Docteur De L'ecole

Nationale Superieure

D E Mecanique

' Aerotechnique

Marina Pecora

Marco Gigliotti

Olga Smerdova

La Commission D'examen

Professeur, ENSAM Colin Xavier

Professeur Gauthier Christian

Amélie Benjamin

Ravi Raj

Purohit Purushottam

Raj Purohit

obtention du Grade de

come

Development of a Cyclic Indentation Method for the Characterisation of Material Gradients in Polymers and

Polymer Composites Due to Thermal Aging

Introduction

The interest in the research subject of this thesis is strictly related to the growing use of polymer-matrix composites in the aircrafts, in the aim to built more lightweight structures. In fact, the average weight loss obtained by using polymer-matrix composites in aircraft components is of 20%, and can reach up to 50%. The market of polymer-matrix composite materials for the realisation of "cold" (-55 • C < T < RT ) structural parts is going towards saturation and aircraft manufacturers foresee the employment of 3D interlock carbon-fiber reinforced epoxy resin-matrix composites in "warm" aircraft structural parts (nacelles, turbo-engines). In these "warm" zones, the temperature range is between 150 and 300 °C that, together with the presence of oxygen, create environmental conditions conducive to degradation phenomena of the polymer matrix over long time of exposure, while carbon fibers do not show any degradation in this temperature range.

For temperatures lower than glass transition temperature, the main degradation phenomenon in the epoxy resins is thermal-oxidation. Depending on the aging conditions and polymer material, the oxidized layers can reach a thickness of several hundreds of micrometers. Because of the relatively small size of the affected zone, a convenient way to study the gradients of mechanical properties caused by the thermal oxidation is the instrumented indentation test. Instrumented indentation experiments have been used in previous studies to characterize the mechanical properties of the oxidized layer, revealing the presence of elastic indentation modulus gradients in this layer, while the core of the polymer sample did not show significant change in local mechanical properties with respect to the material in its virgin state [START_REF] Pochiaraju | Evolution of stress and deformations in high-temperature polymr matrix composites during thermo-oxidative aging[END_REF][START_REF] Johnson | Investigation of oxidation profile in pmr-15 polymide using atomic force microscope (afm)[END_REF][3][4][5][START_REF] Jiang | Enhanced mechanical and thermal properties of monocomponent high performance epoxy resin by blending with hydroxyl terminated polyethersulfone[END_REF][START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF]. However, the classical indentation procedure previously used is not able to capture viscous nature of polymer behaviour. That is why the experimental protocol should be adapted to study the polymer behaviour in a more complete way, focusing not only on the elastic unloading, but also on the time-dependent aspects.

Moreover, concerning the effects of aging on carbon fiber reinforced epoxy matrix composites, previous works showed that fiber debonding and micro-cracks due to oxidative aging of the outer polymer layers offer new paths to oxygen to penetrate into the sample core [START_REF] Ho | Endommagements de composites carbone/époxy induits par la thermooxydation de la matrice en présence de contraintes mécaniques[END_REF][START_REF] Vu | Endommagement Induits par la Thermo Oxydation dans les Composites Carbone/Epoxy Unidirectionnels et Stratifiés[END_REF][START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF]. However, there is a lack of local mechanical properties characterisation in the matrix of aged composites.

To tackle all these issues, the first objective of this work is the development of a new instrumented indentation method that emphasizes the whole complexity of the polymer behaviour. This technique will then be employed to study the material gradients in a thermoset epoxy resin subjected to thermal aging and to characterise the behaviour of the same epoxy matrix within a 3D interlock textile composite in both virgin and thermally aged states.

This thesis work is structured as follow:

• In the first Chapter a review of the research in the domain of instrumented indentation and on the effect of thermal aging on polymers and polymer matrix composites will be presented;

• In the second Chapter, the polymer materials and the testing machine used in this study will be described. Moreover, the sources of error that can affect the instrumented indentation measurements and the methods used to correct or avoid them will be outlined;

• In the third Chapter, the cyclic indentation method developed for testing the time-dependent behaviour of polymers will be described. This testing protocol is inspired by macroscopic fatigue tests and has been at first applied on High Density Polyethylene (HDPE), for which the response to macroscopic cyclic tension and shear loading is already known. Moreover, since this material is above glass transition temperature at room temperature (T g ≈ -120 • C) it is expected to show a pronounced time-dependency;

• In the fourth Chapter, the effects of thermal aging on local time dependent properties of an epoxy resin (PR520) used as matrix in carbon fiber reinforced composites for aircraft applications, will be analysed by studying the response change to cyclic indentation load with increasing aging time and for different oxidative environments;

• In the last Chapter, local mechanical behaviour of the polymer matrix within the 3D interlock carbon-fiber reinforcements will be characterised in its "as received" and aged state and compared with that of the neat polymer.

Chapter 1

Literature review

The literature review proposed in this Chapter is divided in three main parts. The first part concerns the instrumented indentation test, that is the main experimental technique that will be used in this thesis work. Emphasis will be put on the limitations resulting from its application in its classical form to polymer materials. The instrumented indentation techniques more appropriate to describe time-dependent properties of polymers commonly used in the literature will thus be presented. A review of the works in which instrumented indentation has been employed to characterize polymer matrix in composites materials will follow.

In the second part, the behaviour of polymers under macroscopic cyclic solicitation will be described.

The third part will concern the effects of thermal aging on polymers and polymer matrix composites at different scales.

The instrumented indentation test

Instrumented indentation test is an experimental technique that has encountered a large success thanks to its non-destructive character and because of the possibility it offers to characterize the material locally, that proves to be particularly interesting for the characterization of thin-films or nano and micro structured heterogeneous materials.

In its classical form, that derives from traditional Vickers hardness tests, instrumented indentation consists in driving a hard tip, usually made of diamond, which shape is known, into the surface of the material under either load or displacement control. From load and displacement data recorded during the test, the indentation curve is plotted. It is usually composed by two phases: loading and unloading. Figure 1.1a depicts a typical load-displacement curve and Figure 1.1b the corresponding geometrical parameters for a load-unload cycle with a pyramidal tip. The loading curve, for elasto-plastic materials, contains the transition from elastic contact to plastic behaviour in the specimen beneath the indenter and is well described by a power law function of the type [START_REF] Tabor | Hardness of metals[END_REF]:

F = C h n (1.1)
In which C is a constant of the material and n = 2 for a conical indenter. Unlike the loading curve, the beginning of unloading curve consists mostly of elastic recovery. The application of the Oliver and Pharr analysis method [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] allows to determine two quantities representing the material behaviour: the Elastic Indentation Modulus E * and the Hardness H. This method of analysis has been developed on elasto-plastic materials and is based on Sneddon's analysis [START_REF] Sneddon | The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile[END_REF] of elastic contact between a rigid indenter and a flat surface. The application of the Oliver and Pharr analysis method requires the fitting of the unloading curve, between the 20% and the 98% of the maximum load, with a power law function of the type:

F = A (h -h p ) m (1.2)
In which the coefficient A, the residual depth h p and the power law exponent m are the fitting parameters. According to Sneddon's analysis, values of the exponent m for common indenter geometries are: 1 for flat punch, 2 for cone and 1.5 for sphere in the limit of small displacements and for paraboloid of revolution. The derivative of the fitting function at the maximum load represents the material Stiffness S :

S = dF dh = A m (h max -h p ) m-1 (1.
3)

The reduced indentation modulus (E r ) could be calculated from the Stiffness as (Equation 1.4):

E r = √ πS 2β A p (1.4)
Where β is a coefficient depending on the shape of the indenter (1.012 for the Vickers tip) and A p is the projected area of contact. In depth sensing indentation, as the contact area is too small, it can not be directly measured through imaging as the error could be too high, but it is deduced form the contact depth h c assuming that the contact surface deforms elastically during unloading, calculated as (Eq. 1.5):

h c = h max -ε F max S (1.5)
Where ε is the so called "intercept correction" and is a constant equal to 0.75 for a pyramidal tip. For a perfect Vickers indenter, the relationship between contact area and contact depth is (Equation 1.6):

A p = (4.95 h c ) 2 (1.6)
Finally, the elastic indentation modulus is calculated form the reduced modulus by accounting for the effect that the diamond tip is not perfectly rigid (Equation 1.7):

E * = 1 1 Er - (1-ν 2 i ) E i (1.7)
Where ν i and E i are the Poisson's ratio and the Young's modulus of the indenter. The material Hardness H, or mean contact pressure, is given by the ratio between the maximum applied load and the contact area at maximum load:

H = F max A p (1.8)
The Hardness for elasto-plastic material is, according to Tabor [START_REF] Tabor | Hardness of metals[END_REF], proportional to the material yield stress in compression through a material constant called "constraint factor".

Application of instrumented indentation test to polymer materials

Even if it has been developed on material exhibiting elasto-plastic behaviour, the Oliver and Pharr analysis method is widely used in the literature to derive elastic indentation modulus and hardness of polymers. Many studies have highlighted the difficulties that arise from the application of Oliver and Pharr analysis method on load-displacement curves resulting from polymeric surfaces indentation [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF][START_REF] Vanlandingham | Nanoindentation of polymers: an overview[END_REF][START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF][START_REF] Tranchida | Mechanical characterization of polymers on a nanometer scale through nanoindentation. a study on pile-up and viscoelasticity[END_REF][START_REF] Jin | Effects of experimental varaibles on pmma nano-indentation measurements[END_REF][START_REF] Beyaoui | Analysis of nanoindentation curves in the case of bulk amorphous polymers[END_REF][21]. While in the case of elastoplastic materials the indentation modulus is close to the Young's modulus, in the case of polymers it results to be overestimated. These difficulties are mainly caused by the complex viscoelastic-plastic response of such materials. Being the polymer response to a solicitation time-dependent, the values of hardness and elastic indentation modulus often depend on the loading time conditions imposed during the test. Moreover, the hypothesis of elastic unloading, on which the Oliver and Pharr analysis method is based, turns out to be too strong on unloading curves of polymers, in which creep phenomena occurs because of their viscoelastic nature. Creep phenomena during unloading are reflected in the round shape of the unloading curves at the peak load (Figure 1.2), known in the literature as "nose" effect [START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF]. Load-displacement curves obtained on four different polymers [START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF] Viscoelastic creep during unloading could lead to a poor power law fit (Equation 1.2) of the unloading curve and to negative values of the stiffness S in extreme cases [START_REF] Vanlandingham | Nanoindentation of polymers: an overview[END_REF].

Moreover, many authors [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF][START_REF] Vanlandingham | Nanoindentation of polymers: an overview[END_REF][START_REF] Beyaoui | Analysis of nanoindentation curves in the case of bulk amorphous polymers[END_REF] observed that for nonlinear least square power law fit of polymers unloading curves, the value of the exponent m was always grater than 2. Sneddon's analysis [START_REF] Sneddon | The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile[END_REF] on the relation between load and displacement for the indentation problem of an elastic material with a cone, suggests that the theoretical value of m is 2. Based on the fact that this theoretical value is hardly ever obtained during experimental tests on elasto-plastic material, Pharr and Bolshakov [START_REF] Pharr | Understanding nanoindentation unloading curves[END_REF] developed the concept of the effective indenter shape. In fact, Sneddon's analysis is based on the contact between a rigid cone and a flat half-space, while in reality the contact at the unloading is between a rigid cone and an already deformed half-space. To simplify the problem, they introduced the concept of "effective shape" (schematized in Figure 1.3) such that, when pressed into a flat surface, it gives the same normal displacement that would be produced by a rigid cone pressed into an already deformed surface. Concept used to define the effective indenter shape [START_REF] Pharr | Understanding nanoindentation unloading curves[END_REF] Following this theory, the authors [START_REF] Pharr | Understanding nanoindentation unloading curves[END_REF] achieved, by Finite Element simulations, values of m ranging from 1.3 to 1.6, which are in good agreement with their experimental observations on aluminum, tungsten, sapphire, soda lime glass and fused silica. The values of m that result from power law fitting of polymers unloading curves are inconsistent with both Sneddon's [START_REF] Sneddon | The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile[END_REF] and Pharr and Bolshakov's [START_REF] Pharr | Understanding nanoindentation unloading curves[END_REF] analysis.

In the literature some tips are proposed to minimize the viscoelastic effects on experimental load-displacement curves of polymers. One of these tips consists in introducing a holding period at maximum load [START_REF] Hochstetter | Strain-rate effects on hardness of glassy polymers in the nanoscale range. comparison between quasi-static and continuous stiffness measurements[END_REF] in order to allow the material reaching a mechanical equilibrium before the unloading begins. Figure 1.4 shows that the introduction of a 10 s holding period allowed to eliminate the nose on load-displacement curves of PMMA [START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF]. Load-displacement curves at the loading-unloading peak evidencing that the introduction of a 10 s holding period at the maximum load allows to eliminate the nose [START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF] Even if the creep effects upon the unloading seem to be eliminated by the introduction of a holding period, many authors observed the dependency of indentation modulus and hardness upon the duration of the holding period [START_REF] Tranchida | Mechanical characterization of polymers on a nanometer scale through nanoindentation. a study on pile-up and viscoelasticity[END_REF][START_REF] Jin | Effects of experimental varaibles on pmma nano-indentation measurements[END_REF][START_REF] Beake | Nanoindentation and nanoscratch testing of uniaxially and biaxially drawn poly(ethylene terephthalate) film[END_REF][START_REF] Lagoudas | Nanoindentation of CNT reinforced epoxy nanocomposites[END_REF]. The evolution of indentation modulus and hardness of PMMA as a function of holding time [START_REF] Jin | Effects of experimental varaibles on pmma nano-indentation measurements[END_REF] is reported in Figure 1.5. Another tip proposed in the literature to minimize viscoelastic effects during unloading is to use high loading and unloading rates [START_REF] Cheng | Determining the instantaneous modulus of viscoelastic solids using instrumented indentation measurements[END_REF]. However, as shown by Tranchida et al. [START_REF] Tranchida | Mechanical characterization of polymers on a nanometer scale through nanoindentation. a study on pile-up and viscoelasticity[END_REF], the indentation modulus determined was still higher than the respective Young's modulus and the value of the exponent m higher than 2. Moreover, the use of very high loading rates could compromise the instrument accuracy inducing a load overshot.

Other authors propose [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF][START_REF] Cheng | Determining the instantaneous modulus of viscoelastic solids using instrumented indentation measurements[END_REF][START_REF] Ngan | Correcting power-law viscoelastic effects in elastic modulus measurement using depth-sensing indentation[END_REF]] some modifications to the Oliver and Pharr analysis method. For example Hardiman et al. [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF] suggest to evaluate the contact stiffness not at the maximum load, but in the point of minimal displacement drift (Figure 1.6). drift and the point of minimal drift at which stiffness is evaluated [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF] 1. Literature review

In this way, the authors [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF] obtained values of indentation modulus on the 6237 epoxy resin that do not depend on the penetration depth and closer (but not equal) to Young's modulus, they improved the fit quality, but they found a power law fitting exponent m still grater than the expected upper limit of 2.

These experimental and analytical tips, have as their objective to extract from loaddisplacement data of time-dependent materials a single value of indentation modulus independent from testing conditions and as close as possible to the Young's modulus. However, time independent properties like elastic modulus and hardness are not adequate to fully describe the complex visco-elasto-plastic behaviour of polymers.

Other indentation tests and analysis methods for polymers

In an attempt to capture not only elastic and plastic, but also viscoelastic properties of polymers, some authors propose alternative analysis methods of load-unload force displacement curves. For example Oyen and Cook [START_REF] Oyen | Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials[END_REF] developed an analytical viscous elasto plastic model (VEP), based on Maxwell model, to interpret load and unload curves of polymers under sharp indentation. The rheological models associated to loading and unloading curves are represented in Figure 1.7a and 1.7b respectively. The semi-empirical model is constituted by quadratic elastic, viscous and plastic elements in series, so that the load is the same on each element and equal to the total load, while the total displacement is the sum of elastic, viscous and plastic contributions. Moreover, the model assumes that under sharp indentation conditions, plastic deformations can occur during loading. In the case of perfect plasticity, the plastic element is suppressed during the unloading for which the plastic displacement is fixed and equal to the plastic contribution determined from the loading. The differential equations associated to the model of Figures 1.7a and 1.7b are used to fit experimental load-displacement data to determine the hardness, the elastic modulus and the viscosity coefficient of the material. The main limitation of this model is the difficulty in the determination of a unique solution from the fitting procedure. Moreover, the material properties determined from the fitting in a particular loading condition, are hardly able to predict material behaviour under different loading conditions.

Indentation creep and recovery tests

The load-unload indentation test turns out to be poorly suited to derive time-dependent viscoelastic properties of polymers. That is why different testing methods are preferred in the literature. Similarly to the macroscopic polymer characterization, the most used indentation test is the creep test, that allows to derive the creep compliance of the material J C (t). Similarly to strain-time creep curves obtained through tensile creep tests, on displacement-time plot of indentation creep test, a transient stage (primary creep) and a steady state (secondary creep) are distinguishable (Figure 1.8). The tertiary creep, consisting in a rapid increase in strain after steady state that leads to failure in tensile creep tests, is, however, not observed in indentation creep tests. In the literature, a large number of models allowing to interpret indentation creep curves and determine creep compliance have been developed. These models can be divided into two main categories: models that derive from the analytical solution of the contact between a rigid indenter and a linear viscoelastic half-space and models based on different spring and dashpot arrangements.

The analytical solution for the contact between a rigid indenter and a linear viscoelastic half-space has been first derived by Lee and Radok [START_REF] Lee | The contact problem for viscoelastic bodies[END_REF] for the case of spherical indenter, and then extended to conical and paraboloidal geometries by Ting [START_REF] Ting | The contact stresses between a rigid indentor and a viscoelastic half-space[END_REF]. According to the solution for a conical indenter geometry [START_REF] Ting | The contact stresses between a rigid indentor and a viscoelastic half-space[END_REF], under the hypothesis of small strain, the creep compliance can be expressed as:

J C (t) =
8 tan α h 2 (t) π P 0 (1.9) in which α is the semi apex angle of the cone, P 0 is the constant force during creep and h(t) is the measured displacement. According to Equation 1.9, the creep compliance can be directly determined from the experimentally measured displacement values. This formulation assumes linear viscoelasticity and that the creep load is applied instantaneously (step loading). Chatel et al. [START_REF] Chatel | Creep of the contact with a spherical tip and recovery of the imprint on amorphous polymer surfaces[END_REF] performed creep and recovery indentation tests on PMMA with a spherical probe. Thanks to a "home made" experimental device, it was possible to follow in-situ the evolution of the true area of contact during the two phases. The very high loading and unloading rates employed (0.5 and 0.2 s respectively) allowed to consider the loading and unloading phases as instantaneous. The influence of temperature, initial mean contact strain and creep duration on creep and recovery have then been analysed. The representative strain for creep and recovery was defined from the mean geometrical contact strain of Tabor [START_REF] Tabor | The hardness of solids[END_REF] as:

a(t) /R a 0 /R (1.10)
during creep, where a(t) is the evolution of the contact radius with creep time, a 0 is the contact radius at the beginning of the creep and R is the tip radius, and:

a(t) /R(t)
at 0/R (1.11) during recovery, where a(t) is the evolution of the contact radius during recovery, R(t) the evolution of the curvature of the residual print and a t 0 is the contact radius at the beginning of recovery. The initial mean contact pressure has been varied in the range 2 -4%, that are characteristic values of the viscoelastic behaviour of PMMA. From the evolution of the representative creep strain as a function of creep time, it was observed that the Lee and Radok model [START_REF] Lee | The contact problem for viscoelastic bodies[END_REF] is close to the curve corresponding to an initial mean contact strain of 2%, while the model was not able to describe properly the results for the higher values of initial mean contact strain (Figure 1.9). Radok's viscoelastic contact model [START_REF] Chatel | Creep of the contact with a spherical tip and recovery of the imprint on amorphous polymer surfaces[END_REF] This observation was confirmed from the analysis of the recovery phase: the representative strain during recovery tends to zero for an initial contact strain of 2%, while permanent strain is observed for the higher initial contact strain values. Moreover, the recovery for an initial contact strain of 2% was not affected by creep duration, while for an initial contact strain of 4% the creep duration played a role in the recovery kinetics (Figure 1.10a and 1.10b). Thanks to their study, it was possible to identify three types of viscoelastic behaviour in the indentation response of PMMA: a linear elastic behaviour that agrees with the Lee and Radok's solution, a non-linear viscoelastic behaviour that is logarithmic with time, and a highly non-linear viscoelastic behaviour.

As in indentation tests an ideal step loading is difficult to achieve, Lu et al. [START_REF] Lu | Measurement of creep compiance of solid polymers by nanoindentation[END_REF] proposed a correction to Ting's formulation in order to take into account the real ramp loading condition for spherical and Berkovich indenters. They found a reasonably good agreement between creep compliance data measured from nanoindentation and with conventional tension and shear tests in the limit of linear viscoelasticity on PMMA and PC. The limit of linearity has been identified as the applied load that did not induced any permanent print. Even if the condition of linear viscoelasticity is not obtained in the majority of creep response, in particular with the sharp-tip indentation, for which creep compliance is dependent on applied load [START_REF] Vanlandingham | Viscoelastic characterization of polymers using instrumented indentation. i. quasistatic testing[END_REF], according to Tweedie [START_REF] Tweedie | Multiscale chemomechanics of polymer deformation under contact: Predicting structure-property correlations from the bulk to the interphase[END_REF], the analytical separation of nonlinear and plastic deformation from the creep response would enable the application of the linear viscoelastic solution. In the attempt to decouple plastic to linear viscoelastic response, Kato et al. [START_REF] Kato | Multicycle indentation for evaluation of polymer material viscoelastic characteristics[END_REF] proposed to perform a multi-cycle indentation test before creep test, in which indentation tests are performed two or three times at the same sample position. They assumed that the behaviour of the material at the first cycle is elasto-plastic, and consequently an impression with a depth h p is formed. The second cycle is thus carried out on a printed impression under the assumption of material behaviour being elastic. The creep compliance formula under these assumptions become: (1.12) in which the plastic depth h p and the pre-force to indent up to the penetration depth h p are determined from the first indentation cycle, F 0 is the creep load and h (t) is the creep displacement measured on the impression. The validity of equation 1.12 was verified by Finite Element Simulations and the values of creep compliance function experimentally determined was close to that determined through tensile tests on ABS and PBT. Beside these analysis methods of indentation creep tests based on the Lee and Radok [START_REF] Lee | The contact problem for viscoelastic bodies[END_REF] solution, a large number of spring-dashpot arrangements are proposed in the literature to interpret displacement time data of indentation creep tests [START_REF] Yang | Analysis of nanoindentation creep for polymeric materials[END_REF][START_REF] Lu | Measurement of creep compiance of solid polymers by nanoindentation[END_REF][START_REF] Chudoba | Investigation of creep behavior under load during indentation experiments and its influence on hardness and modulus results[END_REF][START_REF] Beake | Modelling indentation creep of polymers: a phenomenological approach[END_REF][START_REF] Oyen | Sensitivity of polymer nanoindentation creep measurements to experimental variables[END_REF][START_REF] Huang | Transient and steady-state nanoindentation creep of polymeric materials[END_REF][START_REF] Mazeran | Determination of mechanical properties by nanoindentation in the case of viscous materials[END_REF]. For example Yang et al. [START_REF] Yang | Analysis of nanoindentation creep for polymeric materials[END_REF] proposed an Elastic-Viscoelastic-Viscous model (EVEV) based on the generalised Kelvin model represented in Figure 1.11 and used to analyse indentation creep by a flat ended punch and then adapted for indentation creep by a Berkovich tip. According to this model, the total displacement during indentation creep can be expressed as (Equation 1.13):

J(t) = 2 tan α π (1 -ν 2 ) h (t) + h p F 0 + F p
h = P 0 K 0 + n i=1 P 0 K i (1 -exp ( -k i t /ϕ i )) + P 0 ϕ 0 t (1.13)
Where n is the number of Kelvin elements in series used in the model. By substituting:

h e = P 0 K 0 =indentation depth at the first spring, h i = P i K i =indentation depth at the i -th Kelvin element, µ 0 = ϕ 0 P 0 =constant related to the viscosity coefficient of the last dashpot and τ i = ϕ i K i =retardation time for the i -th element, one obtains (Equation 1.14):

h = h e + n i=1 h i (1 -exp ( -t /τ i )) + t µ 0 (1.14)
From Figure 1.11 and Equation 1.14, it is evident that this model contains three main contributions: the instantaneous displacement h e associated with deformation during loading, supposed to be instantaneous and, in the case of flat punch, fully elastic; the viscoelastic deformation in the transient stage that is controlled by the exponential terms and the deformation dominated by the viscous flow of macro-molecules that increases linearly with time ( t µ 0 ). The creep compliance can thus be determined as: (1.15) in which h in is an empirically determined length scale. The main disadvantage of such models is that there is not unique solution and thus several sets of fitting parameters can describe the experimentally measured creep compliance. Moreover, the parameters extracted from the fit depend, not only on the material, but also on the form of the rheological model and on the loading conditions (loading rate and creep duration). Finally, the hypothesis of instantaneous elastic loading could be too strong. Despite their limitations, rheological models are a useful tool to compare different polymer materials in the same loading conditions. An alternative way to study the time-dependent behaviour of polymers is to follow the recovery of indentation print through imaging [START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF][START_REF] Chatel | Creep of the contact with a spherical tip and recovery of the imprint on amorphous polymer surfaces[END_REF][START_REF] Tweedie | On the indentation recovery and fleeting hardness of polymers[END_REF]. For example Minervino [START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF] followed the evolution of the residual print resulting from a load-unload sharp indentation test at 5 mN on an epoxy resin through several time decades (up to three months). The heating of the sample for 10 min at 150 °C to accelerate recovery, allowed to completely recover the residual print. This observation proved that the material behaviour was viscoelastic. By combining the results of indentation test and of recovery (as schematised in Figure 1.12), it was possible to determine through a reverse analysis, the parameters of the relaxation spectrum of a viscoelastic behaviour law. 12. Scheme of the experimental procedure used by Minervino [START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF] to determine the relaxation spectrum of an epoxy resin through reverse analysis

J(t) = i J i = h(t) σ 0 h in
The behaviour law determined in this way does not take into account plastic deformations that could occurs for higher loads. Moreover, it is able to reproduce the material behaviour in relaxation that could be different from the creep behaviour.

Dynamic tests

Beside creep and relaxation tests, another testing method that enables the determination of viscoelastic material properties is the dynamic indentation test [START_REF] Kermouche | Extraction of stress-strain curves of elastic-viscoplastic solids using conical/pyramidal indentation testing with application to polymers[END_REF][START_REF] Monclus | In search of validated measurements of the properties of viscoelastic materials by indentation with sharp indenters[END_REF][START_REF] Constantinides | Grid indentation analysis of composite microstructures: Principle and validation[END_REF][START_REF] Asif | Nanoindentation and contact stiffness measurement using force modulation with a capacitive load-displacement transducer[END_REF]. It such tests, a high frequency sinusoidal signal (1 -300 Hz) is superposed to the static signal, as represented in Figure 1.13. [START_REF] Kermouche | Extraction of stress-strain curves of elastic-viscoplastic solids using conical/pyramidal indentation testing with application to polymers[END_REF] Contact stiffness is measured from the unloading of each oscillation or, in other words, continuously during the test. That is why the dynamic test is also known as Continuous Stiffness Measurement (CSM). In dynamic tests, the indenter-sample interaction is modelled by a parallel spring and dashpot arrangement, also known as Voigt element and this implies the assumption of linear viscoelastic behaviour of the material. The application of this method to polymers, allows to determine material storage and loss moduli (E and E " ) as:

E = S √ π 2 A(h) 1 -ν 2 s
(1. [START_REF] Vanlandingham | Nanoindentation of polymers: an overview[END_REF])

E " = ω D s √ π 2 A(h) 1 -ν 2 s (1.17)
where S and D s are the stiffness and the damping of the contact respectively, ω is the oscillation frequency, ν s is the Poisson's ratio of the material and A(h) is the contact area determined at each oscillation. An example of the evolution of storage modulus with penetration depth is represented in Figure 1.14 [START_REF] Monclus | In search of validated measurements of the properties of viscoelastic materials by indentation with sharp indenters[END_REF]. The roughness of the surface together with tip blunt does not allow to properly determine the modulus at shallow indentation depth. Monclus and Jennet [START_REF] Monclus | In search of validated measurements of the properties of viscoelastic materials by indentation with sharp indenters[END_REF], compared the storage and loss modulus determined from dynamic indentation to that determined through single bending DMA tests on POM, PS-1 and PS-3. A good agreement was found between storage moduli determined with the two tests for the three materials. However, the loss modulus agreed only for PS-3. This disparity was attributed to the differences in modelling the tip-sample interaction in the two tests: Voigt model in dynamic indentation and standard linear solid model in DMA. Wright et al. [START_REF] Wright | An improved analysis for viscoelastic damping in dynamic nanoindentation[END_REF] showed through tests on silicon rubber that the standard linear solid model can be applied to dynamic indentation measurements, making results comparable to DMA.

Herbert et al. [START_REF] Herbert | Nanoindentation and the dynamic characterization of viscoelastic solids[END_REF] found a good agreement between storage and loss moduli determined on highly plasticized PVC at ambient temperature through DMA and dynamic nanoindentation with a flat punch in the limit of linear viscoelasticity. The limit of linearity has been identified as the regime for which an increase in applied load or displacement had no effect on the measured storage and loss moduli. Moreover, the advantage in using flat punch is that the contact area does not change even in presence of creep behaviour. According to the authors [START_REF] Herbert | Nanoindentation and the dynamic characterization of viscoelastic solids[END_REF], a successfully application of dynamic nanoindentation to derive storage and loss moduli of a linear viscoelastic solid requires a rigorous characterization of the frequency response of the measurement system. Herbert et al. [START_REF] Herbert | Nanoindentation and the dynamic characterization of viscoelastic solids[END_REF] found that the crucial parameters of the system are: the reference phase angle, the dynamic stiffness and the damping. Moreover, the authors [START_REF] Herbert | Nanoindentation and the dynamic characterization of viscoelastic solids[END_REF] identified a cut-off frequency above which it is not possible to accurately model the instrument frequency response.

Despite of this improvement, the dynamic indentation presents some limitations. First of all, like all model-dependent analysis methods, the material properties are not unique but strictly related to the model chosen to describe its behaviour. Secondly, similarly to the analysis of indentation creep tests, the hypothesis of linear viscoelasticity is not always valid. Third, not all the indentation equipments are able to produce a dynamic solicitation.

Approaches based on energy dissipation

A way to characterize polymers, regardless of linear viscoleasticity assumption and/or any spring and dashpot configuration, based on the analysis of energy dissipated during indentation is proposed by Tweedie et al. [START_REF] Tweedie | Nanomechanical quantification of polymer energy absorption[END_REF].

In materials exhibiting elasto-plastic behaviour, like metals or glasses, the area enclosed by load and displacement curves in a quasi-static indentation test (Figure 1.15) represents the amount of energy dissipated to produce plastic deformations (U p ) [START_REF] Fischer-Cripps | Nanoindentation. Mechanical Engineering[END_REF]. The area enclosed by load and displacement curves is calculated as the difference between the area under the loading curve, that represents the total work of indentation (U t ), and the area under the unloading curve, that represents the elastic energy that is recovered during unloading (U e ).

As reported in Section 1.1, in the case of indentation of elasto-plastic material with a cone, the relation between load and displacement during the loading phase can be expressed as a quadratic function F = C h 2 (Equation 1.1), while during the unloading, the relation between load and displacement is a power law of the type F = A (h -h p ) m [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] (Equation 1.2). The work done in plastic deformation U p , is thus calculated as:

U p = U t -U e = ˆhmax 0 C h 2 dh - ˆhmax hp A (h -h p ) m dh (1.18)
that results in:

U p = C h 3 max 3 -A (h max -h p ) m+1 m + 1 (1.19)
The plastic energy under the unloading curve is often represented in a normalized form with respect to the total energy (η):

η = U p U t (1.20)
and thus:

η = 1 - A C 3 h 3 max (h max -h p ) m+1 m + 1 (1.21)
The ratio between the two coefficients A and C is derived from the equality between load and unload curves at the maximum load:

F = C h 2 max = A (h max -h p ) m (1.22)
Equation 1.21 thus become:

η = 1 - 3 m + 1 1 - h p h max (1.23)
According to Equation 1.23, the ratio of energy dissipated to produce plastic deformations to total energy is proportional to the residual depth to maximum depth ratio. Cheng et al. [START_REF] Cheng | Scaling relationships for indentation measurements[END_REF] have found through finite element simulation that, for hp /hmax > 0.4, the value of η was not dependent on the indenter angle and could be expressed as:

η = 1.27 h p h max -0.27 (1.24)
Cheng et al. [START_REF] Cheng | Scaling relationships for indentation measurements[END_REF] have also determined a linear relation between the elastic energy to total energy ratio and the hardness to indentation modulus ratio:

H E * = κ U e U t (1.25)
in which κ is a constant that depends on the indenter angle and is equal to 5.3 for a conical indenter with a half-angle of 70.3°that is the equivalent cone angle for Berkovich and Vickers tips.

In [START_REF] Tweedie | Nanomechanical quantification of polymer energy absorption[END_REF], the analysis of energy dissipated in quasi-static indentation tests on five polymers (PE, PP, PS, PMMA and PC) showed that energy absorption is dependent on loading rate, deformed volume and physical/structural characteristics of the material. So, besides its direct link to plastic deformations, the energy absorption contains information on the time-dependent character of polymers. Thanks to its "model-free" character, the analysis of energy in indentation is, according to the authors [START_REF] Tweedie | Nanomechanical quantification of polymer energy absorption[END_REF], a promising approach for the description of mechanical properties of polymers.

Study of polymer-matrix composites local properties with instrumented indentation

The mechanical behaviour of composite materials in numerical simulations is often modelled by considering matrix and fibers properties measured separately on each constituent. However, the actual properties of the matrix in the composite (in-situ) could be different from its properties measured on the neat material. In the specific case of thermosetting matrix composites, these differences could arise from the different degree of cure reached by the polymer if it is cured alone or co-cured with fibers.

Instrumented indentation is one of few experimental techniques that allows to measure polymer matrix properties in-situ. In this context, Gregory and Spearing [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF] performed nanoindentation tests in standard and CSM mode in polymer pockets of three different unidirectional carbon fiber reinforced composites in both directions perpendicular and parallel to fibers (Figures 1.16a and 1.16b). Gregory and Spearing [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF] found that the main difficulty in in-situ properties measurement is to decouple the constraint effect induced by stiffer neighbouring carbon fibers to an actual polymer matrix properties change. In order to establish the polymer pocket size required in order to measure polymer properties unaffected by fiber constraint effect, the authors [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF] performed 2D finite element simulations in which the polymer was modelled as elasto-plastic using properties determined through compression tests, the indenter was modelled as a rigid body with a conical shape and the effect of fibers was simulated through boundary conditions preventing movement on both x and y directions. The results of simulations showed that the minimal distance between the indentation point and the constraint in order to obtain unconstrained local properties measurement should be 25 times larger than the maximum indentation depth. By respecting this condition, Gregory and Spearing [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF] found experimentally through indentation tests that the unconstrained in-situ 977-3 epoxy resin indentation modulus was 10-30% higher than the neat modulus, while the hardness ranged from 7% less to 20% more than neat hardness. In the case of FM 300 epoxy resin, the in-situ modulus determined was 30 -40% higher than neat modulus, while in-situ hardness was 20 -26% higher than neat hardness [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF]. However, as shown by Hardiman et al. [START_REF] Hardiman | The effect of fibre constraint in the nanoindentation of fibrous composite microstructures: A finite element invesitgation[END_REF] the morphology of fibers surrounding the resin pockets play a role on the constraint effect they induce. This conclusion arises from a thorough finite element investigation on the effect of fiber constraint on indentation measurements. Together with 2D finite element simulations similarly to [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF], they performed 3D simulations in which the fibers surrounding resin pockets were modelled in different ways: as an encastre boundary, as a circular pattern of closely packed fibers and as a less dense packing arrangement. The results shows that 2D and 3D models in which fibers are represented as an encastre boundary give a similar constrained effect that results to be overestimated with respect to a more realistic configuration in which fibers are modeled as a circular pattern (Figure 1.17). In [START_REF] Hardiman | Fibrous composite matrix characterisation using nanoindentation: The effect of fibre constraint and the evolution from bulk to in-situ matrix properties[END_REF], the same authors experimentally determined on co-cured HTA/6376 composite and 6376 resin that, even for the measurements not affected by fiber constraint, the difference between neat and in-situ indentation modulus depends upon the size of the resin pocket. In all cases the in-situ modulus was higher than the neat modulus and the maximal difference they determined was 19%, and thus neat and in-situ materials are different.

Most of the studies in which instrumented indentation is performed on composite materials aim to determine not only the differences that could exist between neat and in-situ matrix properties, but also the properties of the so called interphase region [START_REF] Vanlandingham | Characterization of nanoscale property variations in polymer composite systems: 1. experimental results[END_REF][START_REF] Gao | Characterization of interphase nanoscale property variations in glass fibre reinforced polypropylene and epoxy resin composites[END_REF][START_REF] Kim | Nanoscale characterisation of interphase in silane treated glass-fibre composites[END_REF][START_REF] Lee | Evaluation of interphase properties in a cellulose fiber-reinforced polypropylene composite by nanoindentation and finite element analysis[END_REF][START_REF] Dowing | Determining the interphase thickness and properties in pomlymer matrix composites uding phase imaging atomic force microscopy and nanoindentation[END_REF]. The interphase is the zone located between matrix and fibers, which is formed during curing process due to the reactions with fiber treatment agents and which properties are supposed to be different from those of each constituent. Kim et al. [START_REF] Kim | Nanoscale characterisation of interphase in silane treated glass-fibre composites[END_REF] performed displacement control indentation tests along the fiber-interphase-matrix of E-glass woven fabric-reinforced vinylester matrix composites. The three constituents (matrix, fibers and interphase) showed different response to indentation, as can be observed from the resulting load-displacement curves represented in Figure 1.18 [START_REF] Kim | Nanoscale characterisation of interphase in silane treated glass-fibre composites[END_REF]. The values of indentation modulus and hardness calculated from load displacement curves, revealed that interphase region has properties intermediate between those for the fiber and matrix. Moreover, from nanoindentation results it was possible to estimate the value of the interphase thickness, even if other experimental methods like nanosctrach and heat capacity jump measurements were able to provide more precise values [START_REF] Kim | Nanoscale characterisation of interphase in silane treated glass-fibre composites[END_REF]. However, according to [START_REF] Vanlandingham | Characterization of nanoscale property variations in polymer composite systems: 1. experimental results[END_REF][START_REF] Dowing | Determining the interphase thickness and properties in pomlymer matrix composites uding phase imaging atomic force microscopy and nanoindentation[END_REF], because of its proximity to fibers, that induce a constrained effect on indentation measurements, the determination of interphase mechanical properties using instrumented indentation is not possible. A way to remove constraint effects from indentation modulus measurement is to normalize the experimental results with finite element simulation results, assuming that the model properly describe all the proper conditions apart interphase contribution [START_REF] Wood | Measuring interphase stiffening effects in styrene-based polymeric thin films[END_REF].

In the literature other experimental techniques than instrumented indentation are used to derive interphase mechanical properties [START_REF] Gao | Characterization of interphase nanoscale property variations in glass fibre reinforced polypropylene and epoxy resin composites[END_REF]. For example Naya at al. [START_REF] Naya | Interface characterization in fiber-reinforced polymer-matrix composites[END_REF], determined shear strength of interphase in carbon and glass-fiber reinforced composites with either thermoset or thermoplastic matrix through fiber push-in tests.

Cyclic behaviour of polymers

Macroscopic cyclic tests can be performed in either stress or strain control. Stress control is preferred as it better reflects the actual working conditions that the structure is under. A cyclic loading in stress control is defined by the stress amplitude σ a = σmax-σ min 2 , the mean stress σ m = σmax+σ min 2 and by the load ratio R = σ min σmax . Due to the growing use of polymers in engineering structures and the well established fact that failure of a material under cyclic load is at a lower stress level than under uniaxial loading condition (fatigue) [START_REF] Sauer | Fatigue of polymers[END_REF], in recent years, a large number of studies have been dedicated to the understanding of the mechanisms of fatigue failure of polymers [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF][START_REF] Zhang | A simple constitutive model for cyclic compressive ratchetting deformation of polyteterafluoroethylene (ptfe) with stress rate effects[END_REF][START_REF] Drozdov | Cyclic viscoelastoplasticity and low-cycle fatigue of polymer composites[END_REF][START_REF] Shen | Cyclic deformation behavior of an epoxy polymer. part i: Experimental investigation[END_REF][START_REF] Ariyama | Cyclic deformation behaviour and morphology of polypropylene[END_REF][START_REF] Rabinowitz | Cyclic deformation and fracture of polymers[END_REF][START_REF] Kang | Uniaxial ratchetting of polymer and polymer matrix composites: Time-dependent experimental observations[END_REF][START_REF] Chen | Ratcheting behaviour of ptfe under cyclic compression[END_REF][START_REF] Bouvet | Fluorescence spectroscopy applied to study cyclic creep behaviour and internal stresses of semicrystalline high-density polyethylene[END_REF][START_REF] Xu | Tensile ratcheting behaviors of bronze powder filled polytetrafluoroethylene[END_REF].

In this Section, we will mainly focus on the results reported in the thesis work carried out in our laboratory by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF]. In Nguyen's research [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] the cyclic behaviour of High Density Polyethilene (HDPE) was studied in both tensile and shear stress. The same material, that will be described in more detail later, is the object of a part of the work presented in this thesis. In order to keep constant the stress rate during the tests, a triangular wave cycle was employed (Figure 1.19a). The maximum normal stress applied in cyclic tensile tests was 17 M P a, while the maximum shear stress in cyclic shear tests was 9.6 M P a. These values were chosen so as not to exceed the yield stress. It is known that the stress-strain response of a polymer to a cyclic load (HDPE in this case) is characterised by hysteresis loops (Figure 1.19b). The hysteresis loops reflect the viscous nature of the polymers for which there is a decay between the applied stress and the strain response. In [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] both frequency and mean stress effects on HDPE response to cyclic tensile and shear solicitation have been evaluated. In order to quantify these effects, three parameters were extracted from hysteresis loops of the stress-strain response that are evidenced in Figure 1.19b. These parameters are:

• Loop area, that is the area enclosed between load and unload curves and is evaluated through a trapezoidal numerical integration;

• Secant elastic modulus, defined as the slope of the straight line that connects the initial point with the point at the maximum strain at each cycle;

• Mean strain, calculated as the mean value between maximum and minimum strain at each cycle.

The effect of frequency and load ratio on the evolution with cycles of loop area, secant modulus and mean strain are represented in Figures 1.20 For the three considered frequencies, both loop area and secant modulus show similar kinetics of evolution with cycles: loop area decreases in the first cycles and then stabilises at a constant value after about 20 cycles, while secant stiffness increases in the first cycles and then stabilises at a constant value after a similar number of cycles. The rate of change of both loop area and secant modulus with cycles becomes independent on frequency after about 4 cycles. Moreover, for higher frequencies, the viscous contribution to the response is minor, as loops are narrower and almost vertical. The same behaviour was observed by Zhang et al. [START_REF] Zhang | A simple constitutive model for cyclic compressive ratchetting deformation of polyteterafluoroethylene (ptfe) with stress rate effects[END_REF] and Rabemananjara at al. [START_REF] Rabemananjara | Caractérisation expérimentale des mécanismes d'endommagement du polyéthylène moyenne densité (pemd) sollicité en fatigue par compression. 36emes Journées de Printemps[END_REF] on cyclic compression tests on PTFE and MDPE respectively. Similar tendency is observed in case of loop area and secant stiffness for different load ratios. In both cases, the rate of change becomes independent of load ratio after about 6 cycles. Moreover, the loop area in the stabilized state is proportional to load ratio. It is interesting to observe that, while loop area and secant modulus reach a stabilized state at which point they become constants, the mean strain continues to increase with cycles for all the frequencies and load ratios considered.

In macroscopic stress controlled cyclic loading, the evolution of strain with cycles is known as ratcheting. Ratcheting consists in an accumulation of secondary inelastic deformation caused by non-zero mean stress [START_REF] Ohno | Recent topics in constitutive modeling for ratcheting[END_REF][START_REF] Kang | Ratchetting: Recent progresses in phenomenon observation, constitutive modeling and application[END_REF] which is superposed on the primary cyclic deformation caused by cyclic loading. It occurs in polymers and metals at ambient and elevated temperature. The ratcheting phenomenon should be taken into account for an accurate prediction of component fatigue life. That is why it has been intensively studied in recent years on different type of loading: tensile, compression, shear or multiaxial [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF][START_REF] Zhang | A simple constitutive model for cyclic compressive ratchetting deformation of polyteterafluoroethylene (ptfe) with stress rate effects[END_REF][START_REF] Drozdov | Cyclic viscoelastoplasticity and low-cycle fatigue of polymer composites[END_REF][START_REF] Shen | Cyclic deformation behavior of an epoxy polymer. part i: Experimental investigation[END_REF][START_REF] Ariyama | Cyclic deformation behaviour and morphology of polypropylene[END_REF][START_REF] Rabinowitz | Cyclic deformation and fracture of polymers[END_REF][START_REF] Kang | Uniaxial ratchetting of polymer and polymer matrix composites: Time-dependent experimental observations[END_REF][START_REF] Chen | Ratcheting behaviour of ptfe under cyclic compression[END_REF][START_REF] Bouvet | Fluorescence spectroscopy applied to study cyclic creep behaviour and internal stresses of semicrystalline high-density polyethylene[END_REF]. In the literature, ratcheting strain is defined in different ways: as the increment of peak strain after each cycle [START_REF] Ohno | Recent topics in constitutive modeling for ratcheting[END_REF], as the evolution of the minimum strain per cycle [START_REF] Chen | Ratcheting behaviour of ptfe under cyclic compression[END_REF] or as the evolution of the mean strain per cycle [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF]. In all cases, ratcheting strain increases with cycles and depends on a lot of factors like loading rate, mean stress, stress amplitude and loading history [START_REF] Kang | Ratchetting: Recent progresses in phenomenon observation, constitutive modeling and application[END_REF]. Because of its analogy to the evolution of strain with time in creep tests, ratcheting is also known under the name of cyclic creep [START_REF] Chen | Ratcheting behaviour of ptfe under cyclic compression[END_REF]. For example, Lissenden at al. [START_REF] Lissenden | Experimental investigation of cyclic and time-dependent deformation of titanium alloy at elevated temperature[END_REF] showed that ratcheting curve of a Titanium alloy at 650 • C was similar to creep curve having primary, steady state and tertiary regions. However, magnitudes of ratcheting strain were higher than those of a creep for a constant stress equal to the mean stress applied in the cyclic stressing. The same conclusion was drawn by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] from the comparison of ratcheting strain and creep strain on HDPE at room temperature: cycling promotes the strain increase cycle by cycle. Moreover, it was shown in [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] that for a constant creep stress equal to the maximum cyclic stress, creep strain was higher than cyclic ratcheting strain. According to the author [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF], ratcheting in polymers could be due to non-linear viscoelastic effects or to viscoplastic flow reinforced by the maximal stress reached cycle by cycle. However, as almost complete recovery after the cyclic tests was observed by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF], the author concluded that no plastic deformations have occurred and the material thus follows a non-linear viscoelastic law.

Cyclic indentation tests

Even in the case of load-displacement curves obtained on elasto-plastic materials, the hypothesis of perfectly elastic unload, on which the Sneddon [START_REF] Sneddon | The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile[END_REF] and the further Oliver and Pharr analysis [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] are based, is not always valid. Oliver and Pharr [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF], in fact, suggested to use a load-time sequence consisting in three triangular wave load-unload cycles, followed by a hold period at the minimum load and a last load-unload cycle including a holding period at the maximum load (Figure 1.22). [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] This suggestion come from the fact that, if the material behaviour would be perfectly elasto-plastic, unload and reload curves should overlap. That was not the case for the metals they tested, in which a very small hysteresis loop was observed between unload and reload curves (Figure 1.23). The non overlapping of unload and reload curve is attributed to reverse plasticity [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF][START_REF] Shuman | Calculating the elastic modulus from nanoindentation and microindentation reload curves[END_REF][START_REF]Reverse Plasticity in Nanoindentation[END_REF]. The reverse plasticity in nanoindentation on crystalline materials has been thoroughly studied by Huang at al. [START_REF]Reverse Plasticity in Nanoindentation[END_REF]. The authors [START_REF]Reverse Plasticity in Nanoindentation[END_REF] found that larger deviation between unload and reload curves is observed on material with smaller elastic modulus and glass transition temperature. Moreover, it was observed that this deviation decreases with increasing number of cycles and with increasing loading rate. The loading path proposed by Oliver and Pharr [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] (Figure 1.22) allows to diminish time dependent plastic deformations and the use of the unloading curve obtained after some cycles for the calculation of indentation modulus, allows to minimize the inaccuracies due to reverse plasticity. However, Shuman et al. [START_REF] Shuman | Calculating the elastic modulus from nanoindentation and microindentation reload curves[END_REF] found on steel, copper, nickel, iron and aluminium that, even if evaluated after some cycles, the Elastic indentation modulus calculated from the unloading curve stiffness, is quite different from their Young's modulus and the scatter among the results obtained at each cycle is significant. So, they proposed to calculate indentation modulus from the reloading curves by applying the same calculation procedure. In this way they found indentation modulus values closer to Young's modulus and the scatter among the results was reduced. According to [START_REF] Shuman | Calculating the elastic modulus from nanoindentation and microindentation reload curves[END_REF], reverse plasticity occurs due to internal stresses generated in the material under load that become unstable during unload (relaxation of pinned dislocations, adjustment of grain boundaries or phase transitions).

When reloaded in the same point, the material deforms elastically as the indenter suits an already formed impression shape. The contact between tip and material could be better during reload and provide a more homogeneous stress distribution [START_REF] Shuman | Calculating the elastic modulus from nanoindentation and microindentation reload curves[END_REF].

Several studies are available in the literature in which material behaviour is studied through a cyclic indentation method [START_REF] Wei | Retardation of cyclic indentation creep exhibited in metal alloys[END_REF][START_REF] Xu | Study of the ratcheting by the indentation fatigue method with a flat cyclindrical indenter: Part i. experimental study[END_REF][START_REF] Collin | Use of spherical indentation data changes to materials characterization based on a new multiple cyclic loading protocol[END_REF][START_REF] Abudaia | Spherical indentation fatigue cracking[END_REF][START_REF] Arciniegas | Thermoelastic phase transformation in tini alloys under cyclic instrumented indentation[END_REF][START_REF] Chorfa | Mechanical behaviour of glass during cyclic instrumented indentation[END_REF][START_REF] Deriano | The indentation fracture toughness (kc) and its parameters: the case of silica-rich glasses[END_REF][START_REF] Matsuzawa | Damage accumulation caused by cyclic indentation in zirconia ceramics[END_REF]. Most of these studies have been performed on metals and glasses. For example, Collin et al. [START_REF] Collin | Use of spherical indentation data changes to materials characterization based on a new multiple cyclic loading protocol[END_REF] showed that the analysis of total energy and plastic to total energy (Eq. 1.20) and elastic to total energy ratios in a multi-cycle indentation test with spherical indenter (Figure 1.24), allowed to accurately determine the hardening law parameters of the three Ludwig equation parameters on two different steels from a numerical study. In [START_REF] Abudaia | Spherical indentation fatigue cracking[END_REF][START_REF] Chorfa | Mechanical behaviour of glass during cyclic instrumented indentation[END_REF][START_REF] Deriano | The indentation fracture toughness (kc) and its parameters: the case of silica-rich glasses[END_REF][START_REF] Matsuzawa | Damage accumulation caused by cyclic indentation in zirconia ceramics[END_REF] fatigue curves and fracture toughness values are determined through the analysis of crack growth and damage accumulation induced by repeated indentations on glasses and metals with either spherical or sharp indenters. In [START_REF] Arciniegas | Thermoelastic phase transformation in tini alloys under cyclic instrumented indentation[END_REF], loading-unloadingreloading indentation test was employed to detect phase transformations in austenitic and martensitic TiNi shape memory alloys.

In [START_REF] Xu | Study of the ratcheting by the indentation fatigue method with a flat cyclindrical indenter: Part i. experimental study[END_REF], the cyclic indentation method with a flat cylindrical indenter is used to study ratcheting of a copper alloy. In this work, the effect of loading frequency, maximum indentation load and load amplitude on the indentation response have been studied. They observed that the indentation depth-cycle curves are characterized by a transient stage in which the indentation depth sharply increases and a steady-state in which the indentation depth per cycle dh dN reaches a constant value. They found that experimental parameters for the steady state stage in uniaxial fatigue tests can be obtained by cyclic indentation with flat ended punch.

Thermal aging of polymers and composites

As suggested by its appellation, aging refers to an evolution of material properties over the time. This evolution can involve a modification of the chemical structure of the macromolecules. In this case, we talk about chemical aging, to be distinguished from the physical aging in which only macromolecular spatial configuration or material composition are affected [START_REF] Fayolle | Vieillissement physique des matériaux polymères[END_REF]. In the specific case of thermal aging, the evolution of polymer properties is combined with a change in macromolecular structure due to the interaction with oxygen and it is promoted by relatively elevated temperatures. For aging temperatures lower than glass transition temperature, the material degradation is due to pure thermal-oxidation, while for aging temperatures higher than the glass transition, secondary degradation phenomena could also occur. These secondary degradation phenomena (thermolysis), are anaerobic, which means that no oxygen is involved and can thus be produced in inert environments and initiate with the polymer decomposition in free radicals and volatile products [START_REF] Pascault | Thermoetting Polymers[END_REF].

The most evident effect of thermal oxidation of polymers is the surface colour change. In particular, as can be observed from Figure 1.25 the thermally aged sample appears darker than the virgin one and this change is more significant as the aging conditions are stricter. This darker layer, is the result of macromolecular chain modifications induced by the reaction of the oxygen with the polymer and concerns not only the external surface as for the most of inorganic solids, but extends in the thickness for several hundred of micrometers.

Three main zones can be distinguished on the cross section of thermo-oxidized polymer samples: the oxidized layer, the transition region and the unoxidized core [START_REF] Tandon | Thermo-oxidative behavior of high-temperature pmr-15 resin and composite[END_REF]. This three zones are higlighted in Figure 1.26, that represents the microscopic image of the cross section of the PMR-15 epoxy resin aged for 175 h at 288 °C in atmospheric air [START_REF] Tandon | Thermo-oxidative behavior of high-temperature pmr-15 resin and composite[END_REF]. aged for 175 h at 288 °C in atmospheric air [START_REF] Tandon | Thermo-oxidative behavior of high-temperature pmr-15 resin and composite[END_REF] The thickness of the oxidized layer evolves with aging time up to a limit value, while the thickness of the transition region is constant during aging [START_REF] Tandon | Thermo-oxidative behavior of high-temperature pmr-15 resin and composite[END_REF]. The kinetics of development of the oxidized layer is regulated by two main phenomena: the oxygen diffusion and the oxygen reaction [START_REF] Pochiraju | Modeling thermo-oxidative layer growth in hightemperature resins[END_REF]. From one side the penetration of oxygen through the sample thickness is promoted by the diffusion, but, at the same time, it is consumed by the reaction with free radicals. The law that describes the growth of the oxidized layer, can thus be expressed as a balance between diffusion and reaction:

δC δt = D δ 2 C δz 2 -r ox (1.26)
where C(x, y, z, t) is the local oxygen concentration; D is the diffusion coefficient that depends on temperature and r ox is the rate of oxygen consumption. The rate of oxygen consumption depends on the reactions involved that in turns are related to the material structure and to aging conditions. As reported by Ernault [START_REF] Ernault | Thermo-oxydation de résines époxy/amine[END_REF], in epoxy networks the oxidation preferential sites are the C-H positioned near nitrogen or oxygen atoms. When oxygen molecules attack these sites, hydroperoxides (POOH) are formed (initiation). The O-O link of hydroperoxides has a weak dissociation energy, that makes the POOH chain scission and the formation of free radicals. The free radicals are highly reactive and enhance the epoxy degradation (propagation). In this phase, when a peroxyde radical attacks the material oxydable site, new hydroperoxides are formed and thus the oxidation process is self-sustained. In a close loop mechanistic scheme [START_REF] Colin | A new method for predicting the thermal oxidation of thermoset matrices. application to an amine crosslinked epoxy[END_REF], the last phase of thermal aging (termination) is characterized by recombination of free radicals and the formation of inactive products. Among all the possible combinations of free radicals, some of them could promote crosslinking [START_REF] Zahra | Thermo-oxidative aging of epoxy coating systems[END_REF].

The macromolecular modifications, chain scission (S) or crosslinking (X), occurring during the three phases of thermal aging are reflected in a change of sample weight [START_REF] Courvoisier | Multi-scale and multi-technique analysis of the thermal degradation of poly(ether ether kerotene)[END_REF]:

S -X = 1 M n - 1 M n0 (1.27)
where M n0 and M n are the number average molecular masses before and after aging.

According to Equation 1.27, a predominance of chain scission leads to a molecular weight loss while a predominance of crosslinking leads to a mass gain. A simpler way to monitor the change in macromolecular structure is to follow the evolution of glass transition temperature (T g ) with aging time. In fact, thanks to the Fox-Flory's relationship [START_REF] Courvoisier | Multi-scale and multi-technique analysis of the thermal degradation of poly(ether ether kerotene)[END_REF], it is possible to relate the mass change to the glass transition temperature change:

T g -T g0 = -k F F 1 M n - 1 M n0 (1.28)
where k F F is the Fox-Flory's constant. By combining Equations 1.27 and 1.28 a relations between glass transition temperature change and the number of crosslinking and chain scission events is obtained:

T g -T g0 = k F F (X -S) (1.29)
According to Equation 1.29, the glass transition temperature increases during aging if crosslinking events predominate, while it decreases with the number of chain scissions. In this context, [START_REF] Ernault | Thermal-oxidation of epoxy/amine followed by glass transition temperature changes[END_REF] monitored the glass transition temperature change of three different epoxy resins differing by the nature of prepolymer and hardener and they found that different epoxy resins showed different macromolecular modification when aged in air at 110 °C and 200 °C, and under 50 bar of oxygen pressure at 110 °C, as reported in Figure 1.27. In particular, the DGEBA/IPDA system (initial T g = 166°C) showed a decrease of T g , suggesting that oxidation induced chain scission. On the contrary, on the DGEBA/TTDA system (initial T g = 69 °C), the T g increased with aging time, suggesting that oxidation induced crosslinking. Finally, on the DGEBU/IPDA system (initial T g = 60 °C), the glass transition temperature at first slightly decreased and then, for higher aging time, increased, suggesting that sites with lower reactivity are activated with the progress of oxidation [START_REF] Ernault | Thermal-oxidation of epoxy/amine followed by glass transition temperature changes[END_REF]. In the three cases, aging under 50 bar of oxygen showed a decrease in glass transition temperature from the beginning of thermal aging. In the same attempt to follow macromolecular change by glass transition temperature evolution during thermal aging, Courvoisier et al. [START_REF] Courvoisier | Multi-scale and multi-technique analysis of the thermal degradation of poly(ether ether kerotene)[END_REF] found on PEEK at the rubbery state that crosslinking predominates for all the aging temperatures (between 250 and 320 °C) and partial oxygen pressures (between 0.21 and 20 bar) employed. Without entering into the detail of the specific complex reactions that induce chain scission and crosslinking during thermal aging, that are, moreover, different from a material to another and that depend on aging conditions, we can generally state that chain scission occurs mainly in the propagation phase of thermal aging, while crosslinking results from the combination of some free radicals [START_REF] Zahra | Thermo-oxidative aging of epoxy coating systems[END_REF].

The chemical reactions and the macromolecular modifications resulting from the polymer exposure over the time at elevated temperatures and in presence of oxygen, compromises the macroscopic failure properties [START_REF] Tandon | Accelerated aging of pmr-15 resin at elevated pressure and/or temperature[END_REF][START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF]. As an example, the evolution with aging time of tensile failure strain of PMR-15 aged in air at atmospheric pressure and under 0.414 M P a at 288 °C [START_REF] Tandon | Accelerated aging of pmr-15 resin at elevated pressure and/or temperature[END_REF] and of four-point bending failure stress of PR520 epoxy resin (T g ≈ 160°C) aged at 120 °C in air and under 2 bar of oxygen [START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF] are reported in Figure 1.28. In a recent work, Ernault et al. [START_REF] Ernault | Origin of epoxies embrittlement during oxidative aging[END_REF] found that the sub-transition (or β transition) reduction with aging is the only molecular modification that can be associated to embrittlement for two epoxy systems, DGEBA/IPDA and DGEBA/TTDA, that undergoes chain scission and crosslinking respectively during thermal aging, as represented in Figures 1. This finding shows that embrittlement is not directly related to chain scission, but to the reduction of the secondary transition to which plastic deformations of epoxy networks are associated [START_REF] Heux | Dynamic mechanical and 13c n.m.r. investigations of molecular motions involved in the beta relaxation of epoxy networks based on dgeba and aliphatic amines[END_REF].

According to what previously said, the reduction of failure stress and strain of oxidized samples is mainly attributed to the oxidized layer, as the core of the sample is not affected by thermal oxidation. Being the characterization of mechanical properties gradient due to thermal aging one of the objective of this thesis, a literature review of this specific aspect will be proposed in the following Section.

Mechanical property gradients in thermally aged polymers

As reported in Figure 1.26, thermal oxidation causes the formation of an oxidized layer that extends from the external surface directly exposed to the oxidative environment through the sample thickness for several hundred of micrometers. The evolution of material elastic local properties in the oxidized layer of various polymers and aging conditions has been studied in numerous previous works [START_REF] Pochiaraju | Evolution of stress and deformations in high-temperature polymr matrix composites during thermo-oxidative aging[END_REF][START_REF] Johnson | Investigation of oxidation profile in pmr-15 polymide using atomic force microscope (afm)[END_REF][3][4][5][START_REF] Jiang | Enhanced mechanical and thermal properties of monocomponent high performance epoxy resin by blending with hydroxyl terminated polyethersulfone[END_REF][START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF]. In these studies [START_REF] Pochiaraju | Evolution of stress and deformations in high-temperature polymr matrix composites during thermo-oxidative aging[END_REF][START_REF] Johnson | Investigation of oxidation profile in pmr-15 polymide using atomic force microscope (afm)[END_REF][3][4][5][START_REF] Jiang | Enhanced mechanical and thermal properties of monocomponent high performance epoxy resin by blending with hydroxyl terminated polyethersulfone[END_REF][START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF], thermal agings were performed at temperatures lower than glass transition, and thus in a regime of pure thermal-oxidation. The experimental technique mainly used is the instrumented indentation test, that allows to trace the evolution of elastic indentation modulus from the external surface to the core of the sample, highlighting the presence of a property gradient through the sample thickness. An example is reported in Figure 1.30a. As in the most of the cases, an increase of indentation modulus is observed near the surface directly exposed to the oxidative environment, where it reaches its maximum value. Then, the indentation modulus decreases in the oxidized layer and reaches a value close to that of the virgin material at the sample core. The increase in indentation modulus at the sample surface is higher for stricter aging conditions. Moreover, the indentation modulus profiles allow to estimate the thickness of the oxidized layer. In Figure 1.30b, the profiles of the concentration of the oxidation products, obtained numerically solving the mechanistic model of thermal oxidation below glass transition temperature of highly cross linked epoxy resins proposed by Colin et al. [START_REF] Colin | A new method for predicting the thermal oxidation of thermoset matrices. application to an amine crosslinked epoxy[END_REF], are represented. By plotting the elastic indentation modulus as a function of concentration of oxidation products, Olivier et al. [3] found that all the points are situated on the same master curve, regardless of the position through the thickness and the aging time (Figure 1.31). This result shows that elastic indentation modulus, in a regime of pure thermal oxidation, is an indicator of the oxidation degree.

Vu [START_REF] Vu | Endommagement Induits par la Thermo Oxydation dans les Composites Carbone/Epoxy Unidirectionnels et Stratifiés[END_REF] showed that the use of moderate oxygen pressures allows to accelerate the degradation phenomena due to thermal oxidation, including the increase in indentation modulus near the sample surface. In Figure 1.32a, the maximum variation of the indentation modulus with respect to that of the virgin material (defined as γ max by Minervino [START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF]) is reported as a function of aging time, in the case of aging in air and under 2 bar of oxygen pressure [START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF]. From Figure 1.32a, it can be observed that, in the case of aging under oxygen pressure, lower aging time allows to reach higher values of γ max . The comparison of the two curves of Figure 1.32a, allows to determine an acceleration factor λ(p) from which the equivalent aging time t * can be calculated as:

t * = t λ(p) (1.30)
The equivalent aging time is the time that would be needed to reach a given γ max if the aging would be performed in air instead of oxygen pressure, at the same temperature. Thanks to this equivalent time of aging in air, is it possible to trace a master curve of the evolution of γ max allowing to predict degradation for longer aging duration, as reported in Figure 1.32b. Minervino [START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF] showed that, during thermal oxidation, residual strain of chemical origin develops in the oxidised layer. When the specimen is cut crosswise, the residual stress field relaxes and causes a shrinkage of the material at the oxidized layer. This shrinkage is higher for longer or more severe aging conditions, as shown in Figure 1.33, where it is plotted against the distance from the exposed surface for different aging duration, obtained on Tactix epoxy resin samples through interferometric microscope measurements.

The effects of thermal aging on polymer matrix carbon fiber reinforced composites

The residual stress field of chemical origin due to thermal oxidation that develops in the polymer [START_REF] Ho | Endommagements de composites carbone/époxy induits par la thermooxydation de la matrice en présence de contraintes mécaniques[END_REF][START_REF] Vu | Endommagement Induits par la Thermo Oxydation dans les Composites Carbone/Epoxy Unidirectionnels et Stratifiés[END_REF][START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF], results in a matrix shrinkage among fibers, schematized in Figure 1.34, that increases with aging time. The evolution of matrix shrinkage with aging time and for different oxygen pressures has been extensively studied in [START_REF] Vu | Experimental characterization of thermo-oxidation-induced shrinkage and damage in polymer-matrix composites[END_REF] through observations with Scanning Electron Microscopy (SEM) and interferometric microscopy of the surfaces of unidirectional [0] 40 carbon fiber epoxy matrix composites in the direction perpendicular to fibers. As an example, Figure 1.35 depicts the interferometric microscopy images of the same composite sample zone, at the virgin state and for different aging time at 150 °C under 5 bar of oxygen. From Figure 1.35 it can be observed that the matrix shrinkage increases with aging time and that is no uniform on the whole observed zone, but increases with the distance among fibers. Moreover the shrinkage increases if the partial oxygen pressure employed during aging increases [START_REF] Vu | Experimental characterization of thermo-oxidation-induced shrinkage and damage in polymer-matrix composites[END_REF], as can be observed from Figures 1.36a and 1.36b where the maximum matrix shrinkage depth is plotted against the distance between fibers for a partial oxygen pressure of 1.7 bar and 5 bar respectively. Matrix shrinkage, together with its embrittlement and the increase of stress field of chemical origin could lead, with the increase in aging time, to a fiber/matrix debonding phenomenon, as oberved by Vu [START_REF] Vu | Endommagement Induits par la Thermo Oxydation dans les Composites Carbone/Epoxy Unidirectionnels et Stratifiés[END_REF] on unidirectional carbon fiber reinforced polymer matrix composites after 1000 h of aging in air at 150 °C. The fiber/matrix debonding is clearly visible on SEM images of the surface of the composite for fiber-to -fiber distances higher than 15 µm (Figure 1.37). Fiber/matrix debonding phenomenon could create new paths to oxygen to penetrate in the core of the composite sample. In this case, not only the external layers of the sample would be affected by thermal-aging, but degradation phenomena could be observed also in the inner part of the sample.

Conclusions

The literature review proposed in this Chapter allows to draw some conclusion and to define the objectives of this thesis work:

1. Time-independent properties like elastic indentation modulus and hardness derived from load-displacement indentation curves are not enough to describe the complex time-dependent behaviour of polymers. For that reason, indentation creep tests and dynamic indentation tests, that allows to derive material creep compliance and storage and loss moduli respectively, are preferred to a simple load-unload indentation cycle to characterize polymers. However, the interpretation of the results require the assumption of linear viscoelasticity;

2. For a distance adequately high from fibers, indentation tests are one of the few techniques that allow to determine actual matrix properties in-situ. The studies in which indentation tests have been applied to polymer matrix composites are limited to the comparison of time-independent properties (indentation modulus and hardness) between neat and in-situ material. Moreover, the carbon-fiber reinforced composite materials employed in previous works are unidirectional.

3. Cyclic tests seem relevant to obtain and characterize polymer response at different time-scales: instantaneous, one-cycle and long time corresponding to the whole test of many cycles;

4. Up to now, at our knowledge, cyclic solicitation in indentation tests have been mainly applied on glass and metals to study crack propagation and evaluate material toughness;

5. Thermal oxidation induces in polymer materials macromolecular modifications that compromise their failure properties. These modifications are concentrated in the oxidized layer that is developed on the sample surface and extend through the thickness for several hundred of micrometers. Because of the relatively small size of the affected zone, here again indentation tests turned out to be the most suitable experimental technique to measure property gradients. However, up to now, only time-independent elastic indentation modulus property gradients have been evaluated. Moreover, the most of the studies on this subject were carried out for aging temperatures lower than glass transition temperature and thus in a regime of pure thermal oxidation;

6. The degradation phenomena occurring in thermally aged polymers will therefore affect polymer matrix composite materials subject to severe environmental conditions. The main topological effects of thermal aging that have been observed on polymermatrix composites are matrix shrinkage, fiber-matrix debonding and matrix cracking. However, there is a lack of characterization of local mechanical properties in the matrix of thermally aged composites.

The main objectives of this thesis work are therefore:

1. To develop an indentation test method able to capture the time-dependent behaviour of polymers; 2. To characterize the environmental effects on the local-time dependent behaviour of an epoxy resin for an aging temperature closer to real operating conditions, that could be higher than the glass transition temperature; 3. To characterize local time dependent properties of the same material as a matrix of a 3D interlock carbon fiber reinforced composite and to study the effect of thermal aging on the local properties of the resin in-situ.

Chapter 2

Setting up of an instrumented indentation experiment for polymers

On the basis of the literature review proposed in the previous Chapter, it is clear that one of few techniques that allows for an experimental local characterization of mechanical property gradients in thermal aged polymers and of the matrix in virgin and thermal aged polymer-matrix composites is the instrumented indentation. However, in order to obtain reliable instrumented indentation test results, some precautions must be taken in the test setting up to avoid the many sources of error that can affect the measurements. In this Chapter, the material and the instrumented indentation testing machine used in this study will be at first presented.

Then, the sources of error that can affect the results will be outlined and the methods used to correct or avoid them described. The sources of error considered are: tip blunt, instrument compliance, thermal drift, surface roughness and tip/sample misalignment.

In the last Section, we will focus on the application of instrumented indentation on polymers. The results of indentation tests on a semi-crystalline polymer, the High Density Polyethylene, carried out at different loading rates, will be presented and the loading and unloading curves analysed to highlight how the time-dependency of material behaviour reflects on indentation curves.

Polymer materials

The polymer materials that will be studied in this thesis work are the High Density Polyethylene (HDPE) and the PR520 epoxy resin.

The HDPE (PE100) is used for the production of pipes for pressurised fluids, an application that requires material properties stability over the time. The HDPE is a semi-crystalline polymer (presenting a crystalline and an amorphous phase) with a simple chemical structure. HDPE samples for instrumented indentation tests have been cut from tensile test samples and have a thickness of about 2 mm. Its degree of cristallinity, determined from Differential Scanning Calorimetry (DSC), is 61% and its melting temperature is The PR520 epoxy resin is a thermoset polymer used as matrix of carbon fiber reinforced composites, commonly employed in the aerospace industry. It has been provided by Cytec Engineered Materials in the form of thick plates (about 1 cm) from which several samples with a thickness of about 3.5 mm have been obtained.

The main PR520 epoxy resin mechanical properties at room temperature and its glass transition temperature, extracted from the manufacturer technical data sheet [103], are reported in Table 2

Presentation of the equipment

The instrumented indentation test machine used in this study is the Fischerscope H100C, presented in Figure 2.1, and equipped with a base square pyramid diamond tip Vickers. This equipment allows to perform force controlled tests in a range between 0.1 and 1000 mN with an uncertainty of ±0.02 mN . The maximal measurable depth is 700 µm and the uncertainty on the displacement values is ±2 nm. During the tests, load and displacement data are recorded every 0.1 s and the maximum allowed testing time is about 6000 s.

Instrumented indentation test at microscopic and nanoscopic scale is a sensitive technique that could be affected by many sources of error. In the following, some of these sources will be presented and the methods to correct or avoid them will be outlined.

Area function calibration

In Chapter 1, it has been shown that the estimation of the projected area of contact is necessary to calculate both material elastic indentation modulus and hardness (Equations 1.7 and 1.8). However, if the real shape of the indenter deviates from the ideal one, for example because of tip blunt, the use of Equation 1.6 to calculate the projected area of contact could lead to some error in the further evaluation of indentation modulus and hardness, especially at low depths. This results in modulus and hardness values that increase with decreasing indentation depth. This phenomenon is known in the literature as Indentation Size Effect (ISE) and has been observed on both polymers [START_REF] Chong | Strain gradient plasticity effect in indentation hardness of polymers[END_REF][START_REF] Tang | Accurate measurement of tip-sample contact size during nanoindentation of viscoelastic materials[END_REF][START_REF] Han | Indentation size effects in polymers and related rotation gradients[END_REF][START_REF] Alisafaei | Characterization of indentation size effects in epoxy[END_REF] and metals [START_REF] Nix | Indentation size effects in crystalline materials: a law for strain gradient plasticity[END_REF][START_REF] Ma | Size dependent hardness of silver single crystals[END_REF][START_REF] Fleck | Strain gradient plasticity: Yheory and experiment[END_REF]. Indentation Size Effect is due, not only to a bad estimation of the projected area of contact due to tip blunt, but also to surface roughness, which will be discussed later, and to the energy dissipated at the surface [START_REF] Zhang | Surface effects on nanoindentation[END_REF]. The ISE is due to large strain gradients present in small indentations that causes geometrical necessary dislocations in the case of metallic materials [START_REF] Fleck | Strain gradient plasticity: Yheory and experiment[END_REF] and an increase in rotational gradients in the case of polymers [START_REF] Alisafaei | Characterization of indentation size effects in epoxy[END_REF], resulting in an increase in hardness.

In order to avoid inaccuracies due to tip blunt on the evaluation of the projected contact area, it is necessary to determine a new equation that relates A p and h c that takes into account the real shape of the indenter. The indenter area function calibration procedure has been carried out by following the method suggested by Oliver and Pharr [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] as well as the standard ISO 14577-2. It consists of making a series of indentation in the same point at different load levels in a reference material that doesn't exhibit material dependent indentation size effects and for which the Elastic Indentation Modulus and Poisson's ratio are known. The loading history used in our procedure is showed in Figure 2.2. As suggested by Oliver and Pharr [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF], loading/unloading rate was changed from a load level to another in order to keep the loading/unloading time constant at 2s and to have the same number of data points in each unloading curve. In order to evaluate thermal drift, a 60s hold period at 25mN was introduced at the end of the test. In our case, the BK7-glass was used as reference material for which Young's modulus and Poisson's ratio were found in the literature to be respectively 82 GP a and 0.21 [START_REF] Antunes | Ultramicrohardness testing procedure with vickers indenter[END_REF].

Knowing these values, it's possible to calculate the projected contact area for each load level (Equation 1.4), plot it as a function of the contact depth and fit these points with a function in such a form (Equation 2.1):

A p = C 0 h 2 c + C 1 h c + C 2 h 1 /2 c + . . . + C 7 h 1 /64 c + C 8 h 1 /128 c (2.1)
The values of the coefficients C i of Equation 2.1 determined from fitting are reported in The experimental A p vs h c data obtained from the application of loading history of Figure 2.2 (black data points) on BK7-glass, together with their fitting function of Equation 2.1 (red line) and the ideal projected area of contact for a perfect Vickers tip, calculated with equation 1.6 (blue line) are represented in From Figure 2.3 it can be noted that considering the shape of the indenter as a perfect Vickers (blue line), leads to an underestimation of projected area of contact and, consequently, to an overestimation of Hardness and Indentation Modulus. Moreover, it can be observed that ideal and real area functions tend to overlap for higher contact depth values. It must be underlined that the calculated area function should not be used for penetration depths lower than the minimum penetration depth used for fitting in the calibration procedure that, in this case, corresponds to a contact depth of about 0.2 µm.

To validate the procedure, some tests have been carried out on another material that does not exhibit material dependent indentation size effect, the Fused Silica [START_REF] Parr | Fused Silica sample certificate[END_REF], using the same loading path of Figure 2.2. The Elastic Indentation Modulus calculated before (red data points) and after calibration (blue data points) for each maximum penetration depth are shown in Figure 2 From Figure 2.4 it can be observed that the important Indentation Size Effect on the results before calibration, disappeared after calibration and the average calculated modulus value (70.2 ± 0.4 GP a) is close to the certified one for our sample (73.5 ± 0.3 GP a) [START_REF] Parr | Fused Silica sample certificate[END_REF].

Verification of instrument compliance

When the indenter is loaded, an equal and opposite reaction force is applied on the load frame, resulting in a deflection of the whole instrument [START_REF] Fischer-Cripps | Nanoindentation. Mechanical Engineering[END_REF]. This deflection is measured by depth sensor and is added to the depth measurement on the sample. The effect of instrument compliance on the experimental data is more important as the sample is stiffer. The instrument compliance is usually modelled as a linear spring and is thus represented by a single value. One of the methods allowing to determine the instrument compliance consists in plotting the reciprocal values of the indentation Stiffness (Eq. 1.3), obtained from a series of at least 5 measurements at different loads on a reference material for which the value of elastic modulus is known, against the reciprocal of the root square of the contact area A p . The plot of 1 S vs 1 √ Ap yields a straight line whose intercept with y axis is the compliance of the instrument.

For the testing machine used in this study, displacement data are supposed to be already corrected for instrument compliance by the software. However, the compliance of our instrument has been verified by using the results obtained by applying the loading path for area function calibration (Figure 2. From Figure 2.5 it can be observed that the 1 S vs 1 √ Ap data are well fitted by a straight line whose intercept is the origin of the axis, meaning that the frame compliance of our instrument is well corrected by the software and negligible with respect to the Fused Silica stiffness and thus, a fortiori, for the less rigid polymer samples.

Thermal drift measurement and correction

Thermal drift is almost unavoidable if the temperature of the environment surrounding the instrument is not carefully controlled [START_REF] Fischer-Cripps | Nanoindentation. Mechanical Engineering[END_REF]. In fact, any temperature change in the environment causes the expansion or the contraction of the indenter, the sample and the apparatus. As the depth sensor is usually very sensitive, any variation in size is recorded during the test and added to the displacement data, causing a large error on the loaddisplacement curves and, consequently, on the quantities deduced from the indentation curves.The best way to avoid thermal drift is to place the indenter in an insulated chamber. However, even in this case, a temperature change deriving from the machine operating may occur.

In the specific case of the testing machine used in this study, thermal drift represents an important source of error as the indentation apparatus is exposed to the environment which temperature is difficult to control. Moreover polymers, that are the subject of this thesis, are more sensitive to temperature change because of their high thermal expansion coefficient. For example, the coefficient of thermal expansion of High Density Polyethylene is 120 In order to evaluate thermal drift, the standard ISO 14577-2 proposes to introduce a hold period -that should last at least the same time of load application-at the maximum load or at the end of unloading, perform a linear fitting of the displacement vs time plot and use the slope of this line (thermal drift parameter T D) to correct displacement values h i at each time t i as follows:

h i T D = h i -t i T D (2.2)
This correction method is appropriate for metals and glasses which do not exhibit time dependent behaviour. In fact, for this kind of materials, any change in depth during constant load hold period is due to thermal drift. However, in the case of polymers, material time dependent phenomena occur while keeping the load constant: creep at maximum load and recovery after the unload. It would be thus preferable to evaluate thermal drift by introducing a hold period at very low load, before the loading.

A careful study has been conducted on different materials (polymers and not) in order to choose the load level, the duration of the holding period and the fitting function of time-displacement data to be used for thermal drift evaluation. The choice was the best compromise between the reduction of the standard deviation among the results and the applicability to a large range of tests of different duration. The load value retained is 0.1 mN and the duration 600 s. The displacement-time data collected during holding at From Figure 2.6 it can be observed that thermal drift is significant and varies widely from a test to the other. For this set of tests, the trend of thermal drift is negative and reaches negative displacement values, which should correspond to an expansion of the sample and/or a contraction of the tip. Moreover, one can see that the evolution of displacement during holding with time is not exactly linear. It has been actually observed that the use of a power law fitting instead of a linear fitting of displacement-time data during holding, results in a better correction for the longer tests. In fact, as in the linear correction the corrected displacement increases linearly with time, for long testing time, the entity of correction could exceed the measured displacement values, resulting in a high deviation of the results. The procedure that has been retained for thermal drift correction in our study is, therefore:

1. Measure thermal drift through a load-hold period of 600 s at 0.1 mN before loading, 2. Fit the time-displacement data collected during holding with a power-law function of the type:

h = a t b + c (2.3)
where a, b and c are the parameters determined through a non linear last square fitting procedure. To allow for a viscoelastic stabilisation, the first 20 s are excluded from the fitting.

3. Propagate the correction to the whole collected displacement data as:

h i T D = h i -(a t b i + c) (2.4)
4. Shift the curve to cancel the displacement accumulated during holding by matching the first point of loading with the last point of holding. From Figure 2.7a it can be observed that, before the application of thermal drift correction, the displacement-time data of the 9 tests are highly scattered. On the contrary, when the correction is applied (Figure 2.7b) the scatter is very low and is virtually constant for the whole indentation creep test. Based on these results, our thermal drift correction procedure is validated and applied for all the tests presented in the manuscript.

Roughness effect and sample preparation

The factors affecting instrumented indentation measurement that have been described so far, derive from defects of the instrument or environmental effects. However, even the sample surface state could impact the experimental results. In fact, if the roughness is too high, the contact area is no longer a cross section because of its irregular perimeter and a high scatter on the results is observed. To avoid the effect of roughness, according to the Standard ISO 14577:1, the average roughness (R a ) should not exceed 1 20 of the penetration depth. To improve the surface state, the polymer samples have been mechanically polished by grinding and finishing. During the grinding phase, SiC sheets with incrementally finer grades have been used with water as lubricant. For finishing, micro cloth sheets on which a diamond suspension has been vaporised have been used. In this stage, a specific lubricant has been employed. The different steps followed for sample preparation are reported in Using this protocol a shiny smooth surface has been obtained for polymer and composite materials.

Misalignment between indenter and sample surface

In the previous Section it has been shown that the surface state, and in particular roughness and planarity, could be improved by using an adequate polishing protocol. However, it may occur that roughness and tilt are an inherent characteristic of the sample and could not be avoided. That is the case for thermally aged polymers and polymer matrix composites, that will be the subject of this thesis. In fact, as observed in previous works [START_REF] Ho | Endommagements de composites carbone/époxy induits par la thermooxydation de la matrice en présence de contraintes mécaniques[END_REF][START_REF] Vu | Endommagement Induits par la Thermo Oxydation dans les Composites Carbone/Epoxy Unidirectionnels et Stratifiés[END_REF][START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF] and as reported in Chapter 1, during aging, a residual stress field of chemical origin develops in the polymer, that results in a matrix shrinkage among fibers that increases with aging time (Figure 1.34). Even in the case of thermally aged pure polymers (not reinforced), it has been shown [START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF] that when the specimen is cut, the stress field of chemical origin due to aging, relaxes and causes a shrinkage of the material at the oxidized layer. Polymer and polymer matrix shrinkage can lead to some errors during indentation tests as the surface of the sample is locally not perpendicular to the indenter. As found by [START_REF] Shafirstein | Error analysis in nanoindentation[END_REF][START_REF] Xu | Effect of sample tilt on nanoindentation behaviour of materials[END_REF][START_REF] Pelletier | Mechanical Characterization of Glassy Polymers using Instrumented Indentation[END_REF][START_REF] Kashani | Sources of error in relating nanoindentation results to material properties[END_REF], the tip/sample misalignment, produces an higher contact area than that calculated with Eq. 1.6 or 2.1, due to the effects shown in Figures 2.8a A misaligned contact results in a translation of the load-displacement curves and an overestimation of hardness and elastic modulus, similarly to the effects produced by a blunt indenter. The standard ISO 14577-1 allows a maximum tilt angle between indenter and sample surface of 1°, while some authors found that acceptable results could be obtained even with higher tilt angles, namely 3°for hard materials and 4°for soft materials [START_REF] Xu | Effect of sample tilt on nanoindentation behaviour of materials[END_REF]. In fact the entity of the error on the area function depends not only on the inclination but also on the sample material ( E /H ratio) and on the type of indenter (geometry and semi-cone angle). In order to evaluate the effects of sample tilt on our indenter/material system, some tests have been realized by fixing the sample on a tilt platform and varying the inclination between specimen and tip.

Then, using the interferometer Talysurf CCI, the topography of resin pockets on the surface of aged composites and on the cross section of aged polymers is obtained. The local slopes are measured on the profiles traced on the matrix among fibers (in the case of composites) and on the profiles from the exposed surface to the sample core of the aged pure polymer to evaluate this possible error.

Effect of sample tilt on instrumented indentation measurements

Thanks to a tilt platform (showed in Figure 2.9) it has been possible to perform tests by varying the inclination angle between sample and indenter. The inclination on x and y axis, that contain the plane of the sample, is adjusted through two micro-metric screws on the bottom of the platform. The inclination angle was calibrated with the displacement of the screws. Some tests have been performed on virgin PR520 epoxy resin sample by varying the inclination angle on x (0°, 1.8°, 3.7°, 5.6°a nd 7.5°) and on y (0°, 1°, 2°, 3°, 4°and 5°) separately. For these experiments, maximum load has been set to 10 mN and loading and unloading rate to 5 mN s . A holding period of 20 s at maximum load has been introduced in order to reduce viscoelastic effects during unloading [START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF]. The indentation modulus calculated for different rotation angles about x (green) and y (red) axis is represented in Figure 2.10. From Figure 2.10 it can be observed that, for both x and y axis, the indentation modulus starts to deviate to that evaluated for a 0°inclination angle, from a misalignment of 4°. So, for this specific sample/ indenter system, the maximum acceptable inclination angle during indentation tests is 3°. In fact, even if the modulus calculated for an inclination of 4°is just slightly higher, some problems have occurred during the tests at this angle; in some cases we had to discard the results among the set of data as the indenter probably slipped on the surface without penetrating it, resulting in a set of data in which the displacement increases while the force remains almost zero. Some comparative tests have been performed on Fused Silica by using the same loading path and by varying the tilt angle from 0°to 5°both on x and y. In Figure 2.11, is represented the percentage variation of E * from the value obtained for a 0°tilt, for PR520 (green symbols) and Fused silica (red symbols). and Fused Silica (red symbols)

From Figure 2.11 it can be observed that, while the indentation modulus percentage variation for PR520 is below 1% up to a tilt angle of 3°, in the case of Fused Silica for a 3°i nclination, the variation is already about 2%, confirming that the effects of stage rotation depend also on the type of material tested and in particular on the E /H ratio.

The results presented in this Section showed that, to avoid errors in contact area estimation and to guarantee a good contact between sample and indenter, tilt angle for the PR520 epoxy resin should not exceed the value of 3°.

Relevance of polymer and polymer-matrix shrinkage due to thermal aging on instrumented indentation tests

In this Section, the slopes existing on the surfaces of thermally aged polymers and polymer matrix composites will be quantified in order to estimate the distance from fibers (in the case of composite surfaces) and from exposed surface (in the case of polymer cross section) for which the slope is lower than the maximum acceptable tilt angle of 3°.

Surface topographies have been measured with the interferometric microscope Talysurf CCI 6000, developed by Taylor and Hobson. A magnification of 50× have been used, allowing to observe a 360 × 360 µm 2 surface with a spatial resolution of 0.35 µm. With this kind of magnification, the maximum measurable slope with respect to the horizontal plane is 27.7°.

For the microscopic observations and the further instrumented indentation tests, the aged samples have been cut and the two sides embedded together and polished, as shown in Figure 2.12. Figure 2.12b depicts the topography of the cross section of the PR520 epoxy resin aged for 72 hours in air at 180 °C. In the upper part of Figure 2.12b, the embedding resin, that penetrated between the two sides of the sample, is visible. Moreover, it is possible to observe a gradient of colour representing the shrinkage (along the z direction) near the sample edge, namely through the oxidized layer, that is constant on the y direction. No colour gradient is actually observed towards the sample core. In order to quantify the slope angles, a profile has been traced along the cross section, following the arrow represented in Figure 2.12b, from the edge to the core, as represented in Figure 2.13. From Figure 2.13 it can observed that, as expected, the maximum depth is at the edge of the sample and is of about 1.7 µm. Then, it decreases and stabilizes at 0 µm at a distance of about 60 µm on the x direction, which should correspond to the thickness of the oxidized layer for the considered aging condition. The slope has been evaluated by dividing the profile in segments equally spaced on x (vertical lines in Figure 2.13) and measuring the angle between the segment and the x axis. The distance between the middle of each segment along x is of 11 µm, which corresponds approximately to the diagonal of the Vickers print on PR520 for the most used testing load in our study of 10 mN . For the segment with the higher slope, the measured tilt angle is of 0.75°. This value is lower of both the upper limit prescribed by the standard and the maximum admissible tilt angle of 3°for our system found in the previous section. The shrinkage of the cross section of PR520 epoxy resin aged samples due to relaxation of residual stress of chemical origin, should not pose problem of misalignment between sample and tip.

The same method has been used to measure the slopes on the surfaces of aged composites due to matrix shrinkage among fiber bundles. The topography of a resin pocket on the surface of a composite sample aged at 180°C in air for 96 hours is depicted in Figure 2.14. The four profiles represented in Figure 2.15a are virtually superposed and, here too, the slope has been measured on segments equally spaced on the x direction (parallel to the profile) of 11 µm. The evolution of the slope as a function of the x coordinate is represented in Figure 2.15b. It can be observed that, close to fiber bundle, the slope exceeds the maximum admissible value of 3°. Then, it gradually decreases up to an almost null value at the center of the pocket, reaching the value of 3°at a distance from fiber bundle of about 50 µm. This distance from fibers for "safe" instrumented indentation tests will be retained in the following in which less severe aging conditions will be employed, and thus supposed to produce a lower shrinkage.

Instrumented indentation tests on polymers

In this Section, the effect of loading rate on load-displacement curves of polymers will be studied. The material used is the High Density Polyethylene (HDPE), since it is above glass transition temperature at room temperature (T g = -116°C) and expected to show more pronounced time-dependency with respect to the epoxy resin.

The loading rates used are 0.2, 2 and 20 mN s and the maximum load was set at 10 mN for the three rates. Data has been corrected for thermal drift according to the procedure described in Section 2.5. In Figure 2.16 the force-displacement mean curves of a set of at least six tests for each loading rate are represented. From Figure 2.16, the first evidence is that the use of lower loading rates leads to higher penetration depths for the same maximum applied load: the higher the loading time is, higher the time-dependent deformations are. The loading curve is thus no longer described by a quadratic relation as for elasto-plastic material (Equation 1.1), but by a more general expression of the type (Equation 2.5)

F = C h n (2.5)
in which C is still a constant related to the material, while n depends on loading rate.

The loading curves with their fitting functions obtained on HDPE for the three loading rates, together with the theoretical curve in which n = 2, are represented in Figure 2.17. From Figure 2.17 is can be observed that, as expected, the loading curve for the higher loading rate is quite close to the theoretical one, with an exponent n resulting from the fitting equal to 1.94 ± 0.04. As loading rate decreases, the curves deviates from the theoretical one and the values of the exponent n decrease: n = 1.74 ± 0.02 for 0.2 mN s and n = 1.71 ± 0.02 for 0.02 mN s . The second observation from Figure 2.16 concerns the round shape of the unloading curve at the correspondence of the peak load. As reported in Chapter 1, many authors [START_REF] Vanlandingham | Nanoindentation of polymers: an overview[END_REF][START_REF] Beyaoui | Analysis of nanoindentation curves in the case of bulk amorphous polymers[END_REF][START_REF] Hochstetter | Strain-rate effects on hardness of glassy polymers in the nanoscale range. comparison between quasi-static and continuous stiffness measurements[END_REF][START_REF] Oyen | Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials[END_REF][START_REF] Chudoba | Investigation of creep behavior under load during indentation experiments and its influence on hardness and modulus results[END_REF] observed this phenomenon on polymer load-displacement curves, known in the literature as "nose" effect. A phenomenon similar to the "nose" effect, has also been observed on macroscopic cyclic compression tests of polymers [START_REF] Zhang | A simple constitutive model for cyclic compressive ratchetting deformation of polyteterafluoroethylene (ptfe) with stress rate effects[END_REF][START_REF] Rabemananjara | Caractérisation expérimentale des mécanismes d'endommagement du polyéthylène moyenne densité (pemd) sollicité en fatigue par compression. 36emes Journées de Printemps[END_REF]. It has been quantified by defining the viscoelastic delayed strain as the difference between the "real" maximum strain and the strain at the maximum stress (Equation 2.6):

∆ε d = ε T max -ε T σmax (2.6)
The authors [START_REF] Zhang | A simple constitutive model for cyclic compressive ratchetting deformation of polyteterafluoroethylene (ptfe) with stress rate effects[END_REF][START_REF] Rabemananjara | Caractérisation expérimentale des mécanismes d'endommagement du polyéthylène moyenne densité (pemd) sollicité en fatigue par compression. 36emes Journées de Printemps[END_REF] found on PTFE and MDPE respectively that ∆ε d is significant in cyclic compression tests and is sensitive to loading rate. In order to quantify the "nose" in indentation tests, a viscoelastic delayed displacement could be defined similarly to the viscoelastic delayed strain in macroscopic compression tests (Equation 2.7):

∆h d = h max -h Fmax (2.7)
The values of ∆h d calculated on the unloading curve for the loading rates of 20, 2 and 0.2 mN s are reported in Figure 2.18. From Figure 2.18 it results that the "nose" is more pronounced for the slower test, while it is absent in the test performed at 20 mN s . This result is coherent with the suggestion of many authors [START_REF] Vanlandingham | Nanoindentation of polymers: an overview[END_REF][START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF][START_REF] Oyen | Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials[END_REF] to perform indentation tests using high loading rates in order to minimize creep effects during unload.

Load-displacement curves have been then analysed according to the Oliver and Pharr analysis method [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] in order to estimate the indentation modulus (Equation 1.7). A script has been implemented in Matlab®for the analysis of experimental data. For the fitting procedure, the non linear least square method has been used. The unloading curve from 20% to 98% was fitted as suggested by the standard ISO 14577-2. To improve the fit quality, start point, lower bound and upper bound have been fixed for the three fitting constants and their values are reported in Table 2 The E * values for the three loading rates are reported in Figure 2.19. The indentation moduli are different in the three cases, however there is no correlation with loading rate. This observation suggests that the difference among the E * is related to the uncertainties induced by the fitting procedure of the unloading curve and is thus not exactly an effect of loading rate.

To confirm this statement, the unloading curves are plotted with their fitting functions in From Figure 2.20 we can observe that the fitting is poor close to the maximum load, where the Stiffness is evaluated. This is not only for the lower rates in which the unloading curves are affected by the nose, but also for the higher rate in which creep during unloading is absent. The three fitting coefficients A, h p and m of Equation 1.2 are represented in Figures 2.21a From Figures 2.21a, 2.21b and 2.21c, a very high scatter in the fitting coefficients for the higher rate is observed and that is probably due to the low number of data points recorded during the test and used for the fitting. From Figure 2.21b it results that the mean value of the final unloading depth determined from power law extrapolation ranges between 0.02 µm and 0.1 µm depending on rate. However, looking at the experimental load-displacement curves of Figure 2.16 it results that the final depth should range between about 0.6 µm and 1.7 µm. Finally, the power law exponent m (Figure 2.21c) is similar for the three loading rates and its mean value is 6.6 ± 0.8. The value of the exponent m determined from fitting is higher than its theoretical upper limit of 2 determined by Sneddon [START_REF] Sneddon | The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile[END_REF], consistently with what have been found on polymer materials by other authors [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF][START_REF] Tranchida | Mechanical characterization of polymers on a nanometer scale through nanoindentation. a study on pile-up and viscoelasticity[END_REF][START_REF] Beyaoui | Analysis of nanoindentation curves in the case of bulk amorphous polymers[END_REF].

Investigation of material pile-up

The failure of the application of Oliver and Pharr analysis method [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] on polymer loaddisplacement indentation curves is commonly attributed not only to viscoelastic effects but also to the so called material pile-up. In fact, the Sneddon's analysis [START_REF] Sneddon | The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile[END_REF] is based on the assumption that the surface of the sample is drown downward during indentation or, in other words, that the material "sink-in". However, the imaging of the residual prints [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF][START_REF] Jin | Effects of experimental varaibles on pmma nano-indentation measurements[END_REF][START_REF] Kranenburg | Challenges and progress in high-throughput screening of polymer mechanical properties by indentation[END_REF][START_REF] King | Mechanical properties of graphene nanoplatelet/epoxy composites[END_REF][START_REF] Silva | Synthesis and characterisation of poly (lactic acid)/halloysite bionanocomposite films[END_REF] revealed that, in some cases, the material bulged out from the surface or, in other words, it "pile-up". If the material pile-up, the real contact area is higher than that predicted by the area function (Eq. 2.1) obtained through calibration on a non piling up material (BK7 in our case). For a better comprehension of piling-up and sinking-in phenomena, a scheme is proposed in Figure 2.22. Pile-up may occur either on metals or polymers. In an attempt to correct the projected area of contact from pile-up, Hardiman et al. [START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF] investigated the residual print of a 5 µm indentation test on 6376 epoxy resin. The Scanning Probe Microscope (SPM) images of residual prints showed contradictory results: the value of the projected area of contact directly measured from the images predicts sink-in behaviour because of its inward curvature, while some pile-up, even if relatively small, is visible around the residual impression. According to them, this behaviour is probably due to recovery of the material occurring after load remove and it makes problematic the estimation of the contact area and of the real amount of pile-up under load from imaging of the residual print. Oliver and Pharr [START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF] found by finite element simulations on elasto-plastic materials that pile-up occurs in materials with large Young's modulus to yield stress ratio and little capacity for work hardening. They have also observed that, no matter the work hardening behaviour of the material is, pile-up is not significant if the ratio of final depth to maximum depth is below 0.7. This criterion has been applied by Tranchida et al. [START_REF] Tranchida | Mechanical characterization of polymers on a nanometer scale through nanoindentation. a study on pile-up and viscoelasticity[END_REF] to investigate pile-up on two polymers, PC and mesomorphic iPP, at different load levels. They found that the final depth to maximum depth ratio was below the threshold of 0.7 on both PC and iPP for penetration depths below 100 nm. However, they found a h f hmax ratio of 0.8 on iPP for penetration depths in the range 100 -1900 nm showing that pile-up could affects the results at larger loads, being related to the amount of plastic flow.

According to this method, the final depth to maximum depth ratio has been calculated from unloading curves of HDPE (Figure 2.20) and for the three loading rates to find out if pile-up could have occurred during contact. The result is reported in Figure 2.23, in which the dashed black line represents the threshold for onset of significant pile-up according to Oliver and Pharr [START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF]. From Figure 2.23 it can be observed that the ratio

h f
hmax increases as the loading rate decreases, and that it is still lower than the threshold value of 0.7 even for the lower rate. It is thus possible to neglect piling-up phenomenon on HDPE.

We will anticipate here that even for the PR520 epoxy resin, that will be subject of study in Chapters 4 and 5, pile-up phenomenon should not be significant as the calculated h f hmax ratio, for a maximum load of 10 mN and a loading rate of 2 mN s , is 0.63 ± 0.02 µm that is slightly lower than the threshold value. It must be underlined that this conclusion is drown from a criterion arising form finite element simulations performed on material exhibiting elasto-plastic behaviour for values of yield stress to Young's modulus ratios that were varied in the range of typical values found on metallic materials. Further investigations are thus needed to verify its applicability to material exhibiting a visco-elasto-plastic response and for values of σy E closer to that experimentally found on polymers.

Conclusions

In this Chapter, the attention has been focused on the instrumented indentation testing technique and its application for polymer materials.

2. A method for the correction of thermal drift on long indentation tests adapted for polymers has been proposed;

3. A polishing protocol has been established to minimize surface roughness;

4. The maximum allowable surface tilt angle for our material/instrumented system is 3°. This threshold value is largely higher than the slopes measured on the oxidized layers of aged epoxy samples. However, in the case of aged composites, profile slopes can reach 10 • near fiber bundles. To avoid tilt during indentation of aged composites, the minimum distance from fiber bundles should be at least 50µm;

5. The indentation curves obtained on a semi-crystalline polymer, the High Density Polyethylene, for three different loading rates have been analysed. It resulted that, as expected, the material response changes with loading rate and the calculated values of indentation moduli are rate dependent, this being a material exhibiting time-dependent behaviour.

Chapter 3

Development of a cyclic indentation method for characterizing the time-dependent behaviour of polymers

The aim of this Chapter is to achieve the first objective outlined in Section 1.5, that is the development of an indentation test method suitable for the study of time-dependent behaviour of polymer materials and that will be used in the following Chapters to capture property gradients due to thermal aging and/or to the presence of fibers in 3D interlock carbon-fiber reinforced composites. According to the literature review proposed in Chapter 1, the unloading curves analysis method developed for elasto-plastic materials, that allows to determine elastic indentation modulus and hardness, assumes that the material recovers elastically during unloading. It has been shown that this hypothesis is not valid for materials exhibiting a time-dependent behaviour like polymers. Moreover, the complex time-dependent behaviour of polymer is not fully described by time-independet parameters like elastic modulus and hardness. The main indentation methods designed for the analysis of polymers (indentation creep test and dynamic indentation tests) allow, therefore, to derive creep compliance and storage and loss moduli only in a regime of linear viscoelasticity. This condition rarely occurs on polymers, especially if a sharp indenter is used. Sharp indenters (Vickers in this study) in fact, usually produce a permanent print because of the high stress concentration induced in the material [START_REF] Fischer-Cripps | Nanoindentation. Mechanical Engineering[END_REF].

In Chapter 1, it was also shown that stress-strain curves of macroscopic cyclic tests under different kind of loading (tensile, compression, shear...) of polymers present some peculiarities which reflect the time-dependency of material response. In particular, stressstrain cyclic curves are characterized by large hysteresis loops, a particular feature of time-dependent behaviour for which there is a delay between the applied stress and the respective strain response.

Inspired by macroscopic fatigue tensile tests, an instrumented indentation cyclic test is thus proposed here to study polymers behaviour at the microscopic scale and has been at first applied on HDPE, for which the response to macroscopic cyclic tension and shear solicitation is already known [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF].

In the first part of this Chapter, the cyclic indentation protocol and the parameters extracted from load displacement curves will be presented. The evolution with cycles of the experimental parameters obtained on HDPE for one loading rate will also be shown.

The response of HDPE to cyclic indentation is supposed to have some similarities with its cyclic macroscopic response. However, because of the differences in which the two tests are performed and in the volume of the material concerned by the loading, some differences could arise:

1. Macroscopic tests are stress controlled, which means that the maximum stress does not change from a cycle to the next one. Cyclic indentation tests are, instead, force controlled, which means that the maximum force does not change from a cycle to the next one, but if the contact area changes, the maximum stress under the tip will change as well during cycles.

2. The indentation curves are force-displacement, while in the macroscopic case the results of cyclic tests are expressed as the strain as a function of the applied stress.

3. In Nguyen's thesis work [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF], the maximum stress level was chosen in order to yield stress not to be exceeded during successive loadings while it is highly possible that yield stress is exceeded during indentation with a sharp tip.

4. The amount of material volume tested through instrumented indentation is of the order of few hundreds of µm 3 , while in macroscopic tests the whole cross section is affected by the same mean stress. This could induce high local stress gradients during sharp indentation that are not found in macroscopic stress controlled cyclic tests.

Despite of these differences, it is interesting to perform a qualitative comparison between macroscopic and microscopic cyclic behaviour of HDPE. After the test presentation, this comparison will be discussed through the evaluation of the effect of loading frequency on the cyclic indentation response.

Cyclic indentation test for polymers

The cyclic indentation test protocol proposed in this study is schematised in Figure 3.1. The loading path (Figure 3.1) consists of an initial load holding phase (0 -A) of 600 s at 0.1 mN allowing to evaluate and correct the thermal drift according to the procedure described in the previous Chapter. Then, a series of load-unload cycles between 0.5 and 10 mN at constant and similar loading and unloading rate follows. A holding period at 0.1 mN has been introduced at the end of some tests (D -E) to evaluate permanent and recoverable parts of the total displacement accumulated during the successive loadings.

It should be mentioned that ideally a recovery test is performed at zero load. However, this condition can not be achieved by the instrument setup. That is the reason why the recovery load was set to the smallest load applicable experimentally. In order to allow the recovery of the most of time-dependent viscoelastic deformations, the duration of the recovery phase has been set approximately to the same duration of the cyclic test (B -C ≈ D -E). Three loading frequencies of 1, 0.1 and 0.01 Hz have been used, which correspond to loading rates of 20, 2 and 0.2 mN s respectively. The duration of the test is also limited by the equipment at about 6000 s, which limits a maximum number of cycles to several thousands at 1 Hz, several hundreds at 0.1 Hz and just 60 at 0.01 Hz. The maximum number of cycles is divided by two if the recovery is studied. Since the cyclic behaviour is the main focus of this study, in most of the tests the recovery phase is not applied.

The cyclic indentation was firstly applied to an elasto-plastic material, the Fused Silica, for which the indentation response is well known. This material does not show any rate or indentation size dependency and is thus commonly used for the calibration of instrumented indentation equipments [START_REF] Fischer-Cripps | Nanoindentation. Mechanical Engineering[END_REF]. Three loading frequencies of 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) are applied and the results are reported in Figure 3.2. For this test, the maximum load was set to 10 mN while the number of cycles is 10 for the three frequencies. Unload and reload indentation curves of Fused Silica (Figure 3.2) are characterized by very small hysteresis loops, probably due to reverse plasticity as observed by other authors [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF][START_REF] Shuman | Calculating the elastic modulus from nanoindentation and microindentation reload curves[END_REF]. Moreover cyclic indentation response of Fused Silica is not affected by loading rates, as curves obtained at different frequencies are virtually superposed and the indentation modulus and hardness evaluated from the unloading curves are independent from both cycle number and loading rate.

A very different response to cyclic indentation loading is instead obtained on polymers, as can be observed from Figure 3.3, in which cyclic load-displacement curve of HDPE (300 cycles) at the intermediary frequency of 0.1 Hz is presented . The load-displacement curves obtained on HDPE (Figure 3.3) are characterized by large hysteresis loops that evolve with cycles as in the macroscopic tensile and shear tests shown in Section 1.2. As previously said, this is a typical feature of time dependent materials in which there is a lag between the applied load and the displacement response. The hysteresis loops of force-displacement cyclic indentation curves will be analysed in a similar way to that used by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] on shear and tensile macroscopic cyclic tests on the same material.

Parameters extracted from cyclic load -displacement indentation curves

The way to extract the parameters that will be used for the analysis of cyclic indentation curves is outlined in Figure 3.4 and described in the following. • The unload modulus (E * unload ) and the reload modulus (E * load ). They are calculated from unload and reload stiffness (S unl and S load ) respectively, according to the Oliver and Pharr [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] analysis method described in Chapter 1. Reload and unload stiffness are the slopes of unload and reload curves, evaluated at the maximum load from the power law fitting functions.

• The area of the hysteresis loop between load and unload curves at each cycle. The areas under load and unload curves are evaluated through a trapezoidal numerical integration. Loop area in load-displacement indentation curves is also known as the irreversible work of indentation [START_REF] Fischer-Cripps | Nanoindentation. Mechanical Engineering[END_REF] and represents the amount of energy dissipated at each cycle.

• The maximum, minimum and mean displacement accumulation with cycles (or time). They describe the long-term viscoelastic response, at the scale of the whole test, and are evaluated from each unloading curve.

• Assuming the stabilised displacement signal after the recovery duration, the residual depth after recovery (Point E in Figure 3.1) represents the permanent depth. It is supposed to be due to the plastic time independent deformations produced during the first loading as the yield stress has been exceeded (that is likely to happen during sharp indentation) and to time dependent plastic deformations (viscoplastic) that could be induced by the cyclic loading.

In the following Section, the parameters extracted from cyclic load-displacement indentation curves will be qualitatively compared to their analogous in tensile tests reported by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF]. In particular, unload and reload moduli will be compared to secant modulus, the area enclosed between load and unload in force-displacement plot will be compared to that in stress-strain plot and the displacement accumulation with cycles will be compared to the evolution of mean strain.

Cyclic behaviour of HDPE: indentation vs tension

In this Section, the parameters extracted from 0.1 Hz cyclic indentation load-displacement curves of HDPE (Figure 3.3) will be presented and qualitatively compared to that obtained by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] from stress-strain curves of cyclic tensile macroscopic tests at the same frequency. At this stage, recovery phase will not be analysed. The evolution with cycles of reload and unload indentation moduli (a), loop area (b) and displacement accumulated during cycles (c) are illustrated in Figures 3.5. Here and everywhere later in this document, the mean values of at least 6 indentation tests are presented with the error bars representing ± one standard deviation. However, in many graphs the scatter is so small that error bars are not visible. For an easier comparison, the evolution of secant stiffness (a), loop area (b) and ratcheting strain (c) with cycles extracted from stress-strain curves of cyclic tensile tests on the same material and for the same loading frequency of 0.1 Hz [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] are reported in Figure 3.6. From Figure 3.5a, it can be observed that the evolution of reload modulus with cycles is composed by a transient stage, in which it increases more quickly, followed by a steady state in which its rate of change is lower and almost constant. It is about 1 GP a at the first cycle and 1.3 GP a at the last cycle. Its overall increase from the first to the 300 th cycle is thus of about 30%. The unload modulus is, conversely, virtually constant with cycles and much more scattered. The modulus evaluated from reloading curves is about 1 GP a lower than the modulus calculated from unloading and is closer to the secant modulus determined from macroscopic tensile tests [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] for the same loading frequency of 0.1 Hz, represented in Figure 3.6a. The evolution with cycles of the secant modulus determined from macroscopic tensile tests, as can be observed from Figure 3.6a, is characterized by a transient and a steady state similarly to the evolution of indentation reload modulus. However, contrary to what observed on reload modulus, the macroscopic secant modulus is constant with cycles in the steady state and equal to about 1.3 GP a, that is, as previously observed, quite close to reload indentation modulus in the steady state.

Hysteresis loop area determined from indentation curves is also characterized by a transient stage in which it decreases quickly through cycles, and a steady state in which decreases slowly and with a constant rate (Figure 3.5b). It can be moreover observed a rapid drop in loop area from the first to the second cycle, that introduces a discontinuity in the trend. The energy dissipated during the first cycle is not reported on the loop area vs cycles plot of macroscopic tensile tests (Figure 3.6b). However, from the second cycle, the loop area decreases and, similarly to what observed on secant modulus, stabilizes after about 20 cycles.

Concerning indentation depth, from Figure 3.5c it can be observed that both maximum and minimum displacement (and mean displacement consequently) increase with cycles: quickly in the first cycles, then slowly and almost linearly in the further cycles. However, the increasing trend of minimum displacement in the first cycles is more pronounced than that of the maximum displacement, while the two curves are virtually parallel in the linear part of the plot. The mean strain evaluated from macroscopic tensile tests evolves with cycles in a similar way to indentation depth. Contrarily to secant stiffness and loop area, the mean strain in the steady state is not stabilized but continues to increase linearly. This is a typical feature of creep strain and creep displacement evolution with creep time observed in macroscopic and indentation creep tests. Some similarities between indentation depth-cycle curves and strain-cycle curves of uniaxial cyclic testing have also been observed by Xu and Yue [START_REF] Xu | Study of the ratcheting by the indentation fatigue method with a flat cyclindrical indenter: Part i. experimental study[END_REF] on copper, as reported in Chapter 1. In [START_REF] Xu | Study of the ratcheting by the indentation fatigue method with a flat cyclindrical indenter: Part i. experimental study[END_REF], cyclic indentation tests were performed at high load, between 500 and 700 N , with a flat punch. Xu and Yue [START_REF] Xu | Study of the ratcheting by the indentation fatigue method with a flat cyclindrical indenter: Part i. experimental study[END_REF] found that the parameters for the steady state stage of uniaxial fatigue tests can be obtained by the cyclic indentation method with a flat punch.

The comparison presented in this Section shows many analogies between the material response to macroscopic cyclic tensile tests and indentation cyclic tests. All the evolution with cycles of the parameters considered, related to the time-dependent nature of the polymer, are characterized by a transient stage, in which they change rapidly, and a steady state. During the steady state, secant modulus and loop area of macroscopic tensile tests are constants, while their analogous in indentation tests (reload modulus and loop area) continue to change with an almost constant rate. It is not surprising to observe discrepancies on macroscopic and microscopic cyclic behaviour because of the differences in the way the two tests are performed and the volume of material affected by the solicitation previously evoked.

In the following Section, the mechanical response of HDPE at different time scales will be studied through the evaluation of the effect of frequency on the parameters extracted from hysteresis load displacement loops.

Study of the time-dependent response of HDPE through several time decades

In this Section, the results of cyclic indentation tests on HDPE will be presented. The time dependency of the polymer behaviour will be studied through the evaluation of the effect of frequency on cyclic indentation results. As previously said, tests have been performed at three different frequencies: 1 Hz, 0.1Hz and 0.01Hz, while the loading ratio R = F min Fmax was kept constant and equal to 0.05. The number of cycles employed depends on the loading rate: 1200 for the highest, 300 for the intermediary and 60 for the lowest. As observed by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] on tensile tests on HDPE and by Zhang at al. [START_REF] Zhang | A simple constitutive model for cyclic compressive ratchetting deformation of polyteterafluoroethylene (ptfe) with stress rate effects[END_REF] and Rabemananjara at al. [START_REF] Rabemananjara | Caractérisation expérimentale des mécanismes d'endommagement du polyéthylène moyenne densité (pemd) sollicité en fatigue par compression. 36emes Journées de Printemps[END_REF] on compression tests on PTFE and MDPE respectively, hysteresis loops become thinner and vertical with cycles: the material ideally tends to elastic behaviour. Moreover, the effect of viscosity is less significant for the higher frequency as the loops are thinner and the material is stiffer. This assessment is confirmed from the evolution of the viscoelastic delayed displacement, calculated, according to Equation 2.7, as the difference between the actual maximum displacement and the displacement at the maximum load, with the number of cycles for the three frequencies, represented in Figure 3.8. From Figure 3.8 it results that, for the frequency of 1 Hz (blue curve), the delayed displacement is constant and equal to zero from the first cycle. For the intermediary frequency of 0.1 Hz (green curve), it decreases quickly in the first four cycles and then stabilizes to a null constant value. For the slower test (red curve) it decreases quickly in the firsts four cycles, then slowly tending to zero. In this case, a stabilisation to a constant value is not observed within the 60 cycles. It should be specified that values of delayed displacement lower than 0.002 µm must be taken with caution as below the depth sensor accuracy.

In the following sections, the effect of frequency on unload and reload indentation moduli, on loop area and on displacement accumulation will be presented and their analogies with macroscopic behaviour will be discussed.

Unload and reload indentation moduli

Unload and reload indentation moduli are representative of the instantaneous response of the material to the solicitation. Even if the limitation of the application of the Oliver and Pharr analysis method [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] on unloading indentation curves of polymer have been outlined in Section 4.2, the evolution of both indentation load and unload moduli with cycles will be presented here.

Looking at the unload and reload curves obtained on HDPE (Figures 3.7) it can be observed that their shape is quite different: unloading presents a round shape at the peak load while reloads are almost a straight line, suggesting that maybe the evaluation of indentation modulus from reloads could lead to results consistent with Sneddon's theory.

The evolution with cycles of the unload and reload indentation moduli for the loading frequencies of 1 Hz (blue), 0.1Hz (green) and 0.01Hz (red) is reported in Figures 3.9a The evolution of unload modulus (Figure 3.9a) confirms what already observed on one cycle curves presented in Section 2.8: it changes with frequency but there is not a simple relation between loading rate and modulus. Moreover, the kinetics of variation with cycles is not clear. On the contrary, the effect of frequency on the reload modulus (Figure 3.9b) is clear and consistent with what observed by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] on secant modulus in the macroscopic tensile tests: it increases with loading frequency. That is because, as previously said, at higher frequency loops are thinner and more vertical and the stiffness is thus higher.

To study more thoroughly the evolution of reload modulus with time, the rate of change of this parameter for the loading frequencies of 1 Hz (blue), 0.1Hz (green) and 0.01Hz (red) is depicted in Figure 3.10. The correspondence between cycles and time could be calculated according to Equation 3.1

t i = t 0 + ∆t cycle • i (3.1)
Where t i is the time corresponding to the cycle i, t 0 is the time at the end of first loading and ∆t cycle is the time for a complete load-unload cycle. From Figure 3.10 it can be observed that the rate of change of load modulus with time is, in the transient stage, higher for higher frequencies. In the steady state the rate is low and almost constant with time. Steady state is reached after about 50 s and, after that time, the rate of change vs time plots are superposed for the three frequencies. This means that they are affected by long-time relaxation processes similarly and independently on frequency. However, despite what was observed by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] on secant modulus determined in macroscopic tensile and shear tests, the load indentation modulus, as previously observed, continues to grow even in the steady state.

The results presented in this Section suggest that the reload indentation modulus is more appropriate to describe time and frequency dependent phenomena occurring during cyclic loading. Even the power law fitting parameters determined from reloading curves are less scattered and consistent with experimental observations and Sneddon's solution. Figure 3.11 shows the evolution with cycles of the exponent of the fitting function m for the firsts 60 cycles, for the three frequencies and for both reload (empty markers) and unload curves (filled markers). The value of m determined on loading curves (empty markers of Figure 3.11) is quite constant with cycle, rather similar for the three frequencies and lower than the theoretical upper limit according to Sneddon's solution [START_REF] Sneddon | The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile[END_REF]. That is not the case for the power law exponent determined from unloading: it is much higher than 2 for all frequencies and all cycles and it is more scattered. This observation confirms that the power law fitting function is not adequate to reproduce the unloading curve of HDPE, even after a high number of load-unload cycles. Furthermore, it is interesting to observe that the power law exponent of reloading curves is close to 1, meaning that the relation between force and displacement is linear during reloadings. According to the analysis of the contact between a flat surface and punch of arbitrary profile [START_REF] Sneddon | The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile[END_REF], a linear relation between load and displacement is obtained in the case of flat-punch in which the contact is fully elastic. The explanation that we suggest is that, once the permanent print is formed during the first loading, the contact area between the indenter and the already formed permanent print in the material is constant during the further reloadings. Based on these observations, it can be assessed that the computation of the elastic indentation modulus during reloading phases is more meaningful. The evolution of elastic indentation modulus with cycles observed in Figure 3.9b reflects a change in elastic properties with time, that is, as previously said, a typical feature of time-dependent viscoelastic materials.

Having regard to the outcome of load and unload indentation moduli analysis, it is clear that the modulus determined from unloading curves is uncertain and difficult to interpret and so it will not be considered in the following.

Figure 3.12a depicts the evolution of load indentation modulus as a function of loading frequency in a semi-logarithmic plot. As indentation modulus is representative of the instantaneous time-dependent response to the cyclic loading, each curve in the Figure 3.12a represents the modulus evaluated at the same testing time for the different loading frequencies, which corresponds to a different cycle number. From Figure 3.12a it results that there is a logarithmic relation between the reload indentation modulus and the loading frequency for each time considered. The slopes of the fitting curves are plotted in Figure 3.12b as a function of testing time. It is evident that the slopes are constant, meaning that the effect of frequency on reload modulus is not dependent on the considered time and thus showing that short and long term responses are governed by the same viscoelastic regime. It is interesting to compare the evolution of reload indentation modulus with frequency to the evolution of elastic modulus with frequency obtained by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] on the same material through DMA tensile tests, in a frequency range from 1 to 60 Hz at ambient temperature. The results of DMA tensile tests are represented in Figure 3.13b. For a better comparison, the evolution of reload modulus with frequency for one testing time (50 s) has been reported again in Figure 3.13a using the same scale on y axis of DMA test results. From Figure 3.13b it can be observed that elastic modulus slightly increases with frequency with a slope that decreases from a decade to the next one. It is interesting to observe the continuity between the reload indentation modulus vs frequency plot (Figure 3.13a), in the range from 0.01 to 1 Hz, and the elastic modulus vs frequency plot that starts from 1 Hz. It would be interesting to investigate further the correspondence between storage modulus and reload indentation modulus in the frequency spectrum obtained by DMA and cyclic indentation tests respectively. A new DMA test in the range of frequencies employed for cyclic indentation test 0.01 -1 Hz could be useful for this further analysis.

Energy dissipation in load-unload cycles

Hysteresis loops are a characteristic feature of the response to a cyclic load of a timedependent material. They are due to a time delay between the applied load and displacement response, and quantify the amount of energy dissipated to produce irreversible (plastic) and not immediately reversible (viscoelastic) deformations. Hysteresis loop area represents the short term time-dependent response at the scale of one cycle. The evolution of hysteresis loop area with cycles for the three frequencies of 1 Hz (blue), 0.1Hz (green) and 0.01Hz (red) is represented in Figure 3.14. From Figure 3.14 it can be observed that, as expected, the loop area decreases with the number of cycles for the three frequencies. Moreover, lower frequencies give a higher amount of dissipated energy, confirming that viscous deformation is more significant in slower tests. However, in contrast to what observed in tensile and shear cyclic tests [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF], loop area does not reach a plateau even after a high number of cycles. Moreover, it is interesting to observe the rapid drop in loop area from the first to the second cycle. This fall is attributed to the differences in deformation mechanisms between the first and the further cycles. In fact, according to what previously said, during the first loading, irreversible deformations, that generates the loop area, are due to both plastic and viscoelastic deformations. From the second cycle, only time-dependent deformations, that could be recoverable (viscoelastic) and permanent (viscoplastic) are instead produced and the energy that is dissipated is due to the delay in the recovery of reversible part of these time-dependent deformations. Moreover, since from the second cycle and up to the end of the test, the evolution of loop area with cycles follows the same law, it is reasonably to think that the time-dependent dissipation mechanisms do not change with cycles. In other words, even if hysteresis loops are bigger for lower frequencies, their reduction in size from one cycle to the next one does not depend on loading rate. That is the case for macroscopic tensile cyclic tests also [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF], in which steady state regime is reached after the same number of cycles for different loading frequencies.

Figure 3.16a depicts in a semi-logarithmic plot, the evolution of loop area with loading frequency for 1st, 2nd, 5th, 10th, 30th and 60th cycle. From Figure 3.16a it can be observed that from the second cycle, the loop area vs frequency curves are well described by a logarithmic law. The slopes of the fitting functions are plotted in Figure 3.16b as a function of the cycle number. In this graph, it can be observed that the slope increases in the first cycles, then the loop area vs frequency curves become parallel after 10 -20 cycles. So, even if it has been observed that loop area does not stabilize within the number of cycles considered in our study, from 10 -20 cycles, that is in the steady stage, the effect of frequency on loop area is not dependent on the cycle number as in the macroscopic tests.

Effect of frequency on indentation mean displacement

The evolution of the mean displacement with cycles for the first 60 cycles and with testing time for the three frequencies of 1 Hz (blue), 0.1Hz (green) and 0.01Hz (red) are represented in Figures 3.17a and 3.17b respectively. From Figure 3.17a it can be observed that indentation mean displacement progressively increases with cycles for the three frequencies and that is higher for lower frequencies, confirming that the material is less hard as loading rate decreases.

Moreover the curves related to the three frequencies are parallel in the steady state, suggesting that the kinetics of evolution of mean displacement accumulated with cycles does not depend on loading frequency in the range considered in this study. These trends are similar to that observed by Chen and Hui [START_REF] Chen | Ratcheting behaviour of ptfe under cyclic compression[END_REF] on the ratcheting in cyclic compression tests on PTFE, for which the lower loading rate (longer test) gave higher ratcheting strain and that for loading rates higher than 40 N s , the ratcheting curves became parallel. Concerning the evolution of mean displacement with time, from Figure 3.17b, it results that the three frequencies compose the same master curve in the four time decades of the log time scale experimentally accessible from our test. In other words, even if the mean displacement values for the three frequencies are different for a given cycle number, when the curves are plotted in function of time, they are superposed for all frequencies. This observation confirms that the process of indentation depth accumulation with cycles is similar to creep and independent on time of cycle.

Net indentation displacement accumulated with cycles

From Figure 3.5c it was observed that, during indentation cyclic tests, both minimum and maximum displacements evolve and, consequently, their mean value. In the previous Section, we have focused on the effect of frequency on the mean displacement accumulated during cycles to allow a comparison with mean strain evolution with cycles in macroscopic tensile tests [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF]. In this Section, we will focus on the analysis of the whole displacement accumulated during cyclic test and recovered during the load hold phase introduced at the end of the test.

As it has been previously observed that the kinetics of displacement accumulation during cycles is similar to the evolution of displacement with time in indentation creep tests, a qualitative comparison between the two kind of tests will be proposed as well in this Section. For such comparison, two different indentation creep tests on HDPE will be considered: one in which the maximum load was set to 10 mN , that is the same maximum load as cyclic tests and the other in which the maximum load was set to the mean load of cyclic indentation tests, that is 5 mN . In both creep tests the holding time is 3000 s that is the same test duration of the cyclic test to which they will be compared. The average of at least nine tests on a displacement-time plot of both cyclic and creep tests is represented in Figure 3.18. Moreover, a recovery phase has been introduced after the cyclic tests and the 10 mN creep tests, as can be observed from Figure 3.18. The recovery phase will be analysed later. Figure 3.18 confirms that the kinetics of evolution with time of creep displacement and cyclic displacement are quite similar. In particular they are characterized by a rapid increase in the first stages, followed by a steady state regime in which the displacement rate of change is virtually constant. However, the slope of the 5 mN creep curve in the steady state is much lower than that of the 10 mN creep curve and that of the accumulated displacement in cyclic tests (maximum, minimum and mean). Concerning the displacement amplitudes, from Figure 3.18 it can be observed that the displacement-time cyclic curve is collocated between the two creep curves.

For a more detailed discussion on the analogies and differences between the displacement accumulated during cyclic tests and creep tests, one must identify which cyclic displacement (minimum, maximum, mean or other) represents the net displacement accumulated during the whole test. In fact, during cyclic indentation, the displacement accumulated at each loading, is partially recovered during the following unloading. The difference between the maximum and the minimum displacement evaluated at the unloading, represents the amount of displacement recovered. The net displacement accumulated at the cycle i, is thus calculated as the difference between the displacement accumulated during the i -th loading and the displacement recovered during the i -th unloading:

h i max -(h i max -h i min ) = h i min (3.2)
From this simple calculation, it results that the net displacement accumulated at the i -th cycle, is represented by the minimum displacement at the same i -th cycle. Even if the comparison between displacement accumulated during cycles and during creep remains qualitative, it was deemed appropriate to consider the net displacement accumulated during cycles (h min ) as the element to be compared. As the kinetics governing both phenomena is quite similar, a model developed for the analysis of indentation creep tests have been applied to both creep and cyclic tests. As reported in Chapter 1, several empirical models allowing to fit displacement data during indentation creep are proposed in literature [START_REF] Yang | Analysis of nanoindentation creep for polymeric materials[END_REF][START_REF] Lu | Measurement of creep compiance of solid polymers by nanoindentation[END_REF][START_REF] Chudoba | Investigation of creep behavior under load during indentation experiments and its influence on hardness and modulus results[END_REF][START_REF] Beake | Modelling indentation creep of polymers: a phenomenological approach[END_REF][START_REF] Oyen | Sensitivity of polymer nanoindentation creep measurements to experimental variables[END_REF][START_REF] Huang | Transient and steady-state nanoindentation creep of polymeric materials[END_REF][START_REF] Mazeran | Determination of mechanical properties by nanoindentation in the case of viscous materials[END_REF].

In most of these models, the material is described by an arrangement of springs and dampers similarly to the models used to describe the strain-time relation in macroscopic creep and relaxation tests [START_REF] Brinson | Polymer Engineering Science and Viscoelasticity -An Introduction[END_REF]. For the description of the evolution of the cyclic and creep displacement with time, we referred to the semi-empirical model reported by Yang et al. [START_REF] Yang | Analysis of nanoindentation creep for polymeric materials[END_REF]. The Elastic-Viscoelastic-Viscous model (EVEV) proposed is based on the generalised Kelvin model described in Section 1.1.2.1 and represented in Figure 1.11. We remind here the displacement-time relation associated to this model:

h = h e + n i=1 h i (1 -exp ( -t /τ i )) + t µ 0 (3.3)
From Figure 1.11 and Equation 3.3, it is evident that this model contains three main contributions: the instantaneous displacement h e associated with deformation during loading, supposed to be instantaneous and, in the case of flat punch considered by the authors [START_REF] Yang | Analysis of nanoindentation creep for polymeric materials[END_REF], fully elastic; the viscoelastic deformation in the transient stage that is controlled by the exponential terms and the deformation dominated by the viscous flow of macro molecules that increases linearly with time ( t µ 0 ) and thus allows to reproduce the steady state regime with constant rate in the displacement-time plot. The hypothesis of elastic instantaneous displacement at the first loading is plausible in the case of indentation by a flat punch with a very high loading rate. In the case of sharp Vickers tip it has been shown that plastic deformations have occurred during loading. Moreover, it has been shown that even though a relatively high loading rate was used, time-dependent deformations have occurred. Finally, the plastic and time-dependent deformations produced during the first loading are likely to be different if the maximum load reached is different. Therefore, as the main aim of this Section is to compare the indentation depth evolution with time during creep and during cycles, the displacement at the end of loading phase for t = t 0 , indicated here as h 0 , is removed from the fitting function that becomes:

h -h 0 = n i=1 h i (1 -exp ( -(t-t 0 ) /τ i )) + t -t 0 µ 0 (3.4)
Equation 3.4 have been used to fit displacement-time data of creep tests and cyclic tests. It should be pointed out that, because of the limitations in the use of these kind of models to identify material properties previously evoked in Chapter 1, the fitting of displacement-time curves has been performed only to compare curves obtained from different tests. The number of exponential terms used in our study is three as suggested in [START_REF] Yang | Analysis of nanoindentation creep for polymeric materials[END_REF]. The fitting has been performed using the Matlab®curve fitting toolbox and the robustness of the solution was tested by varying upper and lower bounds of fitting coefficients and verifying that the solution did not change. The upper and lower bounds of time constants τ i have been fixed in order to have a time constant for each time decade approximately. The evolution of h -h 0 for creep at 10 mN (dashed red curve), and at 5 mN (dashed green curve) and cyclic displacement (dashed blue curve) with time, together with their fitting curves (solid lines) are represented in Figure 3.19. From Figure 3.19 it can be observed that both creep and cyclic displacement evolution with testing time are well described by the generalized Kelvin model. The seven fitting parameters for the three tests together with the R-square and the value of displacement at the end of loading (h 0 ) are reported in Table 3.1. From Table 3.1, we can observe that the coefficient representing the slope of the displacement time-plot in the steady state, µ 0 is quite similar for creep at 10 mN and cyclic displacement. However, for the creep at 5 mN this term is one order of magnitude lower, as can be observed from Figure 3.20, in which the term t µ 0 is represented as a function of testing time for the three tests. The weight of the exponential terms, h 1 , h 2 and h 3 , are, as expected, the highest in the case of creep at 10 mN , and the lowest for the creep at 5 mN , while that of cyclic displacement are intermediate between the two, as can be observed from Figure 3.21a where the ratios between the h i of cyclic and creep curves are represented. The comparison of retardation times τ 1 , τ 2 and τ 3 allows to discuss the evolution of the three curves in the transient stage. For this purpose, the ratios between the τ i of cyclic and creep displacement are reported in Figure 3.21b. From the ratios between the τ i of cyclic and creep at 10 mN (red curve in Figure 3.21b), it can be observed that the accumulation of displacement in cyclic tests, not only is lower in intensity than creep at the same maximum load, but is also delayed in time. The delay is higher for the first phase of the transient stage (governed by τ 1 ) and progressively decreases in the successive phases. Then, once the phase governed by τ 3 is exhausted, the system gets to the steady state in which the rate of change ( 1 µ 0 ) of both cyclic and creep indentation displacement is constant and quite similar in the two cases as previously observed. Instead, from the ratios between the τ i of cyclic and creep at 5 mN (green curve in Figure 3.21b) it can be observed that during creep at the mean load, the displacement evolution is still faster than cyclic displacement in the first part of transient stage. Then, in the second part the two constant times are quite similar, while in the third part, creep displacement accumulation become slower than cyclic displacement accumulation. Once the system gets to the steady state, the rate of change ( 1 µ 0 ) of creep indentation displacement at 5 mN becomes much lower than cyclic displacement. In other words, cyclic and both creep displacements show a different behaviour during transient stage, even if their evolution follow an exponential decay in the three cases. In the steady state, the accumulation of cyclic displacement has a similar rate of change than that of creep at the same maximum load, while that of creep at the mean load is much lower. This phenomenon could be due to non linear viscoelastic and/or viscoplastic mechanisms that are not activated for the lower load. These observations confirm that caution must be taken in the use of the parameters determined from the fitting of displacement-time curves to determine time-dependent material properties: different time constants and different rates of change in the steady state are obtained form the fitting of indentation creep curves at different load.

h 0 h 1 τ 1 h 2 τ 2 h 3 τ 3 µ 0 R 2 (µm) (µm) (s) (µm) (s) (µm) (s) (s • µm -1 ) (-)
As reported in Section 1.2, analogies between the so-called ratcheting strain and creep have also been observed on macroscopic tensile tests [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF][START_REF] Chen | Ratcheting behaviour of ptfe under cyclic compression[END_REF][START_REF] Lissenden | Experimental investigation of cyclic and time-dependent deformation of titanium alloy at elevated temperature[END_REF]. Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] found that magnitudes of ratcheting strain were higher than those of a creep for a constant stress equal to the mean stress applied in the cyclic tests, while for a constant creep stress equal to the maximum cyclic stress, creep strain was higher than cyclic ratcheting strain, similarly to what observed here.

It is interesting to compare the kinetics of displacement recovery after the cyclic tests and after the creep test. As most of the permanent print is produced during the first load, only the recovery of cyclic and creep tests at the same maximum load have been compared. To this end, the displacement-time data recorded during the recovery phases that follow the indentation creep test (red curve) and the cyclic indentation test (blue recovery and after the recovery load removal can also be noted, overall on the recovery curve following the cyclic test, confirming that print is not fully recovered at the end of the recovery phase. Figure 3.22 also reveals a drift occurring at the end of recovery, as the scatter increases and the mean curve slightly diverges from its initial trend. That could be due to thermal drift phenomena that are not taken into account in the applied correction or, as reported by Yang et al. [START_REF] Yang | Analysis of nanoindentation creep for polymeric materials[END_REF], to a load undershot. This phenomenon happens when a relatively high unloading rate is used and results in values of force recorded during holding lower than the prescribed value (0.1 mN here) and that are different form a test to the other; hence the error in the recorded displacement. The analysis of recovery showed that its duration is not enough to establish the amount of permanent depth. At the same time, a longer recovery phase would be affected by instrument drift and thus the results would be difficult to interpret.

As the recovery phase, despite its long duration, did not give the possibility to determine the amount of permanent depth due to plastic deformations, in the following, an alternative method to estimate the amount of plastic depth produced during the first loading will be proposed. This method is based on the analysis of energy dissipated during the loading phases. It allows to determine the time-independent plastic deformation produced during the first loading, but it is not able to establish if there is any time-dependent viscoplastic deformation accumulated during cycles and their amount. In Section 3.2.2 it has been observed on the loop area vs cycle number plot a rapid drop in energy dissipation from the first to the second cycle, while from the second cycle and up to the end of the test, the decreasing trend followed the same law. This observation suggested that the mechanisms of energy dissipation in the first cycle and in the further cycles are different. This difference was attributed to the fact that the amount of energy dissipated during cycles, from the second cycle, is due to time-dependent deformations while at the first cycle a considerable amount of energy is dissipated to produce a permanent print (plastic deformations). Let's consider the amount of energy dissipated during the loading phases, calculated with a trapezoidal numerical integration of the loading curves, depicted in Figure 3.23. In Figure 3.23, grey points represent the experimental data. An abrupt fall of energy is observed from the first to the second cycle. In particular, the decrease in energy dissipated from the first to the second loading is about two times higher than the decrease observed from the second to the 60 th . The energy dissipated at the first loading represents the work made by the indenter to bring the system from the point A to the point B or, in other word, to increase the depth from about 0 µm to the maximum displacement at the end of the loading (h 1st max ). This displacement contains a plastic part and a time-dependent part. Furthermore, from Figure 3.23 it can be observed that, from the second cycle, the energy vs cycle plot is well-fitted by an exponential law, represented as a continuous black line in the figure. Thanks to this function, it is possible to extrapolate the value of energy that would be needed to produce only time-dependent deformation at the first loading (red point in Figure 3.23). The difference between the energy dissipated at the first loading experimentally measured (U 1st tot ) and that extrapolated from the fitting function (U 1st v ) represents, thus, the amount of energy dissipated to produce the permanent depth (U 1st p ). The three energy values determined for the test under consideration are reported in Table 3.2, together with time-dependent to total and plastic to total energy ratios. From the last two columns of Table 3.2, it results that about the 61% of the total work of indentation is employed to produce time-dependent deformations, a part of which will be recovered during the unloading, while the 39% is employed to produce a permanent plastic depth. In Section 1.1.2.3 it has been shown that, according to Cheng et al. [START_REF] Cheng | Scaling relationships for indentation measurements[END_REF], the relation between the plastic to total energy ratio and the plastic to maximum penetration depth for a pyramidal tip and in the case of elasto-plastic materials, can be expressed as:

U 1st tot (nJ) U 1st v (nJ) U 1st p (nJ) U 1st v U 1st
U p U tot = 1.27 h p h max -0.27 (3.5)
From the estimated value of the ratio Up Utot at the first cycle and from the measured value of h max at the end of loading phase, it is possible to calculate, with Equation 3.5, the plastic depth produced by indentation at 10 mN with a Vickers tip, that is of about 1.34 µm. According to our assumption, this value is time and frequency independent. So, to validate the procedure, it has been applied to the other frequencies employed in this study, 1 Hz and 0.01 Hz, and the estimated values of plastic depth h p , in µm, for the three frequencies are reported in Table 3 A good agreement is found between the estimated plastic depths for the frequencies of 0.1 Hz and 0.01 Hz for which there is a difference of only 3%, while for that determined for the higher frequency is about 20% lower. This high difference is probably due to an underestimation of the area under the first loading curve for the higher frequency. In fact, the very low number of data points (5) recorded does not allow to correctly reproduce the curvature of the first loading curve and the use of the trapezoidal numerical integration to estimate the area under this curve could be too gross. However, the low number of data points should not be a problem for the further reloading that are, as previously observed, almost straight lines.

Conclusions

The results presented in this Chapter revealed that cyclic instrumented indentation tests are able to capture the time-dependent nature of polymer response to the loading. The analysis of load displacement curves of cyclic indentation tests at different loading frequencies allows to draw some conclusions on the time-dependent behaviour of HDPE:

1. Load-displacement curves of cyclic indentation tests are characterised by large hysteresis loops that evolve with cycles becoming thinner and more vertical. For higher frequencies, loops are more close and vertical as the viscous contribution is lower at higher loading rates. The same effect of cycles and frequency on hysteresis loops have been observed on macroscopic tensile and shear tests by Nguyen [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF]. However, while in the macroscopic tests the loop area decreases quickly in the first 10 cycles and stabilises after abut 20 cycles for all frequencies, in the case of indentation curves, loop area continues to decrease even after a high number of cycles.

2. Loop stiffness is evaluated from the unloading and reloading curves through the application of the Oliver and Pharr [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] analysis method. However, the application of this method on the unloading curves, showed some limitations. For that reason, only the indentation modulus evaluated from the reload curves (E * load ) has been held for the following. The evolution of E * load with cycles exhibits an increasing trend for all frequencies considered in our study. Moreover, the load indentation modulus is higher for higher loading rates. The rates of change of E * load with time for the three frequencies are well aligned. Even in this case, the trends are analogues to those observed on secant modulus in macroscopic tests. However, in the steady state of macroscopic tests, the secant modulus is constant, while the indentation modulus continues to increase with a constant rate.

3. The mean displacement for the three frequencies composes the same master curve in the four time decades of the log time scale experimentally accessible from our tests. Similar trends have been observed on ratcheting strain in macroscopic tension and shear [START_REF] Nguyen | Caracterisation experimentale et modelisation thermo-mecanique de l'accommodation cyclique du polyéthylene[END_REF] cyclic tests on HDPE and compression cyclic tests on PTFE [START_REF] Chen | Ratcheting behaviour of ptfe under cyclic compression[END_REF].

4. The indentation displacement accumulated during cycles (maximum, minimum and mean) shows a transient stage and a steady state, similarly to the evolution with time of indentation depth in indentation creep tests. However, during cycles, there is a competition between displacement accumulated during loading phases and recovered during the following unloadings. The net displacement accumulated at the i -th cycle is thus given by the difference between the maximum displacement reached at each loading (h i max ) and that recovered during the unloading (h i max -h i min ).

5. The net displacement accumulated in cyclic indentation tests has been compared to that accumulated during indentation creep tests in which the maximum load was set to the maximum cyclic load (10 mN ) in the first case and to the mean cyclic load (5 mN ) in the second case. The duration of the three tests was similar and of about 3000 s. It was observed that the curve representing the net displacement accumulated during cycles as a function of testing time is situated between the displacement-time curves of the two creep tests. Moreover, while the evolution of the three curves in the transient stage was quite different, it has been observed that, in the steady state, the cyclic displacement curve and the creep curve at the same maximum load were parallel. However, the slope of the creep curve at 5 mN in the transient stage was an order of magnitude lower. That is probably due to non-linear viscoelastic and/or viscoplastic phenomena that are not activated at the lower creep load.

6. The analysis of the recovery phase introduced at the end of cyclic tests and 10 mN creep tests revealed that its duration (about the same duration of the test) was not enough to establish the amount of permanent depth produced during the solicitation. Moreover, it was observed that the recovery of displacement accumulated during cycles is lower than that of creep displacement at the same maximum load.

7. The analysis of energy dissipated in cycles suggests that the mechanisms involving a dissipation of energy in the first and in the further cycles are different. It has been thus supposed that, during the first loading, a large amount of energy is dissipated to produce time-independent plastic deformations once the yield stress is exceeded. This observation allows to estimate the amount of plastic depth from the comparison of the energy dissipated in the first cycle with that of the further cycles. However, it was not possible to establish if any time-dependent plastic deformation (viscoplastic) are also produced during the successive loadings. Only the analysis of the residual depth after a full recovery would allow to answer this question.

The testing protocol proposed in this study and its analysis method highlight the complex local time dependent response of polymers. According to the scheme proposed in Figure 3.24, the analysis of the hysteresis load-displacement loops allows to deduce the timedependent response at three different time scales: instantaneous, represented by the reload modulus; short time (at the time-scale of the cycle), represented by the energy dissipated from the second cycle; long time (at the time-scale of the test), represented by the displacement accumulated during cycles. Even if further investigations are needed to establish if the deformations accumulated during cycles are fully recoverable (viscoelastic) or partially viscoplastic, the analysis of energy dissipation showed that most of the permanent print is produced during the first loading and has been associated to time-independent plastic deformations. The large number of information provided by this test could be useful to determine by reverse analysis a complex local behaviour law for polymer materials.

Chapter 4

Application of cyclic indentation to study the effects of thermal aging on the local time-dependent behaviour of PR520 epoxy resin

In the previous Chapter, it has been shown that the cyclic indentation method allows to capture the time-dependent character of polymer response. In this Chapter, the effects of thermal aging on local time-dependent properties of the PR520 epoxy resin will be analysed by studying the response change to cyclic indentation load with increasing aging time and for different oxidative environments.

At First, the material, the aging conditions and the protocol of samples preparation will be described.

Then, the response to cyclic indentation of the PR520 epoxy resin in its virgin state will be presented, by analysing the evolution with cycles of the different parameters extracted from load-displacement curves described in the previous Section. In particular, we will focus on the indentation modulus calculated from reload curves, representative of the instantaneous response; on the energy dissipated during cycles, representative of the short-time response (at the time scale of on cycle) and on the displacement accumulated during cycles h min , representative of the long time material response (at the time scale of the test). In this Chapter, the energy dissipated will be represented in a normalized form with respect to the total energy (η) as it is more appropriate for a comparative study.

Afterwards, the results of cyclic indentation tests performed on aged PR520 samples will be presented. In this part, three different cases will be analysed: the effect of aging on the local time dependent behaviour, the evolution of local properties with aging time and the gradient of properties in the oxidized layer. A discussion about the possibility of use oxygen pressure to accelerate degradation phenomena will follow.

In the last part, some complementary results will be presented in order to integrate the discussion about the mechanisms occurring during thermal aging. These are the macromolecular modifications being detected through glass transition temperature measurements.

Material, aging conditions and sample preparation

The material used in this study is the PR520 epoxy resin, which mechanical properties provided by the manufacturer have been reported in Section 2.1.

The PR520 epoxy resin glass transition temperature has been experimentally measured by Differential Scanning Calorimetry (DSC). In this technique, the difference in the amount of heat flow necessary to increase the temperature of the sample and of a reference material is measured as a function of temperature. When the material undergoes a physical transformation, more or less heat than the reference material will be needed to kept both to the same temperature. Physical transformations, like glass transition, can thus be distinguished on Heat flow vs Temperature plots. To determine glass transition temperature of PR520, the sample has been heated in an inert environment from ambient temperature to 200 °C with a rate of 10 °C min and then cooled down at the same rate. The resulting heat flow vs temperature plot is represented in From Figure 4.1, a step on the heating curve can be observed, which corresponds to the glass transition. The glass transition temperature is determined by the software as the middle point on the slope of the step. The value measured on two different samples is about 146 °C with a scatter of less than 0.1 °C. This value is about 15 °C lower than that reported by the manufacturer on the technical data sheet [103], determined by DSC too and reported in Table 2.2. Some differences could exist among the molecular structure of the same material belonging to distinct production batches, due, for example, to different composition or to differences in the infusion, homogenization and curing temperatures effectively employed during manufacturing and that would explain the gap between the glass transition temperature provided by the manufacturer and measured on our samples. Moreover, slight modifications on the molecular structure could have occurred during the elapsed time between the production and the testing of the material. In fact, as samples have been exposed to atmospheric air and thus in presence of oxygen, the material could have undergo "pre-aging". For these reasons, the value that we have measured will be retained as the effective glass transition temperature of our material in its reference state.

For the sample conditioning, three different environments have been used: air at atmospheric pressure, 2 bar of Oxygen and 2 bar of Nitrogen. In this Chapter, we will mainly focus on the effect of temperature on local mechanical properties of the material in air at atmospheric pressure. The aging in air has been performed in a ventilated oven Fisherbrand, represented in Figure 4.2. The oven Fischerbrand allows an increase in temperature in the chamber up to 200°C and its maintain at the target value with a constant feedback control by a thermocouple placed close to the samples. The environment in the chamber is air at the atmospheric pressure.

As reported in Chapter 1, previous studies showed that the use of moderate oxygen pressures allows to accelerate thermal aging [START_REF] Vu | Endommagement Induits par la Thermo Oxydation dans les Composites Carbone/Epoxy Unidirectionnels et Stratifiés[END_REF]. To establish how much the use of oxygen pressure accelerate the evolution of local time dependent properties, some PR520 epoxy resin samples have been aged under 2 bar of oxygen at the same temperature. The chamber used for this kind of aging is the Topindustrie, represented in The chamber Topindustrie allows to perform aging in air, oxygen and nitrogen. The maximum allowed pressure is 5 bar while the maximum temperature in the chamber is 350 °C. In order to minimize the heat exchanges with the external environment, the chamber is embedded into an insulating shell. Moreover, to avoid air infiltration in the chamber that would modify the environment and the pressure, the chamber lid is fixed with 8 screws that are cross screwed with a torque wrench for four times. Furthermore, to avoid any atmospheric air present in the chamber after its closure, the chamber has been filled with the gas at the maximum allowed pressure and emptied consecutively for 10 times in order to allow the air in the chamber to completely evacuate.

The effects of thermal aging on the mechanical properties of PR520 epoxy resin were the object of previous studies [START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF][START_REF] Guigon | Vieillissement par cyclage thermique de composites interlock 3D à matrice polymère[END_REF]. In [START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF][START_REF] Guigon | Vieillissement par cyclage thermique de composites interlock 3D à matrice polymère[END_REF], the attention was mainly focused on the evolution of elastic modulus in the oxidized layer evaluated from the unloading curve of simple load-unload indentation tests and on the degradation of macroscopic failure properties with aging time evaluated through four-point bending tests, without taking into account the time-dependent nature of polymer behaviour. Moreover, the aging temperature used in [START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF][START_REF] Guigon | Vieillissement par cyclage thermique de composites interlock 3D à matrice polymère[END_REF] was 120 °C, that is a value lower than both glass transition temperatures provided by the manufacturer and measured by DSC. According to what reported in Chapter 1, the degradation phenomenon occurring for aging temperatures lower than the glass transition could be considered as pure thermal-oxidation. However, as explained in the Introduction, the main reason why the effect of thermal aging on the mechanical properties of PR520 epoxy resin are studied, is linked to the possibility to introduce carbon fiber reinforced polymer matrix composites in the "warm zones" of aircraft structures. In these zones temperatures are of the order of 150 °C. To study the effect of a temperature closer to that of the real operating conditions, the aging temperature chosen for our study is, therefore, 150 °C. It must be underlined that this value is slightly higher than the glass transition temperature determined by DSC, so it is probable that secondary degradation phenomena (for example thermolysis) other than pure thermal-oxidation would occur.

The last aging environnement considered in this study is that made in an inert environment (Nitrogen) and will allow to conclude about the aging mechanism occurring at the temperature chosen. In fact, if aging is only due to thermal oxidation, no mechanical properties change is expected on samples aged in an inert environment. For such a aging, the same chamber used for conditioning under oxygen pressure (Figure 4.3) has been used, under a pressure of 2 bar and the same temperature of 150 °C. The aging conditions employed in this study are summarized in Table 4 As reported in Chapter 1, the more evident effect of thermal oxidation is the sample surface color change. In particular, aged samples appear darker than the virgin one and the color change is more important as the aging conditions are stricter. When the sample is transversely cut, it can be seen that the color change affects the external layer for a few hundred of microns, while the core of the sample keeps its original color. In order to evaluate the mechanical properties change due to thermal aging, instrumented indentation tests have been carried out on the surface and on the cross section, at different distances from the exposed surface, of the conditioned samples. Two samples for each aging condition were thus needed: one to be tested on the surface that has been preliminary polished and then conditioned, the other to be tested through the thickness that has been prepared following the scheme reported in

Cyclic behaviour of virgin epoxy resin

In order to evaluate the local mechanical properties change of thermal aged resin samples, it is necessary to analyse the indentation cyclic behaviour of the PR520 epoxy resin in its virgin state. The testing method employed for the analysis presented in this Chapter is similar to that used in the previous Chapter (depicted in Figure 3.1), in which the number of cycles has been reduced to allow time saving. Moreover, being here the aim to evaluate the effect of thermal aging on the material response, just one loading frequency will be considered that is the same for all the aging conditions. The testing conditions used for this study are summarized in Table 4.2. Thermal drift has been corrected according to the procedure described in Section 2.5. The resulting load-displacement loops for the 1st, 2nd, 5th, 10th, 20th and 40th cycle are shown in The cyclic load-displacement curves obtained on PR520 are characterized by large hysteresis loops that tend to close and to become vertical with cycles, similarly to what previously observed on HDPE. According to the cyclic indentation analysis method presented in Chapter 3, a number of parameters can be calculated from load-displacement curves, each of which is related to a particular aspect of the time-dependent response of the polymer. In particular, on the basis of the scheme proposed in Figure 3.24, three time scales can be distinguished on the cyclic indentation response of the material: an instantaneous response, characterized by the indentation modulus calculated from the reloading curves (E * load ); a short term response, characterized by the ratio of the dissipated energy (area enclosed between load and unload curves) with respect to the total energy (area under the loading curve) η, and a long term response, characterized by indentation displacement accumulation during cycles. The evolution with cycles of these three indicators are depicted in 4.6b it can be seen that the amount of energy dissipated in the first load-unload cycle is more than 50%, then it drops to about 25% in the second cycle and decreases following an exponential law reaching a value of just about 4% at the last cycle. The rapid drop in energy dissipation from first to second cycle is, as for HDPE, due to the different mechanisms of energy dissipation between the first and the subsequent cycles. In fact, as discussed in the previous Chapter, during the first cycle, energy is dissipated to produce plastic and time-dependent deformations, while from the second cycle the energy is dissipated to produce time-dependent deformations only. The value of η at the first cycle will be retained for the evaluation of the effects of thermal aging on the plastic behaviour of the material in the following. Concerning the displacement accumulation through indentation cycles, it has been shown in Chapter 3 that, the penetration depth at each loading (h max ), is partially recovered during the subsequent unloading (h max -h min ). The net viscoelastic displacement accumulation during the test is thus represented by the minimum displacement (h min ). From Figure 4.6c it can be observed that the minimum displacement increases quickly in the firsts cycles, then slowly and almost linearly, similarly to the evolution of displacement with time that is usually observed in indentation creep tests.

The first conclusion that arises from Figures 4.6 is a good repeatability of the tests which results in small error bars on the graphs. Moreover, similarly to what observed on HDPE, the evolution with cycles of the considered parameters is characterized by a transient stage in which they rapidly change and a steady state in which their rate of change is lower and almost constant. However, it should be mentioned that, contrarily to what observed on HDPE, the evolution of indentation depth with cycles is evident only if one observes the minimum displacement. The maximum displacement, in fact, changes very little in the first 4 cycles and then it is virtually constant for the further cycles. It is like the maximum penetration depth is upper limited, while the depth at the minimum load increases cycle by cycle.

Study of the effect of thermal aging on local mechanical properties of PR520 through cyclic indentation

In this Section, the effects of thermal aging on the local time-dependent behaviour of PR520 epoxy resin will be studied. At first, the consequences of thermal aging on the cyclic behaviour will be analysed. In this case, for the same distance from the exposed surface, it will be shown how the indentation modulus, the amount of energy dissipated and the displacement accumulation evolve with cycles for the different aging conditions.

Then, we will focus on the evolution of instantaneous, short term and long term response with aging time, for the same distance from the exposed surface and for the same testing time. Finally, it will be shown how local properties evolve in the oxidized layer, for a given testing time and for the different aging conditions. The Section will be concluded with a discussion on thermal aging mechanisms involved in these conditions.

Cyclic behaviour of aged samples

The question of this Section is how the evolution with cycles of load indentation modulus, dissipated energy and accumulated displacement change for the different aging conditions and for the same distance from the exposed surface. The results that will be presented are referred to tests performed according to the protocol reported in Table 4.2, on the surface directly exposed to the environment of samples conditioned at 150 °C in air under atmospheric pressure for 220 h, 520 h and 1005 h. The load displacement curves for the three aging times (red, green and violet curve, from the lower to the higher aging time respectively) together with that of the virgin sample (blue curve) are presented in From Figure 4.7, we can observe that large hysteresis loops are still distinguishable on force displacement curves of aged samples. The first conclusion on the effect of conditioning is that the material become "harder" with aging time as the load displacement curves are shifted to lower displacement values. The evolution with cycles of reload indentation modulus (E * load ), of the amount of energy dissipated through cycles with respect to the total energy (η) and of the net displacement accumulated (h min ) for the virgin and aged samples are represented in Figure 4.8a, 4.8b and 4.8c respectively. From Figure 4.8 the first evidence is that the trend of evolution with cycles of the three parameters is the same for the virgin and for the aged samples. From Figure 4.8a, it can be observed that the reload indentation modulus increases with aging time, passing from 3.8 GP a for the virgin sample to about 5.5 GP a for the higher aging time at the first cycle, and from 4.4 GP a to about 6.3 GP a at the last cycle. The difference between the reload modulus at the last and at the first cycles highlight that its increase with cycles is more important for the aged samples.

The irreversible part of indentation work (Figure 4.8b) seems to be not affected by thermal aging except for the first cycle. According to what has been reported in the previous Section, from the second cycle, the amount of energy dissipated with respect to the total energy is mainly due to time-dependent deformations while at the first cycle, it is due to both permanent (plastic) deformations and time-dependent deformations. So, upon the Figure 4.8b, it can be concluded that the amount of energy dissipated to produce time-dependent deformations, with respect to the total energy, remains the same for the virgin and aged samples, while the one dissipated to produce permanent depth decreases with aging time.

Regarding the long time response, evaluated through the displacement accumulation with cycles, from Figure 4.8c it can be observed that, as expected from load-displacement curves of Figure 4.7, the evolution of minimum displacement with cycles is shifted downward as aging time increases consistently with the fact that the material become harder with thermal aging.

It has been previously assessed, by relying on Figures 4.8, that the evolution with cycles of the different parameters is qualitatively unchanged with aging. While it is clear that η is not affected by aging from the second cycle as the curves are superposed, it is difficult to state from the observation of Figures 4.8a and 4.8c if the evolution of reload modulus and minimum displacement with time is affected or not by aging. So, to evaluate the impact of aging on time-dependent behaviour, the evolution of experimental h min -h min 0 data for the virgin (blue symbols) and the aged samples (red, green and violet symbols) with testing time (t -t 0 ) have been traced and reported in Figure 4.9. For a simpler comparison, the minimum displacement curves have been shifted to zero by subtracting the value measured at the first cycle, indicated with the zero subscript. In this way, the plastic depth produced at the first cycle will be subtracted as well. The experimental data have been fitted with the function describing the displacement accumulation with time used in Chapter 3 (Eq. 3.4) with two exponential terms (black solid lines in Figure 4.9), that we remind here: From Figure 4.9, it can be observed that the points representing the h min -h min 0 values for all aged samples overlap, showing that conditioning time affects only the indentation depth at the first cycle and so that the curves of minimum displacement with time for the aged samples represented in Figure 4.8c, are just shifted but remain parallel. This is probably due to the fact the long time properties reached a steady state for an aging time lower than the lowest aging duration employed (≤ 220 h). However, there is a difference between the time dependent behaviour of h min -h min 0 for virgin and aged samples. As the h min -h min 0 experimental points for thermal aged samples are virtually superposed for the three aging times, just one curve has been used to approximate data (black solid line in Figure 4.9) and the fitting parameters for a two exponential term law (Eq. 3.4) for virgin and aged samples are reported in Table 4 The scatter values reported in Table 4.3 represents the 95% confidence bounds determined from the fitting procedure. From Table 4.3, it can be noted that the retardation times τ 1 and τ 2 , related to the dashpots of the two Kelvin elements (Figure 1.11), are very similar for both cases of virgin and aged samples, meaning that the transient stage is governed by the same characteristic times. Moreover, also the parameter µ 0 , that represents the rate of change in the steady state, is quite close for the two sets of fitting parameters. So, the difference in the two curves is mainly attributed to the h i coefficients that result to be lower for the law describing the displacement accumulation of the conditioned material. We remind here that h i represents the ratio between the applied load P 0 and the spring constant of the i -th Kelvin element (K i ). As the "equivalent" creep load P 0 is the same in all tests, a decrease of the h i in aged samples involves an increase of the spring constants of the Kelvin elements and thus the stiffness of the material. The ratio between the amplitudes h i of virgin and aged fitting curves is of about 1.13 and 1.16 for h 1 and h 2 respectively.

h -h 0 = h 1 (1 -e - t-t 0 τ 1 ) + h 2 (1 -e - t-t 0 τ 2 ) + t -t 0 µ 0 (4.1)
The analysis of displacement accumulation of virgin and aged samples, showed that its kinetics of evolution with cycles is not significantly affected by thermal aging in these conditions. This statement is supported by the fact that the irreversible part of energy from the second cycle is not affected by thermal aging (Figure 4.8b). According to the hypothesis made in the previous Section, the existing difference on the energy dissipated during the first cycle is only due to a difference in the plastic behaviour induced by thermal aging.

Concerning the evolution with testing time of reload indentation modulus, to facilitate the comparison among different curves, the experimental E * load -E * load 0 data for virgin and aged samples have been also fitted with an exponential function with two exponential terms:

E * load -E * load 0 = E 1 1 -e - t-t 0 τ 1 + E 2 1 -e - t-t 0 τ 2 (4.2)
To have a representation more akin to that usually employed to describe the evolution of elastic modulus with time in viscoelastic materials, for example Prony's series, the linear term that increases proportionally to µ 0 has been omitted in this case. The experimental E * load -E * load 0 data for virgin (blue symbols) and aged (red, green and violet symbols) samples, together with their fitting functions (Eq. 4.2) (solid lines) are represented in values for the three aging times are superposed and just one fitting function has thus been used to describe their evolution with testing time. The set of parameters found from the fitting procedure using the analytical expression of Equation 4.2 with two exponential terms is reported in Table 4.4 for both virgin and aged material. The comparison of the values reported in Table 4.4 shows that, even in this case, the retardation times that provide a good fit for both curves are quite similar and that the main effect of conditioning arises from the amplitude of E 1 and E 2 . Both amplitudes are higher for the curve related to the aged samples and the ratios between the E i for aged and virgin reload modulus fitting curves are 1.19 and 1.08 for E 1 and E 2 respectively. From the results presented in this Section it has been observed that the evolution of the amount of energy dissipated to produce viscous deformations with respect to the total energy (η from the second cycle) with cycles is well superposed for the virgin and the aged samples for all the aging durations. The evolution with testing time of the two other parameters (minimum displacement and reload modulus) is well described by an exponential law with two exponential terms in which the characteristic times are similar for virgin and aged samples while the amplitudes change. In particular, it has been observed that the indentation depth decreases with aging time, meaning that the material become harder, while the reload modulus increases, meaning that the material become stiffer. In the following Section, the evolution with aging time of the different parameters for a fixed cycle and for a fixed distance from the surface, will be studied more in detail.

E 1 (GP a) τ 1 (s) E 2 (GP a) τ 2 (

Mechanical properties change with aging time

In the previous Section it has been possible to qualitatively establish how thermal aging affects the indicators calculated from load-displacement hysteresis loops. In this Section we will analyse more in detail the local properties change with aging time, here again for tests performed on the exposed surface. The results presented in this Section derive from the same tests presented in the previous one.

It is well known from the literature that an increase in aging time causes an increase in indentation modulus calculated from the unloading curves of load-unload indentation tests [START_REF] Pochiaraju | Evolution of stress and deformations in high-temperature polymr matrix composites during thermo-oxidative aging[END_REF][START_REF] Johnson | Investigation of oxidation profile in pmr-15 polymide using atomic force microscope (afm)[END_REF][3][4][5][START_REF] Jiang | Enhanced mechanical and thermal properties of monocomponent high performance epoxy resin by blending with hydroxyl terminated polyethersulfone[END_REF][START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF]. That is also the case for the indentation modulus calculated from reloading curves of PR520 in the aging conditions employed in this study as depicted in Figure 4.11, where it is plotted against the aging time. For more clarity, just few cycles are represented on the diagram: the 1 st (red), the 2 nd (blue) and the last (green). For the three cycles considered, and therefore reasonably for the others, the reload indentation modulus of the cycle increases with aging time (t ox ) following an exponential law with two exponential terms (dashed lines in Figure 4.11) of the type:

E * load = E * loadv + a (1 -exp(- t ox τ a 1 )) + b (1 -exp(- t ox τ a 2 )) (4.3)
In which E * loadv represents the reload modulus of the virgin material for the considered cycle; a and b are the amplitudes of the exponential terms and τ a 1 and τ a 2 are the aging time constants. This kind of analytical expression allows to qualitatively follow the evolution of the thermal oxidation reaction with time: initiation, propagation and saturation [START_REF] Minervino | Effets de la thermo oxydation sur le comportement mécanique de matériaux composites pour applications aéronautiques[END_REF][START_REF] Colin | A new method for predicting the thermal oxidation of thermoset matrices. application to an amine crosslinked epoxy[END_REF]. In particular, the initiation is governed by the first exponential term, the propagation by the second exponential term and, once both are exhausted, the saturation is reached. The values of the fitting coefficients for the three cycles considered are reported in Table 4.5. The fitting parameters have been at first determined for the first cycle. Then, the same time constants have been imposed for the fitting of the other cycles without compromising the fit quality. From Table 4.5 it can be observed that the amplitude of the first exponential term a slightly increases with cycles, while that of the second exponential term, that lead to the stabilisation, is virtually constant from a cycle to the other. However, it must be specified that the value of τ a 1 reported in the Table 4.5 is just an intermediary value of all the possible time constants that allow to provide a good fit of the experimental data, ideally going from 1 h to about 70 h. The experimental value of reload modulus as a function of aging time for the first cycle together with the fitting functions in which τ a 1 has been varied, while the other coefficients have been kept constant, are represented in Figure 4.12. The time constant represents the time for the exponential term to reach 1 -1 e of its amplitude, which corresponds to about the 63%. Then it continues to increase and, for a time equal to 3τ , it reaches the 95% of its amplitude. After that time, its contribution to the growing trend is negligible. In the specific case of τ a 1 , it can be noted that in the three cases represented in Figure 4.12, the exponential term reaches the 95% of its amplitude a after an aging time lower than 220 h, that is the first experimental data available after the virgin one. So, to establish more precisely the value of τ a 1 , at least one intermediary point between 0 h and 220 h of aging is needed. However, independently on the exact value of τ a 1 , it is possible to estimate the value of reload indentation modulus that corresponds to the saturation of the thermal oxidation phenomenon, ideally reached after an infinite time, but practically, according to what previously said, after a time corresponding to 3τ a 2 . The calculated value of the "saturated" reload modulus at the first cycle is about 5.7 GP a, while at the last cycle is about 6.5 GP a.

E * loadv (GP a) a (GP a) τ a 1 (h) b (GP a) τ a 2 (
It has been shown from Figure 4.8b that the amount of energy dissipated with respect to the total energy during the first cycle decreases with increasing aging time, while from the second cycle, the evolution for virgin and aged samples are virtually superposed. Figure 4.13 depicts the evolution of η with aging time for the 1 st (red), 2 nd (blue), 3 rd (yellow), 5 th (black), 10 th (purple) and last (green) cycle. The decreasing trend of η of the first cycle with aging time can be, even in this case, described by an exponential law with two exponential terms, similar to Equation 4.4 (dashed red line in Figure 4.13):

η 1st = η 1st virgin -c 1 -exp - t ox τ a 1 -d 1 -exp - t ox τ a 2 (4.4)
In Equation 4.4, η 1st virgin represents the value of η of the first cycle of the virgin material, c and d are the amplitudes of the exponentials, τ a 1 and τ a 2 the characteristic times, representing here the time needed to drop from the initial amplitude value to 1 -1 e of that value. It has been found that the same characteristic times determined from the fitting of the evolution of the reload with aging time, are able to provide a good fit also for the evolution of η at the first cycle. The same considerations about the variability of the first time constant From Table 4.6, it results that the value of η 1st which corresponds to the "saturated" state is about 0.4. As it has been assumed that the value of η at the first cycle is related to the amount of plastic deformations, it is possible to conclude that both elastic and plastic behaviour evolves with aging time in a similar way. Moreover, the decrease in energy dissipated to produce plastic deformations is consistent with the fact that the material become more brittle, and thus less prone to undergo plastic deformations, with thermal aging as observed by other authors [START_REF] Tandon | Accelerated aging of pmr-15 resin at elevated pressure and/or temperature[END_REF][START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF][START_REF] Ernault | Origin of epoxies embrittlement during oxidative aging[END_REF]. For the second cycle, the value of η of the virgin material is slightly higher than that of the aged samples, for which it is constant for any aging time. This gap is not observed from the third cycle and, up to the last cycle, the value of η is constant with aging time. This observation suggests that probably, a small amount of energy is dissipated in the second cycle because of reverse plasticity. As reported in Section 1.3, reverse plasticity causes in metallic materials a small hysteresis loop between unload and reload indentation curves that disappears after few cycles [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF][START_REF] Shuman | Calculating the elastic modulus from nanoindentation and microindentation reload curves[END_REF][START_REF]Reverse Plasticity in Nanoindentation[END_REF]. So, from the evolution of η with aging time, it can be concluded that aging mainly affected the plastic behaviour of the material that arises at the first (plasticity) and at the second cycle (reverse plasticity). However, from the third cycle, for which energy dissipation is mainly due to a delay between the loading and the respective displacement response, the aging time has no influence.

τ
Finally, the effect of aging time on the net time-dependent displacement accumulated during cycles, calculated as the difference between the minimum displacement at the last and at the first cycle, will be considered. Its evolution with aging time is represented in Figure 4.14a. From Figure 4.14a it can be observed that the time-dependent displacement accumulated decreases from the virgin to the aged samples of about 0.02 µm (∼ 10%) but it varies very little for all the aging time considered. This gap of 0.02 µm is directly linked to the difference in the amount of energy dissipated at the second cycle between the virgin and the aged samples. This statement is confirmed by the fact that the difference between the minimum displacement at the last and at the second cycle is constant with aging time, as shown in Figure 4.14b.

Property gradients in aged polymers

The results presented in the previous Sections allowed to conclude about the impact of thermal aging on the indentation cyclic behaviour of the epoxy resin and about the evolution of reload modulus, dissipated energy and displacement accumulation with aging time. For these analysis, it was necessary to consider the results obtained at the same distance from the surface exposed to the environment for all the aging conditions. However, as showed in Section 1.4, the oxidized layer affects the material for a thickness of a few hundred of micrometers [START_REF] Pochiaraju | Evolution of stress and deformations in high-temperature polymr matrix composites during thermo-oxidative aging[END_REF][START_REF] Johnson | Investigation of oxidation profile in pmr-15 polymide using atomic force microscope (afm)[END_REF][3][4][5][START_REF] Jiang | Enhanced mechanical and thermal properties of monocomponent high performance epoxy resin by blending with hydroxyl terminated polyethersulfone[END_REF][START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF]. Thanks to the experimental procedure described in Section 4.1, it is possible to evaluate the local properties change in the oxidized layer and to trace the property gradients as a function of the distance from the external surface for the different aging conditions. In this Section, the results of 40 cycles indentation tests obtained on samples aged in atmospheric air will be considered. At first glance, the cyclic behaviour of the three indicators is qualitatively similar to that of the virgin sample for all the distances considered. The evolution of reload indentation modulus with cycles obtained at a distance of 240 µm from the surface, is virtually superposed to that obtained on the surface of the virgin sample. As the distance decreases, the curves are vertically shifted to higher modulus values, reaching the maximum for the test performed on the external surface (orange curve) that is, however, just slightly higher than that obtained for a distance of 45 µm (red curve).

The irreversible part of indentation work (Figure 4.16b), similarly to what observed for different aging times at the same distance from the surface, is affected by the distance at which the measurement is done mainly at the first cycle. The energy dissipated at the first cycle, that, we remind here, is supposed to be mainly due to plastic deformations, decreases when approaching the surface. From the second cycle, the evolution of η with cycles determined at different distances appears to be virtually superposed.

Finally, the minimum displacement increases with cycles in all cases, and is vertically shifted to lower values as the distance from the surface decreases, being similar to the virgin curve for the higher distance, and to the curve obtained on the surface for the lower distance considered on the cross section.

In the analysis of the effects of thermal aging on the indentation cyclic behaviour, presented in Section 4.3.1, in order to establish if the evolution with time of reload modulus and minimum displacement was the same for different aging time and thus just shifted on the y axis, the E * load -E * load 0 and h min -h min 0 values, where the subscript 0 indicates the value measured at the first cycle, were plotted against the testing time t -t 0 . It was found that all the experimental values related to the aged samples for the three aging times were superposed, but different form the curve of the virgin material.

To check how the time dependent behaviour is affected by the distance from the exposed surface, the E * load -E * load 0 and h min -h min 0 values are, here too, plotted against testing time for the different distances considered and for the virgin material, as depicted in Figure 4.17a and 4.17b respectively. 4.17b two main different time dependent behaviours can be distinguished: one of the external layers that is close to that obtained on the external surface, and one of the internal layers that is close to that obtained on the surface of the virgin sample. In other words, through the oxidized layer, there is a transition from the time dependent behaviour of the oxidized surface to that of the virgin sample observed in Figures 4.9 and 4.10 that, as observed in Section 4.3.1, can be described by an exponential law with two terms in which the amplitudes varies from virgin to aged, while the time constants are the same.

To conclude this Section, the gradients of local properties in the oxidized layer for the three aging times and for the virgin sample will be presented. In Figure 4.18, the reload modulus is plotted against the distance from the exposed surface for virgin (blue) and aged for 220 h (red), 520 h (green) and 1000 h (violet ) samples. For a better visualitation of the results presented in Figure 4.18, the reload modulus has been normalized with respect to the mean value measured on the virgin sample as:

E * load -E * load virgin E * load virgin .100 (4.5)
Moreover, only the modulus evaluated from the first reload is considered for this comparison. From Figure 4.18 it is possible to observe a similar behaviour to that obtained on different epoxy resins and for different aging conditions by other authors on the indentation modulus evaluated from unloading curve of a simple load-unload indentation cycle [START_REF] Pochiaraju | Evolution of stress and deformations in high-temperature polymr matrix composites during thermo-oxidative aging[END_REF][START_REF] Johnson | Investigation of oxidation profile in pmr-15 polymide using atomic force microscope (afm)[END_REF][3][4][5][START_REF] Jiang | Enhanced mechanical and thermal properties of monocomponent high performance epoxy resin by blending with hydroxyl terminated polyethersulfone[END_REF][START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF]. In particular, the modulus of the virgin material is virtually constant through the sample thickness, while that of the aged samples is higher at the surface and it progressively decreases in the oxidized layer approaching the value measured on the virgin sample. The value measured on the surface of aged samples is from about 40% to 50% higher than that of the virgin material, depending on the aging time. Then it decreases, quickly for the less severe aging condition and slowly for the higher aging time. Moreover, the profiles of the samples aged 520 h and 1000 h are virtually superposed from a distance from the external surface of about 70 µm. From Figure 4.19 it can be observed that η of the aged samples, measured at the surface, is about 15 -20% lower than that measured on the virgin sample, depending on the aging time. Then, it increases progressively approaching the value of the virgin material. Moreover, it can be noted that, similarly to what observed on the profiles of reload modulus, η measured through the oxidized layer of the less aged sample increases more quickly with increasing distance from the surface, while the profiles for the samples aged 520 h and 1000 h are quite close from a distance from the surface of about 80 µm. Furthermore, the gradients related to the longer aging durations stabilise at a value of η that is 4 -5% lower than that measured on the virgin sample.

No change of η from the second cycle is expected through the oxidized layer, relying on No appreciable variation of η through the thickness of the aged sample is observed for the second and last cycle (and reasonably for the others). Moreover, the η values are very close to that obtained on the virgin sample, except at the surface for the second cycle where, as already discussed in Section 4.3.2, the amount of energy dissipated in the virgin sample is slightly higher than that of the aged sample.

Furthermore, the profiles of property gradients allow to approximately estimate the thickness of the oxidized layer. It has been assumed that as the difference between the measured reload modulus and that of the virgin sample is lower than 4%, the core of the specimen is reached. The thickness of the oxidized layer estimated from the reload modulus profiles as a function of the aging time is depicted in As the other properties presented in Section 4.3.2, even the evolution of the thickness of the oxidized layer with aging time can be described with an exponential function with two exponential terms (dashed line in Figure 4.21) in which the two characteristic aging times τ a 1 and τ a 2 are still the same to those representing the evolution with aging time of the reload modulus and the irreversible part of work of indentation at the first cycle presented in the previous Section.

The thickness of the oxidized layer found form the profiles of reload modulus through the sample thickness, are significantly lower than that found by Pecora et al. [START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF] on the same material aged in air at 120 °C for 500 h and 1000 h. For an easier comparison, the thickness of the oxidized layers determined from the indentation modulus gradient profiles on the same material but for different aging temperatures are reported in Table 4 However, the authors [START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF] found a significantly lower variation of unload indentation modulus with respect to that of the virgin material: for the lower distance to the exposed surface, the difference determined was 6.4% for the sample aged 500 h and 9.8% for the sample aged 1000 h. In other words, for almost the same aging duration, oxygen penetrated more deeply in the case of aging at 120 °C, but it has induced a much lower change in indentation modulus than that found at an aging temperature of 150 °C. This observation suggests that the increase in aging temperature from 120 °C, that is about 25 °C lower that the glass transition temperature determined by DSC, to 150 °C, that is few degrees above T g has modified the kinetics of oxygen diffusion in the material and, maybe, has induced some secondary degradation phenomena different from pure thermal oxidation.

Aging under oxygen pressure

Previous studies carried out on different epoxy resins subject to thermal-oxydation, showed that the use of moderate oxygen pressures allows to accelerate the consequent degradation phenomena [START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF][START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF][START_REF] Vu | Experimental characterization of thermo-oxidation-induced shrinkage and damage in polymer-matrix composites[END_REF]. In other words, if oxygen pressure is used for aging, less time with respect to aging in air under atmospheric pressure would be needed to reach the same degradation level. For example, Minervino at al. [START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF] found on Tactix epoxy resin that the use of 3 bar of oxygen allows to accelerate the increase in indentation modulus deriving from thermal oxidation of about 7 times. Pecora et al. [START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF] found on PR520 epoxy resin aged at 120°C that the use of 3 bar of oxygen accelerates the increase in indentation modulus of about 5 times, while the decrease in failure stress determined from four-point bending tests was accelerated by a factor 10. Thanks to the acceleration factor, it is possible to build a master curve of the evolution of the considered property as a function of an "equivalent" time of aging in air. In this Section, the evolution with aging time of local properties determined from cyclic indentation tests performed on the surface of PR520 samples aged at 150 °C under 2 bar of Oxygen will be presented in order to establish the possibility to accelerate the degradation phenomena and to understand if the same "acceleration factor" could be applied to all the parameters.

Figure 4.22 depicts the evolution of the normalised reload modulus evaluated at the first cycle as a function of aging time for samples aged in air (red) and under 2 bar of oxygen (blue). From Figure 4.22, it can be observed that, as expected, the increase in reload modulus is higher for the samples aged under oxygen pressure. Moreover, whereas in the case of aging in air the reload modulus is not stabilised after 1000 h of aging, for the samples aged under 2 bar of oxygen, it reaches a plateau after 100 h of conditioning. To determine the acceleration factor, the time constants of the exponential law (Eq. 4.3) resulting from the fitting of experimental data obtained on samples aged in the two environments, are compared. The values of τ a 1 and τ a 2 and of the amplitudes a and b, determined from the fitting under oxygen pressure are reported in From Table 4.8 it can be observed that the amplitude of the first exponential term determined from the fitting of experimental data related to aging under oxygen pressure is almost zero. This means that just one exponential term is sufficient to describe the evolution of reload modulus with aging time in these conditions, according to experimental data available. It is interesting to observe that the sum of the amplitudes a and b related to aging in air and under oxygen pressure is virtually the same, meaning that the aging in the two environments lead to the same "saturated" value of reload modulus. This observation suggests that the use of oxygen pressure can, actually, accelerate the increase in reload modulus, but from Table 4.8 it is clear that in the first stages of aging, the degradation phenomena are different to those leading to propagation and stabilization and are accelerated in different ways from the use of 2 bar of oxygen. An estimation of the acceleration factor of the first stages of aging is not possible, because of the difficulties in the determination of τ a 1 in the case of aging in air, already discussed in Section 4.3.2. However, the ratio between τ a 2 of air and oxygen reported in Table 4.8, gives an estimation of the acceleration of the last stages of thermal oxidation that is equal to about 27. In Figure 4.23, the normalised reload modulus is plotted against the "equivalent" aging time. The "equivalent" aging time is equal to the effective time for the samples conditioned in air, while it is In fact, for the two lower aging durations under oxygen pressure (first two blue data points), an acceleration factor of 27 times is too much. These two points are out of the master curve (dashed line in Figure 4.23) and, in particular are shifted to the right. However, for the longer aging duration, the determined acceleration factor seems to be appropriate as the last three points follow well the master curve.

It is interesting to observe the effect of the use of oxygen pressure on the other parameters extracted from cyclic indentation load-displacement curves. Figure 4.24 depicts η calculated at the first, the second and the last cycle, as a function of aging time for samples conditioned in air (red) and under 2 bar of oxygen (blue). From Figure 4.24 , for the first cycle, it seems that the use of oxygen to condition the samples has slightly accelerated the decrease of η with aging time in the second stage of aging, that leads to propagation and stabilisation. At the second cycle, the value of η is slightly higher for the virgin and for the sample aged 19 h under oxygen pressure. This phenomenon has been already observed in the analysis of the property change with aging time in air (Section 4.3.2) and has been related to some reverse plasticity occurring during the firsts few cycles that is affected by thermal aging. For the further cycles there is not appreciable difference on η for all the aging times and for both environments. The parameters determined from the fitting of η 1st as a function of aging time with Equation 4.4 for the two environments, are reported in Table 4 From Table 4.9 it can be observed that no appreciable difference exists in the kinetics of evolution of η 1st with aging time in the first stage of aging for the two environments, as the characteristic times τ a 1 are quite similar for the two cases. The comparison of the two constant times that govern the second stage of aging (τ a

2 ) shows that a factor of 3 allows to bring the experimental data for an aging time under oxygen pressure > 100 h to the same master curve, as can be observed in Figure 4.25. Even for η calculated at the first cycle, aging under oxygen pressure leads to the same "saturated" value, but the acceleration factor is significantly lower than that found for the reload modulus.

The results presented in this Section showed that the use of oxygen pressure can, actually, accelerate the degradation phenomena caused by thermal aging in air: the increase in the reload modulus and the decrease of amount of energy necessary to produce plastic deformations. However, caution must be taken as not all the phenomena are accelerated in the same way. This observation is consistent to what reported by Ernault et al. [START_REF] Ernault | Thermal-oxidation of epoxy/amine followed by glass transition temperature changes[END_REF]. According to the authors [START_REF] Ernault | Thermal-oxidation of epoxy/amine followed by glass transition temperature changes[END_REF], in fact, the use of high oxygen pressure leads to different macromolecular modifications than aging in air under atmospheric conditions. Moreover, from the analysis of reload modulus, it was found that the different stages of aging are not governed by the same phenomena as they are accelerated in a different way from the use of oxygen pressure.

Discussion on thermal aging

In this Section, the results obtained from complementary tests performed on aged samples will be presented. In particular, the macromolecular modifications induced by thermal aging will be discussed. According to what reported in Section 1.4, from glass transition temperature change resulting from conditioning it is possible to establish whether cross linking or chain scission phenomena predominates during thermal aging.

The PR520 surface color change with aging time and for the different aging conditions employed in this study is represented in From Figure 4.26 it can be observed that, while the surface of the virgin material is white, the aged samples changed their color from yellow, orange, brown and finally black. Moreover, it can be observed that the surface of samples aged in air became black after about 500 h, while in the case of aging under Oxygen pressure the samples are already black after about 100 h. It can also be observed that aging under Nitrogen pressure has caused a change in surface color as well.

The material changes at the macromolecular scale induced by thermal aging have been investigated by evaluating the modification of glass transition temperature in aged samples.

As for the virgin material, already presented in Section 4.1, glass transition temperature has been measured through DSC on specimens drown from the oxidized layer and from the core of oxidized samples by grinding the bulk aged samples. Only aging in air has been considered for this analysis, in the knowledge that aging under oxygen pressure would be worthy of further investigations. Figure 4.27 depicts the evolution of glass transition temperature measured on the oxidized layer (orange) and at the core (blue) as a function of aging time. From Figure 4.27, it results that an increase in glass transition temperature is detected either on the oxidized layer and at the core of the aged samples. Even if this increase is higher on the exposed surface, this result shows that the whole sample has undergone macro molecular modification during its exposure at 150 °C. For the longest aging duration, the difference in T g measured on the oxydized layer is about 10%, while is about 3% on the core.

As reported in Section 1.4, the increase in glass transition temperature shows that, during thermal aging in air at 150 °C, crosslinking phenomena predominate in the PR520 epoxy resin. It is interesting to observe that this trend is in contrast to that found by Guigon [START_REF] Guigon | Vieillissement par cyclage thermique de composites interlock 3D à matrice polymère[END_REF] on the same material aged at 120 °C in air and under 3 bar of oxygen. In fact, Guigon [START_REF] Guigon | Vieillissement par cyclage thermique de composites interlock 3D à matrice polymère[END_REF] measured a decrease in glass transition temperature with aging time in both environments, meaning that the material undergoes mostly chain scission in these conditions. However, for both aging temperatures, the main effects of thermal aging on the mechanical behaviour are an increase in indentation modulus and an embrittlement of the material. This is consistent with what reported by Ernault at al. [START_REF] Ernault | Origin of epoxies embrittlement during oxidative aging[END_REF] in which material embrittlement has been observed on both samples that undergoes chain scission or crosslinking during thermal aging. As explained in Section 1.4, according to Ernault et al. [START_REF] Ernault | Origin of epoxies embrittlement during oxidative aging[END_REF], material embrittlement is not related to chain scission modifications, but to a reduction of the β transition (antiplastification). These observations suggests that the use of an aging temperature slightly higher than the glass transition temperature, could have induced in the material degradation phenomena other than pure thermal-oxidation. This statement is confirmed by the slight colour change showed by the sample aged in an inert environment (2 bar of Nitrogen) for 100 h at the same temperature of 150 °C. On this sample, an increase in reload indentation modulus of about 8% with respect to the virgin material has been measured on the exposed surface. This difference is much lower that that measured on the samples aged in air and under 2 bar of oxygen, suggesting that thermal-oxidation is still the main degradation phenomenon that occurs for this temperature. The effect of aging under Nitrogen pressure has not been investigated further in this thesis work.

Conclusions

In this Chapter, the effect of thermal aging on time dependent local properties of PR520 epoxy resin has been studied through cyclic indentation tests. At first, the cyclic indentation behaviour of the material in its virgin state has been analysed. The material shows a response to local cyclic load similar to that observed on HDPE in Chapter3, highlighting the time-dependent nature of the epoxy resin response even if less pronounced than the semi crystalline polymer. Then, the effects of thermal aging on cyclic indentation response have been presented. The aging conditions considered were thermal aging in air at 150 °C for different durations. The main conclusions that arise from the analysis presented in this Chapter are:

1. The kinetics of evolution of reload indentation modulus (E * load ), irreversible part of indentation work (η) and net displacement accumulated (h min ) with cycles is similar for virgin and thermally aged samples;

2. The evolution of elastic (reload modulus) and plastic (irreversible part of indentation work dissipated at the first cycle) material response with aging time, evaluated on the surface exposed to the environment, can be described by an exponential law with two exponential terms. This law was chosen because it allows to describe the three stages of thermal oxidation: initiation, propagation and saturation. The characteristic aging times that govern the kinetics of evolution of both reload modulus and η 1st with aging time are similar;

3. The amount of energy dissipated to produce time-dependent deformations (η from the second cycle) is not affected by thermal aging. The slight difference observed between η evaluated at the second cycle for virgin and aged samples has been related to reverse plasticity;

4. The effect of decreasing distance from the surface on load displacement curves for the same aging time, is similar to that observed for the same distance and increasing aging time;

5. Both reload modulus and η at the first cycle evolve in the oxidized layer approaching the value measured on the virgin sample, while η form the second cycle is constant through the sample thickness and close to the value measured on the virgin sample;

6. The thickness of the oxidized layer found from the profiles of reload modulus through the sample thickness, are significantly lower than those found in another work [START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF] on the same material aged in air at 120 °C for similar durations. This means that oxygen penetrated more deeply in the case of aging at 120 °C, but it has induced a much lower change in indentation modulus than that found for an aging temperature of 150 °C;

7. The use of 2 bar of oxygen pressure can accelerate the degradation phenomena caused by thermal aging in air: the increase in the reload modulus and the decrease in the amount of energy necessary to produce plastic deformations. However, caution must be taken as not all the phenomena are accelerated in the same way;

8. The increase in glass transition temperature with aging time, at the core and at the surface by 7°C and 15°C respectively, shows that, during thermal aging in air at 150 °C, cross-linking phenomena predominate in the PR520 epoxy resin;

9. An increase in reload indentation modulus of about 8% with respect to the virgin material has been measured on the exposed surface of a sample aged at 150°C in an inert environment (2 bar of Nitrogen), suggesting that the use of an aging temperature slightly higher than the glass transition temperature, could have induced in the material degradation phenomena other than pure thermal-oxidation.

Chapter 5

Local time-dependent properties of PR520 epoxy resin matrix of 3D interlock carbon fiber reinforced composites

In this Chapter, another application of cyclic indentation tests, aiming to capture gradient of local properties, will be proposed. The gradient of properties to which we will refer in the first part of this Chapter are that caused by the presence of fibers in PR520 epoxy resin matrix of 3D interlock carbon-fiber reinforced composites. In fact, curing of polymer matrix around the fibers, as in Resin Transfer Molding process, introduces thermal residual stresses in the material and could modify the crosslinking of polymer matrix. Cured this way matrix could demonstrate different local mechanical behaviour (in-situ) compared with that of the polymer without fiber reinforcement (neat). Moreover, the composite manufacturing process could induce differences in local resin degree of cure because of the complex thermal paths in 3D interlock. For that reason, we have studied not only the differences between neat and in-situ mechanical properties but also the heterogeneity that could exist among different resin pockets in the matrix, in the core as well as on the external surface of the sample. In this Chapter, the material employed and the resin pockets on which instrumented indentation tests have been performed, will be, at first, presented. A discussion on the constraint effect of surrounding fibers on indentation measurements will follow. Then the heterogeneity of mechanical properties of the PR520 epoxy resin matrix of a 3D interlock carbon-fiber reinforced composite and their deviation from neat polymer will be studied by means of cyclic indentation loading. In order to understand if the difference among the results is significant from a statistical point of view, the Student t-test [START_REF] Livingston | Who was student and why do we care so much about hist-test[END_REF] has been applied to compare the mean values of different data sets. In the last part of this Chapter the property gradients in the oxidized layer of thermally aged composites will be presented and compared to that measured on the neat material.

Material and resin pockets

The composite material used in this study is an advanced polymer matrix composite reinforced with 3D fiber architecture. The manufacturing process of this kind of composite materials consists of first feeding the warp yarns into the weaving loom [START_REF] Mouritz | Review of applications for advanced three-dimensional fibre textile composites[END_REF], after which the warp tows are separated with a lifting mechanism and weft yarns are inserted into this space at a specific angle. A third element, the binder yarns, can also be introduced in the 3D pattern and can be aligned in the warp direction or inserted in the weft direction. An example of interlock woven fiber architecture in which warp, weft and binder yarns are highlighted is represented in Figure 5.1. Once the fiber preform is created, it is dried and placed into a mold. Then, the polymer which constitutes the matrix (the PR520 epoxy resin in this case), previously heated to the optimal temperature allowing to minimize its viscosity, is injected in the mold at high pressure. Finally the material is cured following the specific procedure of the resin matrix. This kind of manufacturing process is known as Resin Transfer Molding (RTM) and it has been shown to be the most adapted for 3D woven composites [START_REF] Carronier | Approche intégré du RTM (Resin Transfer Molding)[END_REF]. One important advantage of 3D weaving is that preforms with a complicated geometry can be made to the near-net-shape. Moreover they show a higher impact resistance, higher strain to failure values and higher mode I interlaminar fracture toughness and are less prone to delamination with respect to unidirectional carbon fiber reinforced epoxy laminates [START_REF] Mouritz | Review of applications for advanced three-dimensional fibre textile composites[END_REF].

Another advantage in the use of this kind of composite architecture, specific to our application, is the high number of large resin pockets available in the sample in which the polymer matrix can be locally tested by instrumented indentation. The surfaces on which cyclic indentation tests have been performed have been prepared by mechanically grinding and finishing, following the same protocol used for the neat virgin and aged epoxy resin and reported in Table 2.4. As announced in the Introduction, the analysis of time-dependent local properties of epoxy matrix is divided into three main parts: in the first, the heterogeneity of the matrix will be studied by comparing cyclic indentation results obtained in different resin pockets located on the external surface and in the sample core of the composite plate. In this context, we have distinguished between resin pockets in the core that are located in the plan perpendicular to the weft direction and to the warp direction. In the second part, the properties measured on the resin matrix in-situ will be compared to the properties measured on the neat resin. In the last part, the property gradients measured on thermally aged neat and in-situ epoxy resin will be compared. In this last comparison, the resin pockets located on the external surface will not be considered.

A scheme of the 3D fiber architecture of a composite material similar to the one used in our study is represented in Figure 5.2. In Figure 5.2 the location of resin pockets on the external surface, in the core on the plane perpendicular to the weft direction and in the core on the plane perpendicular to the warp direction have been highlighted. The microscopic images of the 4 resin pockets located in the plan perpendicular to the weft direction, in which instrumented indentation tests have been performed, are represented in Figure 5.3. The red circles highlight the 9 indentation prints performed in each pocket. From the microscopic images of Figure 5.3 the minimal distance between indentation prints and the edge of the closer fiber bundle have been measured and the values are reported in Table 5 The microscopic images of the 5 resin pockets in the plan perpendicular to the warp direction are reported in Figure 5.4. From Figure 5.5 it is possible to observe that the characteristic dimension of resin pockets located on the external surface is of the order of several millimeters, that is much higher than that of the pockets located in the sample core. Because of the high magnification necessary to observe both resin matrix and fiber bundles, the indentation prints are not well visible and have thus been highlighted in red. From Figure 5.5 it can be observed that the indentation prints are very far away from the fiber bundles and that the lower distance is higher than 500 µm.

The matrix shrinkage near the fiber bundles, that can induce a tip/sample misalignment during indentation tests as shown in Section 2.7.2, is, in the case of virgin composite samples, only due to cut and polishing procedures. According to what found by other authors [START_REF] Vu | Experimental characterization of thermo-oxidation-induced shrinkage and damage in polymer-matrix composites[END_REF] and reported in Section 1.4.2, the shrinkage is negligible on virgin samples and shall not affect the indentation tests.

Constraint effect on in-situ measurements

As reported in Section 1.1.3, previous works conduced on unidirectional carbon-fiber polymer-matrix composites showed that the main difficulty in matrix characterization with instrumented indentation is to decouple the constraint effect of surrounding fibers from a proper change of polymer matrix properties [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF][START_REF] Hardiman | The effect of fibre constraint in the nanoindentation of fibrous composite microstructures: A finite element invesitgation[END_REF][START_REF] Hardiman | Fibrous composite matrix characterisation using nanoindentation: The effect of fibre constraint and the evolution from bulk to in-situ matrix properties[END_REF]. Two-dimensional finite element simulations of elasto-plastic composite matrix of unidirectional carbon fiber reinforced composites showed that, to avoid any constraint effect, the resin pockets should be at least 50 times larger than the maximal penetration depth [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF]. For our testing procedure, this theoretical distance is about 33 µm that is significantly lower than the values reported in Tables 5.1 and 5.2. That means that our instrumented indentation measurements should be unconstrained.

However, in the 3D interlock composite considered in our study, the constraint effect of fiber bundles could be different than that of unidirectional fibers. Moreover, the constraint effect could be due not only to fiber bundles surrounding resin pockets, but also by bundles that could pass below the tested surface. As reported by Gregory and Spearing [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF], unconstrained loading curve should follow a power law, also known as Kick's law, already described in Section 1.1 that we remind here (Eq. 1.1):

F = C h n
Since tests performed on neat resin are obviously unconstrained, any deviation of the in-situ loading curve fitting parameters (n and C) from that of neat material is, according to the authors [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF], due to a fibers constraint effect. Gregory and Spearing [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF] considered as unconstrained, tests for which the deviation between neat and in-situ fitting parameters were lower than 10%. In order to verify if properties that we have measured on the external surface and in the core in both warp and weft direction of the composite sample are not affected by any constraint effect, the first loading curves have been fitted with Kick's law (Eq. 1.1) and the in-situ fitting coefficients C and n have been compared to that of the neat resin. The results are summarized in Figure 5.6 in which error bars represent ± one standard deviation. From Figure 5.6 is evident that there is virtually no difference among neat and in-situ exponent n values and a slight difference on curvature C values that, however, does not exceed 5.5%. We can thus consider that properties measured in the composite are unconstrained and so that any deviation observed between different in-situ data sets and neat and in-situ data sets is due to a proper material change.

Statistical analysis of the local properties of composite materials

In order to establish if any apparent local property change observed on measurement performed on different resin pockets in the composite and on neat material is significant and representative of a true property change, a statistical analysis of experimental data has been carried out using the Student t-test [START_REF] Livingston | Who was student and why do we care so much about hist-test[END_REF]. The Student t-test consists in calculating the value of the Z parameter from the mean values (X), standard deviations (σ) and number of tests (N ) of two data sets (Eq. 5.1):

Z = |X 1 -X 2 | σ 2 1 N 1 + σ 2 2 N 2 (5.1)
The calculated value of Z is then compared to its critical value Z crit found in tables [4] for a given level of confidence and for a number of degree of freedom d f calculated as (Eq. 5.

2):

d f = N 1 + N 2 -2 (5.
2)

The comparison between the two values of Z allows to accept (Z > Z crit ) or to reject (Z < Z crit ) the null hypothesis that the two data sets are drawn from different populations with a probability that depends on the level of confidence chosen. The parameters considered for the comparison are: the indentation modulus calculated from reload curves, the displacement accumulated during cycles (h min ) and the irreversible part of indentation work (η).

Heterogeneity of polymer matrix in the composite

As previously mentioned, because of complex thermal paths generated in Resin Transfer Molding process, resin matrix could present different curing degrees and thus could show different local properties in different sample zones. To verify the homogeneity of polymer matrix, several cyclic indentation tests have been performed on different resin pockets among fiber bundles found on the external surface and in the core of the sample as described in Section 5.1. At first, the cyclic indentation results obtained on different pockets on the surface perpendicular to the weft direction were compared by applying Student t-test on the mean values of each pair of pockets of Figure 5.3. The test revealed no statistically significant difference for an interval of confidence of 99.9%. The application of the Student t-test for the comparison of mean values of the results obtained in each pair of pockets on the surface perpendicular to warp direction (Figure 5.4) showed no statistically significant difference with a probability of 99.9% as well. A similar result was obtained from the comparison of the average value of all pockets on the plane perpendicular to the weft direction with that of all pockets on the plane perpendicular to the warp direction. It is thus possible to conclude that the unconstrained matrix far from fibers is homogeneous in the sample core.

Moreover, the comparison of each pair of pockets on the external surface of the sample (Figure 5.5) through the Student t-test for an interval of confidence of 99.9% has not revealed statistically significant differences, showing that the matrix is homogeneous also on the surface.

Finally, the average of all resin pockets in the core and the average of all resin pockets on the external surface have been compared. The evolution with cycles of reload indentation modulus, minimum displacement and dissipated to total energy ratio (η) of in-situ measurements on the core (squares) and on the external surface (circles) are represented in Figures 5.7a From Figures 5.7 it results that the cyclic behavior of matrix surface and core are similar and follow the same trend of neat PR520 epoxy resin presented in the previous Chapter. While cyclic behaviour is similar, a slight difference on core and surface mean values is observed. In particular, reload modulus is higher in the sample core, minimum displacement is higher on the surface while the dissipated to total energy raio η is higher on the surface for the first cycle, and slightly higher in the core for the further cycles. Moreover, the total time-dependent displacement accumulated during cycles, given by the difference between minimum displacement at the last and at the first cycle, is the same in the two cases and equal to 0.16 ± 0.01 µm.

The Student t-test has been applied to compare the two data sets. The values of Z From Figure 5.9 we can observe that, except for η, the percent difference between the two data sets is almost constant with cycles, confirming that the cyclic behaviour of the two materials is similar. This difference is about 2% on reload indentation modulus and 8% on minimum displacement. The difference on η at the first cycle is about 8%, while for the subsequent cycles is between 4 and 5%.

The observed gap between surface and core properties remind the so called "skin-core" structure observed on injected thermoplastic parts. The "skin-core" structure reveals different morphological characteristics, orientation and stress in outer layer and inner part [START_REF] Utsumi | Effects of calcium stearate addition of ultralight molecular weight polyethylene in direct compression molding[END_REF]. As the "skin-core" structure could affect the quality of the ready product, this phenomenon has been the subject of many researches in an attempt to optimize the process parameters such as injection speed, mold temperature, flow rate and cooling rate [START_REF] Gao | The structure and rheology of molten ultra-light-molecularmass polyethylene[END_REF], or to characterize the mechanical properties of skin and core [START_REF] Persson | Characterizing the mechanical properties of skin-core structure in polymer molding process by nanoindentation[END_REF][START_REF] Chu | Characterization of the Micro Injection Molding Process and Its Products[END_REF][START_REF] Karger-Kocsis | Skin -core morphology and failure of injectionmolded specimens of impact -modified polypropylene blends[END_REF]. In that regard, Gao and Mackley [START_REF] Gao | The structure and rheology of molten ultra-light-molecularmass polyethylene[END_REF] characterized by nanoindentation the skin and core local properties of HDPE parts produced by injection molding. They found that the reduced modulus and hardness measured at different distances from the edge are higher on the skin, lower in the core and minimum in the intermediate region. An opposite trend was observed by Chu [START_REF] Chu | Characterization of the Micro Injection Molding Process and Its Products[END_REF] in a thesis work on the characterization of the Micro Injection Molding process. In fact, according to the author [START_REF] Chu | Characterization of the Micro Injection Molding Process and Its Products[END_REF], reduced modulus and hardness of HDPE are higher in the core region than for the skin and trans-crystalline layer. The main cause of "skin-core" structure observed on thermoplastics is the difference in crystals orientation due to thermal gradients from the mold walls towards the center: a skin layer with high orientation is formed because of the hot polymer melt contacting the cold mold. In cross linked thermoset polymers, like epoxies, the molecular structure is not crystalline but amorphous, so that the observed "skin-core" structure is more likely to be due to a different crosslinking degree probably caused, even in this case, by thermal gradients in the mold. To our knowledge, there is a lack of studies on skin effect on epoxy matrix composites produced by resin transfer molding. This phenomenon is thus worthy of further investigations in the future.

Neat vs. in-situ properties of PR520 epoxy resin

In the previous Section, it has been shown that the unconstrained local mechanical properties of the matrix measured in large resin pockets among fiber bundles are homogeneous in the sample core as well as on the sample external surface. However, a statistically significant difference exists between properties measured on the surface and on the sample core. In this section, in-situ properties will be compared with that of the neat material. For this analysis, a total of 25 tests has been performed on the polished surface of the neat polymer. The evolution with cycles of reload indentation modulus, minimum displacement and η of neat (triangles), in-situ in the core (squares) and in-situ on the external surface (circles) PR520 epoxy resin are represented in Figure 5.10a, 5.10b and 5.10c respectively. From Figure 5.10, it results again that the evolution with cycles of the different parameters is similar in the three cases, however a slight difference in mean values exists. In particular, neat reload modulus is pretty close to in-situ modulus measured on the surface, while it is slightly lower than the in-situ modulus measured in the sample core. The neat minimum displacement is, closer to in-situ minimum displacement measured on the core, but slightly higher, while it is lower than the in-situ minimum displacement measured on the surface. Finally, neat η of the first cycle is close to in-situ η measured on the core and lower than in-situ measured on the external surface. From the second cycle, neat η is quite close to both in-situ on the core and on the surface and its evolution with cycles is between the two in-situ measurements. In order to understand if this difference is significant form a statistical point of view, the Student t-test have been applied once more. The neat properties have been compared at first to average in-situ properties evaluated in the composite sample core. The values of Z calculated at each cycle (Eq. 5.1) are represented in Figure 5.11 as a function of the number of cycles for all the parameters. The dashed red lines represent the values of Z crit for an interval of confidence of 99.9% (upper line) and 95% (lower line). The result of statistical test suggests that, except for the irreversible part of work of indentation, there is a 99.9% of probability that the values of the parameters neat and in-situ in the core belong to different populations. The percent difference has been quantified and the results are reported in Figure 5.12. In Figure 5.12, we can observe that the percent difference between neat and in-situ in the core mean values is quite small, that it lies between 1% and 2.5% and that it is almost constant with cycles (except for η).

The last comparison that has been considered is between neat and in-situ measured on the external surface properties. The values of Z of all parameters as a function of the number of cycles is represented in Figure 5.13. In this case, there is not statistically significant difference on reload indentation modulus as its Z values are close to the Z crit corresponding to an interval of confidence of 50% (lower red dashed line). The other parameters belong, however, to different populations for an interval of confidence of 99.9% (upper red dashed line). Their percent difference is about 4% for minimum displacement, 7% for η at the first cycle and 5% for η from the second cycle as represented in the histogram in Figure 5.14. As reported in Section 1.1.3, very few studies are available in the literature that aim to compare experimental local constituent neat and in-situ properties in composite materials. In the particular case of 997 -3 epoxy resin/carbon fiber composite system, Gregory and Spearing [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF] found that in-situ modulus is 20 -30% higher than neat modulus while there was virtually not difference in the measured hardness values. Hardiman et al. [START_REF] Hardiman | Fibrous composite matrix characterisation using nanoindentation: The effect of fibre constraint and the evolution from bulk to in-situ matrix properties[END_REF] found on co-cured neat and in-situ 6376 epoxy resin that, even for unconstrained values, the in-situ modulus depends on matrix pocket radius. That is probably caused by a far-field interphase effect and/or to a difference in local curing conditions near fibers. The authors [START_REF] Hardiman | Fibrous composite matrix characterisation using nanoindentation: The effect of fibre constraint and the evolution from bulk to in-situ matrix properties[END_REF] concluded that the difference between mean values of unconstrained in-situ modulus and neat modulus of 6376 epoxy resin is about 10% while the difference on Hardness is about 6%. In both cases [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF][START_REF] Hardiman | Fibrous composite matrix characterisation using nanoindentation: The effect of fibre constraint and the evolution from bulk to in-situ matrix properties[END_REF], tests were performed on unidirectional carbon fiber reinforced composites, fabricated using an autoclave cure, in the direction parallel to fibers. The results of the comparison between neat and in-situ in the core and on the external surface is significantly lower than those reported by [START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF][START_REF] Hardiman | Fibrous composite matrix characterisation using nanoindentation: The effect of fibre constraint and the evolution from bulk to in-situ matrix properties[END_REF]. However, as previously mentioned, Hardiman at al. [START_REF] Hardiman | Fibrous composite matrix characterisation using nanoindentation: The effect of fibre constraint and the evolution from bulk to in-situ matrix properties[END_REF] found that even unconstrained indentation modulus depends on pocket size and, even if the average unconstrained modulus is 10% higher than the neat modulus, for the larger resin pockets the difference between neat and in-situ values is about 2% that is consistent with our result obtained considering in-situ in the core values. It must be specified that the analysis proposed here does not allow to conclude about the properties of matrix near fibers, in the so called "interphase". It is maybe in this zone that a more significant difference in local properties exists. However, according to what reported in Section 1.1.3, instrumented indentation is not an adequate technique to obtain local properties of interphase because of its proximity to fibers that induces a constraint effect on measurements.

Characterization of the gradient of local mechanical properties in aged composites

In this Section, the property gradients measured in the oxidized layer of in-situ PR520 epoxy resin aged for 1005 h in air at 150 °C will be presented and compared to that obtained on the neat material, already discussed in Section 4.3.3. In this Section, the external surface of the composite plates, in which a "skin-effect" has been identified in the previous Sections, will not be considered. So, all the results presented here are referred to indentation tests performed on the sample sides located in the warp and weft directions (Figure 5.2). In order to perform indentation tests on the oxidized layer of composite materials, the same procedure used to test the neat aged material, described in Figure 4.4 has been followed. The composite material has been thus preliminary aged, then cut, embedded in a clear epoxy and polished according to the protocol reported in Table 2 According to what observed in Chapter 4, thermal aging in these conditions is not expected to modify time-dependent properties. For that reason, only the gradients of elastic (reload indentation modulus at the first cycle) and plastic (η of the first cycle) part of the material response will be considered. We remind that the first point of the profiles, referred to a distance of 0 µm from the exposed surface, are performed in resin pockets located in warp and weft direction (defined as sample core in previous Sections) and not on the external surface of the composite plate. The resin pocket located on the surface perpendicular to the weft direction directly exposed to the environment on which indentation tests have been performed is represented in Figure 5.16. The shadow near fiber bundles in Figure 5.16 clearly shows the matrix shrinkage induced by thermal aging previously discussed in Sections 1.4.2 and 2.7.2. In order to avoid errors due to tip-sample misalignment in the sample zones where matrix shrinkage is more significant (near fibers), it has been shown in Section 2.7.2 that indentation tests should be performed at a distance from fiber bundles of at least 50 µm. However, to be more safe, only tests performed at a distance > 100 µm were considered. The evolution of reload modulus and dissipated to total energy ratio η, both referred to the first cycle, as a function of the distance from the surface exposed to the aging environment for neat (filled red triangles) and in-situ (unfilled red triangles) PR520 epoxy resin aged for 1005 h in air at 150 °C are represented in Figure 5.17a and 5.17b respectively. The mean values obtained on neat (filled blue circles) and in-situ (unfilled blue circles) virgin material have been reported as well in both Figures. a good agreement can be observed between the property gradient profiles measured in the oxidized layer of neat and in-situ aged PR520 epoxy resin. In particular, both neat and in-situ reload moduli of the aged material decrease as the distance from the exposed surface increases, and stabilise to a value close to that of the virgin neat material at the same distance from the exposed surface. Similarly, η of the first cycle increases while going away from the sample edge for both neat and in-situ aged material and stabilizes to a value close, but slightly lower, than that measured on the virgin samples for the same distance from the exposed surface. However, for the test performed on the surface directly exposed to the environment, a gap between neat and in-situ values is observed. This gap on reload modulus is quite similar to that measured between neat and in-situ virgin material, while it is higher on η.

The percent difference between neat and in-situ reload modulus (yellow) and η (blue) of the first cycle measured at the same distance from the exposed surface of aged samples is represented in the histogram of Figure 5.18. The gap between neat and in-situ virgin material is reported as well in the left part of the graph. .18 confirms that the percent difference between neat and in-situ reload modulus measured on the surface of aged samples (about 2.5%) is similar to the difference evaluated between neat and in-situ virgin samples (about 2.2%). However, the difference on neat and in-situ η values of aged samples is much higher (about 6%) than that measured between neat and in-situ virgin samples (less than 1%). That is probably due to a more far field interphase effect induced by thermal-aging. It is interesting to observe that in the oxidized layer, for distances of 45, 120, 200 and 280 µm from the exposed surface, the difference between neat and in-situ values is less than 1%. This observation suggests that the differences in curing degree that have probably caused the slight gap between neat and in-situ properties of the material at the virgin state, were remedied by keeping the samples at elevated temperature for a certain time. To verify if this "homogenisation" of neat and in-situ properties is limited to the oxidized layer or concerns also the inner part of the samples, results obtained in the core of aged samples have been compared and are reported on the right part of the histogram of Figure 5.18. This results are referred to in-situ indentation tests performed in the pocket 2 circled in red in Figure 5.15 (far from the sample edge) and to neat tests performed at a distance from the exposed surface > 300 µm (much higher than the thickness of the oxidized layer). From Figure 5.18, it results that also in the core of aged samples, the difference between neat and in-situ properties is lower than 1%.

Conclusions

In this Chapter, the unconstrained local properties of PR520 epoxy resin matrix of 3D interlock carbon -fiber reinforced composites have been studied through cyclic instrumented indentation. In the first part of the Chapter, the homogeneity of the resin matrix have been discussed through the comparison of reload indentation modulus, ratio of energy dissipated with respect to the total energy (η) and displacement accumulated during cycles, obtained from cyclic indentation tests performed in different resin pockets located on the external surface of the composite plate and in the core of the sample. In order to establish if any difference on local properties is significant from a statistical point of view, the Student t-test has been applied. In the second part of the Chapter, the resin properties measured in the composite (in-situ) were compared to that of the neat material. In the last part, neat and in-situ property gradients due to thermal aging have been compared. The main conclusions that arise from this analysis are:

1. The resin matrix is, far from fiber bundles, homogeneous in the core as well as on the sample surface;

2. The in-situ local properties measured in the composite sample core are significantly different from that measured on the composite sample surface, from a statistical point of view. This difference has been attributed to a phenomenon similar to the "skin" effect observed on the injected thermoplastic parts. The neat properties are significantly different from both the in-situ properties measured on the core and on the external surface. The difference in percentage between PR520 properties in the different configurations considered are summarized in Table 5.3: The difference between the two profiles for the same distance from the exposed surface is < 1%;

E * load h min η 1st η in-situ core -in-situ surface 2% 8% 8% 5% neat -in-situ core 2% 1.5% < 1% < 1 neat -in-situ surface < 1% 7% 6% 6%
4. The permanence of the material at elevated temperature for a certain time have remedied the gap, probably induced by a different degree of cure, measured on the neat and in-situ virgin materials.

The cyclic indentation to test the local time-dependent behaviour of polymers

The cyclic indentation testing protocol proposed in this study and its analysis method highlight the complex local time dependent response of polymers. The analysis of the hysteresis load-displacement loops allows deducing the time-dependent response at three different time scales: instantaneous, represented by the indentation modulus; short time (at the time-scale of the cycle), represented by the energy dissipated from the second cycle; long time (at the time-scale of the test), represented by the displacement accumulated during cycles. Even if further investigations are needed to establish if the deformations accumulated during cycles are fully recoverable (viscoelastic) or partially viscoplastic, the analysis of energy dissipation showed that most of the permanent print is produced during the first loading and has been associated to time-independent plastic deformations. Moreover, even if loop stiffness can be evaluated from the unloading and reloading curves through the application of the Oliver and Pharr, the application of this method on the unloading curves, showed some limitations while its application on reloading curves is consistent with experimental observations and Sneddon's solution. That is why the reload modulus has been chosen to represent the viscoelastic instantaneous response.

The effect of thermal aging on local time-dependent response of PR520 epoxy resin

The application of cyclic indentation tests to thermally aged PR520 epoxy resin samples at 150°C (temperature slightly higher than glass transition measured by DSC) in atmospheric pressure for different durations (up to 1000 h) showed that aging in these conditions mainly affected the elastic and plastic response, while the time-dependent behaviour was not modified. The evolution of elastic (reload modulus at a defined cycle) and plastic (irreversible part of indentation work dissipated at the first cycle) material response with aging time, evaluated at the surface exposed to the environment, is described by an exponential law with two exponential terms. This law was chosen because it allows to describe the three stages of thermal oxidation: initiation, propagation and saturation. The characteristic aging time that governs the kinetics of evolution of both reload modulus and dissipated to total energy ratio of the first cycle (η 1st ) with aging time are similar. The effect of decreasing distance from the surface on load displacement curves for the same aging time, is similar to that observed for the same distance and increasing aging time. Both reload modulus and η at the first cycle evolve in the oxidized layer approaching the value measured on the virgin sample, while η from the second cycle (representative of the time-dependent response) is constant through the sample thickness and close to the value measured on the virgin sample.

It was moreover shown that the use of oxygen pressure (2 bar) can accelerate the degradation phenomena caused by thermal aging in air: the increase in the reload modulus and the decrease in the amount of energy necessary to produce plastic deformations. However, caution must by taken as not all the phenomena and the stages of thermal aging are accelerated in the same way.

The glass transition temperature change with aging time for thermal aging in air allowed to conclude that crosslinking phenomena predominate in the PR520 epoxy resin during exposure in the conditions used in this study.

An increase in reload indentation modulus of about 8% with respect to the virgin material has been measured on the exposed surface of a sample aged at 150°C in an inert environnement (2 bar of Nitrogen), suggesting that the use of an aging temperature slightly higher than the glass transition temperature, could have induced in the material secondary degradation phenomena.

Local time-dependent properties of PR520 epoxy resin matrix of 3D interlock carbon-fiber reinforced composites

The application of cyclic indentation tests in big resin pockets located on the external surface and in the core of 3D interlock carbon-fiber reinforced composites showed that the unconstrained material is homogeneous on the exernal surface and at the sample core. The in-situ local properties measured in the composite sample core are, however, significantly different, from a statistical point of view, from that measured on the composite sample surface, highlighting a phenomenon similar to the "skin effect" usually observed on thermoplastic injected parts. Moreover, the neat properties are significantly different, from a statistical point of view, from both the in-situ properties measured on the core and on the external surface. However, this difference, although statistically significant, is very low (about 2% on reload modulus).

The property gradient profiles measured on the oxidized layer of neat and in-situ aged samples are well superposed, except for the point measured on the surface directly exposed to the environment, suggesting that permanence of the material at elevated temperature for a certain time during aging have remedied the gap, probably induced by a different degree of cure, measured on the neat and in-situ virgin materials.

Perspectives From the conclusions of this work, some suggestions for future research in this domain can be outlined:

• The large number of information provided by cyclic indentation tests could be useful to determine by reverse analysis a complex local behaviour law of the polymer. This behaviour law should include a plastic and a time-dependent part. However, in order to establish if the deformations accumulated during cycles are fully recoverable (viscoelastic) or partially permanent (viscoplastic), the experimental data base should be integrated with a study of the long-term recovery of the residual print.

As the duration of indentation recovery test is limited in time by the equipment and by thermal drift, the analysis of recovery should be performed trough different experimental techniques like, for example, the imaging of the residual print through several time decades.

• As polymer matrix composites are foreseen for relatively high temperature applications, it would be interesting to evaluate the matrix local properties change not only a posteriori after a certain exposure time, but also during the exposure. Thanks to a new indentation equipment recently acquired by our laboratory, it would be possible to perform instrumented indentation tests at elevated temperatures and thus evaluate the local mechanical properties of the neat and in-situ polymer in the actual operating conditions.

• The indentation method proposed in this study allowed to conclude that the difference between neat and in-situ epoxy resin properties far from fiber bundles is very low. However, it does not allow to conclude about the properties of matrix near fibers, in the so called "interphase", in both virgin and aged state. Le Chapitre se poursuit avec une section dédiée au comportement cyclique des matériaux polymères. Dans cette partie, l'accent a été mis sur les résultats présentés dans une thèse réalisée au sein de notre laboratoire par Nguyen, dans laquelle le comportement cyclique en traction et cisaillement du Polyéthylène haute densité (PEHD) a été traité. Dans la partie expérimentale de son travail, Nguyen a évalué les effets de la fréquence et du rapport de charge sur la réponse cyclique à travers trois indicateurs extraits à partir des courbes contraintes-déformations :

• Aire de la boucle d'hystérèse, représente l'aire entre la courbe de charge et de décharge et est évaluée par intégration numérique par trapèzes ;

• Module élastique sécant, défini comme la pente de la ligne qui relie le point initial au point de déformation maximale à chaque cycle ;

• Déformation moyenne, calculée comme la moyenne entre les déformations minimale et maximale à chaque cycle.

Les trois paramètres évoluent avec les cycles et cette évolution dépend de la fréquence de chargement et du rapport de charge. En particulier : l'aire de boucle diminue dans les premiers cycles et se stabilise après une vingtaine de cycles ; le module sécant augmente au cours des premiers cycles et se stabilise après un nombre de cycle similaire ; la déformation moyenne qui, par contre, ne se stabilise pas, mais continue à augmenter au cours des cycles pour toutes les fréquences et rapports de charge considérés.

Les trois paramètres évoluent avec les cycles et cette évolution dépend de la fréquence de chargement et du rapport de charge. En particulier : l'aire de boucle diminue dans les premiers cycles et se stabilise après une vingtaine de cycles ; le module sécant augmente au cours des premiers cycles et se stabilise après un nombre de cycle similaire ; la déformation moyenne qui, par contre, ne se stabilise pas, mais continue à augmenter au cours des cycles pour toutes les fréquences et rapports de charge considérés. La dernière partie de ce Chapitre est consacrée au phénomène qui sera l'objet de cette étude : le vieillissement thermique des polymères. Il s'agit d'un vieillissement chimique, qui comporte donc l'évolution de la structure moléculaire du polymère dans le temps et qui est dû au couplage des phénomènes de diffusion et de réaction de l'oxygène. Le vieillissement thermique induit la formation d'une couche oxydée au départ de la surface du matériau directement exposée à l'environnement et qui se développe dans son épaisseur sur quelques centaines de micromètres. Des études précédentes ont montré que cette couche oxydée affecte les propriétés à rupture des matériaux. Il en résulte que le matériau vieilli est plus fragiles que matériau sain. Comme la zone affectée est relativement petite, l'essai d'indentation instrumentée se révèle être la méthode plus appropriée pour tester localement les propriétés mécaniques de la couche oxydée. Dans ce contexte, des études précédentes ont mis en évidence la présence de gradients de module élastique d'indentation dans l'épaisseur des éprouvettes vieillies. En particulier, le module élastique atteint sa valeur maximale à proximité de la surface extérieure et diminue dans l'épaisseur jusqu'à atteindre un plateau pour une valeur similaire à celle mesurée sur le matériau à l'état vierge. A la lumière de la revue bibliographique, les objectifs de ce travail ont été établis : • Complaisance de l'instrument. L'équipement d'indentation a sa propre complaisance qui, si elle n'est pas prise en compte, se rajoute à la rigidité de l'éprouvette en faussant ainsi la valeur de module mesurée.

• Rugosité de surface. Si la rugosité de surface est trop importante, l'aire de contact a une forme irrégulière est la dispersion des résultats peut être très importante, surtout pour des profondeurs de pénétration faibles. Pour minimiser l'effet de l'état de surface, un protocole de polissage approprié aux matériaux polymères a été mis au point. Le Chapitre se poursuit avec une comparaison qualitative entre la réponse cyclique macroscopique et microscopique. Pour cela, l'effet de fréquence de chargement a été également pris en compte, en réalisant des essais cycliques d'indentation à 1 Hz , 0.1 Hz et 0.01 Hz. Comme la durée d'essai est limitée par l'équipement, le nombre de cycles a été modifié selon la fréquence de chargement utilisée. L'évolution des paramètres, avec les cycles et avec la fréquence, extraits des courbes d'indentation présente certaines similarités avec son analogue en traction. En particulier, l'aire de boucle diminue avec les cycles et avec la fréquence. Par contre, si l'aire de boucle macroscopique se stabilise à une valeur constante après une vingtaine de cycles, en indentation, elle continue à diminuer tout au long de l'essai. Le deuxième paramètre, le module d'indentation, peut être évalué à partir de la rigidité des courbes de décharge (comme dans la méthode classique d'Oliver et Pharr) ou de recharge. Par contre, l'application de la théorie du contact élastique aux courbes de décharge a révélé des incohérences, alors qu'elle est très consistante aux recharges. Pour cette raison, on a décidé de retenir uniquement les modules évalués aux recharges comme indicateur de la réponse élastique instantanée pour la suite du manuscrit. Similairement au module sécant des courbes de traction macroscopiques, le module d'indentation en recharge augmente avec les cycles et avec la fréquence. Par contre, comme dans le cas de l'aire de boucle, si le module sécant macroscopique se stabilise à une valeur constante après une vingtaine de cycles, le module d'indentation continue à augmenter tout au long de l'essai. Enfin, le déplacement moyen évolue dans le même sens que la déformation moyenne de traction. En particulier, les deux augmentent avec les cycles et avec la fréquence et ne se stabilisent pas à une valeur constante dans les deux cas. De plus, il a été observé que les évolutions sont similaires à celles du fluage à l'échelle correspondante. En particulier, le déplacement accumulé pendant les cycles (maximale, minimale et moyen), est caractérisé par un régime transitoire, suivi par un régime stationnaire, comme pour l'évolution du déplacement avec le temps dans les essais de fluage en indentation. Par contre, au cours des cycles, il y a une compétition entre le déplacement accumulé en phase de charge (h i max ) et le déplacement recouvré à la décharge (h i max -h i min ) . Le déplacement net accumulé à chaque cycle est, donc, donné par la différence entre déplacement accumulé en charge et recouvré en décharge, soit le déplacement minimal (h i min ). Le déplacement net accumulé pendant les cycles a été, dans la suite du Chapitre, comparé aux déplacement accumulé pendant des essais de fluage en indentation dans lesquels la charge a été fixée à la charge cyclique maximale (10 mN ) dans un cas et à la charge cyclique moyenne (5 mN ) dans l'autre. La durée des trois essais est similaire et d'environ 3000 s. Il a été observé que le déplacement net accumulé pendant les cycles, représenté en fonction du temps d'essai, se situe entre les deux courbes de fluage. De plus, si les trois courbes sont différentes dans le régime transitoire, dans la partie stationnaire le déplacement cyclique et de fluage à 10 mN sont parallèles. La pente de la courbe de fluage à 5 mN dans le régime stationnaire est, en revanche, un ordre de grandeurs plus faible. Pour établir si le déplacement accumulé pendant les cycles est entièrement recouvrable (viscoélastique) ou en partie permanent (viscoplastique), une phase de recouvrance à une charge très faible a été introduite à la fin de certains essais cycliques. L'analyse de l'évolution du déplacement avec le temps de recouvrance a montré que sa durée (environ la même durée de l'essai) n'a pas été suffisante pour permettre l'identification de la partie permanente de la déformation, n'étant le déplacement pas stabilisé à la fin de cette phase. Le Chapitre se termine avec une analyse sur l'énergie dissipée pendant les cycles représentée par l'aire de boucle. Son évolution avec les cycles montre que les mécanismes de dissipation d'énergie entre le premier cycle et les autres sont différents. Cette différence a été attribuée au fait que, pendant la première charge, une grosse partie de l'énergie est dissipée pour produire des déformations plastiques indépendantes du temps qui se produisent lorsque la contrainte limite d'élasticité est dépassée. A partir du deuxième cycle l'énergie est dissipée pour produire des déformations dépendantes du temps uniquement. Cependant, à ce stade, il n'a pas été possible d'établir si ces déformations sont entièrement recouvrables ou pas. Grâce à ce protocole d'indentation et à sa méthode d'analyse il est possible de mettre en évidence le comportement complexe dépendant du temps des matériaux polymères. En particulier, l'analyse des boucles d'hystérèse, permet de déduire la réponse pour trois échelles de temps différentes : instantanée, représentée par le module de recharge ; temps courts (à l'échelle du cycle), représentée par l'aire de boucle qui dans la suite sera proposée en forme normalisée par rapport à l'énergie totale ; temps longs (à l'échelle de l'essai), représentée par le déplacement accumulé. Le Chapitre se termine avec une discussion sur le vieillissement thermique à cette température. Pour cela, la température de transition vitreuse de la couche oxydée et du coeur des échantillons vieillis a été mesurée par DSC. Selon des études conduites en précédence, le suivi de la T g pendant le vieillissement permet de conclure sur les modifications moléculaires qui ont eu lieu pendant l'exposition. En effet, une augmentation de la température de transition vitreuse, comme dans notre cas, indique une prédominance de réticulations par rapport aux coupures de chaine. En particulier, une augmentation de 15 °C et 7 °C a été mesurée sur la couche oxydée et à coeur respectivement, pour la durée de vieillissement plus élevée sous air. Enfin, les résultats des essais d'indentation obtenus sur la surface de l'échantillon vieilli sous 2 bar d'Azote, ainsi que son léger changement de couleur, montrent que, probablement, des phénomènes de dégradation secondaires anaérobie ont eu lieu pendant l'exposition. En particulier, le module de recharge mesuré sur cet échantillon est 8% plus élevé de celui mesuré sur le matériau vierge. Cette augmentation est, de tout manière, négligeable devant l'augmentation mesurée sur l'échantillon vieilli sous air pour la même durée (∼ 40%), ce qui montre que la thermo-oxydation reste le phénomène de dégradation prédominante à cette température.

Chapitre 5. Propriétés mécaniques locales de la matrice époxy des composites tissés 3D à fibres de carbone

Le dernier Chapitre est dédié à l'étude du comportement locale de la matrice époxy d'un composite tissé 3D à fibres de carbone. Dans ce type de matériau composite, conçu pour des applications aéronautique dans les zones tièdes (50 °C ≤ T ≤ 300 °C) , les fibres sont regroupées en torons qui sont, ensuite, tissés entre eux pour créer le "squelette" du composite. Une fois le tissé de fibres séché, il est placé dans un moule où la résine à l'état liquide est injectée. Ce procédé de fabrication est appelé Resin Transfert Molding (RTM). A cause des trajets thermiques complexes dans le moule induits, par ailleurs, par la présence des fibres, la résine peut atteindre différents degrés de réticulation dans les différentes zones du matériau. C'est pour cette raison que, avant de comparer les propriétés de la matrice dans le composite à celles de la résine pure, on a voulu étudier l'hétérogénéité de la matrice. Pour cela, de nombreux essais ont été réalisés dans des différents poches de résine entre les torons, situées sur la surface extérieure de l'échantillon et à coeur. L'architecture du composite permet de repérer de nombreuses poches de résine suffisamment larges pour pouvoir réaliser des essais à des distances des fibres élevées, permettant de négliger les effets de retrait. Les différentes poches de résine, leurs images microscopiques et les distances minimales du bord des torons ont été présentés dans la première partie du Chapitre. Ensuite, une méthode analytique proposée dans la littérature a été appliquée aux données force-déplacement obtenues dans les différentes poches pour établir si les mesures étaient affectées par des effets de confinement des torons de fibres. Pour établir si les différences mesurées sont statistiquement significatives, le test t-Student a été appliqué à la valeur moyenne et à l'écart type calculés entre chaque couple de poches de résine. Il a été trouvé que les poches sont homogènes à coeur et sur la surface extérieure de l'échantillon, alors qu'une différence statistiquement significative a été mesurée entre la valeur moyenne des essais réalisés à coeur et en surface. Cette différence est constante avec les cycles et d'environ 2% pour le module de recharge, 8% pour le déplacement et l'énergie dissipée au premier cycle et environ 4% pour l'énergie dissipée à partir du deuxième cycle. Cela a été attribué à un phénomène similaire de l'effet de "peau" observé sur les parties injectées en matériau thermoplastique, probablement dû à un gradient thermique dans le moule qui induit un différent niveau de réticulation entre coeur et surface extérieure. Les propriétés locales de la résine, mesurées dans le composite, ont été, par la suite, comparées à celle de la résine pure. Les résultats de cette comparaison montrent que la résine pure a des propriétés légèrement différentes de la résine dans le composite. En particulier, les propriétés de la résine sont intermédiaires entre celle du composite mesurées à coeur et sur la surface. La différence est, dans les deux cas, très faible et, pour certains paramètres, pas significative d'un point de vue statistique. Dans la dernière partie du Chapitre, les gradients de propriétés mesurés dans la couche oxydée du composite vieilli sous air pour 1000 h ont été comparés à ceux mesurés sur la résine vieillie dans les mêmes conditions et présentés dans le Chapitre précédente. Les profils du module de recharge et de l'énergie normalisée du premier cycle de résine et composite sont bien superposées. En particulier, pour des distances depuis la surface extérieure comparables, la différences entre les deux gradients est inférieure à 1% . Cette différence est moins importante que celle mesurée sur les matériaux vierges et de celle mesurée sur la surface exposée à l'environnement des matériaux vieillis. La superposition des gradients montre aussi que l'épaisseur de la couche oxydée est très similaire dans les deux configurations.

Conclusions

Grâce à l'étude réalisée dans cette thèse, il a été tout d'abord possible d'adapter la norme d'indentation ISO 14577, développée pour les matériaux métalliques, aux matériaux polymères. Il a été également montré que le protocole d'essai et sa méthode d'analyse qui existent pour les métaux, ne sont pas appropriés pour décrire un comportement dépendant du temps complexe tel que le comportement mécanique des polymères. Pour cette raison, le coeur de ce travail a été le développement d'une méthode d'indentation cyclique, capable de mettre en évidence ce type de comportement particulier. Cette méthode a été ensuite appliquée pour caractériser les effets du vieillissement thermique sur le comportement locale et dépendent du temps d'une résine époxy employée pour la fabrication de composites tissée 3D, conçu pour applications à températures relativement élevées. Enfin, la même méthode a permis de étudier l'hétérogénéité de la même résine dans le composite, de comparer ses propriétés à celle de la résine pure et de comparer les gradients de propriétés dus au vieillissement de la résine et du composite. Le marché des matériaux composites à matrice organique (CMO) pour la réalisation de pièces structurales "froides" (-55°C < T < Tamb) arrive à saturation et l'industrie aéronautique vise à utiliser les CMO tissés 3D dans les pièces structurales dites "chaudes" (50°C < T < 300°C) des avions (nacelles, turbomoteurs). Ces conditions environnementales peuvent entrainer des phénomènes de dégradation à long terme. L'action de l'environnement et la complexité de la microstructure peuvent conduire au développement de gradients de propriétés dans les matériaux. À haute température, la matrice polymère peut présenter un comportement complexe dépendant du temps. The market of organic matrix composite (OMC) materials for the realisation of "cold" (-55°C < T < RT) structural parts is going towards saturation and aircraft manufacturers foresee the employment of 3D reinforced OMC in "warm" (50°C < T < 300°C) aircraft structural parts (nacelles, turbo-engines). These environmental conditions may lead to degradation phenomena over long time. The action of environment and the complexity of the material microstructure may lead to the development of material property gradients. At high temperature, the organic polymer matrix may exhibit complex time-dependent behaviour. Therefore, there is a need to develop an experimental technique able to characterise the material behaviour at local scale and to capture material gradients and timedependent behaviour. The present work proposes the development of a cyclic instrumented indentation test to tackle all these issues. Instrumented indentation is a popular testing technique: its basic version, appropriate for elasto-plastic materials, includes the realisation of a single loading/unloading test, the measurement of the material hardness through the analysis of the indentation print, and the analysis of the unloading curve based on the assumption of elastic unloading behaviour to obtain the indentation modulus. This analysis is inappropriate for materials exhibiting time-dependent behaviour, which leads to the need of a new method optimized for polymer materials. Several indentation methods are available for polymer materials (indentation creep, nanoDMA), but are limited to some specific loadings. The instrumented indentation cyclic test proposed in this work tries to emphasize the whole complexity of the polymer behaviour, is inspired by macroscopic cyclic tests and is analysed similarly. The technique is first set up and developed by testing a HDPE thermoplastic polymer, for which the response to macroscopic cyclic tension and shear loading is known. By following, at different frequencies, the evolution with time (with cycles) of the principal indicators of the cyclic behaviour -that is, the indentation modulus, the hysteresis loop area and the indentation depth accumulation -it is possible to highlight the time-dependent response of the material and to perform a proper -though qualitative -comparison with the macroscopic behaviour.

List of Figures

The cyclic indentation protocol is then employed to study the material gradients in a thermoset PR520 epoxy resin subjected to thermal aging at 150°C under air at atmospheric pressure (up to 1000h), 2 bar O2 (for 400h) and N2. It is shown that the evolution with cycles of the indentation modulus and the cyclic creep is not affected by thermal aging. However, their absolute values vary from the surface to the core of polymer indicating the presence of gradients. The hysteresis of the first cycle is different through the gradient: from the second cycle, however, the hysteresis is similar for all aging conditions and distances from the exposed surface. Moreover, the time-dependent behaviour stays unchanged. Results from samples aged under 2 bar O2 and 2 bar N2 allow to conclude that the aging kinetics is not related to pure thermo-oxidation phenomena. The cyclic indentation method is finally applied to characterise the behaviour of PR520 epoxy matrix within a 3D interlock textile composite, in its virgin state and thermally aged at 150°C under air at atmospheric pressure (up to 1000h). The polymer in large matrix pockets between the fibrous reinforcements is studied in this work, so the constraining effect coming from the reinforcement is negligible. The study of the composite in virgin state reveals that the behaviour of polymer matrix close to the external surface is different from that located in internal zones of the composite and from the neat polymer. The comparison between the thermally aged neat polymer and matrix in internal zones of the composite shows that the environment-induced property gradients are similar.

Figure 1 . 1 .

 11 Figure 1.1. Typical load-displacement curve resulting from instrumented indentation tests (a)and schematic representation of the corresponding geometrical parameters (b)[START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF]

Figure 1 . 2 .

 12 Figure 1.2.Load-displacement curves obtained on four different polymers[START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF]

Figure 1 . 3 .

 13 Figure 1.3.Concept used to define the effective indenter shape[START_REF] Pharr | Understanding nanoindentation unloading curves[END_REF]

Figure 1 . 4 .

 14 Figure 1.4.Load-displacement curves at the loading-unloading peak evidencing that the introduction of a 10 s holding period at the maximum load allows to eliminate the nose[START_REF] Briscoe | Nano-indentation of polymeric surfaces[END_REF]

Figure 1 . 5 .

 15 Figure 1.5. Indentation elastic modulus and hardenss of PMMA as a function of holding time [19]

Figure 1 . 6 .

 16 Figure 1.6. Scheme of load-displacement data highlighting the areas of positive and negativedrift and the point of minimal drift at which stiffness is evaluated[START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF]

Figure 1 . 7 .

 17 Figure 1.7. VEP indentation model on loading (a) and unloading (b)[START_REF] Oyen | Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials[END_REF]

Figure 1 . 8 .

 18 Figure 1.8. Displacement-time data during indentation creep of PET, PMMA and epoxy resin [29]

Figure 1 . 9 .

 19 Figure 1.9. Evolution of the representative strain as a function of creep time for different values of initial mean contact strain and comparison of the resluts with the Lee andRadok's viscoelastic contact model[START_REF] Chatel | Creep of the contact with a spherical tip and recovery of the imprint on amorphous polymer surfaces[END_REF]

Figure 1 . 10 .

 110 Figure 1.10. Evolution of the representative strain as a function of the recovery time normalized by the creep time for an initial mean contact strain of 2% (a) and 4% (b) [32]

Figure 1 . 11 .

 111 Figure 1.11. Generalised Kelvin model used by Yang et al. [29] to fit displacement data during indentation creep tests

Figure 1 .

 1 Figure 1.12. Scheme of the experimental procedure used by Minervino [10] to determine the relaxation spectrum of an epoxy resin through reverse analysis

Figure 1 . 13 .

 113 Figure 1.13. Load-displacement curves in dynamic indentation tests[START_REF] Kermouche | Extraction of stress-strain curves of elastic-viscoplastic solids using conical/pyramidal indentation testing with application to polymers[END_REF]

Figure 1 . 14 .

 114 Figure 1.14. Evolution of storage modulus with indentation depth[START_REF] Monclus | In search of validated measurements of the properties of viscoelastic materials by indentation with sharp indenters[END_REF]

Figure 1 . 15 .

 115 Figure 1.15. Load-displacement curve in which the energy dissipated due to plastic deformations (U p ) is highlighted [51]

Figure 1 . 16 .

 116 Figure 1.16. Indentations performed in resin pockets perpendicular (a) and parallel (b) tofibers direction[START_REF] Gregory | Nanoindentation of neat and in situ polymers in polymer-matrix composites[END_REF]

Figure 1 . 17 .

 117 Figure 1.17. Actual Young's modulus to unconstrained Young's modulus ratio plotted againstthe distance from fibers obtained through finite element simulations with different models. In all the simulations the penetration depth was kept constant to 1 µm[START_REF] Hardiman | The effect of fibre constraint in the nanoindentation of fibrous composite microstructures: A finite element invesitgation[END_REF]

Figure 1 . 18 .

 118 Figure 1.18. Load-displacement curves for fiber, matrix and interphase at a maximum penetration depth of 60 nm [58]

Figure 1 . 19 .

 119 Figure 1.19. Triangular-wave stress-time plot used for cyclic tensile and shear tests (a) stressstrain response to cyclic tensile solicitation (b) [64]

and 1. 21 .Figure 1 . 20 .Figure 1 . 21 .

 21120121 Figure 1.20. Evolution of loop area (a), secant modulus (b) and mean strain (c) with cycles for macroscopic cyclic tension tests performed on HDPE at 0.1 Hz (blue), 0.05 Hz (green) and 0.01 Hz (red) [64]

Figure 1 . 22 .

 122 Figure 1.22. Load-time sequence proposed by Oliver and Pharr[START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF]

Figure 1 . 23 .

 123 Figure 1.23. Expanded view of the unloading-reloading portion at the peak load of load-displacement curves performed on aluminium single crystal[START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF]

Figure 1 . 24 .

 124 Figure 1.24.Multicycle indentation test protocol used in[START_REF] Collin | Use of spherical indentation data changes to materials characterization based on a new multiple cyclic loading protocol[END_REF]

Figure 1 . 25 .

 125 Figure 1.25. Epoxy resin (T g = 190 -250°C) sample in the virgin state (a) and after 500 h of thermal oxidation in air at 150°C [10]

Figure 1 . 26 .

 126 Figure 1.26. Microscopic image of the cross section of the PMR-15 epoxy resin (T g ≈ 345°C)aged for 175 h at 288 °C in atmospheric air[START_REF] Tandon | Thermo-oxidative behavior of high-temperature pmr-15 resin and composite[END_REF]

Figure 1 . 27 .

 127 Figure 1.27. Glass transition temperature change with aging time for DGEBA/IPDA (a), DGEBA/TTDA (b) and DGEBU/IPDA (c) [96]

Figure 1 . 28 .

 128 Figure 1.28. Evolution with aging time of tensile failure strain of PMR-15 aged in air at atmospheric pressure and under 0.414 M P a at 288 °C [97] (a) and of four-point bending failure stress of PR520 epoxy resin aged at 120 °C in air and under 2 bar of oxygen [98] (b)

29

 29

  .

Figure 1 . 29 .

 129 Figure 1.29. Evolution of β transition peak with aging time for DGEBA/IPDA (a) and DGEBA/TTDA (b) and evolution of tensile strain at break as a function of β transition amplitude (c) [99]

Figure 1 . 30 .

 130 Figure 1.30. Elastic indentation modulus (a) and concentration of oxidation products (b)profiles obtained on 977-2 epoxy resin samples aged in air at 150 °C for different durations[3]

Figure 1 . 31 .

 131 Figure 1.31. Elastic indentation modulus as a function of the concentration of oxidation products [3]

Figure 1 . 32 .

 132 Figure 1.32. Maximum variation of the indentation modulus with respect to that of the virgin material ( γ max ) as a function of aging time (a) and of the equivalent aging time (b) for Tactix epoxy resin samples aged in air and under 2 bar of oxygen pressure [10]

Figure 1 . 33 .

 133 Figure 1.33. Surface profiles measured through interferometric microscopy on the cross sections of virgin and thermally aged Tactix epoxy resin samples [10]

Figure 1 . 34 .

 134 Figure 1.34. Schema of the matrix shrinkage among fibers resulting from composite thermal aging [9]

Figure 1 . 35 .

 135 Figure 1.35. Evolution of matrix shrinkage with aging time evaluated through interferometric microscopy images of the same sample zone [101]

Figure 1 . 36 .

 136 Figure 1.36. Maximum matrix shrinkage depth as a function of the distance between fibers for sample aged at 150 °C under 1.7 bar (a) and 5 bar of oxygen pressure [101]

Figure 1 . 37 .

 137 Figure 1.37. Fiber/matrix debonding phenomenon observed on SEM images of the surface of unidirectional carbon fiber reinforced polymer matrix composites after 1000 h of aging in air at 150 °C [9]

Figure 2 . 1 .

 21 Figure 2.1. Instrumented indentation test machine Fischerscope H100C

Figure 2 . 2 .

 22 Figure 2.2. Loading history used for the area function calibration

Figure 2

 2

Figure 2 . 3 .

 23 Figure 2.3. A p vs h c experimental data points obtained on BK7-glass (black points), toghether with their fitting function of equation 2.1 (red line) and the ideal projected area of contact calculated according to Equation 1.6

  .

  4.

Figure 2 . 4 .

 24 Figure 2.4. Indentation modulus calculated on Fused Silica at different load levels before (red data points) and after (blue data points) area function calibration

1 √

 1 2) on the Fused Silica sample. The plot of 1 S vs Ap is reported in Figure 2.5.

Figure 2 . 5 .

 25 Figure 2.5. Verification of instrument compliance

0. 1

 1 mN in 9 tests performed on High Density Polyethylene are represented in Figure 2.6.

Figure 2 . 6 .

 26 Figure 2.6. Displacement-time data collected during the 600 s holding period at 0.1 mN at the beginning of the tests for a set of 9 tests carried out on HDPE

Figure 2 .Figure 2 . 7 .

 227 Figure 2.7 shows the displacement-time data collected during a long indentation creep test of about 6000 s at a constant load of 5 mN on HDPE before (left) and after (right) the application of the thermal drift correction.

Figure 2 . 8 .

 28 Figure 2.8. Scheme of the tip/sample misalignment (a) and of the contact area resulting from a misaligned tip/sample contact (b)[START_REF] Pelletier | Mechanical Characterization of Glassy Polymers using Instrumented Indentation[END_REF]

Figure 2 . 9 .

 29 Figure 2.9. Tilt platform used to perform instrumented indentation tests at different inclination angles

Figure 2 . 10 .

 210 Figure 2.10. Evolution of the reduced modulus with the rotation angle about x (green) and y (red) axis.

Figure 2 . 11 .

 211 Figure 2.11. Percentage variation of E * from the value at 0°tilt for PR520 (green symbols)and Fused Silica (red symbols)

Figure 2 . 12 .

 212 Figure 2.12. Image of the cross section of the embedded aged sample (a) and topography of an area of the cross section near the edge (b)

Figure 2 . 13 .

 213 Figure 2.13. Profile traced along the z axis of the topography of the thermally aged resin sample cross section

Figure 2 . 14 .

 214 Figure 2.14. Topography of the surface of a thermally aged composite sample

Figure 2 . 15 .

 215 Figure 2.15. Four parallel profiles traced on the topography of the surface of a thermally aged composite sample from the edge of the fiber bundle to the center of the resin pocket (a) evolution of the local slope toward the profile (b)

Figure 2 . 16 .

 216 Figure 2.16. Effect of loading rate on HDPE one cycle load-displacement curves

Figure 2 . 17 .

 217 Figure 2.17. Loading curves obtained on HDPE for different loading rates of 20 mN s (blue), 2 mN s (green) and 2 mN s (red) with their power law fitting curve (continuous black lines) and the theoretical loading curve for elasto-plastic materials with the exponent n = 2 (dashed black line)

Figure 2 . 18 .

 218 Figure 2.18. Evolution of the viscoelastic delayed displacement as a function of loading rate

Figure 2 . 19 .

 219 Figure 2.19. Evolution of the reduced indentation modulus as a function of loading rate

  Figure 2.20.

Figure 2 . 20 .

 220 Figure 2.20. Unloading curves obtained on HDPE for different loading rates of 20 mN s (blue), 2 mN s (green) and 0.2 mN s (red) with their power law fitting curve (continuous black lines)

  , 2.21b and 2.21c respectively as a function of loading rate.

Figure 2 . 21 .

 221 Figure 2.21. Fitting coefficients A (a), h f (b) and m (c) of unloading curve as a function of loading rate

Figure 2 . 22 .

 222 Figure 2.22. Scheme of material pile-up and sink-in under contact[START_REF] Hardiman | The effects of pile-up, viscoelasticity and hydrostatic stress on polymer matrix nanoindentation[END_REF]

Figure 2 . 23 .

 223 Figure 2.23. Ratio of final to maximum depth against loading rate for HDPE. The dashedblack line represents the threshold for the onset of significant pile-up according to[START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF]

Figure 3 . 1 .

 31 Figure 3.1. Cyclic instrumented indentation tests loading path

Figure 3 . 2 .

 32 Figure 3.2. 10 cycles load-displacement curves obtained on Fused Silica at loading frequencies of 1 Hz, 0.1 Hz and 0.01 Hz

Figure 3 . 3 .

 33 Figure 3.3. 300 cycles load-displacement curve obtained on HDPE at a loading frequency of 0.1 Hz

Figure 3 . 4 .

 34 Figure 3.4. Parameters extracted from cyclic force-displacement indentation curves

Figure 3 . 5 .Figure 3 . 6 .

 3536 Figure 3.5. Evolution with cycles of unload and reload indentation moduli (a), loop area (b) and maximum, minimum and mean displacement (c) for a 300 cycles indentation test performed on HDPE at a loading frequency of 0.1 Hz

  Figures 3.7a

  , 3.7b and 3.7c depict the evolution of hysteresis loops with cycles for the three frequencies. For clarity, just few cycles are represented in Figures 3.

  [START_REF] Minervino | The effect of thermo-oxidation on the mechanical behaviour of polymer epoxy materials[END_REF]

  .

Figure 3 . 7 .

 37 Figure 3.7. Load-displacement hysteresis loops evolution with cycles for tests performed at loading frequencies of 1 Hz (a), 0.1 Hz (b) and 0.01 Hz

Figure 3 . 8 .

 38 Figure 3.8. Delayed displacement calculated on each unloading curve of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycle number

  and 3.9b respectively.

Figure 3 . 9 .

 39 Figure 3.9. Unload (a) and reload (b) indentation moduli calculated at each cycle of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycle number

Figure 3 . 10 .

 310 Figure 3.10. Rate of change of indentation reload modulus with time as a function of testing time for HDPE tested at: 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red)

Figure 3 . 11 .

 311 Figure 3.11. Exponent of the power law fitting function m of unloading (filled markers) and loading (empty markers) curves as a function of cycles and for the three frequencies of 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red)

Figure 3 . 12 .

 312 Figure 3.12. Evolution of reload indentation modulus as a function of loading frequencies for different testing times (a) and slopes of indentation modulus versus frequency lines obtained for different testing times (b).

Figure 3 . 13 .

 313 Figure 3.13. Evolution of reload indentation modulus with frequency obtained through cyclic indentation tests for 50 s of testing time (a) and evolution of Young's modulus with frequency obtained through tensile DMA tests at ambient temperature (b) [64]

Figure 3 . 14 .

 314 Figure 3.14. Hysteresis loop area calculated at each cycle of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycle number

Figure 3 .

 3 15 depicts in a semi-logarithmic diagram, the rate of variation of loop area per cycle as a function of the cycle number. The rate of change of loop area at the ith cycle is calculated as the absolute value of the difference between the loop areas at the cycle i and i -1, divided by the loop area at the cycle i -1.

Figure 3 . 15 .

 315 Figure 3.15. Hysteresis loop area rate calculated at each cycle of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycle number

Figure 3 . 16 .

 316 Figure 3.16. Evolution of loop area as a function of loading frequency for different cycles (a) and slopes of loop area versus frequency lines obtained for different cycles (b)

Figure 3 . 17 .

 317 Figure 3.17. Mean displacement calculated on each unloading curve of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycles (for the first 60 cycles) (a) and of testing time (b)

Figure 3 . 18 .

 318 Figure 3.18. Displacement-time plot of indentation cyclic test (blue curve), indentation creep test at 10 mN (red curve) and indentation creep test at 5 mN (green curve)

Figure 3 . 19 .

 319 Figure 3.19. Evolution of creep at 10 mN (dashed red curve) and at 5 mN (dashed green curve)and cyclic minimum displacement (dashed blue curve) with testing time, together with their fitting curves (solid lines)

Figure 3 . 20 .

 320 Figure 3.20. Evolution of t µ 0 as a function of testing time for creep at 10 mN (red curve) and 5 mN (green cuve) and cyclic displacement (blue curve)

Figure 3 . 21 .

 321 Figure 3.21. Ratio between the h i (a) and τ i (b) of cyclic and creep curve at 10 mN (red) and 5 mN (green)

Figure 3 . 23 .

 323 Figure 3.23. Energy dissipated during loading phases as a function of cycle number

Figure 3 . 24 .

 324 Figure 3.24. Scheme of time-dependent material response to cyclic indentation

  Figure 4.1.

Figure 4 . 1 .

 41 Figure 4.1. Heat flow vs temperature plot determined through DSC on a virgin PR520 epoxy resin sample

Figure 4 . 2 .

 42 Figure 4.2. Ventilated oven Fischerbrand used for aging in air

  Figure 4.3.

Figure 4 . 3 .

 43 Figure 4.3. Chamber Topindustrie used for aging under oxygen and nitrogen pressure

Figure 4

 4

Figure 4 . 4 .

 44 Figure 4.4. Scheme of aged samples preparation for instrumented indentation tests through the thickness

Figure 4 Table 4 . 2 .

 442 Testing conditions used for the study of thermal aging on PR520 epoxy resin

Figure 4 . 5 .

 45 Figure 4.5. Load-displacement loops for the 1st, 2nd, 5th, 10th, 20th and 40th cycle obtained on a virgin PR520 epoxy resin sample

Figure 4 .

 4 6a, 4.6c and 4.6b respectively. Error bars (where visible) represent ± one standard deviation evaluated through a series of at least 6 tests.

Figure 4 . 6 .

 46 Figure 4.6. Evolution of E * load (a), η (b) and h max , h min and h max -h min (c) with indentation cycles for a virgin PR520 epoxy resin sample

Figure 4

 4

Figure 4 . 7 .

 47 Figure 4.7. Load-displacement curves resulting from cyclic indentation tests performed on the surface of virgin and thermal aged PR520 epoxy resin samples

Figure 4 . 8 .

 48 Figure 4.8. Evolution with cycles of E * load (a), η (b) and h min (c) for the virgin sample (blue circles) and for the samples aged at 150 °C in air for 220 h (red squares), 520 h (green triangles) and 1005 h (violet rhombus)

Figure 4 . 9 .

 49 Figure 4.9. Evolution of experimental h min -h 0 data (points) together with their fitting functions (solid lines) with testing time for virgin and conditioned PR520 epoxy resin

Figure 4

 4

Figure 4 . 10 .

 410 Figure 4.10. Evolution of experimental E * load -E * load 0 data (points) together with their fitting functions (solid lines) with testing time for virgin and conditioned PR520 epoxy resin

Figure 4 . 11 .

 411 Figure 4.11. Evolution of reload modulus with aging time for the 1 st (red), 2 nd (blue) and last (green) cycle

Figure 4 . 12 .

 412 Figure 4.12. Evolution of reload modulus with aging time for the 1 st cycle with three different fitting functions obtained by varying the retardation time τ a 1 of Equation 4.3 and keeping constant the other coefficients

Figure 4 . 13 .

 413 Figure 4.13. Evolution of η with aging time for the 1 st (red), 2 nd (blue), 3 rd (yellow), 5 th (black), 10 th (purple) and last (green) cycle

Figure 4 . 14 .

 414 Figure 4.14. Evolution of displacement accumulated in cycles calculated as the difference between the minimum displacement at the last and at the first cycle (a) and between the minimum displacement at the last and at the second cycle (b) with aging time

Figure 4 .

 4 15 depicts the load displacement curves obtained from cyclic indentation tests performed at different distances from the exposed surface, according to the scheme proposed in Figure4.4, of a sample aged for 1000 h.

Figure 4 . 15 .Figure 4 . 16 .

 415416 Figure 4.15. 40 cycles load-displacement curves obtained at different distances from the exposed surface of a sample aged in air at 150 °C for 1000 h

Figure 4 . 17 .

 417 Figure 4.17. Evolution with testing time of E * load -E * load 0 (a) and h min -h min 0 (b) for the virgin sample (dark blue) and for different distances from the exposed surface ranging from 0 µm (orange) to 240 µm (light blue) of a sample aged in air at 150 °C for 1000 h

  and

Figure 4 . 18 .

 418 Figure 4.18. Evolution of the reload modulus, normalized with respect to the mean value measured on the virgin sample and evaluated from the first reload, as a function of the distance from the exposed surface for the virgin sample (blue) and for the samples aged in air at 150 °C for 220 h (red), 520 h (green) and 1000 h (violet)

Figure 4 .

 4 [START_REF] Jin | Effects of experimental varaibles on pmma nano-indentation measurements[END_REF] depicts the dissipated to total energy ratio (η) of the first cycle change through the oxidized layer with respect to its mean value measured on the virgin sample.

Figure 4 . 19 .

 419 Figure 4.19. Evolution of the irreversible part of work of indentation η, normalized with respect to the mean value measured on the virgin sample and evaluated at the first cycle, as a function of the distance from the exposed surface for the virgin sample (blue) and for the samples aged in air at 150 °C for 220 h (red), 520 h (green) and 1000 h (violet)

Figure 4 .

 4 16b. This is confirmed form the evolution of η of the second and last cycle as a function of the distance from the exposed surface depicted in Figure4.20. For more clarity, only the result of the virgin material (blue) and of the sample aged 1000 h are reported in Figure4.20.

Figure 4 . 20 .

 420 Figure 4.20. Evolution of the irreversible part of work of indentation η evaluated at the second and last cycle, as a function of the distance from the exposed surface for the virgin sample (blue) and for the sample aged in air at 150 °C for 1000 h (violet)

  Figure 4.21.

Figure 4 . 21 .

 421 Figure 4.21. Evolution of the thickness of the oxidized layer, estimated from the reload modulus gradient profiles, as a function of aging time

Figure 4 . 22 .

 422 Figure 4.22. Evolution of the normalised reload modulus as a function of aging time for samples aged in air (red) and under 2 bar of oxygen pressure (blue)

2

 2 times the effective time for sample conditioned under oxygen pressure.

Figure 4 . 23 .

 423 Figure 4.23. Evolution of the normalised reload modulus as a function of the equivalent aging time for samples aged in air (red) and under 2 bar of oxygen pressure (blue)

Figure 4 .

 4 Figure 4.23 confirms what previously said. In fact, for the two lower aging durations under oxygen pressure (first two blue data points), an acceleration factor of 27 times is too much. These two points are out of the master curve (dashed line in Figure 4.23) and, in particular are shifted to the right. However, for the longer aging duration, the determined

Figure 4 . 24 .

 424 Figure 4.24. Evolution of η as a function of aging time for the samples aged in air (green) and under 2 bar of oxygen (blue) for the first, the second and the last cycle

Figure 4 . 25 .

 425 Figure 4.25. Evolution of η of the first cycle as a function of the equivalent aging time for the samples aged in air (green) and under oxygen pressure (blue)

  Figure 4.26.

Figure 4 . 26 .

 426 Figure 4.26. PR520 samples surface color change with aging time for the different environments used in this study: air, Oxygen and Nitrogen

Figure 4 . 27 .

 427 Figure 4.27. Evolution of the glass transition temperature with aging time of the oxidized layer (orange) and of the sample core (blue)

Figure 5 . 1 .

 51 Figure 5.1.Example of interlock woven fiber architecture[START_REF] Mouritz | Review of applications for advanced three-dimensional fibre textile composites[END_REF]

Figure 5 . 2 .

 52 Figure 5.2. Scheme of a 3D interlock carbon-fiber reinforcement similar to the one used in thisstudy[START_REF] Schneider | A meso-fe voxel model of an interlock woven composite[END_REF]

Figure 5 . 3 .

 53 Figure 5.3. Microscopic images of the 4 resin pockets located in the plan perpendicular to the weft direction

Figure 5 . 4 .

 54 Figure 5.4. Microscopic images of the 5 resin pockets located in the plan perpendicular to the warp direction

  .

Table 5 . 2 .

 52 Minimal distance between indentation prints and the edge of the closer fiber bundle for the 5 resin pockets in the warp direction Finally, a total of 3 pockets have been tested on the sample external surface and 30 indentations have been performed in each pocket as represented in Figure5.5.

Figure 5 . 5 .

 55 Figure 5.5. Microscopic images of the 3 resin pockets located on the external surface of the composite sample

Figure 5 . 6 .

 56 Figure 5.6. C and n fitting coefficients of the (first) loading curves for neat PR520 epoxy resin and in-situ measured in the core and on the external surface

Figure 5 . 7 .

 57 Figure 5.7. Evolution of reload indentation modulus (a), minimum displacement (b) and irreversible part of work of indentation (c) with indentation cycles for in-situ measured on the core ( ) and in-situ measured on the external surface ( ) PR520 epoxy resin properties

Figure 5 . 10 .

 510 Figure 5.10. Evolution of reload indentation modulus (a), minimum displacement (b) and irreversible part of work of indentation (c) with indentation cycles for the neat (A), in-situ measured in the core (0) and in-situ measured on the external surface ( ) PR520 epoxy resin properties

Figure 5 . 11 .

 511 Figure 5.11. Evolution with cycles of the parameter Z for the comparaison between neat and in-situ in the core mean values of reload indentation modulus (]), mean displacement (-) and η (×) and of Z crit for an interval of confidence of 99.9% (upper red dashed line) and 95% (lower red dashed line)

Figure 5 . 12 .

 512 Figure 5.12. Histogram representing the percent difference between neat and in-situ in the core mean values of reload indentation modulus (yellow), minimum displacement (black) and η (green) for 1st, 10th, 20th, 30th and 40th cycle

Figure 5 . 13 .

 513 Figure 5.13. Evolution with cycles of the parameter Z for the comparaison between neat and in-situ on the surface mean values of reload indentation modulus (]), minimum displacement (-) and η (×) and of Z crit for an interval of confidence of 99.9% (upper dashed red line) and 50% (lower dashed red line)

Figure 5 . 14 .

 514 Figure 5.14. Histogram representing the percent difference between neat and in-situ on the surface mean values of reload indentation modulus (yellow), minimum displacement (black) and η (green) for 1st, 10th, 20th, 30th and 40th cycle

. 4 .

 4 The cross section of the thermally aged composite sample is represented in Figure5.15.

Figure 5 . 15 .

 515 Figure 5.15. Cross section of a composite sample aged for 1005 h in air at 150 °C. The resin pockets near the edge and at the sample core on which indentation tests have been performed are circled in red

Figure 5 . 16 .

 516 Figure 5.16. Resin pocket located on the surface perpendicular to the weft direction directly exposed to the aging environment of a composite sample conditioned for 1005 h in air at 150 °C

Figure 5 . 17 .

 517 Figure 5.17. Reload indentation modulus (a) and irreversible part of work of indentation (b) gradients measured in the oxidized layer of neat (filled red triangles) and in-situ (unfilled red triangles) PR520 epoxy resin aged for 1005 h in air at 150 °C. The mean values measured on the neat (filled blue circles) and in-situ (unfilled blue circles) virgin material are represented as well. The gradients are referred to the values calculated at the first cycle

  ,

Figure 5 . 18 .

 518 Figure 5.18. Histogram representing the percent difference between neat and in-situ reload modulus (yellow) and η (blue) for different distances from the edge of samples aged for 1005 h in air at 150 °C. The difference measured on the virgin materials is reported on the left part of the graph

Figure 5

 5 Figure 5.18 confirms that the percent difference between neat and in-situ reload modulus measured on the surface of aged samples (about 2.5%) is similar to the difference evaluated between neat and in-situ virgin samples (about 2.2%). However, the difference on neat and in-situ η values of aged samples is much higher (about 6%) than that measured between neat and in-situ virgin samples (less than 1%). That is probably due to a more far field interphase effect induced by thermal-aging. It is interesting to observe that in the oxidized layer, for distances of 45, 120, 200 and 280 µm from the exposed surface, the difference between neat and in-situ values is less than 1%. This observation suggests that the differences in curing degree that have probably caused the slight gap between neat and in-situ properties of the material at the virgin state, were remedied by keeping the samples at elevated temperature for a certain time. To verify if this "homogenisation" of neat and in-situ properties is limited to the oxidized layer or concerns also the inner part of the samples, results obtained in the core of aged samples have been compared and are reported on the right part of the histogram of Figure5.18. This results are referred to in-situ indentation tests performed in the pocket 2 circled in red in Figure5.15 (far from the sample edge) and to neat tests performed at a distance from the exposed surface

Chapitre 4 .

 4 Application de l'essai cyclique d'indentation à l'étude des effets du vieillissement thermique sur le comportement local dépendant du temps de la résine époxy PR520

46 2. 1 . 53 2. 3 . 89 3. 10 .

 4615338910 (a) and parallel (b) to fibers direction [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17. Actual Young's modulus to unconstrained Young's modulus ratio plotted against the distance from fibers obtained through finite element simulations with different models. In all the simulations the penetration depth was kept constant to 1 µm [54] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18. Load-displacement curves for fiber, matrix and interphase at a maximum penetration depth of 60 nm [58] . . . . . . . . . . . . . . . . . . . . . . . . 1.19. Triangular-wave stress-time plot used for cyclic tensile and shear tests (a) stress-strain response to cyclic tensile solicitation (b) [64] . . . . . . . . . . 1.20. Evolution of loop area (a), secant modulus (b) and mean strain (c) with cycles for macroscopic cyclic tension tests performed on HDPE at 0.1 Hz (blue), 0.05 Hz (green) and 0.01 Hz (red) [64] . . . . . . . . . . . . . . . . 1.21. Evolution of loop area (a), secant modulus (b) and mean strain (c) with cycles for macroscopic cyclic tension tests performed on HDPE with R = 0.5 (blue), R = 0.2 (green) and R = 0.5 (red) [64] . . . . . . . . . . . . . . . . 1.22. Load-time sequence proposed by Oliver and Pharr [13] . . . . . . . . . . . 1.23. Expanded view of the unloading-reloading portion at the peak load of load-displacement curves performed on aluminium single crystal [13] . . . 1.24. Multicycle indentation test protocol used in [82] . . . . . . . . . . . . . . . 1.25. Epoxy resin (T g = 190 -250°C) sample in the virgin state (a) and after 500 h of thermal oxidation in air at 150°C [10] . . . . . . . . . . . . . . . . 1.26. Microscopic image of the cross section of the PMR-15 epoxy resin (T g ≈ 345°C) aged for 175 h at 288 °C in atmospheric air [90] . . . . . . . . . . . . . . . 1.27. Glass transition temperature change with aging time for DGEBA/IPDA (a), DGEBA/TTDA (b) and DGEBU/IPDA (c) [96] . . . . . . . . . . . . 38 1.28. Evolution with aging time of tensile failure strain of PMR-15 aged in air at atmospheric pressure and under 0.414 M P a at 288 °C [97] (a) and of four-point bending failure stress of PR520 epoxy resin aged at 120 °C in air and under 2 bar of oxygen [98] (b) . . . . . . . . . . . . . . . . . . . . . . . 39 1.29. Evolution of β transition peak with aging time for DGEBA/IPDA (a) and DGEBA/TTDA (b) and evolution of tensile strain at break as a function of β transition amplitude (c) [99] . . . . . . . . . . . . . . . . . . . . . . . 40 1.30. Elastic indentation modulus (a) and concentration of oxidation products (b) profiles obtained on 977-2 epoxy resin samples aged in air at 150 °C for different durations [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.31. Elastic indentation modulus as a function of the concentration of oxidation products [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.32. Maximum variation of the indentation modulus with respect to that of the virgin material ( γ max ) as a function of aging time (a) and of the equivalent aging time (b) for Tactix epoxy resin samples aged in air and under 2 bar of oxygen pressure [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.33. Surface profiles measured through interferometric microscopy on the cross sections of virgin and thermally aged Tactix epoxy resin samples [10] . . . 44 1.34. Schema of the matrix shrinkage among fibers resulting from composite thermal aging [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.35. Evolution of matrix shrinkage with aging time evaluated through interferometric microscopy images of the same sample zone [101] . . . . . . . . . . 45 1.36. Maximum matrix shrinkage depth as a function of the distance between fibers for sample aged at 150 °C under 1.7 bar (a) and 5 bar of oxygen pressure [101] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.37. Fiber/matrix debonding phenomenon observed on SEM images of the surface of unidirectional carbon fiber reinforced polymer matrix composites after 1000 h of aging in air at 150 °C [9] . . . . . . . . . . . . . . . . . . . Instrumented indentation test machine Fischerscope H100C . . . . . . . . . 51 2.2. Loading history used for the area function calibration . . . . . . . . . . . . A p vs h c experimental data points obtained on BK7-glass (black points), toghether with their fitting function of equation 2.1 (red line) and the ideal projected area of contact calculated according to Equation 1.6 . . . . . . . 2.4. Indentation modulus calculated on Fused Silica at different load levels before (red data points) and after (blue data points) area function calibration . . 2.5. Verification of instrument compliance . . . . . . . . . . . . . . . . . . . . . 2.6. Displacement-time data collected during the 600 s holding period at 0.1 mN at the beginning of the tests for a set of 9 tests carried out on HDPE . . . 2.7. Displacement-time data collected during 9 long indentation creep tests on HDPE of about 6000 s at a constant load of 5 mN before (a) and after (b) the application of thermal drift correction . . . . . . . . . . . . . . . . . . 2.8. Scheme of the tip/sample misalignment (a) and of the contact area resulting from a misaligned tip/sample contact (b) [118] . . . . . . . . . . . . . . . . 2.9. Tilt platform used to perform instrumented indentation tests at different inclination angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10. Evolution of the reduced modulus with the rotation angle about x (green) and y (red) axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11. Percentage variation of E * from the value at 0°tilt for PR520 (green symbols) and Fused Silica (red symbols) . . . . . . . . . . . . . . . . . . . 2.12. Image of the cross section of the embedded aged sample (a) and topography of an area of the cross section near the edge (b) . . . . . . . . . . . . . . . 2.13. Profile traced along the z axis of the topography of the thermally aged resin sample cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14. Topography of the surface of a thermally aged composite sample . . . . . 2.15. Four parallel profiles traced on the topography of the surface of a thermally aged composite sample from the edge of the fiber bundle to the center of the resin pocket (a) evolution of the local slope toward the profile (b) . . . 2.16. Effect of loading rate on HDPE one cycle load-displacement curves . . . . 2.17. Loading curves obtained on HDPE for different loading rates of 20 mN s (blue), 2 mN s (green) and 2 mN s (red) with their power law fitting curve (continuous black lines) and the theoretical loading curve for elasto-plastic materials with the exponent n = 2 (dashed black line) . . . . . . . . . . . 2.18. Evolution of the viscoelastic delayed displacement as a function of loading rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.19. Evolution of the reduced indentation modulus as a function of loading rate 71 2.20. Unloading curves obtained on HDPE for different loading rates of 20 mN s (blue), 2 mN s (green) and 0.2 mN s (red) with their power law fitting curve (continuous black lines) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.21. Fitting coefficients A (a), h f (b) and m (c) of unloading curve as a function of loading rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.22. Scheme of material pile-up and sink-in under contact [15] . . . . . . . . . 73 2.23. Ratio of final to maximum depth against loading rate for HDPE. The dashed black line represents the threshold for the onset of significant pile-up according to [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1. Cyclic instrumented indentation tests loading path . . . . . . . . . . . . . 79 3.2. 10 cycles load-displacement curves obtained on Fused Silica at loading frequencies of 1 Hz, 0.1 Hz and 0.01 Hz . . . . . . . . . . . . . . . . . . . . 80 3.3. 300 cycles load-displacement curve obtained on HDPE at a loading frequency of 0.1 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.4. Parameters extracted from cyclic force-displacement indentation curves . .82 3.5. Evolution with cycles of unload and reload indentation moduli (a), loop area (b) and maximum, minimum and mean displacement (c) for a 300 cycles indentation test performed on HDPE at a loading frequency of 0.1 Hz 84 3.6. Evolution with cycles of secant modulus (a), loop area (b) and mean strain (c) extracted from tensile macroscopic cyclic tests performed on HDPE at a loading frequency of 0.1 Hz [64] . . . . . . . . . . . . . . . . . . . . . . . . 85 3.7. Load-displacement hysteresis loops evolution with cycles for tests performed at loading frequencies of 1 Hz (a), 0.1 Hz (b) and 0.01 Hz . . . . . . . . . 87 3.8. Delayed displacement calculated on each unloading curve of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycle number . . . . . . . . . . . . . . . . . . . . . . 88 3.9. Unload (a) and reload (b) indentation moduli calculated at each cycle of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycle number . . . . . . . . . . . . . . . Rate of change of indentation reload modulus with time as a function of testing time for HDPE tested at: 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11. Exponent of the power law fitting function m of unloading (filled markers) and loading (empty markers) curves as a function of cycles and for the three frequencies of 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) . . . . . . . . 3.12. Evolution of reload indentation modulus as a function of loading frequencies for different testing times (a) and slopes of indentation modulus versus frequency lines obtained for different testing times (b). . . . . . . . . . . . 3.13. Evolution of reload indentation modulus with frequency obtained through cyclic indentation tests for 50 s of testing time (a) and evolution of Young's modulus with frequency obtained through tensile DMA tests at ambient temperature (b) [64] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14. Hysteresis loop area calculated at each cycle of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycle number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15. Hysteresis loop area rate calculated at each cycle of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycle number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16. Evolution of loop area as a function of loading frequency for different cycles (a) and slopes of loop area versus frequency lines obtained for different cycles (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17. Mean displacement calculated on each unloading curve of cyclic indentation tests performed on HDPE at 1 Hz (blue), 0.1 Hz (green) and 0.01 Hz (red) as a function of cycles (for the first 60 cycles) (a) and of testing time (b) . 3.18. Displacement-time plot of indentation cyclic test (blue curve), indentation creep test at 10 mN (red curve) and indentation creep test at 5 mN (green curve) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19. Evolution of creep at 10 mN (dashed red curve) and at 5 mN (dashed green curve) and cyclic minimum displacement (dashed blue curve) with testing time, together with their fitting curves (solid lines) . . . . . . . . . . . . . 3.20. Evolution of t µ 0 as a function of testing time for creep at 10 mN (red curve) and 5 mN (green cuve) and cyclic displacement (blue curve) . . . . . . . . 4.13. Evolution of η with aging time for the 1 st (red), 2 nd (blue), 3 rd (yellow), 5 th (black), 10 th (purple) and last (green) cycle . . . . . . . . . . . . . . . 4.14. Evolution of displacement accumulated in cycles calculated as the difference between the minimum displacement at the last and at the first cycle (a) and between the minimum displacement at the last and at the second cycle (b) with aging time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15. 40 cycles load-displacement curves obtained at different distances from the exposed surface of a sample aged in air at 150 °C for 1000 h . . . . . . . . . 4.16. Evolution with cycles of E * load (a), η (b) and h min (c) for the virgin sample (dark blue) and for different distances from the exposed surface ranging from 0 µm (orange) to 240 µm (light blue) of a sample aged in air at 150 °C for 1000 h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17. Evolution with testing time of E * load -E * load 0 (a) and h min -h min 0 (b) for the virgin sample (dark blue) and for different distances from the exposed surface ranging from 0 µm (orange) to 240 µm (light blue) of a sample aged in air at 150 °C for 1000 h . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18. Evolution of the reload modulus, normalized with respect to the mean value measured on the virgin sample and evaluated from the first reload, as a function of the distance from the exposed surface for the virgin sample (blue) and for the samples aged in air at 150 °C for 220 h (red), 520 h (green) and 1000 h (violet) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19. Evolution of the irreversible part of work of indentation η, normalized with respect to the mean value measured on the virgin sample and evaluated at the first cycle, as a function of the distance from the exposed surface for the virgin sample (blue) and for the samples aged in air at 150 °C for 220 h (red), 520 h (green) and 1000 h (violet) . . . . . . . . . . . . . . . . . . . . 4.20. Evolution of the irreversible part of work of indentation η evaluated at the second and last cycle, as a function of the distance from the exposed surface for the virgin sample (blue) and for the sample aged in air at 150 °C for 1000 h (violet) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21. Evolution of the thickness of the oxidized layer, estimated from the reload modulus gradient profiles, as a function of aging time . . . . . . . . . . . . 4.22. Evolution of the normalised reload modulus as a function of aging time for samples aged in air (red) and under 2 bar of oxygen pressure (blue) . . . . 143 4.23. Evolution of the normalised reload modulus as a function of the equivalent aging time for samples aged in air (red) and under 2 bar of oxygen pressure (blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.24. Evolution of η as a function of aging time for the samples aged in air (green) and under 2 bar of oxygen (blue) for the first, the second and the last cycle 145 4.25. Evolution of η of the first cycle as a function of the equivalent aging time for the samples aged in air (green) and under oxygen pressure (blue) . . . 146 4.26. PR520 samples surface color change with aging time for the different environments used in this study: air, Oxygen and Nitrogen . . . . . . . . . . . 147 4.27. Evolution of the glass transition temperature with aging time of the oxidized layer (orange) and of the sample core (blue) . . . . . . . . . . . . . . . . . 148 5.1. Example of interlock woven fiber architecture [126] . . . . . . . . . . . . . 154 5.2. Scheme of a 3D interlock carbon-fiber reinforcement similar to the one used in this study [128] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.3. Microscopic images of the 4 resin pockets located in the plan perpendicular to the weft direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.4. Microscopic images of the 5 resin pockets located in the plan perpendicular to the warp direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.5. Microscopic images of the 3 resin pockets located on the external surface of the composite sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.6. C and n fitting coefficients of the (first) loading curves for neat PR520 epoxy resin and in-situ measured in the core and on the external surface . 160 5.7. Evolution of reload indentation modulus (a), minimum displacement (b) and irreversible part of work of indentation (c) with indentation cycles for in-situ measured on the core ( ) and in-situ measured on the external surface ( ) PR520 epoxy resin properties . . . . . . . . . . . . . . . . . . . 163 5.8. Evolution with cycles of the parameter Z calculated from in-situ measurements in the sample core and on the external surface of: reload indentation modulus (]), minimum displacement (-) and η (×) and of Z crit for an interval of confidence of 99.9% (upper red dashed line) and 95% (lower red dashed line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9. Histogram representing the percent difference between in-situ in the core and in-situ on the external surface mean values of reload indentation modulus (yellow), minimum displacement (black) and η (green) for 1st, 10th, 20th, 30th and 40th cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10. Evolution of reload indentation modulus (a), minimum displacement (b) and irreversible part of work of indentation (c) with indentation cycles for the neat (A), in-situ measured in the core (0) and in-situ measured on the external surface ( ) PR520 epoxy resin properties . . . . . . . . . . . . 5.11. Evolution with cycles of the parameter Z for the comparaison between neat and in-situ in the core mean values of reload indentation modulus (]), mean displacement (-) and η (×) and of Z crit for an interval of confidence of 99.9% (upper red dashed line) and 95% (lower red dashed line) . . . . . 5.12. Histogram representing the percent difference between neat and in-situ in the core mean values of reload indentation modulus (yellow), minimum displacement (black) and η (green) for 1st, 10th, 20th, 30th and 40th cycle 5.13. Evolution with cycles of the parameter Z for the comparaison between neat and in-situ on the surface mean values of reload indentation modulus (]), minimum displacement (-) and η (×) and of Z crit for an interval of confidence of 99.9% (upper dashed red line) and 50% (lower dashed red line)168 5.14. Histogram representing the percent difference between neat and in-situ on the surface mean values of reload indentation modulus (yellow), minimum displacement (black) and η (green) for 1st, 10th, 20th, 30th and 40th cycle 5.15. Cross section of a composite sample aged for 1005 h in air at 150 °C. The resin pockets near the edge and at the sample core on which indentation tests have been performed are circled in red . . . . . . . . . . . . . . . . . 5.16. Resin pocket located on the surface perpendicular to the weft direction directly exposed to the aging environment of a composite sample conditioned for 1005 h in air at 150 °C . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17. Reload indentation modulus (a) and irreversible part of work of indentation (b) gradients measured in the oxidized layer of neat (filled red triangles) and in-situ (unfilled red triangles) PR520 epoxy resin aged for 1005 h in air at 150 °C. The mean values measured on the neat (filled blue circles) and in-situ (unfilled blue circles) virgin material are represented as well. The gradients are referred to the values calculated at the first cycle . . . . . . . 173 5.18. Histogram representing the percent difference between neat and in-situ reload modulus (yellow) and η (blue) for different distances from the edge of samples aged for 1005 h in air at 150 °C. The difference measured on the virgin materials is reported on the left part of the graph . . . . . . . . . . 174 Mots-clefs : Indentation des matériaux, Composites polymères, Matériaux--Effets des hautes températures, Microstructure (physique), Polymères--Détérioration, Thermoplastiques, Résines époxydes, Vieillissement thermique, Gradients de propriétés

Table 2 .

 2 131 • C. Its glass transition temperature determined from three point bending Dynamic Mechanical Analysis (DMA) is -116 • C [64]. The main HDPE mechanical properties, extracted from its technical data sheet [102] are reported in Table 2.1.

	Yield stress (M P a)	23
	Yield strain (%)	9
	Maximum elongation (%)	350
	Tensile secant modulus (M P a)	900
	Tensile creep modulus, 1000 h, 2 M P a (M P a)	360
	Flexural creep modulus, 1000 h (M P a)	330
	1. HDPE mechanical properties extracted from the manufacturer technical data sheet
	[102]

  .2.

	Glass transition temperature (°C) 161
	Tensile strenght (M P a)	82.1
	Tensile modulus (GP a)	4.0
	Poisson's ratio	0.398
	Compressive strength (M P a)	128
	Compressive modulus (GP a)	3.7
	Shear strength (M P a)	61.4
	Shear modulus (GP a)	0.8
	Flexural strength (M P a)	153.1
	Flexural modulus (GP a)	3.5
	Table 2.2. PR520 epoxy resin mechanical properties extracted from the manufacturer technical
	data sheet [103]

Table 2

 2

	.3.
	C 0	C 1	C 2	C 3	C 4	C 5	C 6	C 7	C 8
	22.94 0.81 1.23 -5.73 33.21 -70.03 77.10 -44.72 10.30

Table 2 .3. Coefficients

 2 C i of area function (Equation 2.1) determined from fitting

Table 2 .

 2 4.

		Paper grade time (min)
		500	1
	grinding	1000 2000	2 3
		4000	4
	finishing	3 µm 1 µm	5 5

Table 2 .

 2

4. Polishing protocol used for PR520 virgin and aged samples preparation

  .5.

		A	h p	m
	Start Point	0	0.1	1
	Lower Bound	0	0	0
	Upper Bound 100 h max 20

Table 2 . 5 .

 25 Constraints used for the unloading curve fitting

Table 3 .1. Fitting

 3

parameters for a EVEV model with three exponential terms for creep and cyclic displacement data

Table 3 .

 3

	2. Amount of energy dissipated at the first loading experimentally measured (U 1st tot ), extrapolated from the fitting function (U 1st v ) and to produce the permanent depth (U 1st p )

Table 3 . 3 .

 33 .3. Hz 0.01 Hz 1.11 µm 1.34 µm 1.39 µm Estimated value of plastic depth (h p ) for the loading frequencies of 1 Hz, 0.1 Hz and 0.01 Hz

	1 Hz	0.1

Table 4 .

 4

1. PR520 aging conditions used in this study

Table 4 . 3 .

 43 .3. Fitting parameters for a two exponential term law (Eq. 3.4) describing the evolution of h min -h min 0 in the case of virgin and conditioned samples

	h 1	τ 1	h 2	τ 2	µ 0
	(µm)	(s)	(µm)	(s)	(s • µm -1 )
	virgin 0.053 ± 0.002 11.2 ± 0.6 0.087 ± 0.002 59.8 ± 1.4 11920 ± 10
	aged 0.047 ± 0.002 11.7 ± 0.6 0.075 ± 0.002 63.2 ± 1.4 12080 ± 10

Table 4 . 4 .

 44 s) virgin 0.32 ± 0.02 20.5 ± 1.3 0.45 ± 0.01 151.3 ± 10.0 aged 0.38 ± 0.02 24.7 ± 1.2 0.49 ± 0.01 154.2 ± 8.6 Set of parameters found from the fitting procedure using the analytical expression of Equation 4.2 with two exponential terms for virgin and aged samples

Table 4 . 5 .

 45 Set of parameters of the analytical expression of Equation 4.3 allowing to describe the evolution of reload modulus with aging time for the first, the second and the last cycle

	h)

Table 4 . 6 .

 46 a 1 are still valid in this case. The values of the fitting parameters of Equation 4.4 are reported in Table 4.6. Set of parameters of the analytical expression of Equation 4.4 allowing to describe the evolution of η with aging time for the first cycle

	η 1st virgin	c	τ a 1 (h)	d	τ a 2 (h)
	0.504 0.079	35	0.031	815

Table 4 . 7 .

 47 Thickness of the oxidized layer, estimated from the indentation modulus gradient profiles, of the PR520 aged in air for about 500 h and 1000 h at 150°C and 120°C[START_REF] Pecora | Effect of thermo-oxidation on the failure properties of an epoxy resin[END_REF]

	.7.

Table 4 .

 4 [START_REF] Ho | Endommagements de composites carbone/époxy induits par la thermooxydation de la matrice en présence de contraintes mécaniques[END_REF]. For an easier comparison, the fitting parameters related to aging in air, already determined in Section 4.3.2, are reported as well.

		a (GP a)	τ a 1 (h)	b (GP a) τ a 2 (h)
	air	0.35	35 (1 -70)	0.18	815
	2 bar O 2	∼ 0	0.9	0.54	30

Table 4 . 8 .

 48 Set of parameters of the analytical expression of Equation4.3 allowing to describe the evolution of reload modulus with aging time for aging in air and under 2 bar of oxygen

Table 4 . 9 .

 49 .9. Set of parameters of the analytical expression of Equation4.3 allowing to describe the evolution of η of the first cycle with aging time for aging in air and under 2 bar of oxygen

		c	τ a 1 (h)	d	τ a 2 (h)
	air	0.079	35	0.031	815
	2 bar O 2 0.075	35	0.030	270

  .1.

	pocket n°1	2	3	4
	minimal distance (µm) 125 108 376 72

Table 5 . 1 .

 51 Minimal distance between indentation prints and the edge of the closer fiber bundle for the 4 resin pockets in the weft direction

Table 5 . 3 .

 53 Difference in percentage between PR520 properties in different configurations 3. Property gradients due to thermal aging in air at 150°C for 1005 h, measured on the oxidized layer of neat and in-situ PR520 epoxy resin are virtually superposed showing that the thickness of the oxidized layer is quite similar for both materials.

  It is maybe in this zone that a more significant difference in local properties exists. Further investigations are thus needed to establish the local properties in the interphase zone and how they are affected by thermal aging. However, an alternative experimental technique should be foreseen for such characterization since indentation test measurements are affected by the constraint effect of fiber bundles and thus not representative of actual mechanical properties of the matrix. du matériau étudié. De nombreuses études précédentes ont montré les difficultés liées à l'application de cette méthode d'analyse aux matériaux polymères pour lesquels l'hypothèse de décharge élastique est trop contraignante étant donné que la réponse de ce type de matériaux est dépendante du temps. De plus, la méthode d'Oliver et Pharr permet de déterminer deux paramètres indépendants du temps qui ne sont pas appropriés pour décrire un comportement complexe dépendant du temps. Pour cette raison, des méthodes d'essai d'indentation différentes sont proposées dans la littérature pour la caractérisation locale des matériaux polymères. Il s'agit, notamment, des essais de fluage en indentation et des essais d'indentation dynamiques. L'essai de fluage en indentation permet de déterminer, à partir de l'évolution du déplacement dans le temps pour l'application d'une charge constante, la complaisance du matériau J c (t). Les méthodes d'analyse qui permettent d'étudier le fluage en indentation peuvent être divisées en deux catégories: méthodes basées sur la solution analytique du contact entre un indenteur rigide et une surface viscoélastique linéaire et méthodes basées sur des modèles rhéologiques qui consistent en différentes configurations de ressorts et amortisseurs. Dans les deux cas, la valeur de complaisance déterminée est comparable à celle obtenue dans les essais macroscopiques seulement dans un domaine de viscoélasticité linéaire et sous l'hypothèse de chargement instantané. Ces hypothèses sont très rarement vérifiées en phase expérimentale, surtout si des pointes pyramidales, qui induisent une concentration de contraintes très élevée, sont employées. L'essai d'indentation dynamique consiste à superposer une oscillation sinusoïdale au signal de force et cela permet de déterminer le module de conservation et le module de perte du polymère à travers l'application d'un modèle rhéologique (l'élément de Voigt) aux données expérimentales. Comme dans le cas précèdent, cette méthode d'essai est limitée au domaine de viscoélasticité linéaire et, de plus, un équipement spécifique est nécessaire pour produire ce type de sollicitation. La revue bibliographique sur l'indentation instrumentée se termine avec l'application de cette technique aux matériaux composites à matrice polymères. Les auteurs qui ont conduit des études sur ce sujet ont mis en évidence que, pour mesurer les propriétés de la matrice qui ne soient pas affectées par l'effet de confinement des renforts (beaucoup plus rigides), les mesures doivent être réalisées à des distances des fibres 25 fois supérieures à la profondeur de pénétration.

Mise en place d'un essai d'indentation instrumentée pour les polymères

  1. Développer une méthode d'indentation qui soit capable de mettre en évidence la dépendance au temps des matériaux polymères ; 2. Caractériser les effets du vieillissement sur la réponse locale dépendent au temps d'une résine époxy vieillie employée pour la fabrication de composites tissées 3D à fibres de carbone conçus pour l'application dans les zones tièdes des structures aéronautiques ; 3. Caractériser les propriétés de la même résine dans le composites à l'état vierge et vieilli. Arrondissement de la pointe. La méthode d'analyse des courbes d'indentation prévoit le calcul de l'aire de contact à partir de la valeur de déplacement sous contact, grâce à la fonction d'aire qui dépend de la géométrie de la pointe. Si la forme réelle de la pointe est très différente de sa forme idéale, à cause par exemple de l'usure, l'aire calculée avec la fonction idéale est sous-estimée et, par conséquent, les valeurs de module et dureté obtenues seront affectées. Il est cependant possible de minimiser cette source d'erreur grâce à la calibration de la fonction d'aire, en utilisant un étalon (verre BK7 dans notre cas) dont le module élastique d'indentation est connu et qui ne présente pas d'effet de dépendance du module de la profondeur de pénétration.

	Chapitre 2. Dans ce Chapitre, l'équipement d'indentation utilisé -Fischerscope H100C, équipé d'une
	pointe pyramidale Vickers -et les principaux matériaux testés -Polyéthylène haute densité
	pour le développement de la méthode et résiné époxy PR520 pour les applications -sont
	décrits. Ensuite, les étapes à suivre pour réaliser correctement un essai d'indentation
	instrumentée, de façon à minimiser les sources d'erreurs qui peuvent augmenter la dispersion
	sur les résultats, sont détaillées. D'après la norme d'indentation pour les matériaux
	métalliques ISO 14577, les sources d'erreur les plus importantes sont :
	•

Développement d'un essai cyclique d'indentation pour la caractérisation du comportement dépendant du temps des polymères

  Si pour les matériaux métalliques, le défaut de planéité peut être minimisé avec un protocole de polissage approprié, dans le cas de matériaux polymères et composites vieillis, à cause des retraits induits par le vieillissement, le défaut de planéité est une caractéristique intrinsèque des surfaces. D'après la norme, la pente maximale ne doit pas dépasser 1°, alors que d'autres auteurs ont montré que la valeur de pente maximale acceptable dépend de plusieurs facteurs comme la géométrie de la pointe et le matériau. Pour quantifier la pente maximale acceptable pour notre système indenteur/matériau, des essais d'indentation ont été réalisés sur la résine PR520 à l'aide d'une plateforme tournante en variant l'angle d'inclinaison entre 0°et 8°et en évaluant le module élastique d'indentation pour les différentes inclinaisons. Il a été trouvé que le module s'écarte de la valeur mesurée pour une inclinaison nulle à partir de 3 ○ . Une fois la limite maximale acceptable pour notre système établie, les pentes sur les surfaces de la résine et du composite vieillis ont été quantifiées. A partir des images de microscopie interférométrique de la section transversale d'un échantillon de résine vieilli, des profils ont été tracés le long de la couche oxydée et les pentes mesurées sur des segments espacés de environ 10 µm. Il a été conclu que, comme la pente maximale est inférieure à 1ř , le défaut de planéité ne constitue pas un problème dans ce cas. La même procédure a été suivie pour évaluer les pentes sur les surfaces des composites vieillis, sur des profils tracés entre le bord du toron de fibres et le milieu de la poche de résine. Dans ce cas, la pente maximale est supérieure de 8°à proximité du toron, et diminue jusqu'à atteindre une valeur inférieure à 3°pour une distance des fibres d'environ 50 µm. • Dérive thermique. La dérive thermique s'est révélée être la source d'erreur la plus significative dans nos applications. Premièrement parce-que l'équipement est exposé à l'environnement et deuxièmement parce-que les matériaux polymères ont un coefficient de dilatation thermique beaucoup plus élevé par rapport aux matériaux métalliques. En effet, une variation de température dans l'environnement cause une dilatation (ou une contraction) de l'échantillon, de la pointe et de l'équipement qui se traduisent par un écart très important sur les données de déplacement mesurées, surtout pour les essais longs. L'application de la méthode de correction de la dérive thermique décrite dans la norme ne s'est pas révélée efficace pour nos applications. Pour cette raison, on a proposé une méthode différente qui a été validée sur différents matériaux et protocoles d'essai. Cette méthode consiste à mesurer la dérive thermique à travers une phase de maintien de 600 s à très faible charge (0.1 mN ) avant la phase de chargement. Les données déplacement-temps acquis pendant cette phase sont ajustées avec une loi de puissance qui est, ensuite, propagée à toutes les données de déplacement enregistrées pendant l'essai. Le Chapitre se termine avec l'application de l'essai classique d'indentation et de la méthode d'analyse d'Oliver et Pharr (décrite dans le Chapitre 1) sur les courbes force-déplacement obtenues sur un matériau polymère, le PEHD, pour différentes vitesses de chargement. Il a été mis en évidence que la réponse dépend de la vitesse de chargement. En particulier, des vitesses moins élevées induisent des profondeurs de pénétrations plus importantes. Un phénomène de fluage en phase de décharge a également été mis en évidence à travers l'observation du si dit effet "nose" (forme arrondie de la courbe en début de décharge). L'application de la méthode d'analyse d'Oliver et Pharr a montré certaines limitations, principalement à cause du fait que l'hypothèse de décharge élastique n'est pas valable sur les polymères pour lesquels on aurait plutôt un retour viscoélastique. Si dans la littérature certains auteurs suggèrent des modifications au protocole d'essai classique pour minimiser les effets viscoélastiques en phase de décharge, la question de comment décrire de façon appropriée une réponse complexe et dépendante du temps reste ouverte. Dans ce Chapitre, le protocole d'essai et sa méthode d'analyse développée au cours de cette thèse sont décrits. Il s'agit d'un essai d'indentation cyclique, inspiré des essais de fatigue macroscopiques, qui consiste en plusieurs cycles de charge/décharge entre 0.5 mN et 10 mN dans lesquels la vitesse de charge est égale à celle de décharge et constante pour tous les cycles. Ce type de sollicitation permet de mettre en évidence la réponse dépendante du temps des matériaux polymères à différentes échelles. Une phase de recouvrance a été introduite à la fin de certains essais pour essayer d'estimer la partie permanente et recouvrable de la déformation induite par le chargement cyclique. Comme annoncé précédemment, l'essai cyclique a été développé sur le polyéthylène haute densité (PEHD) principalement parce que sa réponse à une sollicitation cyclique macroscopique (traction et cisaillement) a été largement étudiée en précédence dans les travaux de Nguyen et deuxièmement parce-que, par rapport aux résines époxydes, il présente une réponse dépendante du temps plus accentuée, en étant au-dessus de sa température de transition vitreuse à l'ambiante (T g = -116 °C). Si pour les matériaux élasto-plastiques, comme par exemple la Silice Fondue, il a été montré que les courbes de décharge et recharge sont superposées, dans le cas de matériaux polymères, la réponse en termes de courbes force-déplacement est caractérisée par des grosses boucles d'hystérèse qui évoluent avec les cycles. Pour interpréter cette réponse assez complexe, un certain nombre de paramètres ont été extraits à partir des courbes brutes. Ces paramètres sont similaires à ceux typiquement utilisés pour l'analyse de la réponse cyclique macroscopique, en sachant, toutefois, que les deux types d'essai sont différents. Premièrement, les essais macroscopiques sont réalisés en contrôle de contrainte, alors qu'en indentation le contrôle est en force. Par conséquent, les courbes obtenues en indentation sont des courbes force-déplacement et non contraintedéformation. De plus, le volume de matériau concerné par la sollicitation est très différent dans les deux cas, ce qui implique que, en indentation, de forts gradients de contraintes locales se développent, surtout si une pointe pyramidale (comme dans notre cas) est utilisée. Malgré cela, par analogie avec les paramètres des courbes de traction cyclique, trois paramètres ont été extraits des courbes d'indentation cyclique : l'aire de boucle, qui représente l'énergie dissipée à chaque cycle ; les modules d'indentation, évalués à partir des courbes de décharge et de recharge ; les déplacements maximal, minimal et moyen, évalués à partir des décharges.

	Chapitre 3.

• Défaut de planéité. Si la surface de l'échantillon n'est pas perpendiculaire à la pointe, similairement à l'effet d'arrondissement de la pointe, l'aire de contact calculée est sous-estimée et, par conséquent, le module et la dureté seront surestimés.

  La méthode expérimentale développée au cours de cette thèse, et décrite dans Chapitre 3, est appliquée, dans le présent Chapitre, à la caractérisation des effets du vieillissement thermique sur la réponse locale dépendant au temps de la résine époxy PR520. Les vieillissements ont été réalisés dans trois environnement différents et pour différentes durées à la température de 150 °C . Les environnements et les durées de vieillissement sont :• Air à pression atmosphérique pour 220, 520 et 1000 h, réalisés dans le four ventilé Fisherbrand ;Le protocole employé pour les essais présentés dans ce Chapitre consiste en 40 cycles entre 0.5 et 10 mN . Une seule fréquence de chargement (0.1 Hz) a été considérée. Les essais ont été réalisés sur la surface extérieure directement exposée à l'environnement et dans l'épaisseur, pour différentes distances de cette dernière. Dans la première partie du Chapitre, le comportement cyclique en indentation de la résine vierge a été analysé. Le matériau montre une réponse similaire à celle du PEHD analysé dans le Chapitre précédente, mais sa dépendance au temps est moins prononcée par rapport au matériau semi-cristallin. Pour l'étude du vieillissement, trois cas ont été considérés : effets du vieillissement sur la réponse cyclique, évolution des propriétés avec la durée de vieillissement et évolution des propriétés dans l'épaisseur (gradients). Les courbes force déplacements obtenues sur la surface extérieure des échantillons vieillis sont caractérisées, comme pour le matériau vierge, par des grosses boucles d'hystérèse, qui sont décalées vers des valeurs de déplacement moins élevées par rapport à celles du matériau vierge. Il a été montré, en outre, que la cinétique d'évolution du module de recharge, de l'énergie normalisée et du déplacement accumulé est similaire pour le matériau vierge et vieilli pour différentes durées. De plus, les évolutions de l'énergie normalisée avec les cycles pour les différentes durées de vieillissement sont superposées à partir du deuxième cycle. Il a été donc possible de conclure que la quantité d'énergie dissipée pour produire les déformations dépendantes du temps n'est pas affectée par le vieillissement, alors que celle dissipée pour produire des déformations plastiques (premier cycle) est réduite entre le matériau vierge et vieilli. L'évolution du module de recharge (réponse élastique) et de l'énergie normalisée du premier cycle (réponse plastique) avec le temps de vieillissement, mesurés sur la surface directement exposée à l'environnement, peut être décrite avec une loi exponentielle avec deux termes exponentiels. Cette loi permet de reproduire les trois phases du vieillissement thermique : initiation, propagation et saturation. Les temps caractéristiques de vieillissement qui gouvernent la cinétique d'évolution avec le temps de vieillissement sont similaires pour les deux paramètres.Grâce aux résultats des essais réalisés à différentes distances depuis la surface directement exposée à l'environnement, il a été observé que l'effet de l'approche à la surface, pour la même durée de vieillissement, est similaire à celui de l'augmentation de la durée de vieillissement pour la même distance de la surface, sur les courbes force-déplacement. Le module de recharge et l'énergie normalisée du premier cycle évoluent dans la couche oxydée et atteignent une valeur proche de celle mesurée sur le matériau à l'état vierge. A partir du deuxième cycle, l'énergie dissipée est constante dans la couche oxydée. A partir des profils des gradients il a été possible d'estimer l'épaisseur de la couche oxydée. Ces valeurs sont significativement inférieures à celles trouvées dans une autre étude conduite sur le même matériau vieilli à 120 °C pour des durées similaires. Cela signifie que l'oxygène a pénétré plus en profondeur dans le cas de vieillissement à 120 °C , mais il a induit une variation de module moins importante que celle trouvée à 150 °C. L'analyse des résultats obtenus sur la surface des échantillons vieillis sous 2 bar d'oxygène, et en particulier l'évolution des paramètres avec le temps de vieillissement, a permis de conclure que l'emploi d'une pression d'oxygène modérée accélère les phénomènes de dégradation induit par le vieillissement : l'augmentation du module de recharge et la diminution de l'énergie nécessaire pour produire des déformations plastiques. En particulier, les valeurs des deux paramètres à saturation, obtenus pour un temps de vieillissement qui tend à l'infini, sont très similaires dans les deux environnements. Cependant, des précautions doivent être prises car tous les phénomènes et les phases du vieillissement ne sont pas accélérés de la même façon. En particulier, il a été montré que l'augmentation du module de recharge en phase de propagation est accéléré d'environ 27 fois, alors que la diminution de l'énergie du premier cycle normalisée de seulement 3 fois.

• 2 bar d'Oxygène pour

[START_REF] Jin | Effects of experimental varaibles on pmma nano-indentation measurements[END_REF][START_REF] Wright | An improved analysis for viscoelastic damping in dynamic nanoindentation[END_REF][START_REF] Heux | Dynamic mechanical and 13c n.m.r. investigations of molecular motions involved in the beta relaxation of epoxy networks based on dgeba and aliphatic amines[END_REF] 200

et 400 h, réalisés dans l'enceinte Topindustrie.

Le but de ces essais a été de vérifier la possibilité d'accélérer les phénomènes de vieillissement par rapport à ceux produits en air à pression atmosphérique ;

• 2 bar d'Azote pour 100 h, réalisés dans la même enceinte des vieillissements sous Oxygène, avec l'objectif de vérifier si des phénomènes de dégradation se produisent aussi dans un environnement inerte à cette température.

  1.1. Typical load-displacement curve resulting from instrumented indentation tests (a) and schematic representation of the corresponding geometrical parameters (b) [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Load-displacement curves obtained on four different polymers [17] . . . . . 1.3. Concept used to define the effective indenter shape [22] . . . . . . . . . . . 1.4. Load-displacement curves at the loading-unloading peak evidencing that the introduction of a 10 s holding period at the maximum load allows to eliminate the nose [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Indentation elastic modulus and hardenss of PMMA as a function of holding time [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Scheme of load-displacement data highlighting the areas of positive and negative drift and the point of minimal drift at which stiffness is evaluated [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. VEP indentation model on loading (a) and unloading (b) [28] . . . . . . . 1.8. Displacement-time data during indentation creep of PET, PMMA and epoxy resin [29] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. Evolution of the representative strain as a function of creep time for different values of initial mean contact strain and comparison of the resluts with the Lee and Radok's viscoelastic contact model [32] . . . . . . . . . . . . . . . 1.10. Evolution of the representative strain as a function of the recovery time normalized by the creep time for an initial mean contact strain of 2% (a) and 4% (b) [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11. Generalised Kelvin model used by Yang et al. [29] to fit displacement data during indentation creep tests . . . . . . . . . . . . . . . . . . . . . . . . . 1.12. Scheme of the experimental procedure used by Minervino [10] to determine the relaxation spectrum of an epoxy resin through reverse analysis . . . . . 1.13. Load-displacement curves in dynamic indentation tests [44] . . . . . . . . 1.14. Evolution of storage modulus with indentation depth [45] . . . . . . . . . 1.15. Load-displacement curve in which the energy dissipated due to plastic deformations (U p ) is highlighted [51] . . . . . . . . . . . . . . . . . . . . . 1.16. Indentations performed in resin pockets perpendicular

  Il est donc nécessaire de mettre au point une technique expérimentale capable de caractériser le comportement du matériau en fonction du temps à l'échelle locale, pour saisir les gradients des propriétés. Ce travail propose la mise en place d'un essai cyclique d'indentation instrumentée pour répondre à ces questions. L'indentation instrumentée est une technique expérimentale qui a rencontrée un grand succès au cours des dernières années. Dans sa forme classique, développée pour les matériaux à comportement élasto-plastique, elle consiste à réaliser un seul cycle de charge/décharge pour en déduire la dureté du matériau et le module élastique d'indentation. Cette analyse n'est pas appropriée pour les matériaux présentant un comportement dépendant du temps, ce qui nécessite de développer une nouvelle méthode optimisée pour les matériaux polymères. Les méthodes d'indentation pour les matériaux polymères proposées dans la littérature (fluage, dynamique), sont limitées à certaines charges. L'essai cyclique d'indentation proposé dans ce travail vise à mettre en évidence la complexité du comportement du polymère. Il est dérivé des essais cycliques macroscopiques et emploi une méthode d'analyse similaire. La technique est développée sur un polymère thermoplastique, le PEHD, pour lequel la réponse cyclique macroscopique en traction et cisaillement est connue. En suivant l'évolution au cours du temps des principaux indicateurs du comportement cyclique (le module d'indentation, l'aire de la boucle d'hystérésis et l'accumulation du déplacement) pour différentes fréquences, il est possible de mettre en évidence la réponse du matériau et d'effectuer une comparaison qualitative avec le comportement macroscopique. Le protocole d'indentation cyclique est ensuite utilisé pour étudier les gradients de propriétés dans la résine époxy thermodurcissable PR520 soumise à un vieillissement thermique à 150°C sous air à pression atmosphérique (jusqu'à 1000h), sous 2 bar d'O2 et de N2 (pour 400h). Il est montré que la cinétique d'évolution du module d'indentation et du déplacement n'est pas affectée par le vieillissement. Cependant, leurs valeurs absolues varient de la surface au coeur du polymère, ce qui indique la présence de gradients. L'hystérésis du premier cycle est différente à travers le gradient, mais à partir du deuxième cycle, l'hystérésis est similaire pour toutes les conditions de vieillissement et les distances de la surface exposée. Les résultats obtenus sur des échantillons vieillis sous 2 bar d'O2 et de N2 permettent de conclure que la cinétique de vieillissement n'est pas de la pure thermo-oxydation. La méthode d'indentation cyclique est ensuite appliquée pour caractériser le comportement de la matrice époxy PR520 dans un composite tissé 3D, à l'état vierge et vieilli à 150°C sous air à pression atmosphérique. L'étude du composite à l'état vierge révèle que le comportement de la matrice polymère à proximité de la surface externe est différent de celui situé dans les zones internes du composite et du polymère pur. La comparaison entre le polymère pur et la matrice à l'état vieilli montre que les gradients de propriétés induits par l'environnement sont similaires.ABSTRACTKey words: Polymeric composites, Materials at high temperatures, Microstructure, Polymers--Detereioration, Thermoplastics, Epoxy resins, Instrumented Indentation, Thermal ageing, property gradients

The area function has been determined from the calibration procedure and the compliance of the instrument verified;

Acknowledgements

From Figure 3.22, it can be observed that, as the amount of displacement accumulated during creep at 10 mN is higher than that accumulated during cycles, the depth at the beginning of recovery (or the final depth after unloading) is higher for creep test than cyclic test. In particular, it is about 2.5 µm in cyclic tests and about 2.8 µm in creep test. Looking at Figure 3.19 it can be observed that the same difference of 0.3 µm between creep and cyclic displacement at the end of the tests is found on the final depths at the end of unloading, showing that the same amount of displacement has been recovered during the unloading of the two tests. Moreover, it can be observed that the creep displacement not only is accumulated faster, but is also recovered faster. In fact, even if the initial point of recovery curve following the creep test is higher than that following the cyclic test, after a certain recovery time, the two curves intersect and, at the end of the test, the amount of depth recovered seems to be higher in the case of creep test. However, because of the relatively high scatter, it is not possible to exactly establish the gap, if there is, between the two residual depths. As the plastic print is produced during the first loading (that is the same for the two tests compared here), the gap between the two recovery curves should correspond to the amount of viscoplastic displacement accumulated during the tests. Nevertheless, the recovery curves are not stabilized within the time interval considered in the two cases. Moreover a gap between the displacement measured at the end of the calculated for each parameter are represented in Figure 5.8 as a function of the cycle number, together with the critical values of Z for an interval of confidence of 99.9% (upper red dashed line) and of 95% (lower red dashed line). From Figure 5.8 it appears that there is a probability of 99.9% that the measurements performed on the external surface and in the core of the composite belong to different populations. The percent difference of all parameters between external surface and core in-situ mean values has been quantified and the results for 1st, 10th, 20th, 30th and 40th cycle are reported in Figure 5.9.

Conclusions and perspectives

In this Section, the main results obtained in this thesis work will be summarized, allowing to draw up the perspectives as well.

The instrumented indentation test on polymers

In order to obtain reliable instrumented indentation test results, some precautions must be taken in the test setting up. In particular, the sample preparation should ensure a flat and smooth surface. Moreover, if tests with shallow depth of indentation are foreseen, the contact area to depth function should be calibrated in order to take into account the tip blunt. The source of error that mostly affects the results in our configuration is the thermal drift, mainly because the equipment used is not well protected from the environment and because the tested materials (polymers) have an high coefficient of thermal expansion. Moreover, the testing protocols used in this work are quite long, making the thermal drift to increase the scatter on the results. That is why a new method for correcting data for thermal drift has been proposed in this work. The indentation curves obtained on a semi-crystalline polymer, the High Density Polyethylene, showed that the material response changes with loading rate, this being a material exhibiting time-dependent behaviour. The analysis of unloading curves through the Oliver and Pharr method revealed that the fit of unloading curves with power law function was bad because of creep phenomena occurring during unloading and the calculated values of indentation moduli were rate dependent. Moreover, the extraction of a single value of elastic modulus is not adequate to describe the complex time-dependent behaviour of polymers.

Appendix A

Extended abstract in French Développement d'un essai cyclique d'indentation pour la caractérisation des gradients de propriétés dus au vieillissement thermique dans les matériaux polymères et composites

Chapitre 1. Étude bibliographique

Ce Chapitre présente une revue bibliographique sur la technique expérimentale principalement employée et les phénomènes étudiés au cours de cette thèse. A partir de cette revue, il sera possible de définir plus en détail les objectifs de ce travail. Dans la première partie, la technique d'indentation instrumentée dans sa forme classique a été décrite avec un accent sur son application aux matériaux polymères. L'essai d'indentation consiste à enfoncer une pointe de géométrie et matériau connus dans le matériau à tester, et à suivre l'évolution du déplacement en fonction de la charge appliquée. A partir des courbes force-déplacement obtenues pendant un cycle de charge-décharge il est possible, grâce à l'application de la méthode d'Oliver et Pharr (développée à partir de la solution du contact élastique de Sneddon), de déterminer le module élastique d'indentation et la dureté

List of Tables