M Ponçot

F Addiego

A Dahoun

True intrinsic mechanical behaviour of semi-crystalline and amorphous polymers: influences of volume deformation and cavities shape

Keywords: amorphous polymer, semi-crystalline polymer, toughened blend, uniaxial tension test, volume damage, cavities shape, corrective form factor, intrinsic true mechanical behaviour

A model enabling the determination of the intrinsic mechanical constitutive equations of uniaxially stretched polymers is presented. This model takes into account the cavitation-induced volume strain which can occur during the deformation of such materials. In particular, the true intrinsic axial stress and strain depends on the overall volume strain and a form factor depicting the evolution of the voids shape. Based on our model, the true intrinsic behaviour of high-density polyethylene (HDPE), polypropylene / ethylene-propylene rubber (PP/EPR), and polyethylene terephtalate (PET) was assessed in tension. Compared to the overall true behaviour, the intrinsic true behaviour of the materials did not exhibit anomalies at large strain levels with changing experimental parameters (strain rate and temperature), and can be accurately predicted by means of phenomenological constitutive equations as the one proposed by G'sell and Jonas (1979).

Introduction:

Thermoplastic polymers are currently representing a great issue for various actors of the industry to manufacture irregular shaped pieces requiring advanced mechanical properties. This is particularly the case for the automotive industry where the demand in thermoplastics increases due to their high toughness, coupled with their low density that enables to address environmental specifications [START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF]. Depending on its formulation (homo-polymer, copolymer, branching, use of reinforcing agents…) and on its microstructure (amorphous, semi-crystalline...), a given thermoplastic can exhibit a wide range of end-use mechanical properties. Accordingly, it is of high interest to identify, analyse, and predict with precision the thermo-mechanical behaviour of thermoplastics.

The deformation mechanisms of polymers depend on their initial microstructures which are mainly induced by the formulation and the conformation processes. As a result, mechanical behaviour presents significant differences if the polymer matrix is amorphous [START_REF] Boyce | An experimental and analytical investigation of the large strain compressive and tensile response of glassy polymers[END_REF][START_REF] Bartczak | On the plastic deformation of the amorphous component in semicrystalline polymers[END_REF][START_REF] Steenbrink | Void growth in glassy polymers[END_REF][START_REF] Vigny | Constitutive viscoplastic behavior of amorphous PET during plane-strain tension stretching[END_REF]Ponçot et al., 2011), semi-crystalline [START_REF] Sell | Microstructure transformation and stress-strain behavior of isotactic polypropylene under large plastic deformation[END_REF][START_REF] Amornsakchai | On the influence of initial morphology on the internal structure of highly drawn polyethylene[END_REF][START_REF] G'sell | Experimental characterization of deformation damage in solid polymers under tension and its interrelation with necking[END_REF]Addiego et al., 2006;[START_REF] Séguela | On the natural draw ratio of semi-crystalline polymers: review of the mechanical, physical and molecular aspects[END_REF][START_REF] Galeski | Cavitation during Drawing of Crystalline Polymers[END_REF][START_REF] Ponçot | Thermomecanical behaviour of metal/polymer composites along different deformation paths: Determination of intrinsic mechanical behaviour laws on polymer film[END_REF] or filled with rubber particles [START_REF] Bucknall | Rubber-toughening of plastics, part 1: Creep mechanisms in HIPS[END_REF]Pukansky et al., 1994;[START_REF] Plummer | Deformation and entanglement in semi-crystalline polymers[END_REF][START_REF] Van Der Wal | Polypropylene-rubber blends: 5. Deformation mechanism during fracture[END_REF][START_REF] Nitta | Morphology and mechanical properties in blends of polypropylene and polyolefin-based copolymers. Polyolefin Blends[END_REF][START_REF] Mae | Microstructural observation and simulation of micro damage evolution of ternary polypropylene blend with ethylene-propylene-rubber (EPR) and talc[END_REF][START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF]. Chain orientation, shear banding, and damage are the main involved deformation micromechanisms whatever the initial microstructure of the considered material. The prevalence of one mechanism regarding the others induces such kind of mechanical properties differences.

Although the micromechanisms of deformation of polymers are currently well described as a function of the microstructure, a quantitative inelastic modelling is still lacking. Such predictive model would facilitate the understanding of the relationship between macroscopic behaviour and micromechanisms of deformation. Over the past three decades, several constitutive models were proposed to assess the elastic-viscoplastic deformation behaviour of solid polymers: i) Phenomenologically-based models: Due to the difficulty to implement the elementary deformation mechanisms in the constitutive equations, some authors relied on purely phenomenological models (G'sell and Jonas, 1979; G'sell and Dahoun, 1995;[START_REF] Chaboche | Formalisme général des lois de comportement : applications aux métaux et polymères. Introduction à la mécanique des polymères[END_REF][START_REF] Sell | Microstructure transformation and stress-strain behavior of isotactic polypropylene under large plastic deformation[END_REF]1999;[START_REF] Khan | Finite deformation of a polymer: experiments and modelling[END_REF][START_REF] Khan | Thermo-mechanical large deformation response and constitutive modeling of viscoelastic polymers over a wide range of strain rates and temperatures[END_REF][START_REF] Zaïri | Elasto-viscoplastic constitutive equations for the description of glassy polymers behavior at constant strain rate[END_REF]Regrain et al., 2009). These last authors fitted the nonlinear behaviour of semicrystalline polymers by specific mathematical functions taking into account external factors as strain rate and temperature. ii) Physically-based models: As an alternative, the physical approach was recently illustrated by [START_REF] Boyce | Constitutive model for the finite deformation stress-strain behavior of poly(ethylene terephthalate) above the glass transition[END_REF] describing the elastic-viscoplastic behaviour of amorphous polyethylene terephthalate above its glass transition and integrating the strain rate dependence. This model was then extended by Azhi et al. (2003) and [START_REF] Richeton | Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates[END_REF], taking into account the strain-induced crystallization. Other macroscopic inelastic constitutive models for various semi-crystalline polymers were recently proposed by [START_REF] Nikolov | Self-consistent modeling of large plastic deformation, texture and morphology evolution in semi-crystalline polymers[END_REF] and [START_REF] Ayoub | Modelling large deformation behaviour under loading-unloading of semi-crystalline polymers: Application to high density polyethylene[END_REF]2011). In their representation, the material was considered as an aggregate of two-phase layered composite inclusions which are randomly oriented. These models simulated the macromolecular chains orientation but did not consider the volume damage.

However, as it has been expressed first by [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF] then by Tveergaard (1981) and [START_REF] Lemaitre | A course on damage mechanics[END_REF], the determination and the introduction of the volume deformation in the expression of the true stress and strain are essential to provide efficient mechanical constitutive equations. [START_REF] Detrez | Plasticity/damage coupling in semi-crystalline polymers prior to yielding: Micromechanisms and damage law identification[END_REF] used a model based on the damage mechanics framework introducing a macroscopic damage variable and not the void volume fraction as indicator of damage. Investigations based on the Gurson-type description of porous materials provided mechanical behaviours caption first limited to spherical voids [START_REF] Steenbrink | Void growth in glassy polymers[END_REF][START_REF] Laiarinandrasana | Temperature dependent mechanical behaviour of PVDF: experiments and numerical modelling[END_REF][START_REF] Zaïri | Constitutive equations for the viscoplastic-damage behaviour of a rubber-modified polymer[END_REF]2008). Recently [START_REF] Zaïri | A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation[END_REF] developed a model considering the evolution of voids shape. In particular, voids were first supposed spherical at the onset of cavitation, and then with increasing strain level voids evolved into ellipsoids with major axis oriented along tensile direction. The gradual orientation of the macromolecular chains, the voids shape evolution and the overall volume variation were successfully implemented in the mathematical formulation of the constitutive model. Therefore, it is of high importance to accurately describe the anisotropic growth of voids, using physically realistic damage models such as the one proposed by [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals[END_REF].

This paper focuses on the achievement of the true intrinsic expressions of stress and strain during the tensile deformation based on the evolution of void fraction and shape developing in the materials. To this end, three thermoplastics presenting different microstructures were studied: amorphous polyethylene terephthalate (PET), semi-crystalline high-density polyethylene (HDPE) and isotactic polypropylene filled with ethylene-propylene rubber particles (PP/EPR). Void fraction (or volume damage) was calculated in situ during the stretching procedure, while the voids shape evolution was determined from post mortem microscopic observations. The influence of strain rate and temperature on the intrinsic true mechanical behaviour of these three polymers was also investigated. The intrinsic behaviour of the materials was compared to that obtained by simulation using the phenomenologicallybased model introduced by G'sell and Jonas (1979). A comparison between the equivalent true intrinsic mechanical behaviours deduced from different deformation paths using the Von Mises plastic criterion was proposed.

Experimental section:

Materials Polyethylene terephtalate

As amorphous thermoplastics, we selected a grade of polyethylene terephtalate (PET) that was supplied by Eastman (Voridian PET 9921). Its average molecular weight is w M = 51 000 g.mol -1 , while its polydispersity is 2 (Ponçot et al., 2011). It was processed by extrusion into 5 mm-thick plates. Differential scanning calorimetry results, at a standard heating rate of 10 °C.min -1 , show a glass transition temperature of about 75 °C, a melting temperature of 240 °C and a cold crystallization temperature of 155 °C. The initial amount of crystallinity of PET is equal to 0 since the enthalpies of cold crystallization and fusion compensate one each other. Wide-angle X-ray scattering (WAXS) analysis confirms that these PET plates have an amorphous microstructure and an isotropic distribution of the chains whatever the spatial direction.

High density polyethylene

As semi-crystalline thermoplastic polymer, a grade of high-density polyethylene (HDPE) manufactured by DuPont Canada (ref. Sclair 2907) was chosen. Its number and weight average macromolecular weights, determined by previous authors (e.g. [START_REF] Amornsakchai | On the influence of initial morphology on the internal structure of highly drawn polyethylene[END_REF] are equal to n M = 16,800 g.mol -1 and w M = 93,600 g.mol -1 . Solid cylinders, 110 mm in diameter, were extruded by the Plastifab Company (Montréal, Canada). In this process, the material is slowly cooled in air after getting out of the die in order to minimize internal stresses and favour homogeneous microstructure. 7 mm-thick plates are cut out of the core region of the cylinders along planes parallel to the extrusion axis. Differential scanning calorimetry results, at the standard heating rate of 10 °C.min -1 , show a glass transition temperature close to -120 °C, a melting temperature of 135 °C. The weight index of crystallinity is found to be equal to 78 wt. % (hydrostatic weighting and differential scanning calorimetry) or 76 vol. %. The crystalline morphology is characterized by spherulites of about 50 µm in diameter composed with regularly twisted lamellae. The long period of the semicrystalline stacks, as determined by small-angle X-ray scattering, is equal to 27.3 nm that gives, in conjunction with the crystallinity ratio, an average lamella thickness of 20.7 nm. The high crystallinity and coarse microstructure of this HDPE, already investigated by previous authors [START_REF] Wunderlich | Macromolecular physics: Crystal structure, morphology, defects[END_REF][START_REF] Addiego | Caractérisation de la variation volumique du polyéthylène au cours de la déformation plastique en traction et en fluage[END_REF], are due to its very low content in catalytic residues and the absence of nucleating agents.

Polypropylene / ethylene-propylene rubber

As engineering thermoplastic, we selected a polypropylene / ethylene-propylene rubber (PP/EPR) that was supplied by Total Petrochemicals (ref. PPC 3650). The samples were processed by injection moulding into 4 mm-thick plates. The amount of EPR mixed with the PP matrix is 17 wt. %. The final blend presents the following average molecular weights: n M = 69,000 g.mol -1 and w M = 599,000 g.mol -1 [START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF]. Differential scanning calorimetry analysis, at standard heating rate of 10 °C.min -1 , reveals two melting peaks. The first one at 115 °C corresponds to the melting of polyethylene crystals. The second one at 167 °C is attributed to the fusion of PP crystals of the -crystalline phase. The polyethylene lamellae are identified and localized inside the EPR nodules. It means that these particles have a core-shell shape. This characteristic has been previously demonstrated by scanning and transmission electron microscopy [START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF]. The glass transition temperature of the PP matrix is close to 10 °C. The elastomeric phase shows a glass transition at -40 °C. Typical diameter of these nodules is about 10 µm. Furthermore, additional analyses by wide-angle Xray scattering inform that the average crystallinity index of PP matrix is 45 vol. %. The long period of the semi-crystalline PP stacks, as determined by small-angle X-ray scattering (SAXS), is equal to 19.1 nm that gives, in conjunction with the volume index of crystallinity, an average lamella thickness of 8.6 nm.

Measurements

Mechanical testing

The video-controlled mechanical testing method used in this paper, VidéoTraction  , gives access to the true mechanical behaviour of polymers for different deformation paths from video measurements performed locally in the middle of the sample. As a result, the development of necking is taken into account in the measured mechanical behaviour. To ensure the localization of the plastic deformation (necking) at the centre of the sample, a slight geometrical defect is machined out in the middle of the gauge length of the dumbbellshaped specimen (reduction of 6 % of the specimen width). This specific analysed region is called Representative Volume Element (RVE). The axial true strain in the RVE, ε 33 , is obtained by a polynomial interpolation of partial strains measured from the displacement of axial markers using Lagrange Transform and following Hencky's definition. For uniaxial tensile testing, the transverse strains in the RVE, ε 11 and ε 22 , are equal if the strain field is transversally isotropic in the centre of the neck [START_REF] Bucknall | Rubber-toughening of plastics, part 1: Creep mechanisms in HIPS[END_REF][START_REF] Naqui | Review tensile dilatometric studies of deformation in polymeric materials and their composites[END_REF]Pukansky et al., 1994;[START_REF] G'sell | Experimental characterization of deformation damage in solid polymers under tension and its interrelation with necking[END_REF]Addiego et al., 2006;[START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF]. Lastly, the volume strain in the RVE, ε v , is simply computed from the trace of the true strain tensor:

ε v = ε 11 + ε 22 + ε 33 (1)
The volume strain can be positive, null or negative. In particular, when it is superior to zero we call it volume damage (since the "reversible" elastic part keeps weak and can be neglected), and in the opposite case (when it is negative) we consider that a compaction mechanism is active.

The axial true stress (Cauchy stress) is determined in the same RVE as the load per unit actual cross-section:

) 2ε exp( S F σ 11 0 33   , (2)
where S 0 is the initial cross-section of the tensile specimen.

More details about the used methodology can be found in previous papers [START_REF] G'sell | Video-controlled tensile testing of polymers and metals beyond the necking point[END_REF][START_REF] G'sell | Experimental characterization of deformation damage in solid polymers under tension and its interrelation with necking[END_REF]Addiego et al., 2006;[START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF], Ponçot et al., 2011). Note that VidéoTraction methodology enables to perform tensile testing at constant axial strain rate.

All the mechanical tests are performed at 30 °C in case of PP/EPR in order to study the influence of the true strain rate (10 -2 s -1 , 5.10 -3 s -1 and 5.10 -4 s -1 ). For HDPE, tests are done at three different temperatures (25 °C, 40 °C and 45 °C) at a constant strain rate (10 -3 s -1 ). For PET, the selected temperature is 50 °C to stretch samples in the glassy state at three different strain rates (5.10 -3 s -1 , 5.10 -4 s -1 and 5.10 -5 s -1 ). The deformation rates were chosen in such a way to ensure a good regulation of the testing machine actuator and to avoid any self-heating of the material during the mechanical testing.

Scanning electron microscopy

Scanning electron microscopy (SEM) is used to observe the microstructure of the materials, and hence, to identify the strain-induced deformation micro-mechanisms. To this end, Post Mortem micrographies are recorded using an environmental Quanta FEG 600 electron microscope from FEI Company. The advantage of this equipment is its ability to work under a wide range of saturated vapour pressures. Accordingly, no more sample metallization is necessary to avoid charging effect at the surface of insulating material. The selected microscope parameters are: water vapour pressure of 100 Pa and acceleration voltage of 5 kV. Most of the images were recorded with the secondary electrons detection mode (large field detector). Prior observations, all the samples were submitted at their core to a chemical etching in order to reveal their microstructure. This treatment consists in eliminating heterogeneous layer induced by the cutting procedure and all other artefacts [START_REF] Olley | An improved permenganic etchant for polyolefines[END_REF]. The chosen solution is a mixture of sulphuric acid, orthophosphoric acid and potassium permanganate powder. Polymer samples were soaked during 5 hours using a magnetic stirrer. Then, they were rinsed successively in hydrogen peroxide, distilled water and pure ethanol.

Optical microscopy

In transmission mode, this technique is used to observe the volume damage in case of a translucent material such as amorphous PET. Post mortem observations are conducted after recovery of the stretched sample and are performed on 20 µm-thick slices cut using a microtome. These samples are taken of the center of the neck where the RVE is localized.

Experimental results:

True mechanical behaviours of PP/EPR and PET as a function of true strain rates

This study is conducted on the two polymers PP/EPR and PET. Figure 1 shows the true mechanical curves of these two materials obtained by the VideoTraction ™ system in tension for three different true strain rates. For both materials, an increase of stress is observed up to moderate strain levels with increasing the stretching rate. The yield point is higher in the case of PET than in the case of PP/EPR whatever the studied strain rate. After the yield point, different amplitudes of softening appear between these two polymers, since PP/EPR is semicrystalline and stretched at a temperature higher than its glass temperatures (T g PP = 10 °C and T g EPR = -45 °C) and PET is amorphous and stretched in its glassy state (T g = 75 °C). Then, plasticity stage is characterized by a quite large plateau. Finally, strain-hardening takes place up to large strain level. However, during the strain-hardening stage, the stress-strain curves are not parallel anymore for the two materials, as indicated in figure 1. This phenomenon does not enable to predict the tensile behaviour at large strain level using phenomenological constitutive equations as the one proposed by G'sell and Jonas (1979).

True mechanical behaviours of HDPE as a function of temperature

HDPE specimens are uniaxially stretched at three different temperatures (25°C, 40°C and 45°C) in the rubber state since its glass temperature is of about -120°C. Results are presented in figure 2. It is to be noted that the tensile curves of HDPE are similar to that of PP/EPR and includes a weak softening stage. As expected, stress increases with decreasing temperature up to moderate strain level. However, as noted for PET and PP/EPR, for high strain level, true stress-strain curves obtained for the different temperatures cross each other, which does not permit their simulation.

Damage mechanisms: volume strain and voids shape

By means of a 2D video extensometer, the volume strain is evaluated in situ at the macroscopic scale using equation 1. Figures 3 and4 present the curves collected for PP/EPR, PET and HDPE. The influence of strain rate and temperature is quite similar on the development of volume damage for the three different materials (figures 3 and 4). In particular, the volume deformation increases with strain rate and decreases with temperature. In figure 3, for PP/EPR and PET, curves present the same evolution with the axial true strain whatever the strain rate. This means that similar voids in terms of shape develop inside each material. Polymers stretched in the glassy state, such as the amorphous PET, show the highest amount of voids. At very large strain values, we can note differences between the volume strain behaviours of PP/EPR and PET. In the first case (figure 3 a)), we observe a plateau followed by a slight decrease. For PET (figure 3 b)), linearity keeps constant until failure of the tensile specimen.

In the case of HDPE (figure 4), the volume strain is lower than that of the two previous materials whatever the strain level. This is due to the fact that HDPE is stretched in a highly viscoelastic state (T gHDPE = -120 °C). As previously exposed by Addiego et al. (2006), a slight compaction is observed during the first step of plasticity at the highest temperature (45 °C). This unusual phenomenon of compaction is mainly caused by the strain-induced chain orientation of amorphous phase that is facilitated with increasing temperature, while cavitation mechanisms are hindered (Addiego et al., 2006).

To identify the morphology of the cavities developing within the materials during the plastic deformation by stretching, post mortem microscopic observations were made after the viscoelastic recovery. Figure 5 shows the RVE microstructure of PP/EPR after stretching procedure at 30 °C and 5.10 -3 s -1 up to the axial true strains of 0.3, 0.5 and 1.4. These SEM micrographies exhibit quite large cavities induced by the tensile test. It appears that volume damage mostly takes place inside the EPR nodules due to their core-shell microstructure [START_REF] Mae | Microstructural observation and simulation of micro damage evolution of ternary polypropylene blend with ethylene-propylene-rubber (EPR) and talc[END_REF][START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF]. Indeed, stresses concentration localized around the particles are transferred to the extremities of their internal rigid part made of independent PE crystalline lamellae. Voids show an evolution of their shape from a spherical state (micrographies a) and b) of figure 5) to an ellipsoidal state similar to cigars (micrography c) of the figure 5) because it was previously shown that the EPR nodules align along the stretching direction (van der [START_REF] Van Der Wal | Polypropylene-rubber blends: 5. Deformation mechanism during fracture[END_REF]. Close to the sample breaking, the PP/EPR microstructure is highly fibrillar.

In the case of glassy PET, an important dilatation is recorded (Figure 3 b)) and is correlated with the development of crazes as shown in figure 6 and previously enlightened by the past [START_REF] Vigny | Constitutive viscoplastic behavior of amorphous PET during plane-strain tension stretching[END_REF][START_REF] G'sell | Experimental characterization of deformation damage in solid polymers under tension and its interrelation with necking[END_REF]. The nucleation of this kind of damage begins once the yield stress is reached. Figure 6 presents two optical micrographies of the RVE of PET specimens stretched at 50 °C and 5.10 -5 s -1 up to the axial true strain values of 0.45 and 1.2. Crazes oriented perpendicular to the stretching direction are observed. At 45 degrees of the mechanical solicitation direction, we can notice the presence of shear bands which link the extremities of crazes. Note that crazes formed during the test do not show a significant opening until the rupture of the sample. A gradual multiplication and coalescence of the crazes occur until the sample breaking (Ponçot et al., 2011). For this material, we consider that voids have a constant shape whatever the strain level. Voids can be modeled by discs (penny-shaped voids).

Figure 7 is a set of SEM micrographies of HDPE specimens stretched at 25 °C and 10 -3 s -1 up to true strain values of 0.29, 0.48 and 1.3. Due to the specific morphology of spherulites encountered in HDPE (twisted lamellae), the process of cavitation is quickly characterized by a very important amount of spherical voids which finally do not much elongate themselves until the rupture of the sample. So, figure 7 reveals mainly bubble-shaped voids [START_REF] Addiego | Caractérisation de la variation volumique du polyéthylène au cours de la déformation plastique en traction et en fluage[END_REF]Addiego et al., 2006 and2011).

From these numerous micrographies reported in the figures 5, 6 and 7, it is possible to express additional comments on the evolution of the slope of the curves  v = f ( 33 ) of figures 3 and 4. In the case of HDPE (once the compaction phenomenon is over) and PET, the quasi constant and positive slope value means that cavities shape does not significantly evolve during the deformation. Voids keep a spherical shape in the case of HDPE and a penny shape in case of PET whatever the temperature and the strain rate, respectively. On the contrary, in case of PP/EPR, a decrease of this slope is observed for high strain levels. Elongated cavities continue to grow with the deformation but essentially in one direction: the stretching direction. As a consequence, this phenomenon induces a slight closure of the width of the cigars. It is attributed to a highly fibrillar microstructure of the stretched polymer.

Discussion:

Our results show the influence of the true strain rate and temperature on the true mechanical behaviour and volume strain of PP/EPR, PET and HDPE. Many authors reported the strong competition between two main micro-mechanisms of plastic deformation which are the orientation of the macromolecular chains and the volume damage in the case of amorphous and semi-crystalline polymers stretched in tension [START_REF] Ward | Mechanical Properties of Solid Polymers[END_REF][START_REF] Samuels | Plastic Deformation of Polymers[END_REF][START_REF] Samuels | Structured Polymer Properties: The Identification, Interpretation, and Application of Crystalline Polymer Structure[END_REF][START_REF] Plummer | Deformation and entanglement in semi-crystalline polymers[END_REF][START_REF] Dahoun | Etude des mécanismes de déformation et d'endommagement plastique des polymères solides[END_REF][START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF], Ponçot et al., 2011;Martin et al., 2011). For the three studied materials of this paper, true mechanical behaviours curves are well stackable (using a constant shift factor to the stress values) until the plastic plateau. Then, crosses between curves appear which means that one micro-mechanism of plastic deformation prevails over the other one depending on the mechanical test parameters (strain rate and/or temperature). Raw data collected by the VideoTraction ™ system (figures 1 and 2) does not take into account the microstructure evolution induced by the mechanical solicitation. But, the simultaneous observation of the true stressstrain curve and volume straintrue strain curves leads to discuss about the influence of the mechanical test parameters on the involved micro-mechanisms of deformation. Intersections between the curves are due to an increase of the true stress values in the case of the lowest true strain rate and highest test temperature (Figures 1 and2) at the onset of strain-hardening. Regarding figures 3 and 4, this hardening stage is induced by an important orientation of the macromolecular chains along the tension direction when the amount of volume damage is the lowest. Consequently, curves of figures 1 and 2 inform us about the true mechanical behaviour of a new material which can be assimilated to a composite: a polymer matrix "filled" with voids. In order to exclusively characterize the mechanical properties of the polymer matrix alone, it is determinant to take into account the volume deformation in the calculation of the true strain and the true stress variables since specimen section changes during the test. Therefore, the determination of the true intrinsic mechanical behaviours presents a great interest and constitutes the rest of this paper. The influence of the stress triaxiality effect (mechanical concept of necked specimens introduced by Bridgman in 1952 and Beremin in 1980 on metals) is neglected in this work since it is maximal at small strain values close to the yield point and the structural softening of the material, while the volume strain is weak at the same time. A few measurements of this triaxiality phenomenon were obtained on the three studied materials on the basis of the Bridgman hypothesizes. They have shown weak deviations between the true stress and the true effective stress of about 5 %. In this work, we focus our attention on the correction of the curves at large strains once necking is activated and where the volume strain is high. An extension to large strain and damage based model was discussed in a very recent paper of [START_REF] Boisot | Experimental investigations and modeling of volume change induced by void growth in polyamide 11[END_REF], where comparison of the experimental results with finite element computations was carried out. It was shown that the triaxiality ratio gradually decreases from the yield point (onset of necking) to strain-hardening stage (end of necking).

Equations of the true intrinsic stress and strain

As the damage always occurs simultaneously with elastic and plastic strains, their classical variables must be taken into consideration. The strain tensor can be defined as follows:

] ε [ ] ε [ ] ε [ pl el   (3)
where [ el ] and [ pl ] denote the elastic and the plastic strain tensors, respectively.

Each part of this relationship shows a contribution of the volume variation which is important to quantify (Lemaître, 1996):

] ε [ ] ε [ ] ε [ ] ε [ ] ε [ v pl i pl v el i el     (4)
where [ v el ] and [ v pl ] denote the elastic and the plastic volume strain tensors, respectively.

Depending on the situation, a constant volume deformation is called "isochoric deformation", "conservative deformation", "deviatoric deformation" or "intrinsic deformation". The remainder is named "volume deformation" and it can be either isotropic or anisotropic. It is useful to write:

  0 ε ε ε ] ε [ Tr d el 33 d el 22 d el 11 i el     (5)   0 ε ε ε ε ] ε [ Tr v el v el 33 v el 22 v el 11 v el      (6) and   0 ε ε ε ] ε [ Tr d pl 33 d pl 22 d pl 11 i pl     (7)   0 ε ε ε ε ] ε [ Tr v pl v pl 33 v pl 22 v pl 11 v pl      (8)
Elastic volume strain:

Volume deformation in the elastic zone is due to the material compressibility (dilatation and contraction of the macromolecular chain bonds). There is not cavity created during this first stage of deformation. In tension, the elastic volume deformation is expressed by:

K 3 σ K σ ε 33 h el v   ( 9
)
where K is the consistency of the material and  h the hydrostatic stress.

The initial part of the curve  v - 33 is linked to the Poisson's coefficient, , by:

el 33 el v ε ) ν 2 1 ( ε   (10)
Finally, a spherical tensor is obtained since the hydrostatic stress is isotropic and our polymer materials all present a Poisson ratio close to 0.35:

              E /3 σ ν) 2 (1 0 0 0 E /3 σ ν) 2 (1 0 0 0 E /3 σ ν) 2 (1 ] ε [ 33 33 33 el v (11)
where E is the Young's modulus of the material.

Plastic volume strain:

It is considered that the plastically deformed and damaged material is a composite system characterized by two distinct phases: matrix and cavities (figure 8). Then, true stress and strain can be expressed as follows in the particular case of this composite (Lemaître, 1996)

: ] σ [ . f ] σ [ . ) f (1 ] σ [ v v i v    (12) ] ε [ . f ] ε [ . ) f (1 ] ε [ v v i v    (13)
where f v is the cavity fraction created inside the material,  i and  i are the intrinsic true stress and strain (of the polymer matrix without voids), respectively.

The evolution of  33 i as a function of  33 i represents the true intrinsic mechanical behaviour of the solid polymer without cavities. Their expressions, based on the volume deformation and the voids shape, are presented hereafter.

Introduction of a -factor taking into account the cavities shape developing during plastic strain.

It is relevant to introduce a shape factor, , linked to the cavity geometry present inside the polymer matrix based on the relation:

v 33 i 33 γε ε ε   where 1 2β 1 γ   and  = C 3 /C 1 .
C i =1, 2, 3 represents the geometrical dimensions of a cavity in the three directions of space taking the axis 3 as the tensile direction. Since during a tension test, the hypothesis of transversal isotropy is always verified, the equality of the parameters C 1 and C 2 characterising the cavities dimensions in the two transversal directions is systematically verified.

Three main cavities morphologies can be considered as illustrated in figure 9: -Crazes which are developed perpendicularly to the stretching direction (0), -Spherical bubbles (1), -Tubes elongated in the tension test direction ().

In the case of the formation of crazes, the specimen was only stretched in the tension direction without transversal contraction. The volume deformation tensor presents an only non-null component such as:

v v 33 v 22 v 11 ε ε ; 0 ε ε   
. The intrinsic true strain is hence equal to:

v 33 i 33 ε ε ε  
. The section of the specimen which endures the load remains unchanged and the intrinsic true stress is expressed by: 33 i 33 σ σ  .

In case of spherical cavities, the deformation is isotropic and volume strain tensor is composed with: 3

ε ε ε ε v v 33 v 22 v 11    . The intrinsic true strain is 3 ε ε ε v 33 i 33  
. The axial component of the intrinsic true stress is written:

) ε 3 2 ( exp σ S S σ S F σ v 33 i 33 i i 33   
where S and i S are the areas of the unstretched and stretched material section, respectively, and F is the applied load.

The last remarkable shape of cavities is tube-like shape. These elongated voids in the tension direction show a large length C 3 in comparison with their lateral dimensions. It is considered that they do not endure transversal contraction. In this case components of the volume deformation tensor can be written:

0 ε ; 2 ε ε ε v 33 v v 22 v 11   
. The intrinsic true strain in the axial direction is:

33 i 33 ε ε 
. The expression of the intrinsic true stress is given by:

) (ε exp σ S S σ S F σ v 33 i 33 i i 33    .
These relations show that depending on the value of the volume damage, the intrinsic behaviour of the material is more or less different from the overall one. In the case of a positive volume deformation, the polymer has an intrinsic true strain inferior to the global strain and an intrinsic true stress superior to the macroscopic stress.

Figure 10 a) illustrates the influences of the different remarkable values of the  factor on the final intrinsic true mechanical behaviour of the HDPE stretched at three different temperatures. The introduction of the cavities shape in the correction of the true stress and strain should bring a higher accuracy in the assessment of intrinsic behaviour of the material.

It appears that whatever the chosen  factor, cavities shape does not significantly influence the mechanical behaviour until the beginning of the plasticity. It is mainly due to the low values of the volume deformation until this stage (figure 4 a)). So, the first stage of the cavitation process (crazing), when it occurs, can always be neglected. Then, all the curves converge again at the highest true strain values what means that the progressive elongation of the voids in the tensile axis does not affect a lot the final intrinsic mechanical behaviour of HDPE. The most important effect of the cavities shape on the true mechanical behaviour occurs during the plastic plateau and the initiation of the strain-hardening (0.35 <  33 < 1.2).

In the previous work of [START_REF] Addiego | Caractérisation de la variation volumique du polyéthylène au cours de la déformation plastique en traction et en fluage[END_REF] which deals with tension tests of HDPE, a relation of the beta factor as a function of the volume strain was given (figure 10 b)):  = exp(4 v -1). We notice that the voids shape evolves with stretching as previously discussed since beta value changes from 0.4 to 1.4. This result indicates a weak elongation of the voids along tensile direction during the plastic deformation and confirms the fact that taking a constant value of the beta factor to determine the intrinsic mechanical quantities is an acceptable hypothesis.

From these last results dealing with the specific influence of the evolution of the beta factor with stretching in the case of HDPE, we now consider a constant value of the beta factor during the stretching procedure for our three polymer matrices.

Determination of the true intrinsic mechanical behaviours

The microscopic observations of damage after stretching of PP/EPR, PET (Figures 5 and6) and HDPE (Figure 7) enable the choice of the most suitable  factor in the determination of the intrinsic true behaviour. In the case of PP/EPR, a  factor tending toward infinity is used due to the presence of cigar-like voids. For HDPE, a  factor of 1 is retained because of the development of bubble-like voids. For PET, the chosen  factor is 0 due to the presence of crazes having a penny shape. Using the equations described previously, the intrinsic true mechanical behaviours of each polymer are presented in figure 11. These corrections eliminate the anomalies observed in the case of the overall true behaviours directly obtained by means of the VideoTraction ™ system. The curves calculated either at three different true strain rates (PP/EPR and PET) or three different temperatures (HDPE) are finally really stackable. The use of the phenomenological law of G'sell and Jonas (1979) is now suitable for simulation and prediction of the mechanical properties of these thermoplastics because no more curves intersection exists [START_REF] Ponçot | Thermomecanical behaviour of metal/polymer composites along different deformation paths: Determination of intrinsic mechanical behaviour laws on polymer film[END_REF][START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF].

Comparison of the equivalent true intrinsic mechanical behaviours between various deformation paths using the Von Mises plastic criterion:

In order to compare the mechanical behaviours of the materials in uniaxial tension, plane tension and simple shear, we applied the plastic criterion of Von Mises taking the conventional uniaxial tensile test as reference. The deduced expressions of the true intrinsic stress and strain are:  For plane tensile test: Figure 12 presents the differences between the true mechanical behaviours (a) and the equivalent true intrinsic mechanical behaviours (b) in the case of PP/EPR material deformed based on these three deformation paths. The integration of the volume strain taking into account the cavities shape gives results which are in good agreement with the previous simulations [START_REF] Hervé | n-Layered inclusion-based micromechanical modelling[END_REF][START_REF] Chaboche | Formalisme général des lois de comportement : applications aux métaux et polymères. Introduction à la mécanique des polymères[END_REF] and experiments brought by G'sell et al. (1994,1997,1999) in the case of HDPE. The behaviours in uniaxial tension and simple shear are the two extreme ones in terms of stress since all the other simple or complex mechanical paths behaviours are encountered between pure tension and pure shear [START_REF] Dahoun | Comportement plastique et textures de déformation des polymères semicristallins en traction uniaxiale et en cisaillement simple[END_REF][START_REF] Sell | Microstructure transformation and stress-strain behavior of isotactic polypropylene under large plastic deformation[END_REF]1999). This means that damage is maximum in tension and minimum in shear. The behaviour in plane tension is below the one of tension since the hydrostatic pressure remains weak for the kind of sample geometry used to perform the tests. We notice that the curves are similar until the beginning of the strain-hardening. It informs us that the micromechanisms of deformation involved at this early stage of deformation are comparable whatever the mechanical path. It confirms the fact that the real influence of the volume damage on the mechanical behaviour happens during the plastic deformation of the material. Before, deformation of polymer is essentially governed by interlamellar and interchains shearing inducing creation of shear bands [START_REF] Baltá-Calleja | Plastic deformation of polypropylene: VI : Mechanisms and properties[END_REF][START_REF] Kaufman | Lamellar and interlamellar structure in melt-crystallized polyethylene[END_REF] localized in the amorphous phase. Similar results in the case of PET and HDPE were provided (G'sell et al., 1999;[START_REF] Addiego | Caractérisation de la variation volumique du polyéthylène au cours de la déformation plastique en traction et en fluage[END_REF]Ponçot et al., 2011).

Conclusion:

In this paper, we develop a methodology which enables the determination of the true intrinsic mechanical behaviour of thermoplastic polymers from the overall behaviour, volume strain and voids shape factor. To this end, we selected two semi-crystalline polymers (polypropylene/ethylene-propylene rubber, PP/EPR and high-density polyethylene, HDPE) and one amorphous polymer (polyethylene terephtalate, PET). The VidéoTraction ™ system was used in tension to determine the overall behaviour of the materials, i.e. the true stresstrue strain curves and the volume straintrue strain curves. Attention was also focused on the impact of strain rate and temperature on the tensile behaviour of the materials. The void shape of the materials was examined at the post mortem state by scanning electron microscopy and optical microscopy. At high strain rate, volume damage is the most important microstructural modification since polymer chains do not have enough time to accommodate to the macroscopic strain by an orientation process. The higher the temperature (compared to the glass temperature of the material), the more important is the orientation of the macromolecular chains along the stretching direction. Actually, this is due to an increase of chain mobility since the material is in its viscoelastic state. In these last conditions, volume damage is relatively weak. Consequently, there is a strong competition between orientation of the macromolecular chains and damage characterized by the creation of voids inside the polymer. It results that raw tensile curves recorded by means of the optical system are not stackable for large true strain values when strain-hardening occurs. It means that the modification of one test parameter does not affect the true mechanical behaviour in the same manner from the elastic to the plastic regions. Intersections between true stressstrain curves are observed and make impossible the simulation of these curves using phenomenological constitutive equations. Microstructure observations reveal that cavities do not develop in the same manner for all the investigated materials. PP/EPR presents cigars or tubes oriented in the stretching direction, HDPE shows bubbles characteristic of an isotropic volume damage and PET is damaged by crazes perpendicular to the tension direction. Thus, based on our observations, a form factor called  was introduced in the determination of the true intrinsic mechanical behaviours of polymers . The calculated intrinsic tensile curves exhibit no irregularity (stackable curves), and hence, can be easily fitted by a phenomenological model such as the G'sell and Jonas law [START_REF] G'sell | Determination of the plastic behavior of solid polymers at constant true strain rate[END_REF][START_REF] Ponçot | Comportements thermomécaniques de polymères chargés selon différents chemins de déformation et traitements thermiques[END_REF]. The determination of the true intrinsic mechanical behaviours enables the comparison of different deformation paths based on the plastic criterion of Von Mises used to determine the equivalent behaviours. Thus, we calculated the equivalent true intrinsic simple shear and plane tension behaviours based on the true intrinsic tensile behaviour. These three behaviours are similar prior up to yield point. Similarities of the micromechanisms of deformation involved during solicitation of amorphous and semi-crystalline polymers before plasticity of the material are enlightened. Figure 5: SEM micrographies of the RVE of PP/EPR tensile specimens uniaxially stretched at 30 °C and  = 5.10 -3 s -1 (the stretching direction is vertical).

Figure captions:

Figure 6: Optical micrographies taken on 20 µm-thick slices of the RVE of PET specimens uniaxially stretched at 50°C and 5.10 -5 s -1 (the stretching direction is vertical).

Figure 7: SEM micrographies of the RVE of HDPE probes uniaxially stretched at 25 °C and 10 -3 s -1 (the stretching direction is vertical) (Addiego et al., 2006). .

Figure 1 :

 1 Figure 1: True tensile behaviors as a function of the true strain rate in the case of a) PP/EPR and b) PET (tensile anomalies are indicated by an ellipsoid).

Figure 2 :

 2 Figure 2: True tensile behaviors as a function of temperature in the case of HDPE (tensile anomalies are indicated by an ellipsoid).

Figure 3 :

 3 Figure 3: Evolution of the volume strain as a function of the true strain rate in case of a) PP/EPR and b) PET.

Figure 4 :

 4 Figure 4: Evolution of the volume strain in function of the temperature in case of stretched HDPE.

Figure 8 :

 8 Figure 8: Damaged polymers are considered as a two-phase system.

Figure 9 :

 9 Figure 9: Remarkable values of the  factor linked with current cavities shapes observed in uniaxially stretched polymers.

Figure 10

 10 Figure 10: a) Impact of the three different remarkable values of the  factor on the final intrinsic true mechanical behavior of HDPE and, b) experimental evolution of the beta factor as a function of the volume strain.

Figure 11 :

 11 Figure 11: True intrinsic mechanical behaviors in uniaxial tension of a) PP/EPR, b) PET and c) HDPE taking into account volume strain and voids shape ( form factor).

Figure 12

 12 Figure 12: a) Equivalent true mechanical behaviors and b) Equivalent true intrinsic mechanical behaviors of PP/EPR using the plastic criterion of Von Mises. Comparison of three deformation paths: uniaxial tension, plane tension and simple shear.

Acknowledgements:

The previous works of Professor G'sell Christian on polymers plasticity and his precious and judicious ideas are greatly honoured in this paper.


Mouad Bouita

Jean-Philippe Tinnes

Patrice Bourson

Marc Malfois

Marc Ponçot
email: marc.ponçot@univ-lorraine.fr

A NEW RAMAN SPECTROSCOPY BASED METHOD FOR MONITORING THE CRISTALLINITY RATIO OF POLYETHYLENE TEREPHTALATE

Keywords: Polyethylene terephthalate, Differential scanning calorimetry, WAXS, Macromolecular conformations, Crystallinity ratio

Tracking the Raman scattering bands of polyethylene terephthalate (PET) during thermal heating has allowed to establish criteria for monitoring the crystallinity ratio within a few seconds. The existing criteria, based on Raman bands at 998 cm -1 and at 1096 cm -1 , present certain disparities from the crystallinity ratio given by the total heat flow in differential scanning calorimetry (DSC). Based on the evolution of the Raman bands, a newfound method allowing the effective calculation of crystallinity ratio is proposed. For the same, the band at 1727 cm -1 , which is characteristic of the C=O carbonyl bonds of the ester group of PET is used. The new criterion, designated r 1727,2 , was compared and validated by DSC and XRD. This novel approach also provides access to information on the various conformational states of the bonds under study, i.e. their proportions and positions with respect to the plane of the macromolecule aromatic rings. Evidently, this allows to reveal the characteristic thermal behavior of the PET microstructure such as the glass transition, crystallization and melting. This method can be useful for the close detection, quantification, understanding and tracing of the micromechanisms involved at the macromolecular conformations scale during thermal cycles.

INTRODUCTION

Measurement of the crystallinity ratio of semi-crystalline polymers is usually performed using destructive methods such as DSC [1] and densitometry [2] or more sophisticated and nondestructive tests such as X-ray diffraction [START_REF] Masao | X-ray diffraction by polymers[END_REF] and spectroscopic methods such as FTIR, Raman etc. [START_REF] Käppler | [END_REF]5] . Some experiments allow to follow the evolution of the polymer microstructure, during the deformation for example, understanding the involved mechanisms. In particular, tracking the crystallinity ratio during the manipulation requires short acquisition times. This kind of experiment therefore requires high energy light sources provided in synchrotrons which is not suitable for routine studies. Spectroscopic techniques such as Raman spectroscopy, are non-destructive techniques that are capable of giving several structural information at short acquisition times [5] . In the last decades by the use of optical fibers, portable Raman spectrometers were developed. As a result, innovative coupling systems such as DSC/Raman [6] and mechanical test/Raman [7] which allow to acquire several information related to the evolution of the structure of materials collected in real time during thermal and mechanical tests could also be developed [8] . The detailed study of the Raman spectra, provide information which allows the tracking of several mechanisms that are a result of strain and temperature effects [9] .

It should be noted that Raman spectroscopy is very sensitive to the conformational states of PET. These conformations have been linked to the different phases of semi-crystalline polymer [10] . The crystallization of PET is highly dependent on the conformational states of the macromolecule. There are three bonds involved, firstly the C-O bond between the terephthalic group and the ester group. This bond can be present in the 'gauche' or 'trans' conformation. Secondly, the C-C bond of the ethylenic segment, which can also be present in the 'gauche' or 'trans' conformation. And finally, the ester groups of the terephthalic acid unit where both 'trans' and 'cis' conformations are possible. The crystallization of PET can take place only if all conformations are 'trans' [10][11][12] .

Several authors have established criteria for the study of crystallinity, orientation of amorphous and crystalline phases and damage in the case of a range of polymers based only on Raman spectroscopy [START_REF] Martin | Etude par spectroscopie Raman du polypropylène isotactique au cours de sa déformation uniaxiale[END_REF][START_REF] Chaudemanche | [END_REF][15] . These criteria were validated by comparing them to results obtained by other techniques such as X-ray diffraction or infrared spectroscopy. However, the studies have shown that in the case of PET, the different crystallinity criteria established did not report what was really happening in the polymer but gave a qualitative idea of this ratio [15,[START_REF] Donnay | Étude des mécanismes de déformation de membranes polymères poreuses pour applications biomédicales[END_REF] .

The monitoring of low energy modes has been studied [START_REF] Colomban | [END_REF] , and it was possible to follow the crystallinity and separate it from anisotropy using the bands at 60,75,130, and 170 cm -1 for PET. On another hand, the monitoring of the band 1727 cm -1 allowed to associate the change of its width at half height (FWHM) with the density of the material [18,19] , and this gave a primary idea on the crystallinity ratio of the material as this one varies with the density.

Monitoring the FWHM has been the focus of several subsequent studies evaluating important transition temperatures such as the glass transition, cold crystallization and melting at the time of a controlled heating [15,[START_REF] Donnay | Étude des mécanismes de déformation de membranes polymères poreuses pour applications biomédicales[END_REF][START_REF] Hafsia | Identification des micro-mécanismes de déformation du PET amorphe et semi-c ll u u c u 'u m c qu , ph h[END_REF] . Several vibrational bands were found to contain contributions from both the crystalline and amorphous phases, so their deconvolution gave an idea of the amount of the crystalline lamellae in the polymer [START_REF] Hafsia | Identification des micro-mécanismes de déformation du PET amorphe et semi-c ll u u c u 'u m c qu , ph h[END_REF][START_REF] Adar | [END_REF][START_REF] Adar | [END_REF][23][24] . This deconvolution also gives an idea of the conformations present in the material at the time of the measurement. Recently, a criterion for calculating the crystallinity ratio of PET has been established using the bands 1095 cm -1 and 1118 cm -1 (two bands that characterize the crystalline and amorphous phases, respectively) [15,[START_REF] Donnay | Étude des mécanismes de déformation de membranes polymères poreuses pour applications biomédicales[END_REF][START_REF] Hafsia | Identification des micro-mécanismes de déformation du PET amorphe et semi-c ll u u c u 'u m c qu , ph h[END_REF] . It was then noted that there is a significant variation in the crystallinity ratio of the polymer calculated by this criterion compared to the one given by DSC, and an index was proposed to follow more accurately the variation of the crystallinity ratio but this does not give in any case the exact value of the latter.

The subject of this study is to review in the case of PET the different criteria established, understand their limitations and then propose a new criterion to calculate a crystallinity ratio consistent with the results of the DSC and more likely to X-ray diffraction technique. The proposed method will enable us to use Raman spectroscopy instead of the destructive techniques which will shorten the approach time.

MATERIALS AND METHODS

Polyethylene terephthalate was supplied by Goodfellow [START_REF] Polyester | PETP Sheet -Polyethylene terephthalate (ES30-SH-000130[END_REF] and is used as-received without any pre-treatment. It is in the form of a 300x300 mm 2 plate and is 3mm thick. The polymer is in a transparent amorphous initial state, with a density of 1.3g/cm 3 and a specific heat of 1200J/K/kg. Experiments were conducted at the Jean Lamour Institute in Nancy and consisted of thermal tests performed with a TA Q200 calorimeter [START_REF] Höhne | Differential Scanning Calorimetry[END_REF] . The samples are put in capsules without cover. This will allow to make in-situ Raman acquisitions, during a thermal cycle. This can be done by using a coupling system that acts as a lid on the capsule in a way that the spectroscope probe is positioned directly above the sample [START_REF] Donnay | Étude des mécanismes de déformation de membranes polymères poreuses pour applications biomédicales[END_REF][START_REF] Hafsia | Identification des micro-mécanismes de déformation du PET amorphe et semi-c ll u u c u 'u m c qu , ph h[END_REF] . The thermal cycle is p f m w h h g mp f 5K/m g g f m 30 ˚C 280 ˚C. Th R m spectrometer used is an RXN1-785 nm from Kaiser Optical Systems. The laser has a wavelength of 785 nm, a power of 400mW and an optical fiber that allows to delocalize the probe and link it directly to the DSC-Raman coupling system [START_REF] Hafsia | Identification des micro-mécanismes de déformation du PET amorphe et semi-c ll u u c u 'u m c qu , ph h[END_REF] . The acquisition parameters are taken in such a way as to have a minimal time of 30 s for a good signal/noise ratio. And , h ll w h p c um y 2.5 ˚C.

The DSC data are analyzed by the "TA Universal Analysis Software" and the Raman data are processed by a home-made program encoded on the numerical software Matlab®. The spectra were preprocessed by applying a Savitsky-Golay smoothing of window 3 and order 0 [START_REF]An Adaptive Savitsky-Golay Filter for Smoothing Finite Element Computation[END_REF] and then subtracting the baseline using a Wavelet-based algorithm [START_REF] Qian | [END_REF]29] . This will allow the isolation of the different Raman bands for the calculation of the different crystallinity criteria.

It is important to normalize the acquired spectra using the PET normalization band which is the band at 633 cm -1 , a temperature-insensitive band [30] . A fitting procedure that consists of a three sub-bands deconvolution is applied. And because of the asymmetry of the band to be fitted, conditions on the sub-bands made it possible to respect this form: firstly, the condition The modulated DSC (MDSC) was also performed to understand some mechanisms occurring during the heating: a heating ramp was carried out on the same material and under Heat-Iso Temperature Modulation (HITM) conditions [START_REF] Reading | Modulated Temperature Differential Scanning Calorimetry Theoretical and Practical Applications in Polymer Characterisation[END_REF][START_REF] Thomas | [END_REF] . The experimental parameters of the modulated DSC are a heating rate of 5K/min, a modulation period of 60 s and an amplitude of 0.796˚C [START_REF] Thomas | [END_REF] .

Finally, in order to confirm our results, a time-resolved wide angle X-ray scattering (WAXS) experiments (ALBA, Bl-11 NCD Sweet, Barcelona, Spain) coupled to a Linkam calorimeter were performed [START_REF] Saidi | [END_REF] . The monochromatic beam was set at 12.4 keV. The diffraction patterns (Debye-Scherrer chamber) were recorded on a 2-D CDD plate detector (Rayonix LX255-HS) placed at 92 mm behind the sample (permitting to record in transmission mode all the Debye-Scherrer rings/halos). The Linkam device allowed to reproduce the heating ramp of 5K/min.

The patterns were analyzed with Fit2D software, and for each, a 360˚ azimuthal integration was performed to get a texture independent diffractogram. The calculation of the crystallinity ratio was done with a home-made program encoded on the numerical software Matlab®. The used method for the calculation of the crystallinity ratio is given in the supporting information.

RESULTS AND DISCUSSION

The results of the DSC are shown in Figure 1. It highlights the evolution of the heat flow as a function of temperature at a heating rate of 5K/min. The crystallinity ratio calculated by the relation c [1] is also reported, where Hm is obtained after integrating the heat flow and Hm 0 is equal to 140 J/g [35] . In the case of amorphous PET, the crystallinity ratio remains constant and is equal to zero until the cold crystallization temperature (T=122˚C) is reached, and then rises to the value of 25 wt.%. The crystallinity ratio stays constant at this value until the first melting temperature (T=243,5˚C) is attained, thus bringing the degree of crystallinity back to its initial zero value at the totally molten state when the last melting temperature (T m end =250 ˚C).

The processing of the previously acquired spectra is shown in Figure S1 (Supporting information). Some considerable changes are noticed especially at the time of cold crystallization and melting temperatures ranges.

The used zones for the calculation of the different criteria are highlighted using different colors.

The criteria to be studied will be denoted as r 998 , r 1096,1 , r 1096,2 , r 1727,1 and are calculated using the following equations [START_REF] Hafsia | Identification des micro-mécanismes de déformation du PET amorphe et semi-c ll u u c u 'u m c qu , ph h[END_REF]36] :

r 998 (1)
r 1096,1 (2)
r 1096,2 (3)
I(i) designate, here and after, the integral of the band i.

Lastly, the full width at half maximum (FWHM) of the band at 1727 cm -1 allows to establish a criterion of densification and thus of crystallinity r 1727,1 by the following relations [18,19] :

r 1727,1 (4) (5)
Where , and designate respectively the density of PET, its amorphous phase and its crystalline phase. Their values can be found in Ref [39,40] .

Figure 2 shows the evolution of these four criteria applied to our PET during the first heating The band at 998 cm -1 characterizes the 'trans' conformations of the C-C bond of the ethylene segment of PET [10,37] . According to Donnay, these conformations exist mainly in the crystalline phase (94%) [START_REF] Donnay | Étude des mécanismes de déformation de membranes polymères poreuses pour applications biomédicales[END_REF] . We can notice an increase of this criterion then a decrease allowing to follow globally the crystallinity ratio. The criterion does not allow to follow effectively the crystallinity ratio brought by. This is justified by the fact that the band at 998 cm -1 is directly related to the 'trans' conformation of the C-C bond and not to the crystalline phase directly, the proportion of these conformations is then naturally greater than the zero value. This is indeed the case in the molten state as well as in the initial amorphous state. The increase in the criterion, before the cold crystallization temperature is explained by the 'gauche' to 'trans' conformation transition without any significant crystallization process [START_REF] Donnay | Étude des mécanismes de déformation de membranes polymères poreuses pour applications biomédicales[END_REF] .

This transition is only enhanced by the temperature. Additionally, this band has relatively low intensity compared to the overall spectrum. When it decreases in intensity, the measured area depends more and more on the noise of the signal which can be significant.

The second criterion r 1096,1 uses the areas of the two bands at 1096 cm -1 and 1116 cm -1 . The 1096 cm -1 band is characteristic of 'trans' conformations of the C-O bond of the ethylene segment. The band at 1116 cm -1 f u m l z b f h 'gauche' conformations of the same chemical bond [37] . The criterion thus represents the ratio of the 'trans' conformations of the C-O bond and closely follows the evolution of the crystallinity ratio. In the crystallized state the value of the criterion considers the contribution of the 'trans' c f m f h C-O bonds in the amorphous and crystalline phases [15,37] . This is evident from the value of the criterion in the molten and the initially amorphous states. The intensities are exactly the same c f m g qu l g p p 'trans' conformation in the amorphous phase.

The r 1096,2 criterion corrects this by subtracting from the previous one, the contribution of the 'trans' c f m f h m ph u ph wh ch w ll u lly be equal to the area of the vibration band at 1096 cm -1 in the amorphous state (I(1096)a) [15] . However, there remains a non-negligible difference of this criterion with respect to the value of the DSC crystallinity ratio. It should also be noted that DSC measurements may also introduce some uncertainties at the time to choose the right baseline for the peaks area integration which may explain the differences between the proposed criteria and the DSC crystallinity values [1,38] .

Concerning r 1727,1 , the criterion can lead to inaccurate monitoring of crystallinity. Indeed, this criterion for the temperature range 122°C-245°C underestimates the degree of crystallinity while an overestimation out of this range is also observed.

The band at 1727 cm -1 is characteristic of the C=O carbonyl bond of the ester group [18,24] . In p m c y ll z , h b mu b 'trans' c f m h pl f the aromatic ring [12] . The FWHM of the band at 1727 cm -1 was strongly related to the density of the polymer [12,41] . Plotting its derivative in Figure S2 (Supporting information) gives a fairly accurate overview of the DSC thermogram. This makes it possible to identify the glass transition, the cold crystallization and the melting temperatures.

While tracking the 1727 cm -1 band, it should be noted that a narrowing takes place before cold crystallization as shown in Figure S3 (Supporting information). After that, a broadening accompanied by an asymmetry happens at the melting and leads us to the same state as the amorphous initial state. This is explained by a conformational disorder of the out-of-plane rotational states of the carbonyl bonds in addition to the conformations of the ethylenic segment bonds in the amorphous phase [15,[START_REF] Donnay | Étude des mécanismes de déformation de membranes polymères poreuses pour applications biomédicales[END_REF]36] . The decrease in the width of the band is explained then by the resonance stabilization of the terephthatoyl group [36] .

The asymmetry of the band at 1727 cm -1 can be explained by the contribution of several bands. Those bands can highlight the presence and the contribution of the existing conformations [START_REF] Adar | [END_REF][23][24] . Everall et. al [24] have therefore proposed a four-band deconvolution without giving much details about the physical meaning of the deconvolution. Many authors later have proven that the four-band deconvolution cannot be applied to such band [START_REF] Donnay | Étude des mécanismes de déformation de membranes polymères poreuses pour applications biomédicales[END_REF][START_REF] Adar | [END_REF]23] .

This was also not retained in our case because it results in a rather broad residue. Following the example of Adar and al [START_REF] Adar | [END_REF] we apply a three-band deconvolution located at 1717 cm -1 , 1726 cm -1 and 1738 cm -1 as shown in Figure 3. As a first step, it was chosen to use a mathematical function which is the so-called Pearson VII function with the following formula: (6) where M is the form factor of the function and depending on its value, the function can be either Gaussian (M>>8) or Lorentz (M<1), otherwise it can be approximated to a Pseudo-Voigt.

For a first approach, no condition is set on the parameter M.

The evolution of the form factors in Figure 4 allows the mathematical shape of the bands to be determined and are therefore summarized as the following:

1717 cm -1 : Gaussian. 1726 cm -1 : Gaussian. 1738 cm -1 : Pseudo Voigt. The evolution of the proportions of areas of these three sub-bands is reported in Figure 5 It is easily observed that the three proportions vary with the crystallinity ratio. The two bands at 1738 cm -1 and 1726 cm -1 have approximately the same contributions below the cold crystallization temperature and above the melting temperatures. Between these two thermal events, the two bands at 1738 cm -1 and 1717 cm -1 see their contributions decrease in favor of the central band. The criterion r 1727,2 = (6) plotted in Figure 6 allows to track the change in crystallinity ratio given by DSC very accurately. I tot is the sum of areas limited by the three sub-bands. As previously discussed, the crystalline phase is favored by C=O bonds in the plane of the 'trans' aromatic ring. The amorphous phase is described as randomly out-ofplane bonds in an aromatic ring. Band allocation can be suggested as follows:

1726 cm -1 : 'trans' conformations in all planes.

1738 cm -1 : 'trans' conformations outside the aromatic plane.

1717 cm -1 : off 'trans' conformations 'cis' c f m , m ccu ly.

Thus, h c p h p p f 'trans' c f m f h C=O b h

aromatic plane. These suggestions are also supported by the mathematical form of the subbands. Indeed, the sub-band at 1726 cm -1 which characterizes 'trans' conformations in all planes must have a Gaussian character due to the fact that these conformations are scattered throughout space. The sub-band at 1738 cm -1 corresponds to 'trans' conformations outside the plane. This remains in the Gaussian range but tends towards a Lorentz by the suppression of the 'trans' population in the aromatic plane, resulting in a lower M value than at 1726 cm -1 .

It can also be characterized by a Pseudo-Voigt function with a Lo z f c η= 0.2.

The band at 1717 cm -1 also has a Gaussian character and corresponds to 'cis' conformations.

The glass transition can also be highlighted by the slight change in the evolution of the three sub-band proportions around 80˚C, especially for the band at 1727 cm -1 . This is due to the initiation of the macromolecular mobility and the beginning of conformation transitions.

These modifications will be more evident at the cold crystallization and the melting temperatures. However, a significant difference between the criterion and c takes place in the melting zone. This fact can be explained by performing a modulated DSC experiment. The modulated DSC thermograms in Figure S4 (Supporting information), show that the total heat flow can be divided into two independent heat flow called reversing and non-reversing according to the following equation [START_REF] Reading | Modulated Temperature Differential Scanning Calorimetry Theoretical and Practical Applications in Polymer Characterisation[END_REF][START_REF] Thomas | [END_REF][START_REF] Thomas | [END_REF] .

(

) 7
where is the total heat flow equivalent to a standard DSC experiment at the same average heating rate, is usually called the reversing heat flow component and f(T,t) is the kinetic or the non-reversible component of the total heat flow and is calculated from the difference between the overall signal and the heat capacity component.

This result makes it possible to differentiate the different physical phenomena that take place during the heating ramp. On the reversing heat flow, the glass transition takes place at exactly T=79 °C, and melting at a temperature T=247 °C. Following the non-reversing flow, cold crystallization does takes place at T= 122 °C. Two important events also occur: the enthalpic recovery and the crystalline improvement [START_REF] Thomas | [END_REF]42,43] . The latter represents the process of crystals melting at temperatures below the thermodynamic melting point and then crystallizing and melting one or more times with temperature increasing. In the case where crystalline improvement c h c lcul f h c y ll y by DSC (χ c modified), the difference between this crystallinity ratio and the r 1727,2 criterion is greatly reduced as shown in Figure 6. The previous discussion allowed us to understand that the criterion put in place is independent of crystalline perfection. The transitions that occur in this phenomenon does not affect the C=O double bond. The physical explanation of this finding is that during heating, and around the melting point, the melting in the crystalline improvement does not change the states of conformation. the bonds keep the conformations of the crystalline state in their amorphous state. Involving a fast recrystallization which will disappear right after by melting. this phenomenon will be redundant until there is a radical change in the conformations present in the material. The criterion r 1727,2 is therefore sensitive to a cumulative degree of crystallinity. This means that any phenomenon that does not involve changes in the conformation states (e.g the crystalline improvement) will not affect the value of the criterion.

In order to validate the criterion, XRD experiments were performed. After the treatment of the 2D patterns, the evolution of the diffractograms during the heating ramp can be plotted in Figure S5 (Supporting information). Based on those diffractograms, one can get the crystallinity ratio by following the procedure mentioned in the supplementary material. Figure 6 shows also the comparison between r 1727,2 and XRD crystallinity ratio. F gu S6 (Supporting information), the general narrowing of the crystalline peaks can be observed simultaneously to the drop of the crystallinity ratio. These modifications in their shape can be justified by a crystalline lamellae perfecting process. This observation comes to support the fact that the criterion r 1727,2 does not take into account the crystalline perfection improvement as there is a slight difference between the WAXS and Raman crystallinity ratios u l 200˚C.

CONCLUSION

This study provides an accurate overview of the various crystallinity criteria presented in the literature using Raman spectroscopy measurements. For this purpose, a DSC/Raman coupling system has been developed. The comparison between the existing criteria and the crystallinity ratio obtained from DSC showed that the r 998 criterion is related to the 'trans' conformations of the C-C bond in the ethylic segment and not to the crystalline part specifically and thus is not a precise criterion.

The r 1096,2 criterion is more precise and can monitor the crystallinity ratio but has been found to be unsuitable for in situ measurements during a tensile test [START_REF] Hafsia | Identification des micro-mécanismes de déformation du PET amorphe et semi-c ll u u c u 'u m c qu , ph h[END_REF] .

Considering the differences between the established criteria and the crystallinity ratio obtained from DSC, a new approach has been developed to better determine the degree of crystallinity. This approach is based on the contribution of the various morphological configurations present in the material. The new criterion, designated r 1727,2 , was compared and validated by DSC and XRD. I p h m u f 'trans' c f m f h C=O bond in the plane of the aromatic core. It gives access to a cumulative crystallinity rate.

This new method that allows monitoring of the crystallinity ratio, is based on the contribution of the different conformational configurations present in the material. This gives an idea on the effects of rotation of the macromolecular chains with respect to the plane of the aromatic core, the possible stabilization of the bonds and finally, more generally, the effect of the thermal transitions such as crystallization or fusion on the state of the conformations in general.

The suggested criterion, as it stands, may have the potential advantage of being independent of macromolecular orientation. And therefore, can accurately monitor the crystallinity during a tensile test.

Therefore, we can conclude that Raman spectroscopy can provide accurate information to identify and track in situ micro mechanisms that occur with temperature changes and why not when an external load is applied.

  on position [x(1)< x(2)< x(3)] ensures avoiding band over-runs. A second condition written as [A(1)< A(3)< A(2)] allows to keep the asymmetry of the primary band. A(i) is the intensity of the sub-band i and x(i) is its position. The mathematical form of the sub-bands is given and discussed further in the discussion part.

Figure 1 :

 1 Figure 1: Evolutions of the heat flow and the crystallinity ratio for the initially amorphous PET during a heating ramp at 5K/min taking Hm 0 is equal to 140J/g.

Figure 2 :

 2 Figure 2: Comparison between the crystallinity ratio and the crystallinity criteria (a) r 998 -(b) r 1096,1 , r 1096,2 , r 1727,1 for the initially amorphous polyethylene terephthalate during a heating ramp at 5K/min.

Figure 3 :

 3 Figure3: Deconvolution of the 1727 cm -1 band to three sub-bands at 1717 cm -1 , 1726 cm -1 and 1738 cm -1 in the amorphous and semi-crystallized state of PET at (a)-T=30˚C, (b)-T=150 ˚C.

Figure 4 :

 4 Figure 4: Evolution of the factor form M of the three sub-bands during a heating ramp at 5K/min for an initially amorphous PET.

Figure 5 :

 5 Figure 5: Evolution of the proportions of areas of the three bands at 1717 cm -1 ,1726 cm -1 and 1738 cm -1 .

Figure 6 :

 6 Figure 6: Comparison between the XRD, DSC and modified-DSC crystallinity ratios and the r 1727,2 criterion during a heating ramp at 5K/min for an initially amorphous PET.

ACKNOWLEDGEMENTS:

The authors wish to thank the LMOPS laboratory for their technical assistance and invaluable help during experiments. They also acknowledge the ALBA synchrotron for giving the opportunity to perform in situ XRD experiments.