case during a simultaneous deposition in Figure 4.52-(b) where all the plies reach the SFT simultaneously.

Figure 4 . 52 -

 452 Figure 4.52 -Evolution of the temperature with time in the upper elements of Ply 1 to Ply 6 (a)-during the cooling phase after Ply 6 has been placed for a successive deposition simulation and (b)-during the cooling phase in case of a simultaneous deposition

Figure 4 .

 4 53-(a)which leads to thermal stress increment gradient through the laminate thickness. This gradient is irrelevant in case of the simultaneous deposition where the same temperature is applied to the different plies leading to similar ΔT through the laminate thickness c.f.

Figure 4 .

 4 53-(b).

Figure 4 . 53 -

 453 Figure 4.53 -Evolution of the temperature difference ΔT with time in the upper elements of Ply 1 to Ply 6 (a)during the cooling phase after Ply 6 has been placed for a successive deposition simulation and (b)-during the cooling phase in case of a simultaneous deposition

  184

Figure 6 . 2 -

 62 Figure 6.2 -Crystallinity fraction of the nodes plies at the end of the process when using (a) -𝑁 𝐿1 , (b) -𝑁 𝐿2 , (c) -𝑁 𝐿3 and (d) -𝑁 𝐿4 time steps

  Les trois principales sources d'énergie de nos jours sont les combustibles fossiles, le gaz naturel et le pétrole. Ces sources ne sont pas renouvelables, elles s'épuisent rapidement et leurs prix augmentent et leurs prix augmentent considérablement Cela est dû à l'accélération des demandes économiques et aux effets géopolitiques tels que les guerres. Du point de vue industriel, la consommation de ces carburants influence directement les coûts de production. L'effet des carburants ne se limite pas aux prix des produits, en effet leur consommation aggrave les effets néfastes sur l'environnement. L'utilisation excessive de combustibles fossiles est associée au réchauffement climatique et à la pollution. Ainsi, la Commission européenne a présenté une série de propositions visant à aligner les politiques climatiques et énergétiques de l'Union européenne sur l'objectif de réduction des émissions nettes de gaz à effet de serre d'au moins 55% d'ici 2030 par rapport à celles de 1990[1]. À cette fin, les recherches ont été intensifiées pour trouver des solutions d'énergie verte et propre. L'hydrogène est l'un des candidats possibles comme carburant alternatif propre, avec plusieurs avantages comme[2]: C'est un des éléments les plus abondants sur la terre Sa combustion ne produit que de l'eau et de la chaleur donc les émissions des gaz à effet de serre sont réduites voir éliminées Il peut être produit par électrolyse donc l'impact environnemental lié à sa production est réduit L'une des applications potentielles de l'hydrogène peut être comme carburant des véhicules. En effet, Toyota a présenté en 2014 le premier véhicule utilitaire fonctionnant à l'hydrogène fin 2014 c. f. Figure A.1 pour lequel le nombre total de piles à combustible utilisées pèse 56 Kg.

Figure A. 1 -

 1 Figure A.1 -2015 Toyota Mirai[2]

Figure A. 2 -

 2 Figure A.2 -Variation du module de stockage avec la température pour différents polymères[3]

Figure A. 3 -

 3 Figure A.3 -Résistance à la traction et prix correspondants pour les polymères les plus utilisés[5] Les réservoirs composites thermoplastiques pour le stockage de l'hydrogène sont fabriqués par enroulement filamentaire avec consolidation in situ qui ressemble au processus de placement de bande automatisé. Ces procédés comprennent plusieurs étapes de chauffage et de refroidissement à des températures élevées afin d'atteindre l'épaisseur souhaitée de la structure. Ils sont généralement associés à des taux de refroidissement élevés en raison des températures et des vitesses de traitement élevées. De plus, des gradients thermiques et de cristallinité sont présents au cours du procédé. D'autres inhomogénéités sont associées à ces procédés telles que la séquence d'empilement du stratifié, l'échauffement local et les différences géométriques entre les surfaces supérieure et inférieure dues à la présence des mandrins et des rouleaux de compactage. Tous ces facteurs présentent des sources conduisant à la formation de contraintes résiduelles dans la structure composite. Ces contraintes peuvent induire des défauts géométriques par des distorsions provoquées sous le moment fléchissant résultant du gradient de contraintes résiduelles. De plus, ils peuvent affecter les performances de la structure principalement les propriétés régies par le

  Des mesures DSC ont été effectuées sur des échantillons produits avec les deux mandrins. La fraction de cristallinité résultant du procédé avec le mandrin froid est estimée entre 20 et 22 %. L'utilisation d'un mandrin chauffé conduit à une fraction de cristallinité comprise entre 28 et 30 %. Ceci peut être directement lié au refroidissement plus homogène dans le cas du mandrin chauffé et au fait que le matériau est maintenu au-dessus de sa température de transition vitreuse. Ces différences entre les fractions de cristallinité se déduisent des thermogrammes de DSC présentés dans la Figure A.4. Le premier chauffage indique le comportement du matériau après le procédé alors que le second chauffage indique son comportement après le recuit à 380°C.

Figure A. 4 -

 4 Figure A.4 -Thermogrammes DSC obtenus pour le cas du mandrin (a) froid et (b) chauffé à 180°C La rampe de chauffe lors des balayages DSC met en évidence la présence d'un pic exothermique caractéristique de la cristallisation à froid dans le cas du mandrin froid (la courbe noire de la Figure A.4-(a)) qui ne figure pas dans le cas du mandrin chaud (Figure A.4-(b)). L'enthalpie de fusion est cependant supérieure à l'enthalpie de cristallisation à froid. On peut conclure que l'utilisation d'un mandrin froid conduit à un polymère partiellement cristallisé qui peut présenter une cristallinité partielle homogène ou des régions complètement amorphes et des régions complètement cristallines. La cristallinité de ce polymère peut être augmentée lors de traitements thermiques tels que le recuit. Ceci peut être détecté en comparant les cristallinités obtenues durant le premier chauffage (après le procédé) et le deuxième chauffage (après le recuit) pour le mandrin froid dans la Figure A.5-(a). Dans le cas du mandrin chaud, la cristallinité obtenue pour les différents empilements est supérieure à celle dans le cas du mandrin froid en comparant Figure A.5-(a) et (b). La comparaison des histogrammes durant le premier et le second chauffage pour le cas du mandrin chaud dans la Figure A.5-(b) montre une légère augmentation de la cristallinité des échantillons après leur recuit à 380°C ce qui peut suggérer soit la présence de zones qui n'ont pas totalement cristallisés durant le procédé et dont la cristallisation à froid causait une enthalpie assez faible qui n'a pas pu être détectée par le DSC.

Figure A. 5 -

 5 Figure A.5 -Fraction de cristallinité calculée à partir des tests DSC lors du premier et du second chauffage pour les échantillons obtenues avec un mandrin (a)froid et (b)chauffé à 180°C En plus que la fraction de cristallinité, des températures caractéristiques peuvent être obtenus à partir des thermogrammes DSC. Parmi ces températures, la température de transition vitreuse '𝑇 𝑔 ' est considérée comme un paramètre de grand intérêt lors de l'étude des matériaux amorphes et semi-cristallins. En effet, cette température est associée à un changement du comportement du matériau de vitreux à caoutchouteux lors du chauffage et vice versa lors du refroidissement. Une modification des propriétés thermomécaniques est associée à ce changement de comportement. Une comparaison des 𝑇 𝑔 mesurées avant et après le refroidissement depuis l'état fondu est présentée à la Figure A.6. Les éprouvettes obtenues avec un mandrin froid présentent une 𝑇 𝑔 de 140.8°C lors du 1 er chauffage qui augmente à 148°C lors du 2 ème chauffage. La valeur lors de la 2 ème chauffe est proche de celle mesurée pour le cas du mandrin chauffé qui est de 148.19°C et ne varie pas beaucoup lors de la 2 ème chauffe. Les valeurs présentées ici sont conformes à la littérature pour la 𝑇 𝑔 du PEEK/AS4 et du PEEK pur de l'ordre de 140.4-152°C [7],[8],[9],[10].

Figure A. 6 -Figure A. 8 -

 68 Figure A.6 -Les températures de transition vitreuse mesurées à partir des scans DSC pendant le 1 er et le 2 ème chauffage des échantillons obtenus avec les mandrins froid et non chauffé Les mesures DSC ont montré aussi que le comportement en fusion n'est pas affecté par la température du mandrin. Un seul pic de fusion a été observé durant le premier chauffage dans les deux cas (mandrin froid et chauffé). Et pareil pour les deux températures de mandrin, un double

Figure A. 9 -

 9 Figure A.9 -Comparaison de la courbure mesurée pour les différents spécimens après la coupe avant toute charge thermique La comparaison entre les courbures des UD et celles des éprouvettes [90 3 /0 3 ] peut directement donner un aperçu de l'effet de l'histoire thermique sur les contraintes résiduelles et les courbures. Il ressort de la Figure A.10-(a) que l'effet thermique sur la courbure n'est pas négligeable dans le cas du mandrin froid et est capable d'induire des contraintes résiduelles même avec des stratifiés unidirectionnels alors qu'il est négligeable dans le cas du mandrin chauffé cf.Figure A.10-(b).

Figure A. 10 -

 10 (b).

Figure A. 10 -

 10 Figure A.10 -Évolution de la courbure avec la température pour les stratifiés asymétriques [90 3 /0 3 ] vs UD à 6 plis pendant le chauffage et le refroidissement à partir de 120 °C (a) -mandrin froid et (b)-mandrin chaud Pour les différentes éprouvettes testées, le chauffage des éprouvettes jusqu'à 120°C (en dessous de leur transition vitreuse) n'induit aucune modification des courbures au refroidissement. Aucun changement de courbure n'est enregistré pendant l'essai isotherme à 120°C et aucun décalage entre les courbures lors des phases de chauffage et de refroidissement n'a été mesuré. Cette observation est valable quelle que soit la température du mandrin.

Figure A. 11 -Figure A. 12 -

 1112 Figure A.11 -Évolution de la courbure avec la température pour les stratifiés dissymétriques [90 3 /0 3 ] dans le cas du mandrin froid pendant le chauffage et le refroidissement à partir de 150 °CLes courbures lors du refroidissement des différents cycles de recuit de dans le cas du du mandrin froid présentent un décalage par rapport à celles mesurées lors de la phase de chauffage. Ce décalage augmente lorsque la température de recuit passe de 150 à 180°C. Cette augmentation est cependant réduite pour une température de recuit de 200°C contre 180°C. Afin de comprendre les principaux moteurs de cette augmentation, il convient de comparer les courbures dans différents cas :

Figure A. 13 .

 13 De plus, des gradients thermiques se produisent à travers l'épaisseur du stratifié lors de son traitement. En effet, le laser chauffe le pli entrant et la surface du mandrin au-dessus de la température de fusion de la résine PEEK pour assurer leur fusion afin d'assurer leur soudure ensemble. La chaleur laser se propage à travers l'épaisseur du stratifié par conduction et les différents plis du substrat sont chauffés. Leurs températures sont cependant dépendantes de l'épaisseur du substrat. Les mêmes tendances décrites pour Ply 1 se répètent pour le dépôt des couches suivantes avec des valeurs légèrement différentes.

Figure A. 13 -

 13 Figure A.13 -(a) Evolution de la température des noeuds avec le temps lors de la dépose du Ply 1 et (b) -Zoom sur le gradient de température lors du 1 er pic Des gradients de vitesse de refroidissement sont également observés pendant tout le processus et la différence à travers l'épaisseur du substrat devient plus grande à mesure que son épaisseur augmente. Par exemple, on peut voir sur la Figure A.14 à droite qu'une différence maximale autour de -2000°C/s existe entre le haut du substrat (Noeud 1 du Ply 2) et son bas (Noeud 11 du Ply 1). Cette différence se réduit au fur et à mesure de la consolidation mais peut encore atteindre -400°C/s (à la 14ème seconde) entre le haut et le bas.

Figure A. 14 -

 14 Figure A.14 -Évolution des vitesses de refroidissement avec le temps pour le Noeud 1 et le Noeud 11 correspondant aux couches de substrat (Ply 1 et Ply 2) lors du dépôt du Ply 3 L'histoire thermique a des implications majeures sur les propriétés et les cristallinités des matériaux. Par exemple, une vitesse de refroidissement plus lente induit un temps de séjour plus long qui améliore la fusion des cristaux. Entre les températures de cristallisation et de transition vitreuse, une vitesse de refroidissement plus lente conduit directement à des niveaux de cristallinité

Figure A. 15 -

 15 Figure A.15 -Évolution de la température et de la fraction de cristallinité pour les différents noeuds du Ply 1 lors de son dépôt. Les lignes continues (-) présentent la température tandis que les lignes pointillées (-o) présentent la cristallinité En revanche, lors du dépôt du Ply 2 au-dessus du Ply 1, les différents noeuds du Ply 1 sont réchauffés au-dessus de la 𝑇 𝑚 du PEEK mais à des températures inférieures à celles présentées sur la Figure A.15. Le modèle de fusion prédit une refusion des différents noeuds.Le modèle prédit que la cristallinité du noeud 11 diminue de 27 % en fin de pose du Ply 1 à 24 % lors du réchauffage du Ply 1 lorsque le Ply 2 est déposé. Les fractions de cristallinité des différents noeuds ré-

Figure A. 16 -

 16 Figure A.16 -Fractions de cristallinité prédites à travers l'épaisseur des six plis à la fin du procédéLes résultats montrent un comportement similaire pour tous les plis. Les différences en termes de valeurs de cristallinité sont dues principalement aux nombres différents de réchauffage et de refroidissement auquel chaque pli est soumis et aux écarts de température maximum entre les plis.

  Par conséquent, le modèle a utilisé un SFT=143°C pour définir la température à laquelle les déformations et contraintes résiduelles commencent à être bloquées dans le stratifié. Les courbures prédites pendant le chauffage et le refroidissement jusqu'à 120°C ont été comparées à celles mesurées expérimentalement. La comparaison a suggéré une bonne estimation des courbures mesurées dans le cas du chauffage à 120°C en utilisant le modèle proposé et les hypothèses et simplifications 210 envisagées cf. Figure A.17. Au cours du procédé, la courbure est nulle puisque les stratifiés sont fixés au mandrin. Une fois séparé, le stratifié présente, sous l'effet du moment fléchissant, une courbure à 30°C qui a été mesurée égale à 2.58 𝑚 -1 par rapport à une valeur prédite numériquement de 2.72 𝑚 -1 qui est une différence de 8 %. Cette courbure diminue lors du chauffage à 1.05 𝑚 -1 (expérimental) contre 1.04 𝑚 -1 (numérique) à 120°C.

Figure A. 17 -

 17 Figure A.17 -Comparaison entre les courbures mesurées expérimentalement et prédites numériquement de l'échantillon [90 3 /0 3 ] fabriqué à l'aide du mandrin froid pendant le chauffage et le refroidissement à partir de 120 °C Alors que le modèle prédit une décroissance linéaire de la courbure, les courbures mesurées expérimentalement présentent une évolution non linéaire. Cela conduit à des différences entre les deux ensembles de valeurs avec un décalage maximum égal à 0.5 𝑚 -1 se produisant à 60°C. L'objectif principal de la thèse est de présenter un modèle numérique capable de prédire les contraintes résiduelles lors du traitement sur le SPIDE-TP. Cependant, ce modèle se doit d'être le plus simple possible afin de limiter la dépense en temps de calcul. Le modèle présenté précédemment avec ses entrées et ses exigences a été jugé capable de prédire les contraintes résiduelles. Une simulation a nécessité environ 192 heures pour être complétée avec le temps maximum passé sur le modèle mécanique. Le code a été réalisé sous Matlab à l'aide d'un pc DELL Precision 7530 avec un processeur Intel® CoreTM i7-8850H 16 Go DDR4-SDRAM. Afin d'essayer de limiter ce temps de calcul, une étude paramétrique a été menée afin de trouver une forme plus simple du modèle en n'incluant que les principaux paramètres affectant les contraintes résiduelles. La première tentative visait à réduire le nombre d'éléments à travers l'épaisseur de chaque pli alors le nombre de noeuds dans un pli a été réduit de 11 jusqu'à 2. Les résultats obtenus n'étaient pas trop différents de ceux présentés précédemment dans le cas de 11 noeuds.

Figure A. 18 -

 18 Figure A.18 -Composantes de contraintes résiduelles dans le plan 𝜎 𝑥𝑥 (a) et 𝜎 𝑦𝑦 (b) à travers l'épaisseur du stratifié [90 3 /0 3 ] simulées à l'aide de deux éléments pour chaque pli et obtenues à T=30°C avant et après démoulage et après refroidissement à partir de 120°C Un gradient de contrainte résiduelle a été prédit dans le cas du stratifié dissymétrique [90 3 /0 3 ] avec une contrainte de traction maximale de 25,2 MPa à sa surface inférieure. Cette valeur est supérieure au tiers de la résistance à la traction transversale PEEK/AS4 égale à 78 MPa. Elle est donc critique et doit être prise en compte lors de la conception.

Table 1 .

 1

	Temperature (°C)	400	450	500	550	600
	Exposure time	2 hrs 15 min	2 min	15 s	2 s

2 -Suggested maximum exposure times at various processing temperatures for carbon/PEEK composites in non-oxidizing environments

-6 °C-1 )

	Longitudinal	-0.22,-0.66
	Radial	10-30

Table 1 . 3

 13

-Properties of AS4A carbon fiber reported by the manufacturer

[START_REF]Hexcel website[END_REF]

Table 1 . 4

 14

-Tensile and flexural moduli of PEEK, polytetrafluoroethylene (PTFE) and polyimide (PI) at ambient temperature

Table 1 .5.

 1

	Amorphous phase modulus	Temperature condition
	𝐸 𝑎 = 2.7 𝐺𝑃𝑎	T<143°C
	𝐸 𝑎 = 2.7 -0.06(𝑇 -𝑇 𝑔 ) 𝐺𝑃𝑎	143°C <T<182°C
	𝐸 𝑎 = 0.36 -0.002(𝑇 -182) 𝐺𝑃𝑎	182°C <T<340°C

longitudinal thermal conductivity for APC-2 ( W/m°C) 6

	𝑲 𝑻 , longitudinal thermal conductivity for APC-2 ( W/m°C)	0.72
	𝑪 𝑷 , thermal heat capacity (J/Kg°C)	1425
	Density ( kg/m 3 )	1560

2.4.

𝑲 𝑳 ,

Table 1 . 6

 16

-Thermal properties for APC-2 composite

[33]

Table 1 . 7 -

 17 Longitudinal and Transverse CTE for PEEK/AS4 from different studiesThe values of 𝛼 𝐿 and 𝛼 𝑇 from the different studies mentioned in this section are presented in Erreur ! Source du renvoi introuvable.. It can be concluded that 𝛼 𝐿 is low and its dependence on the temperature is negligible. On the other hand, 𝛼 𝑇 presents two values, the 1 st one around 30• 10 -6 °C-1 below 𝑇 𝑔 and the 2 nd one around 75• 10 -6 °C-1 above 𝑇 𝑔 .

	< 𝑇 𝑔	0.5	0.6	0.5	-.0.3
	-6 °C-1 )
	> 𝑇 𝑔	1	0.6	0.5	0.6
	< 𝑇 𝑔	30	29.1	30	34.3
	𝜶 𝑻 (• 10 -6 °C-1 )
	> 𝑇 𝑔	75	80	80	66

Table 1 .8 .

 1

	Temperature (K)	293	323	373	423	473
	𝝂 𝑳𝑻	0.35	0.35	0.34	0.37	0.36

Table 1 . 8

 18

-In-plane Poisson's coefficient for APC-2 variation with temperature

[START_REF] Uematsu | Delamination behavior of a carbon-fiber reinforced thermoplastic polymer at high temperatures[END_REF]

Table 2 . 3

 23

		[𝟗𝟎 𝟑 ]	[𝟗𝟎 𝟔 ]	[𝟗𝟎 𝟗 ]	[𝟗𝟎 𝟑 ]	[𝟗𝟎 𝟔 ]	[𝟗𝟎 𝟗 ]
	Number of specimen	3	3	3	3	3	3
	Length between spans or free-length 𝑳 𝟎 (mm)	20	20	30	20	30	30
	Width 𝒃 (mm)	9.6±0.02	10±0.02	15±0.02	10±0.02	15±0.02	15±0.02
	Thickness 𝒉 (mm)	0.58±0.03	1.05±0.04	1.3±0.05	0.58±0.03	1.05±0.04	1.3±0.05

-Dimensions of the specimens selected for the three-point bending test on the DMA machine

Table 3 .

 3 1 gives the set of measured values for the glass transition 𝑇 𝑔 , the fusion transition 𝑇 𝑓 , the cold crystallization temperature 𝑇 𝑐𝑐 and the melt crystallization temperature 𝑇

	Phase transition temperature	𝑻 𝒎𝒂𝒏𝒅𝒓𝒆𝒍 = 𝑻 𝒂𝒎𝒃𝒊𝒆𝒏𝒕	STD Error	𝑻 𝒎𝒂𝒏𝒅𝒓𝒆𝒍 = 𝟏𝟖𝟎°𝑪 STD Error
		1 st scan	140.8	1.7	148.2	1.1
	𝑇 𝑔 (°C)
		2 nd scan	148.67	1.2	147.8	1
	𝑇 𝑐𝑐 (°C)	1 st scan	172.5	0.5	//	//
		1 st scan	345	1.1	344.8	1.1
	𝑇 𝑚 (°C)
		2 nd scan	343.2	0.2	344	0.2
		1 st scan	304.4	0.2	306.3	0.1
	𝑇 𝑐 (°C)
		2 nd scan	303.7	0.9	305.8	0.2

𝑐 (occurring at high temperature near the fusion). The temperatures in Table

3

.1 correspond to the mean values for similar specimens. The standard error (STD Error) presented in Table

3

.1 and the rest of the paragraph is equal to the standard deviation divided by the square root of the specimens' number.

Table 3 .

 3

1 -The transition temperatures measured with the DSC tests and the maximum standard error

Standard error Temperature (°C) Standard error Temperature (°C)

			𝑻 𝒎𝒂𝒏𝒅𝒓𝒆𝒍 = 𝑻 𝒂𝒎𝒃𝒊𝒆𝒏𝒕	𝑻 𝒎𝒂𝒏𝒅𝒓𝒆𝒍 = 𝟏𝟖𝟎°𝑪
	Laminate sequence
	[90 3 ]	1 st heating	13%	28°C, 171°C	7.5%	132°C
	[90 6 ]	1 st heating	12%	30°C,165°C	8%	173°
	[90 9 ]	1 st heating	15%	31°C	10%	181°C

Table 3 .

 3 2 -Maximum standard error and the corresponding temperatures for DMA tests in the case of unheated and heated mandrels.

Table 4 .

 4

	Property		Value ( Temperature T is evaluated in °C)	Source
	Longitudinal 𝐾 𝐿 and transverse 𝐾 𝑇 thermal	𝐾 𝐿 =46	[7], [4]
	conductivity (W.𝑚 -1 .𝐶	°-1 )	𝐾 𝑇 =0.43	[4]
	Density 𝜌 ( Kg.m³)		𝜌 = -0.26 * 𝑇 + 1619.4	[4],[5]
	Specific heat 𝐶 𝑝 (𝐽. 𝐾𝑔 -1 . °𝐶-1 )	𝐶 𝑝 = 4.49 * 𝑇 + 593	[4]

1 -Thermo-physical properties data of the PEEK/AS4 composite used in the thermal model

  𝑖𝑗 = 𝑄 𝑖𝑗 ̅̅̅ (∆𝜀 𝑘𝑙 -𝛼 𝑘𝑙 ∆𝑇 -𝛽 𝑘𝑙 ∆𝐶) Eq.4.48 ∆σ ij , ∆ε kl , ∆T, ∆C are the stress, strain, temperature and crystallization increments over a time step. The time step is determined from the thermal model in Section 4.2.2.1 and is equal to 0.0133 s. The stress state at the step 'n' is therefore the sum of the stress at step 'n-1' and ∆σ ij using Eq. 4.49. The reduced stiffness matrix 𝑄 𝑖𝑗 ̅̅̅̅ calculation scheme is presented in Appendix C.

	𝜎 𝑖𝑗	𝑛 = 𝜎 𝑖𝑗	𝑛+1 + ∆𝜎 𝑖𝑗	Eq. 4.49

Table 1 -

 1 Summary of experimental devices developed and/or used to study the material behavior and the residual strains

	Technique	Property studied
	DMA	Transverse modulus, Glass transition temperature
	Static 3-Point bending test	Transverse modulus
	DSC	Crystallinity fraction, characteristic temperatures (glass transition, melting, cold crystallization temperature)
	Microscope	Microstructure, presence of defects
	Curvature measurement using a climate chamber and a camera	Evolution of the curvature with temperature

  comprimé. À l'état cryogénique, l'hydrogène liquide doit être maintenu à très basse température car il commence à bouillir à -253°C[2]. Il en résulte la nécessité de le stocker dans des réservoirs bien protégés et isolés pour éviter les fuites d'hydrogène et assurer les basses températures requises.En tant que gaz comprimé, l'hydrogène a une faible densité d'énergie volumétrique par rapport aux combustibles fossiles. Par conséquent, pour une même production d'énergie, la quantité d'hydrogène requise est supérieure à celle des hydrocarbures, ce qui entraîne un besoin de réservoirs plus grands. Avec une taille de réservoir accrue, le coût des matériaux augmente et plus d'hydrogène est nécessaire pour transporter ce réservoir avec la voiture, réduisant ainsi l'efficacité du système.Compte tenu de ce qui précède, l'allègement des moteurs est donc une nécessité pour augmenter le rendement des voitures à hydrogène. Grâce à leur légèreté, leur résistance spécifique élevée et leur résistance à la corrosion, les matériaux composites peuvent être considérés comme une solution à explorer La présence de ces matériaux plus légers n'a cessé d'augmenter au cours des 30 dernières années dans différents secteurs notamment l'aéronautique et le transport.

  Le chapitre bibliographique a montré qu'une importante littérature existe déjà sur les différents sujets concernés tel que les propriétés et le comportement des matériaux, le procédé et les contraintes résiduelles. Les travaux de recherche discutés dans le premier chapitre ont montré que le procédé de dépose de bandes, est généralement associé à des gradients thermiques élevés ainsi qu'à des contraintes géométriques conduisant à la formation de contraintes résiduelles dans la pièce finale. Plusieurs études ont porté sur la caractérisation expérimentale des gradients thermiques souvent en utilisant des thermocouples placés entre les différents empilements. Certains travaux ont tenté de simuler la présence de gradients de cristallinité liés aux gradients thermiques. Cependant, ni la validation expérimentale directe de la présence de ces gradients de cristallinité ni leur lien avec d'autres propriétés mesurables des stratifiés tel que le module transverse n'a été réalisée dans les études mentionnées, ce qui est la première lacune identifiée. La deuxième lacune identifiée concernait la validation expérimentale des modèles numériques simulant la présence des contraintes résiduelles. En effet, deux études complètes pouvaient être identifiées parmi les revues mentionnées, celle de Chapman et celle de Sonmez & Hahn qui ont incorporés les propriétés des matériaux, une histoire thermique et la présence de gradients de cristallinité pour identifier les contraintes résiduelles. Cependant, nulle des deux études n'a pu valider les résultats numériques concernant la dépose de bandes avec des valeurs expérimentales provenant du même procédé et la comparaison a eu lieu via des résultats de moulage à la presse. Afin de pouvoir répondre aux lacunes identifiées dans la littérature, le matériau sélectionné pour cette étude est le PEEK/AS4 fournis sous forme de pré-imprégnés de Tencate® avec un taux volumique de fibres de 59%. Une machine de dépose de bandes appelée SPIDE-TP et basée au Cetim de Nantes a été utilisé pour produire différents stratifiés à partir de ces pré-imprégnés. Les stratifiés produits étaient soit des unidirectionnels ([90 3 ],[90 6 ] et[90 9 ]) soit des empilements croisés non-symétriques ([90 3 /0 3 ] et[(90 3 /0 3 ) 2 ]) ou symétriques ([(90 3 /0 3 ) 2 /90 3 ]). Ces stratifiés ont été produits en deux lots, le premier en utilisant un mandrin froid non chauffé et le deuxième en utilisant un mandrin chauffé à 180°C. Des échantillons ont été découpés pour la caractérisation et l'étude du comportement du matériau et pour la mesure de la variation des courbures avec la température.

Jeronimidis & al.[START_REF] Jeronimidis | Residual Stresses in Carbon Fibre-Thermoplastic Matrix Laminates[END_REF] measured the in-plane Poisson's coefficient for APC-2 using a tensile machine and

6 * ℎEq. 2.2Once the two deflections 𝑠 ′ and 𝑠 ′′ are calculated and their corresponding loads are identified, the flexural modulus 𝐸 𝑓𝑙𝑒𝑥𝑢𝑟𝑎𝑙 can be calculated using Eq. 2.3.

[START_REF] Gao | Cooling rate influences in carbon fibre/PEEK composites. Part 1. Crystallinity and interface adhesion[END_REF]

Jun & Hong[37] studied the curvatures of unsymmetrical laminates with arbitrary lay-up and different number of plies. The results for arbitrary lay-ups agreed with Hyer's results for cross ply laminates on the fact that the classical laminate theory fails to predict the behavior of unsymmetrical laminates starting from a certain critical length/thickness ratio. The modified laminate theory proposed by Hyer, for which the general equations were presented in this paragraph was later used in other works to predict the roomtemperature shapes of unsymmetrical laminates[36].

Comme évoqué précédemment, les travaux de recherche présentés dans ce manuscrit portent sur la compréhension de l'état de contraintes résiduelles et de l'état des matériaux au niveau du pli. Le manuscrit de thèse est divisé en quatre chapitres.Le premier chapitre propose une synthèse de l'état de l'art concernant les composites carbone/PEEK et les procédés d'enroulement filamentaire et de placement de ruban. Il s'intéresse également aux contraintes résiduelles : types, origines, effets ainsi qu'aux méthodes expérimentales et numériques qui ont été présentées dans la littérature pour estimer les contraintes résiduelles lors de tels procédés de fabrication.Le deuxième chapitre présente les techniques de caractérisation mécanique utilisées pour étudier le comportement du matériau après traitement ainsi que la description de la méthode utilisée pour mesurer les déformations résiduelles générées par le procédé.Dans le troisième chapitre, les résultats obtenus à partir des méthodes décrites au deuxième chapitre sont présentés et fournissent la caractérisation expérimentale des propriétés et du comportement du composite en plus des courbures générées dues aux contraintes résiduelles.

List of symbols and abbreviations

APC-2

Overview of the hypotheses and the resolution scheme 4.2.2.1 2-D thermal model solved with PGD

In this part, the model described is based on the work of Barasinski [1] and Perez [2]. The corresponding details can be found in the cited works and a brief description is presented here. The model solve the heat problem in an Eulerian frame where it assumes a fixed roller, laser and deposition head while the material is considered moving at a constant velocity 𝒗 𝒅𝒆𝒑 . Eq. 4.3 is solved in 2-D for a laminate with N plies. The crystallization enthalpy isn't taken into account in the model since it's negligible compared to the laser heat input [9] & [10]. The boundary conditions are presented in Figure 4. 4 according to [1] and [2] as the material flows from the right to the left:

The material enters the domain at a fixed temperature equal to the ambient (25°C) (Dirichlet condition : 𝑇 = 𝑇 𝑎𝑚𝑏𝑖𝑒𝑛𝑡 )

The material exits the domain without exchanging heat with its surrounding ( 𝜕𝑇 𝜕𝑥 = 0) The heat exchange with the air, the roller and the mandrel occurs via convection. The convection coefficients ℎ 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒/𝑎𝑖𝑟 , ℎ 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒/𝑚𝑎𝑛𝑑𝑟𝑒𝑙 and ℎ 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒/𝑟𝑜𝑙𝑙𝑒𝑟 are presented in Table 4.2. 4.2 -Heat exchange coefficients implemented in the PGD solved model [1] 125

The density, specific heat and thermal conductivity implemented to resolve the heat problem correspond to the values presented in Table 4.1 which were identified for PEEK/AS4 composite.

The heat problem was solved using the Proper Generalized Decomposition (PGD). The resulting solution is the temperature distribution along the length (along the deposition direction) and through the thickness of a laminate during the deposition of the final ply. For more details on the PGD and the resolution scheme, the reader may refer to [1], [2] and [11].

The process parameters selected as input to the thermal model are presented in Table 4.3. This study will be limited to the evaluation of the thermal model in the case of a cold mandrel only. The placement speed and the mandrel temperature are consistent with those used to produce the specimen in the two previous chapters. The input power and the laser angle choices were judged the most suitable and closest to the real scenario to simulate the thermal history during the process. In fact, the SPIDE-TP is a temperaturecontrolled process and therefore the access was restricted to the temperature variation. Moreover, the deposition head angle was set as automatic without a precise value as mentioned in Section 2.4. Therefore, several trials were conducted by changing the input power and the angle laser in order to obtain a suitable nip-point temperature. The parameters listed in Table 4.3 are used as direct inputs to study the heat flux in an interface window which gives access directly to the results of the PGD.

Input Power ( W) 1000

Placement Speed ( m/s) 0.15

Laser Angle 30°

Mandrel temperature (°C) 25 The principal drawback of this model is the inability to handle the equations and the code written but rather an interface is used which takes into account the choice for the process parameters and the maximum number of plies that can be chosen is 6. Once the parameters are implemented in the interface, the results are directly generated.

1-D Analytical solution thermal model Theory

In this part, an analytical model based on the work of Weiler [3] is presented. The general formulation is briefly described and for more details, the reader may refer to [3]. Weiler & al. [3] presented an analytical solution for the transient 1-D thermal model of LATP. In order to solve the heating problem, several thermal aspects are studied and well described in [3].

The authors in [3] cite several boundary conditions in addition to thermal aspects regarding the geometry and the materials which are detailed in the article. These simplifications are based on the observation of the process and the material behavior, they include most importantly:

Adapting the analytical equations to suit the processing on SPIDE-TP

.7 -Algorithm of solution to obtain the temperature distribution during processing on the SPIDE-TP using the analytical equations proposed in [12] In this work, the analytical equations are used to predict the temperature distribution for a 6-plies laminate during processing on the SPIDE-TP. They were used in a solution algorithm which takes into account the nip-point temperature choice when the laminates were elaborated and the varying laminate thickness after each deposition of the plies. The general algorithm is presented in The material properties correspond to the mean values of 𝜌, 𝐶 𝑝 , 𝐾 𝐿 and 𝐾 𝑇 described in Table 4.1. 𝑇 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 is based on data from SPIDE-TP showing the variation of the temperature of the tape being placed and the substrate (surface of the plies already placed) as shown in Figure 4.8. The temperature data contains a lot of noise and overshooting so a mean value of 410°C was selected as 𝑇 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 .

[

Eq. 4.28

𝑧 𝑘 and 𝑧 𝑘-1 correspond to the z-ordinate of respectively the upper and lower side of the kth layer with respect to the mid-plane. Eq. 4.25 can be inverted if the objective is to determine the mid-plane strains and the laminate curvatures from {𝑁} and {𝑀} and becomes Eq. 4.29

Using Eq. 4.25-Eq. 4.29, the behavior of the whole laminate is described. In some cases, it seems crucial to determine the stress state in a specific layer. Knowing the forces {𝑁} and moments {𝑀}, the stress state {𝜎} 𝑘 in a location 𝑧 in the kth layer can be obtained using Eq. 4.30

Eq. 4.30

The different equations from Eq. 4.20 to Eq. 4.30 present the classical laminate theory used to describe the simplest problems where only mechanical load act on a given laminate. However, the laminate can be subjected to other loads due to process related aspects including thermal loads, geometrical constraints and hygroscopic effects. Therefore, the laminate theory has been extended to account for the presence of these different parameters stated earlier that could affect the residual stresses and the material behavior. In this matter, Eq. 4.30 becomes Eq. 1.10 to account for the presence of the thermal loads with [𝛼] 𝑘 the thermal expansion coefficients and 𝛥𝑇 is the difference in material temperature

Eq. 4.31 Hyer [34] reported the room-temperature shapes of several unsymmetrical laminates with different stacking sequences and dimensions. In his study, Hyer observed that unsymmetrical thin laminates cure into cylindrical shapes rather than saddle-like shape as predicted by the classical laminate theory. Another observation was made which contradicts the classical laminate theory predictions is the snap-through phenomenon in the case of cylindrical-shaped laminates for which the curvature sign can be changed by a simple snapping. These observations led Hyer to publish two more papers on the subject of cured shape of unsymmetrical laminates [35]& [36]. In these two papers, a theory based on the CLT was presented to describe the behavior of unsymmetrical laminates. The details for this theory can be found in the two papers published by Hyer [35]& [36], however a small briefing of the equations and the assumptions considered will be presented here for the sake of clarity.

The unsymmetrical laminates cylindrical shapes observed in [34] were repeated for different laminates sequences and thicknesses and couldn't be attributed to process related problems or material related defects. The author attributed these deviations from the CLT predictions to geometric nonlinearities as in the case of large deflections of thin flat plates. In order to solve the problem, Hyer proposed a model to reduce the complexities associated with the resolution of non-linear problems. The material properties were considered temperature-independent and no mechanical loads were considered during the resolution.

The strain-displacement relations were defined using the von Kármán equations describing the large deflections of thin plates. Applying the von Kármán displacement field with the Green-Lagrangian strain tensor, the strain-displacement relations were defined in Hyer theory using Eq. 4.32-Eq. 4.34

On the other hand, the PEEK properties were shown to be highly depend on both the crystallinity and the temperature. The major change occurs when the glass transition temperature is reached. These dependencies are simplified in this model. The thermal expansion coefficient 𝜶 𝒎 and the modulus 𝑬 𝒎 are defined in the same manner: they're the result of the combination of both the amorphous and crystalline phases which act in series and therefore can be estimated through a simple mixture law. However, the data on the semicrystalline PEEK is more abundant than that on the fully crystallized PEEK, the law mixture considers the amorphous and semi-crystalline phase rather than the crystalline phase by using the relative crystallinity 𝜒 𝑐𝑟 instead of the absolute crystallinity fraction 𝜒 𝑐 as presented in Eq. 4. 46 𝐸 𝑚𝑐 and 𝐸 𝑚𝑎 are the semi-crystalline and amorphous phases modulus. 𝛼 𝑚𝑐 and 𝛼 𝑚𝑎 are the semi-crystalline and amorphous CTE. linity fraction 𝜒 𝑐𝑟 is calculated during the processing using the crystallization-melting model.

Property Value Source

𝑬 𝒎𝒄 (GPa) Below 𝑇 𝑔 : 6 [25], [42], [43] Above 𝑇 𝑔 : 1.5

Above 𝑇 𝑔 : 0.009

Below 𝑇 𝑔 : 112 [44], [45] Above 𝑇 𝑔 : 38.5

𝜶 𝒎𝒂

Below 𝑇 𝑔 : 220

Above 𝑇 𝑔 : 38.5

Table 4.6 -PEEK resin properties input to the thermo-mechanical model

The dependence of the modulus and the CTE on the temperature is simplified in this model. Two values are considered for each variable below and above the glass transition temperature 𝑇 𝑔 =143°C as presented in Table 4.6. The properties were considered as mean values from those reported in the cited references in the last column of Table 4.6. For the amorphous modulus, a very low value was reported by Chapman [25] and Bas [42] above the glass transition as well as in the DMA tests presented in Section 3.4.2.

Regarding the Poisson's ratio 𝝊 𝒎 , a constant value of 0.38 is considered independent of the temperature and the crystallinity [46], [41] and [47]. The shear modulus 𝑮 𝒎 of PEEK is estimated from the modulus and Poisson's coefficient using Eq. 1.6, therefore it should follow the same dependency on temperature and crystallinity as the modulus since 𝝊 𝒎 is constant.

Finally, the coupled thermal-crystallization-melting models results presented in the previous sections showed that the material is heated above its melting temperature and cooled down very fast. Despite of the deposition of Ply 5 are relaxed under the heating effect. The development of the residual stresses during the cooling of the 6-plies laminate after the deposition of Ply 6 is similar to that previously discussed during the deposition of Ply 5. Therefore, the final residual stresses after placement of Ply 6 are presented in the following before demolding and after demolding i.e. after the out-of-plane displacement is set different than zero. Figure 4.36 presents the in-plane residual stresses before demolding and after demolding. Before demolding along the x-axis corresponding to the transverse one for the 90° oriented plies (with the negative z) and the longitudinal direction for the 0° oriented plies (positive z), the 90° oriented plies are submitted to a tensile stress. The maximum value is located at the bottom of Ply 1 i.e. the one in direct contact with the mandrel and it corresponds to 28.8 MPa. On the other hand, the 0° oriented plies are submitted to a compressive stress with a minimum value of -15.2 MPa located in the lower element of Ply5.

The stresses obtained at the end of the process (before demolding) depend on the process and the gradients present through the thickness of the laminate are directly related to the crystallinity gradients and the thermal gradients discussed earlier.

After demolding: Once the laminate is separated from the mandrel, the uneven distribution of residual stresses through the laminate thickness generates a bending moment and the specimen curvature occurs leading to a redistribution of the stresses through the laminate thickness. The 90° oriented plies remain under tension with a minimum value of 1.37 MPa at the bottom of Ply 2 and a maximum value of 28.6 MPa at the bottom of Ply 1. On the other side, Ply 4 and a part of Ply 5 are under compression with a minimum value of -99 MPa at the bottom of Ply 4. The upper element of Ply 5 and Ply 6 become subjected to a tensile stress with a maximum value of 69.5 MPa at the top of Ply 6.

Curvatures

Figure 4.36 shows a stress gradient along the x-axis and y-axis. This bending moment leads to a curvature once the laminate separates from the mandrel. Figure 4.37 presents the predicted evolution of the cold mandrel manufactured [90 3 /0 3 ] strips' curvature from room temperature i.e. after demolding and its evolution during heating to 120°C and the subsequent cooling along with the experimentally measured ones.

Regardless of the small differences between the predicted and experimentally measured curvatures, the model is considered capable of predicting the specimen curvatures.

The effect of a reduction in the number of nodes through the thickness on the curvatures and the residual stresses

The main objective of the thesis is to present a numerical model able to predict the residual stresses during processing on the SPIDE-TP. However, this model is required to be as simple as possible in order to limit the calculation time expense. The previously presented model along with its inputs and requirements was judged able to predict the residual stresses. One simulation required around 192 hours to be completed with the maximum time spent on the mechanical model. The code was performed in Matlab using a DELL Precision 7530 pc with a processor from Intel® Core TM i7-8850H 16 Go DDR4-SDRAM. In order to try and limit this calculation time, a parametric study was conducted in order to find a simpler form of the model with the main parameters affecting the residual stresses only included. The first attempt aimed to reduce the number of elements through the thickness of each ply. Two elements are then considered by considering the average temperature of the lowest 5 elements and the 6 highest elements as presented in

Continuous lines (-) present the temperature while dashed lines (--) present the crystallinity

The temperature profile has a similar shape to the previously presented results (in the case of 11 elements).

The crystallinity fraction profile, on the other hand, is different. The previously simulated crystallinity gradient through the thickness isn't present and the result presented in Figure 4. 39 show that each ply is divided into two elements: the lower one is amorphous with a crystallinity less than 2% while the second one is semi-crystalline with a crystallinity in the case of Ply 1 equal to 25%.

The simulation in this case using the same method and assumptions as in the case of the previously presented [90 3. /0 3 ] laminate led to the curvatures presented in the case of a heating up to 120°C. The results presented in The curvatures predicted using only two elements for each ply, one of them being amorphous and the other one semi-crystalline are very similar to those previously obtained with a curvature at ambient temperature equal to 2.63 𝑚 -1 which decreases linearly with temperature until reaching 0.99 𝑚 -1 at 120°C. The slope of the line joining the different curvatures is equal to -0.017 𝑚 -1 /°𝐶. The curvatures predicted using the 11 elements are however higher but not very different than the ones predicted using two elements. The maximum and minimum differences between the two sets of curvatures are equal to 0.1 𝑚 -1 and 0.05 𝑚 -1 and occur at 30°C and 120°C respectively. With the time saving occurring ( from 192 hours to 4 hours) using the two-elements per ply scheme and due to the low differences between the 2-elements/ply and the 11-elements/ply schemes, the 2-elements per ply is the one adopted for the rest of this study.

For an SFT equal to the glass transition temperature (143°C), the in-plane residual stresses components 𝜎 𝑥𝑥 and 𝜎 𝑦𝑦 obtained at the end of the process i.e. before demolding, after demolding and after cooling from 120°C are presented in The number of time steps 𝑁 𝐿 originating from the PGD based model is equal to 9006. However, this number induces a long time of calculation when the whole model is applied. Therefore, a convergence study was conducted in order to select a fewer numbers of time steps. This convergence analysis was conducted using

Appendix B

With the increasing use of fiber reinforced polymers, the studies focusing on their properties are numerous.

In fact, the properties of a composite are not only used to simulate or understand their behavior or their response under loading conditions, but these properties of composites can be manipulated through the two important parts of the composite: fiber and matrix. As a result, the composite's properties characterization depend highly on:

Fiber and matrix fraction Fiber spatial layout in the matrix While the fiber and matrix fraction is easy to implement, the spatial layout of fibers in the matrix can be very complex and affects the properties of the composite. Due to the difficulty of implementing a very complex geometry, characterization methods used to estimate the effective properties of the composite consider a simplified geometry.

Homogenization technique is the one in which the heterogeneous material can be presented by an effective equivalent homogeneous body. When an elastic response is considered, the goal of homogenization is to describe the effective elastic properties of the unidirectional lamina function of its constituents.

A unidirectional fiber composite lamina with circular cross-section for fibers, such as PEEK reinforced with carbon fibers, is considered as transversely isotropic material. The isotropic plane is the one transverse to the fiber direction. For a transversely isotropic material, six effective elastic properties need to be studied in this work: longitudinal modulus 𝐸 𝐿 (parallel to the fiber direction), transverse modulus 𝐸 𝑇 (transverse to the fiber direction), in-plane Poisson's ratio 𝜈 𝐿𝑇 , in-plane shear modulus 𝐺 𝐿𝑇 , in-plan longitudinal and transverse thermal expansion coefficients 𝛼 𝐿 and 𝛼 𝑇 respectively.

The determination of each of these effective properties is based on assumptions, even though they were extensively described and studied previously, a brief description will be recalled here for the estimation of each property. Due to the fact that PEEK is an isotropic matrix while carbon fibers are considered transversely isotropic, in all cases below the properties used will correspond to an isotropic matrix and a transversely isotropic fiber. The equations considered are extracted from the work in [53].

Appendix C

The reduced stiffness matrix components for the laminate theory are calculated using the following scheme. The reduced stiffness matrix 𝑄 𝑖𝑗 for each material in the laminate is used. In this work, a single material is used corresponding to the PEEK/AS4 with a fiber fraction equal to 0.59 using Eq. 1.10 composite materials, residual stresses, tape placement, experimental characterizations, numerical simulations, multi-scale analyzes Abstract: The manufacturing processes of composite materials structures such as tape placement and filament winding are inevitably accompanied by the formation of residual stresses and strains. The process conditions and the presence of these stresses directly influence the behavior of the final structure. Two experimental campaigns were carried out. The first aimed to understand the behavior of the material after the process and the influence of the mandrel temperature on this behavior. For this purpose, various mechanical and physical characterization tests were carried out on laminates processes with a cold and a heated mandrels. The second experimental campaign focused on measuring the curvatures of different samples. Then, a methodology was presented In order tosimulate the formation of residual stresses and strains during the process. First, the thermal history was obtained in addition to the variation of crystallinity during the process. The results of the thermal and crystallization models served as input data to simulate the residual stresses and strains during the process. The results of this model showed the occurrence of a bending moment as a result of stress gradients leading to curvature during demoulding. This model was validated by comparing the curvatures obtained numerically with the experimental ones. Once the model was validated, a sensitivity study was carried out to determine the first-order factors influencing the formation residual stresses.