Advanced Controls and Measurements for the Injection Molding of a Short Fiber Reinforced Polymer

G. Saint-Martin (a,*) , F. Schmidt (b) , P. Devos (c) , C. Levaillant (b) (a) Technofan, 10 Place Marcel Dassault, B.P. 53 -ZAC Grand Noble, 31702 Blagnac Cedex, France (b) Ecole des Mines d'Albi-Carmaux, Campus Jarlard, route de Teillet, 81013 Albi Cedex 9, France (c) DRIRE Midi Pyrénées, 12 rue Michel Labrousse, B.P. 1345, 31107 Toulouse Cedex 1, France (*) Corresponding author

Abstract

This study is a collaboration between Technofan, Microturbo, Liebherr Aerospace, CROMeP and LGMT. It is a four years project focused on the optimization of the injection molding process, applied to short fiber reinforced polymers, in order to work on :

-The mechanical properties improvement of the molded parts ; -The reduction and the reinforcement of parts structural defects ; -The prediction of mechanical properties. The aeronautic sector generalizes its policy of aluminum's substitution by reinforced polymers. The main advantages of these materials are a low cost and a low mass. Our study focuses on the Ultem  2300-1000 polyetherimide resin from General Electric Plastics. The injection molding process allows to realize very complex shapes in one step, but can also generates some important defects. The main defects studied in this paper are weldlines and voids. First, we present a detailed characterization of these defects which includes weldlines, meldlines and voids in thick zones. Then, we present the results of some experiment series realized in order to clarify the dependence between the defects and the injection molding process. A reinforcement of the meldlines has been obtained by modifying the symmetry of the mold cavity. This effect is simulated with Moldflow  software, and a quantitative comparison is achieved between experimental and numerical fiber orientation. Then, we apply different non-destructive control methods (like ultrasonic controls or density measurements) in order to retain the most relevant ones for the detection of voids and for their application in the aeronautic industry. Finally, we present an original process control method that can help to detect if there is a risk of void for thick parts at the very first stage of the injection molding production process.

1-Introduction

A policy of weight reduction of airplanes has been followed for the last ten years. Equipment designers have been trying to substitute conventional metallic alloys (steel, titanium, aluminum) by injection molded high properties polymers, for semi-structural airplanes parts. Consequently, there is currently a considerable interest in short fiber reinforced thermoplastic composites. These materials are easily processed and they also offer advantages of low cost and mass or special characteristics (chemical and fire resistance), very useful in the aeronautic field. The injection molding process allows to create very complex shapes without post machining and can even be justified with relatively short series (typically lower than 1000 units per year). Unfortunately, studies over many years [START_REF] Papathanasiou | Flow Induced Alignment in Composite Materials[END_REF][START_REF] Carlsson | Thermoplastic Composite Materials[END_REF][START_REF] Altan | A Review of Fiber-Reinforced Injection Molding : Flow Kinematics and Particle Orientation[END_REF] have shown quite complicated fiber orientations which depend on a lot of parameters and on the cavity geometry. Even if there are good previsions of fibers orientation in simple shapes, almost all-industrial cases are more complicated because of product shapes and mechanical requirements. This is a problem for the part designing phase which need accurate information about mechanical characteristics and fibers orientations. The injection molding process also generates very critical defects such as weldlines, meldlines and voids.

Firstly, we briefly present our injection molding machine layout and the molds we use. These different mold cavities have been designed to study the defects and to realize transpositions to industrial applications.

Meldlines are created when at least two flow fronts join themselves with parallel trajectories (4). It is a common phenomenon which appears during the mold filling stage. The aeronautic industry uses very complex parts (fan wheels, butterfly valves, etc.) which often include a lot of meldlines. This is an important problem acting on the mechanical behavior of the products. Our work consists in a characterization of meldlines morphologies. We study the influence of a deviation volume during the filling stage and on mechanical properties of center-gated ISO tensile test specimens. Then we use Moldflow  software in order to simulate the injection molding cycles. The numerical fiber orientation results will feed a mechanical program calculating the specimen properties. Finally, mechanical and orientation data from experiments and calculations can be compared.

According to literature and to our experiments, voids seem to be initiated during the filling stage. But they are mainly influenced by the packing, holding and cooling stages [START_REF] Akay | Generation of Voids in Fiber Reinforced Thermoplastic Injection Moldings[END_REF]. The substitution of the metal parts with injection molded polymeric parts leads frequently to fill important thickness. This is highly generator of void risk. Therefore, the mechanical behavior can be degraded. As this problem has not been studied a lot, we first realize morphological characterizations of the voids in thick parts. Different non-destructive testing (NDT) methods have been evaluated for this application. Then we use sensors data for the entire injection molding cycle. A void creation empirical model (VC EM ) is then proposed and results are compared to NDT and morphological experimental results.

2-Injection Molding Experiments

The injection molding machine is a DK Codim 5.02/175/600, 175 tons of clamping force, with an oil mold temperature regulation system (up to 230°C), and electric barrel heaters (up to 430°C). A data acquisition system gets signals from the machine (hydraulic pressure, reciprocating-screw parameters, etc.) and from in-mold transducers (temperature and pressure). A modular mold is used in order to realize different part shapes with a minimum cost. Figure 1 shows the injection molding machine and details of the modular mold.

The main mold cavities used for our work are represented in figure 2. All parts are molded by pairs with one instrumented and one for post-molding characterizations. Pair A corresponds to ISO tensile test specimen that can be molded with or without weldline. Pair B corresponds to ISO tensile test specimen that can be molded with different deviation configuration of the meldline. Pair C is a 7mm thick plate that can be molded with or without meldline. Pair D is a derived geometry from an industrial part (fan wheel presented in figure 22) that corresponds to a thick divergent plate including a thin perpendicular pale.

The material used in this study is a 30% short fiber reinforced polyetherimide from General Electric Plastics, commercialized with the label "Ultem  2300-1000". Other grades with different reinforcement levels (from 0% to 40%) are also used for special experiments focused on fibers effect.

Different parameters are studied. The first step consists in processing Taguchi experiment series. They lead to define the parameters that influence the results in terms of defects, mechanical characteristics, fibers orientations, etc. Then, we realize other more classical experiments with refined parameters variation steps.

3-Meldlines Study

The generic term "weldline" includes two types of defects. Weldlines correspond to the junction of two opposite flow fronts. Meldlines correspond to the junction of two non-opposite (frequently almost parallel) flow fronts. Considering the mechanical behavior, weldlines have the worst effect, especially for reinforced polymers. Moreover, it is very difficult to get a significant improvement of the weldline mechanical resistance by acting on process parameters. This point has been largely studied in the literature (4). Meldlines are very interesting because of two aspects. Firstly, considering complex industrial parts, a lot of meldlines are formed during the filling stage (4). Secondly, a modification of the defect formation phase can highly improve the defect morphology, and as well its mechanical behavior.

3.1-Fiber Orientation Measurements

Injection molded ISO specimens (see figure 2-B) are cut in the meldline region. A fine polishing step (1µm roughness disk) until the average plane is processed (see figure 3). The meldline region is then entirely observed with a Scanning Electron Microscope (SEM) at fixed magnification and electronbeam conditions. SEM images are then treated in order to get fibers orientation information represented by the Advani tensor ( 6). An average orientation tensor is finally calculated for each image which are assembled in the part reference (see figure 3).

3.2-Mechanical characterizations

A conventional ISO tensile test is used for the mechanical characterization of meldlines. The equipment is an Instron 4467 of with a 30kN load cell. For each process and mold configuration, ten pairs of specimen are tested. Each pair corresponds to a meldline-included part, and to a no-meldline part (reference). This allows the calculation of meldline factors for each specimen pair. Process parameters we study are : hydraulic holding pressure, mold temperature and polymer temperature (nose temperature). The mold configurations that influence the meldline deviation are presented in table 1 andfigure 4. All specimen gates are cuted before testing in order to have similar section. Considering temperatures (see figure 5), increasing the gap between polymer and mold temperature results in a better specimen rigidity (Young modulus) and in a better tensile strength. The deviation effect also appears in figure 5 between D0w and D3w configurations. The hydraulic holding pressure level has no significant influence on the mechanical properties of the meldline, but has an important effect on the reference specimen (no meldline). On a contrary, the deviation volume configuration has no significant influence on the reference specimen, and a strong effect on the meldline mechanical behavior. These results are represented in figure 6.

Therefore, an important deviation volume rate can lead to the meldline loss (meldline factor reaches 0.98). This result can appear for high holding pressure levels.

3.3-Injection Molding Simulation

We use Moldflow  MPI 2.0 software package for the simulation of the injection molding process. As shown in figure 7, both specimens parts, runners and sprue are simulated. This choice had been done in order to represent as close as possible the experimental conditions for further comparisons.

The mesh is realized using Moldflow  modeler and is refined in the meldline region. The rheological parameters of the Ultem  2300-1000 (GE4106 in Moldflow  database) are set to default values. For our comparisons, we adjust the cycle conditions in order to fit experimental pressure data at points A and B (pressure sensors locations).

The calculated fibers orientation results can be extracted as one Advani tensor for each mesh element. A Krigging interpolation method ( 7) is then applied to both experimental and simulated results in the meldline region. Consequently, we have the capability to quantify the gap existing between experimental and simulated fibers orientations [START_REF] Saint-Martin | Fiber Orientation of Meldlines in Injection Molding of Fiber Reinforced Thermoplastic : A Quantitative Comparison of Experimental and Numerical Results[END_REF].

In a regular case (no meldline), experimental and simulated fibers orientations are in very good agreement. The choice of the meldline region had been done in order to test the limitations of Moldflow  . Figure 9 show the results for D0, D2 and D4 deviation volume configurations. The All configurations present a good qualitative agreement between experimental measurements and calculation results. The average absolute difference is lower than 0.2 on the a 11 value for the majority of the surface. The critical zone is located at the very beginning of the meldline (close to the pawn). The deviation effect that leads to make the a 11 surface higher on the experimental measurements is also reported by the calculations.

4-Voids Study

Voids can appear in injection molded parts when the thickness becomes important. Typically, a thickness greater than 5mm is difficult to mold without void. Geometric particularities like ribs or section changes can also generate some voids. Considering the mechanical resistance of the molded part, this kind of defect is very important. A void leads to a stress concentration and can create some cracks. Voids generation has not been intensively studied in the literature. Anyway, some interesting articles had been published for the last ten years [START_REF] Akay | Generation of Voids in Fiber Reinforced Thermoplastic Injection Moldings[END_REF][START_REF] Beall | Thick Walls Encourage Sink Marks and Voids[END_REF][START_REF] Akay | Generation of Voids in Fiber Reinforced Thermoplastic Injection Moldings[END_REF][START_REF] Jaworski | Using CAE Injection Molding Simulation to Predict Overpack, Voids and Sink[END_REF]. Industrial parts including voids can seriously be damaged during their utilization. When a part region has an important thickness, it is often due to high stresses applied on. Then, if voids exist in this region, a braking can quickly appears. Therefore, the methodology we have is to optimize the process and the part design in order to avoid this type of defect that is not acceptable.

4.1-Non-Destructive Testing and Voids Evaluation

Injection molded thick plates levels (see figure 2-C) are produced using different hydraulic holding pressure. All the other process parameters are kept constant. Figure 10 shows the polymer pressure evolution for different holding levels (holding time remains constant). The pressure sensor is located in the center of the plate.

For voids detection, we have evaluated different non-destructive testing methods (x-ray computertomography, infrared detection, ultrasonic detection, etc.). In our case, the best result in terms of cost and availability has been given by density measurement method (norm ISO 1183). Interesting results has also been given by ultrasonic NDT method [START_REF] Hsu | Ultrasonic Non-Destructive Evaluation of Void Content in CFRP[END_REF] which is very sensitive to density variation. Density measurements give a derived global information about the void content of the part. The ultrasonic C-scan method can provide 2D density cartography across the part [START_REF] Dumont-Fillon | [END_REF]. It is also possible to realize B-scan tests in order to get 3D information. Figure 11 shows the ultrasonic cartography of the thick plate molded with different holding levels. Inspections are processed using a Krautkramer US1P-12 ultrasonic unit linked to a focalized sensor, an Andscan (Defense Research Agency) software, and to an independent electric command. All scans are realized in amplitude variation mode. High density gives high signal amplitude, and as well low density gives low signal amplitude (linear relationship). According to this method, the void content appears to equals zero for hydraulic holding pressure greater than 75 bar. Then, the more the pressure level is low, the more the void content is high. Then we can define an apparent ultrasonic void content parameter VC US from the cartography by quantifying the white surface (the lowest apparent density).

A traditional cutting and observation method has also been processed in order to compare it with the NDT results (ultrasonic control and density measurement). Each plate is cuted along the width (10 samples per plate). Samples are polished and observed using an optical microscope. Voids are then isolated and their total surface is quantified. A pseudo-volume optical void content VC O can then be defined. Figure 12 shows the voids morphology in the plate section for different holding pressure levels. Voids are larger on both sides and are also located in intermediate thickness regions.

The void content parameters VC US and VC O have an almost linear dependence with the holding pressure level under 7.5 MPa (see figure 13). The apparent density ρ app presents a saturation effect for high holding pressure levels.

4.2-Void Creation Empirical Model (VC EM )

Our goal is to define an empirical void creation model which can be used as a quality control tool for industrial productions. According to literature and to our knowledge [START_REF] Akay | Generation of Voids in Fiber Reinforced Thermoplastic Injection Moldings[END_REF][START_REF] Saint-Martin | Fiber Orientation of Meldlines in Injection Molding of Fiber Reinforced Thermoplastic : A Quantitative Comparison of Experimental and Numerical Results[END_REF][START_REF] Beall | Thick Walls Encourage Sink Marks and Voids[END_REF][START_REF] Akay | Generation of Voids in Fiber Reinforced Thermoplastic Injection Moldings[END_REF][START_REF] Jaworski | Using CAE Injection Molding Simulation to Predict Overpack, Voids and Sink[END_REF], all injection molding stages have to be taken into account : filling, packing, holding and cooling.

The thermal regime is defined by the evaluation of the Cameron number :

2 h V aL Ca = (4)
If Ca is lower than 0.01, the regime is adiabatic. If Ca is greater than 1, there is an equilibrium regime. Between theses two values, there is a thermal transition regime [START_REF] Agassant | Polymer Processing[END_REF]. Our case leads to an adiabatic regime (Ca = 0.0028).

Moldflow is used for the evaluation of the temperature profile along the thickness and across the part. Simulation and process parameters are set and adjusted in order to have a good agreement with the experimental pressure measurements (see figure 15).

As Moldflow does not allow to get all calculation results in numerical format, we use temperaturethickness curves for different positions labeled from 0 to 4 (see figure 15). Temperature profiles do not change significantly along the part's width (y coordinate) except in the diverging entrance, but changes significantly along the part's length (x coordinate) (see figure 16). , X = 0 to 4 as shown in figure 15), the next step consists in working on the packing, holding and cooling stages. Theses phases have in common a negligible polymer movement. We adapt Carlslaw and Jaeger cooling model [START_REF] Carslaw | Conduction of Heat in Solids[END_REF]. Sensor temperature data corresponds to the mold temperature ) (t T Mold , sensor pressure data corresponds to the polymer pressure (assumed homogeneous), and the time origin is fixed at the end of the filling stage. Then we are able to calculate the polymer temperature

-3 -2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5 3 3,5
X = 0 X = 1 X = 2 X = 3 X = 4 z [mm] Polymer Temperature [°C]
) ( 2 2 1 cos 2 1 ) 1 ( ) ( ) ( 2 ) , ( 0 4 2 1 2 2 2 t T h z n n t T z T t z T Mold n h at n n Mold x ef x e + ÷ ÷ ÷ ÷ ø ö ç ç ç ç è ae ú û ù ê ë é ÷ ø ö ç è ae + ⋅ ⋅ ÷ ø ö ç è ae + - ⋅ - = å ∞ = ÷ ø ö ç è ae + - π π π (5)
The polymer temperature evolutions during packing, holding and cooling phases are represented in figure 17 for each calculation location (X = 0 to 4).

Since the polymer cools inside the mold cavity, the pressure decreases, depending on the hydraulic holding level (see figure 10). Considering the polymer no-flow temperature T nf (General Electric Plastics data), we can calculate the pressure at the no-flow boundary p*(z). For this calculation, we use the sensor pressure data and we consider that polymer pressure is uniform along thickness. Therefore, we have different p*(z) profiles depending both on the holding level and on the location, as shown in figure 18 in the particular case of location 2.

X = 0 X = 1 X = 2 X = 3 X = 4
The horizontal line corresponds to the solidification boundary.

Cooling phase is represented from t = 0s (end of filling stage) to t = 65s.

The thickness is divided into 21 layers. We assume that the void content rate depends on the volumetric shrinkage applied to the polymer at the no-flow boundary [START_REF] Kumazawa | Prediction of Anisotropic Shrinkage of an Injection Molded Part[END_REF]. In order to evaluate this assumption, we use a pvT Tait model defined using General Electric Plastics data (see figure 19).

When the pressure reaches zero and the polymer is solidified, shrinkage appears. If we consider the no-flow boundary (defined by the temperature), we assume that there is a void formation (instead of shrinkage) in all area where p*(z) = 0. The volumetric shrinkage ( 0 v v sol -) is evaluated using the pvT diagram (see figure 17). The void content value VC EM is then defined as :

( ) ò ⋅ - ⋅ ⋅ = 2 0 0 2 V L nf C EM dz v v h A VC (7)
Where L V is the length of the thickness where p*(z) = 0 (see figure 18). A C is a scale factor introduced to adjust the VC EM value for further comparisons with other void content parameters (VC O and VC US ).

4.3-Comparison between Experimental Measurements and VC EM Results

Figure 20 presents a plot of VC EM (average curve for X = 1 to 4), VC O and VC US versus the apparent part density ρ app . All three parameters present similar evolutions and the same minimum apparent density (1.52) that insure void free injection molded thick plates. This apparent density value corresponds to a holding pressure level of 7.5 MPa. If the holding pressure is increased over this value, the apparent density does not increase a lot (saturation effect presented in figure 13). For holding pressure values under 7.5 MPa, the lower the apparent density is, the higher the void content is. Therefore, it is possible to predict empirically the void formation in injection molded parts. We use sensor data (temperature and pressure) and Moldflow  filling simulation results (temperature) in order to apply the VC EM and it is possible to transpose this calculation to a process control tool that could constitute a quality control method.

= ⋅ = ⋅ + = Þ < ï î ï í ì = ⋅ = ⋅ + = Þ > + ú û ù ê ë é ÷ ÷ ø ö ç ç è ae + ⋅ - ⋅ = ⋅ - ⋅ ⋅ - ⋅ - C b T T p b b T b p T v b T B T b b T v T T p T v b T B T b b T v T T p T v t B p C T v p T v t p b T b t T b s s s t t T b m m m t t s m vnf v0 Tnf Tair (6) Absolute Temperature [K] Specific Volume [cm3/g]
A general layout of the quality control method we want to create is presented in figure 21.

5-Conclusions

Both meldlines and void defects do not have the same impact on the mechanical properties of an injection molded part. On one hand, meldlines should be considered as perfectible defect. The deviation method consisting in the creation of a flow through the defect during the filling stage has a good efficiency. Moreover, injection molding simulation tools like Moldflow  appear to be able to predict the results in terms of fibers orientation. Extended studies can also lead to a prediction of the [START_REF] Haramburu | Short-Fiber-Reinforced Thermoplastic for Semi-Structural Parts : Process-Properties, 8 th Euro-Japanese Symposium[END_REF]. On the other hand, voids are not acceptable and the creation of a prediction method that could be used at the production stage is on the way. Injection molding simulation brings again a precious help for this problem.

6-Future Work and Applications

Concerning the meldline defects, future studies will be focused on :

-Mesh refinement in the meldline region ; -Simulation of the complete injection molding process (including packing and holding).

Fibers orientation data will also be used in order to realize mechanical calculations based on Moldflow  simulations [START_REF] Haramburu | Short-Fiber-Reinforced Thermoplastic for Semi-Structural Parts : Process-Properties, 8 th Euro-Japanese Symposium[END_REF]. Comparisons between experimental and calculated mechanical properties will finally be done. Fair results had been found about the deviation efficiency on the improvement of meldline mechanical properties. This method will be applied to industrial parts (see figure 22) at the mold designing stage. Runner and gates will be optimized in order to have deviated meldlines.

Moldflow  software will also be used in order to generalize this model to the entire part. Several NDT methods have to be applied to industrials cases. Density measurements and ultrasonic controls are mainly studied. Applications to industrial injection molded parts are also planned (see figure 22).

Concerning the void defect, future studies will mainly be focused on voids location along the thickness. If we consider fibers orientations, shear rate during filling, and voids location after cooling, it appears that voids are generated in regions where fibers are perfectly aligned in the (x ; y) plane. We assume that in high shear rates regions, the polymer flow leads to no z-alignment for the fibers (Advani tensor term a 33 ≈ 0), and this means the polymer matrix is the only barrier against voids formation. On a contrary, for lower shear rate regions, z-alignment does not equals zero, and fibers can create a barrier against void formation. Therefore, inside the region where p*(z) = 0, the most breakable zone corresponds to the highest shear rate region during filling. Figure 23 presents a superposition of shear rate, fibers orientation and voids location for the thick injection molded plate. In addition, Kumuzawa [START_REF] Kumazawa | Prediction of Anisotropic Shrinkage of an Injection Molded Part[END_REF] shows that z-linear shrinkage is much higher than x or y one. Consequently, it is the z-linear shrinkage that mainly explains the formation of voids. Complementary experiments will be processed using Ultem  grades having different reinforcement rates (from 0% to 40%) in order to confirm this approach.

Figure 1 -Figure 2 -

 12 Figure 1 -Injection Molding Machine and Modular Mold

Figure 3 -

 3 Figure 3 -Fibers Orientation Experimental Measurements

Pressure / Temperature SensorFigure 4 -

 4 Figure 4 -The ISO Specimen Configurations

Figure 5 -Figure 6 -

 56 Figure 5 -Effect of Mold and Polymer Temperature on the Mechanical Properties of the Meldline

Figure 7 -Figure 8 -

 78 Figure 7 -The D0 Numerical Model for Injection Molding Simulations

Figure 9 -

 9 Figure 9 -Experimental and Simulated Fiber Orientation Results for the Meldline Region

Figure 11 -

 11 Figure 11 -Ultrasonic C-scan of a Thick Plates Molded with Different Holding Pressure Levels

Figure 10 -

 10 Figure 10 -Polymer Pressure Evolution in the Mold Cavity for Different Holding Levels

Figure 12 -Figure 13 -

 1213 Figure 12 -Voids Morphology and Location in Thick Injection Molded Plates

Figure 14 -Figure 15 -

 1415 Figure 14 -Boundary Conditions for the Filling Stage of the Thick Plate

  each time step and position along the thickness.

Figure 16 -

 16 Figure 16 -Simulation Temperature Results at the End of Filling Stage

Figure 17 -

 17 Figure 17 -Polymer Cooling for Different Locations in the Part

Figure 19 -

 19 Figure 19 -The Tait pvT Model of the Ultem  2300-1000

Figure 21 -Figure

 21 Figure 21 -Quality Control Method Based on the VC EM

Figure 22 -Figure 23 -

 2223 Figure 22 -Technofan  Ventilation Fan Wheel

Table 1 -

 1 Deviation Volume Configurations

	D0	D1	D2	D3	D4
	% Deviation Volume 0.00%	19.8%	40.5%	66.6%	84.1%

Table 2

 2

shows holding pressure and mold / polymer temperature levels used for our experiments.

Table 2 -

 2 Experimental Process Parameters Levels

	Holding Pressure [MPa]	5	7.5	10	12.5	15
	Mold Temperature [°C]	140	160	180
	Polymer Temperature [°C]	380	400	420

Acknowledgements

We thank Dr. R. Zheng (Moldflow  Pty. Ltd.) for his helpful advice on the orientation calculations. The work of S. Proust and E. Jourdain (CROMeP) for experimental measurements, and Moldflow  simulations is gratefully acknowledged. We also thank S. Devaux (CRITT of Toulouse) for the helpful availability of his ultrasonic equipment, and O. Darnis (Technofan  ) for his support. Special thank goes to all collaborating companies (Liebherr  , Microturbo  , LGMT). This study is realized with the financial support of the European Community, the Midi-Pyrénées Regional Council and of the French Agency of Technical Research.

Notations


S Andrieu

F Schmidt

Y Le Maoult

Heat conditioning modelling of thermoforming process: comparison with experiments

The goal of this study is to improve the efficiency of infrared heating ovens used in the thermoforming process. The heat conditioning of thermoplastic sheets before forming consists in heating up thermoplastic sheets above the glass transition temperature of the polymer. The thickness distribution of the final product is strongly dependent on the initial temperature distribution inside the sheet. 816 2340 Temperature (K) Quartz tube Tungsten wire X Z

Well-calculated process parameters during the heat conditioning step will allow improvements in, for example, cycle-time reduction. For that, we use different electric heaters that operate in different infrared ranges as: long, medium and short infrared wavelengths respectively. Thermoforming machines are generally equipped with ceramic heaters that operate in long infrared wavelengths. Recently, halogen lamps that are short wavelengths heaters have started to be employed in thermoforming applications. So, an experimental infrared heating set-up has been developed in order to compare the different kinds of heaters and the efficiency of lamp reflectors. The temperature of each infrared heater in the oven is controlled independently in order to optimise the surface temperature distribution of the thermoplastic sheet. For that, an 880 LW AGEMA infrared camera has been used to measure the temperature distribution of different thermoplastic sheets. We have chosen in this study the most commonly employed polymers in thermoforming which are: ABS and PP used for thin gage sheets.

Numerical simulations using software based on control-volume method have been performed and compared with experimental data.

1-Introduction

Thermoforming is an industrial process that involves a heating step before forming. During this heating the thermoplastic sheet temperature increases until a forming temperature is reached. The range of forming temperatures varies from polymer to polymer [START_REF] Throne | Technology of thermoforming[END_REF]. Basically, the range is large with amorphous polymers like polystyrene and narrow with crystalline polymers like polypropylene. The thermoforming machines are generally equipped with ceramic heaters that function in the long-waves infrared (? max 3 µm) [2]. In our study we use halogen lamps that are short-waves infrared (? max ~ 1.2 µm).

The aim of our study is to understand the radiative transfers between infrared heaters and thermoplastic sheets. We want to measure sheet temperature in order to obtain a better temperature distribution over the sheet surface and through the thickness. We want to study thin-gage and heavy-gage thermoplastic sheet. So it's important to avoid an indication of the temperature gradient inside the sheet. To calculate the radiative transfers, PLASTIRAD® software based on a control volume method is used. A comparison between experimental and numerical results will allow validation of the model.

2-Experimental infrared heating set-up

The "infrared heating set-up" (fig. 1) has been developed in order to study the interaction between the infrared radiation and polymers during the heating step of the thermoforming process. Thermoforming machines do not allow measurement of the sheet temperature during the heating. The infrared device makes these measurements possible. Moreover this tool can optimise the temperature distribution: the power of each lamp is adjustable.

The infrared heating device consists of an aluminium structure (1) that supports an infrared oven (2) power-regulated by an electrical power box (3), a metallic frame (4) to fix the thermoplastic sheet [START_REF] Modest | Radiative heat transfert[END_REF]. The aluminium structure has been conceived to change the distance between the sheet and the oven. For the moment this is fixed at 15 cm. A polished aluminium mirror ( 6) is adjusted at 45° under the sheet to measure the sheet temperature on the surface that isn't directly exposed at the infrared radiation ("back surface"). An infrared camera (7) (880 LW AGEMA) is used. A pyrometer (8) has been recently disposed to measure the local sheet temperature on the "front surface". We seek to obtain an indication of the temperature gradient through the thickness of the thermoplastic sheet A "halogen oven" is constituted by eighteen 1000W-halogen lamps (13195Z/98-235 Phlips). A "long-waves oven" is constituted by ten 1000W-ceramic lamps (FSR 1000W-230V Elstein).

3-Comparison between different infrared heaters

We want to study the efficiency of the different infrared ovens. So, ten 1000Whalogen lamps ( =1.2 µm) have been used in the "halogen oven" to compare with the "long-waves oven" ( =3 µm). The time to reach the same temperature is twice longer with the "long waves oven" (fig. 2). This is due to the long inertia of ceramic lamps (about 15 minutes measured with a thermopile 0.6-25 µm). It's seven times longer than for halogen lamps. The polymer used in this comparison is an ABS sheet of surface area 35*35 cm 2 and thickness 0.45 mm. Due to the low inertia of halogen lamps (2 minutes) and their geometry, these infrared heaters have been used to regulate accurately the temperature distribution. For that, we studied the effect of the power regulation of halogen lamps. Each lamp of the "halogen oven" is numbered in the following picture (fig. 3) Table 1-Influence of the lamp power on the temperature distribution

The power of each halogen lamp (HL) is expressed in percentage from 0 to 100%. The electrical power supplied to the oven P t is expressed in kW. The maximal temperature T max , the minimal temperature T min , the average temperature T ave and the standard deviation SD of the "front surface" are indicated for each set. The fifth set is a good compromise because SD is suitable in regards with the T ave obtained. The evolution of the average temperature during the heating and the cooling of the ABS sheet is given for the five sets in figure 4.

The speed of rice in temperature is function of the total power of the halogen lamps and their localisation.

4-Comparison with numerical simulations

Homopolymer polypropylene (HPP) has been already optically studied with an infrared spectrometer [START_REF]Infrared heating modelling of thermoplastic[END_REF]. These properties of transmission and reflectivity are necessary to perform numerical simulations. We carried out temperature measurements of a HPP sheet of surface area 35x35 cm 2 and of thickness 0.45 mm during a heating of 40 seconds. The temperature picture of the "back surface" is given on the following picture. The interval between lamps in the "halogen oven" is 80 mm. The lamp heating length is taken to 272 mm. The temperature of the tunsten wire and the quartz tube of the halogen lamp considered are indicated in table 2. The average temperature inside the thermoplastic sheet is 125.2°C. There is a concentration of the radiation in the centre of the sheet. This is due to the assumption in the model of the absorption that follows a Beer-Lambert law. There is a normalisation of the absorption perpendicularly to the surface of the thermoplastic sheet.

5-Conclusion

The experimental infrared heating set-up is a tool to study and optimise the heating during the thermoforming process. Surface temperature measurements have been successfully performed. Experimental set-up will be useful to perform a zonal heating PLASTIRAD® software will be improved in order to take into account the real path of the radiative beam and the influence of the reflectors with a ray tracing method.

Figure 1 -

 1 Figure 1-Infrared heating set-up

Figure 2 -

 2 Figure 2-Comparison long-waves/ short waves

Figure 3 -

 3 Figure 3-Infrared oven designed with numerical simulations

Figure 4 -

 4 Figure 4-Average temperature evolution

Figure 5 -Figure 6 -

 56 Figure 5-Temperature distribution for HPP

Figure 8 -

 8 Figure 8-Calculated temperatures

Table 2 -

 2 Source temperatures

Acknowledgements

The authors would like to thank our industrial partners Philips Lighting and EDF R&D Companies for technical support and funding.

This document was created with Win2PDF available at http://www.daneprairie.com. The unregistered version of Win2PDF is for evaluation or non-commercial use only.


Lucas Bourreau

Véronique Bouteiller

Franck Schoefs
email: franck.schoefs@univ-nantes.fr

Laurent Gaillet

Benoit Thauvin

Julien Schneider

Samuel Naar

Uncertainty assessment and decision from concrete electrical resistivity measurements on a coastal large bridge

Keywords: reinforced concrete, bridge, marine environment, resistivity NDT measurements, uncertainty

For RC structures exposed to marine environments, chloride-induced corrosion leads to deteriorations that can be associated to different exposure conditions: tidal, splash and atmospheric. The on-site assessment of the corrosion condition of the reinforcement in concrete relies on a corrosion diagnosis which consists in different techniques and among them non destructive techniques such as half-cell potential, resistivity and corrosion measurements. Although the two first techniques are commonly used, variability and uncertainties in NDT condition assessment remain factors that need to be better understood in order to increase the reliability of the diagnosis particularly in the case of marine structures (in comparison with atmospheric structures for which literature is more important).

The objectives of the Project DéCoF-Ré (2014-2017) were to study both the concrete durability and the corrosion state of the piers of the Ile de Ré bridge which is located in the French Atlantic coast. Concerning the corrosion study, visual inspections, rebar localization, half-cell potential mapping, electrical concrete resistivity mapping and corrosion rate measurements were performed during three years in order to analyse the influence of the measurement procedures, the investigated zones (pier number, side, height) and the environments (tide level, climatic conditions). This paper focuses on the resistivity measurements conducted 40th IABSE Symposium, 19-21 September 2018, Nantes, France. Tomorrow's Megastructures with a four electrodes arrangement probe (Wenner type) on the piers of the Ile de Ré bridge using a vessel. Although resistivity values cannot directly provide an information on the passive/active state of the rebar because the measurement qualifies the electrical resistivity of the concrete cover (and not the resistivity of the steel/concrete interface which is usually obtained by Linear Polarisation Resistance measurements), it is a useful non destructive technique to point out the areas where the corrosion might be strongest. The purpose of this paper is to outline the errors made while measuring the resistivity given the uncertainty assessment based on both (i) the repeatability tests and (ii) the material local anisotropy and measurement's variability. Moreover, a model based on the evaluation of Probability of Wrong Assessment is proposed to help the owner of the bridge in taking a decision.

Introduction

In Europe, most civil engineering reinforced concrete (RC) structures have been built during the last 60 years and their ageing is becoming a great challenge in terms of deterioration assessment, repair decision and management strategy [START_REF] Broomfield | Corrosion of steel in concrete -Understanding, investigation and repair[END_REF][START_REF] Cigna | COST 521 -Corrosion of steel in reinforced concrete structures -Final Report: European Communities[END_REF].

For RC structures exposed to marine environments, chloride-induced corrosion leads to deteriorations that can be associated to different exposure conditions: tidal, splash and atmospheric. On-site, the assessment of the corrosion condition of the reinforcement in concrete relies on a corrosion diagnosis which consists in different techniques [START_REF] Montemor | Chloride-induced corrosion on reinforcing steel: from the fundamentals to the monitoring techniques[END_REF][START_REF]Standard test method for halfcell potentials of uncoated reinforcing steel in concrete[END_REF][START_REF]COST 509 -Corrosion and protection of metals in contact with concrete -Final Report: European Commission[END_REF][START_REF]Standard test method for halfcell potentials of uncoated reinforcing steel in concrete[END_REF] and among them non destructive techniques such as half-cell potential [7], resistivity [START_REF] Gonzalez | Considerations on reproducibility of potential and corrosion rate measurements in reinforced concrete[END_REF][START_REF] Hussain | Underwater half-cell corrosion potential bench mark measurement of corroding steel in concrete influenced by a variety of material science and environmental engineering variables[END_REF] and corrosion measurements [START_REF] Caliński | A dendrite method for cluster analysis[END_REF]. Although the two first techniques are commonly used, variability and uncertainties in NDT condition assessment remain factors that need to be better understood in order to increase the reliability of the diagnosis [START_REF] Breysse | Accounting for variability and uncertainties in NDT condition assessment of corroded RC-structures[END_REF][START_REF] Breysse | Méthodologie d'évaluation non destructive de l'état d'altération des ouvrages en béton[END_REF] particularly in the case of marine structures (in comparison with atmospheric structures for which literature is more important).

The objectives of the Project DéCoF-Ré (2014-2017) were to study both the concrete durability and the corrosion state of the piers of the Ile de Ré bridge which is located in the French Atlantic coast. Concerning the corrosion study, visual inspections, rebar localization, half-cell potential mapping [START_REF] Bourreau | On-site corrosion monitoring -reliability[END_REF], electrical concrete resistivity mapping and corrosion rate measurements were performed during three years in order to analyse the influence of the measurement procedures, the investigated zones (pier number, side, height) and the environments (tide level, climatic conditions). This paper focuses on the resistivity measurements conducted with a four electrode arrangement probe (Wenner type) on the piers of the Ile de Ré bridge (Figure 1) using a vessel. Although resistivity values cannot directly provide an information on the passive/active state of the rebar [START_REF] Hornbostel | Relationship between concrete resistivity and corrosion rate -A literature review[END_REF] because the measurement qualifies the electrical resistivity of the concrete cover, it is a useful non destructive technique to point out the areas where the corrosion might be strongest. The purpose of this paper is to outline the errors made while measuring the resistivity given the uncertainty assessment based on the repeatability tests. Another issue was analysed: the material local anisotropy and measurement's variability. It is not detailed but the key results are reminded in the conclusion.

40th IABSE Symposium, 19-21 September 2018, Nantes, France. Tomorrow's Megastructures Figure 1: Figure 1: Mesh and position of the probe for repeatability measurements.

Concrete resistivity assessment with a 4 electrode probe on the bridge

The resistivity measurements were performed using a Proceq RESIPOD® probe which follows the Wenner technique. This probe is composed of four equally spaced electrodes which ensure the electrical contact with the concrete surface. A small alternative current I is passed between the outermost electrodes trough the concrete cover and the resultant potential difference V is measured between the two inner electrodes. The concrete electrical resistivity ρ for a semi-infinite geometry is given by (1), where "a" is the electrode spacing (5cm for the Resipod probe).

ρ = 2π * a * 𝑉 𝐼 (1)
In the case of measurements performed on samples in the laboratory, the error of measurements coming from the device itself, as given by the constructor is ξρ,dev=3* σρ,dev =2.4 kΩ.cm.

Corrosion diagnosis was performed on 15 piers (14 offshore and 1 onshore). The measurements were carried out on two opposite sides of the piers: side C was more exposed to dominant winds and daily sunshine leading to a stronger action of wetting/drying cycles whereas side G was less exposed (Figure 2). The dimensions of the investigated zones were 3m height and 1.85 m width. Measurements were carried out between +3.95 m and +6.95 m Chart Datum (CD). Chart Datum fixes an absolute sea level reference along the French coast and in this paper, La Rochelle -La Pallice harbour was considered as the reference. As the objective of the paper was to assess and to model uncertainty of resistivity measurements, the influence of the corrosion was disregarded, considering a pier, which was diagnosed passive on both sides. Measurements were performed during the rising tide on one non-damaged pier, thus allowing the concrete to dry and to oxygenate during about 2-3 hours for the bottom part (1m high) and about 6-7 hours for the higher part (2-3 meters). 2 Protocols and results

Procedure for the repeatability test

The objective was to determine the repeatability of the resistivity measurements when performed onsite. For this purpose, the probe was always positioned according to direction II (in Red, Figure 3) and 35 consecutive measurements (not readings) on the same location were carried out in each mesh. The twelve meshes of sides C and G were investigated.

Results of measurements

The mean value μρ,rep and the standard deviation σρ,rep were computed from the 35 measurements in each mesh. For each of the 13 meshes of side C (Figure 4), Figure 5 (a) plots the 35 resistivity values measured in one mesh with its mean resistivity value inside the same mesh. Similarly, Figure 5 (b) plots the results for the 13 meshes of side G. For both exposures (sides C and G), results showed that for resistivity measurements in the range 10-160 kΩ.cm, the scatter increases when the mean resistivity increases (10-150 kΩ.cm range).

Uncertainty modelling of the state dependant unvertainty

To quantify this trend, the evolution of the standard deviation with the mean resistivity is presented in Figure §. All the values were fitted with a linear relation (2) which regression coefficient was equal to 0.96. Linear fitting parameters were 0.13 (coefficient of variation) for the slope and near zero for the y-intercept (-2,05 kΩ.cm), which was in the same range of magnitude of the error of the device (+/-2.4 kΩ.cm) indicating that the measurements are not biased. These results mean that when performing resistivity measurements on-site, the scattering of the measurements increases with the mean resistivity value by a factor of about 50. To our knowledge, this result has not been published yet.

3 Discussion on the protocol Les us consider now other orientations of electrodes than the one plotted on Figure 3. That is a measure of the effect of material variability on the measurement. We propose here the protocol and the results.

Protocols for material variability assessment

According to the 4 positions plotted on Figure 6, tests have been carried out to cover the whole surface (10 measurements).

Figure 7: Scheme of the mesh and positions (I, II, III and IV) of the probe.

Modelling

Standard deviation was computed from the 40 measurements on each mesh (4 positions x10 tests) and liked to the condition state.

It is plotted on Figure 8.

Conclusions

This paper aims to evaluate several sources of onsite error based on a dedicated campaign of concrete electrical resistivity measurements performed on a bridge in a marine environment. It is shown that errors of assessment from the resistivity measurements are not negligible. The repeatability error increases linearly with the resistivity and the local anisotropy of the material affects strongly the assessment for high resistivity values (ρ≥80 kΩ.cm): the coefficient of variation of total error reaches 30% around 140 kΩ.cm.

Figure 2 :

 2 Figure 2: Top view of the pier and exposure sides C and G.

Figure 3 :

 3 Figure 3: Mesh and position of the probe for repeatability measurements.

Figure 4 :

 4 Figure 4: Position of steel rebars and of the meshes for sides C (left) and G (right).

Figure 5 :

 5 Figure 5: Repeatability test: sides C (a) and G (b).

Figure 6 :

 6 Figure 6: Evolution of standard deviation with mean resistivity for the repeatability test.

σ

  𝛒,rep = 0,13. µ 𝛒,rep -2,05

Figure 8 :

 8 Figure 8: Evolution of standard deviation with mean resistivity for material and measurement variability's. As presented in Figure 8, the scattering of the resistivity measurements was not increasing with the mean resistivity following a linear relation but an exponential relation (R²=0.96). The equation of the exponential fitting relation given in (3)Erreur ! Source du renvoi introuvable. covers a mean resistivity range from 10 to 140 k.cm.σ 𝛒,𝑡𝑜𝑡 = 1.16 𝑒 0.026.μ 𝛒,𝑡𝑜𝑡

40th IABSE Symposium, 19-21 September 2018, Nantes, France. Tomorrow's Megastructures Acknowledgements The authors would like to thank the team of the DéCoF-Ré Project: B. Godart, W. Traverst, P. Boujard, N. Coulaty-Chin, A. Orcesi, Y. El Rabbih, X. Derobert, M. Sissoko (IFSTTAR); R. Queguiner, S. Pasquiet, P. Boulaire, M. Rebours, C. Naudat (Cerema Ouest); F. Landrin and V. Queyrat (Cerema Ile de France), J-F. Barthélémy (Cerema Dtech ITM); M. Roche, O. Amiri (Nantes University); M. Brouxel, 40th IABSE Symposium, 19-21 September 2018, Nantes, France. Tomorrow's Megastructures S. Naar (SIXENSE-Concrete), A. Audouin-Dubreuil, F. Lavoute, M. Barbier (Département de la Charente-Maritime); F. Gazet (Dekra). They also thank the marine team for their help with the vessel.


E Mathey
email: mathey@enstimac.fr

L Penazzi

F Schmidt

F Ronde-Oustau

Thermal transfer in laminated injection moulds and comparison of the mould assembly techniques

Keywords: laminate tooling, polymer injection, mould cooling, numerical simulation

Recent studies have shown the possibilities to apply rapid prototyping methods for tool making. An advantage of these processes is the ability to obtain complex shapes (that would be impossible to obtain using machining). This aspect is particularly interesting to create cooling circuit in injection moulds. In this paper, conception of cooling circuit in laminated moulds is considered. After a description of steady state and transient heat transfer models in an injection mould, the influence of the cooling circuit geometry, the mould making technique (massive machined mould or laminated mould) and the bonding technique for laminated moulds (glued or brazed) is studied. Results show the advantage of a thermal barrier (which could easily be obtained in laminated tools) on mould thermal regulation. Discontinuities of mould material introduced by the lamination lead to a loss of thermal performance of the mould. However, a brazed mould shows performance almost equal to massive moulds.

INTRODUCTION

Rapid prototyping processes are now applied to the tool making process to reduce development time, fabrication costs and delays. Among them, the Stratoconception TM process consists in building pieces by assembling previously machined sheet laminates. In the case of steel, sheets can be assembled by glue or brazing [START_REF] Bryden | Laminated steel tooling in the aerospace industry[END_REF]. This procedure has been successfully applied to the fabrication of the tools of several processes such as injection, SPF, RTM… [START_REF] Bryden | Laminated steel tooling in the aerospace industry[END_REF][START_REF] Gley | From design to rapid tooling an application of Stratoconception : rapid tooling for foam casting[END_REF][START_REF] Pelaingre | Moule d'extrusion-soufflage, réalisation d'outillage en Stratoconception : application à l'extrusion-soufflage[END_REF] In the case of injection moulds, this process allows free and complex forms, especially for cooling channels following the piece surface [START_REF] Pelaingre | Moule d'extrusion-soufflage, réalisation d'outillage en Stratoconception : application à l'extrusion-soufflage[END_REF]. Conception methods used today for the cooling or heating of injection moulds [START_REF] Menges | How to make injection molds[END_REF][START_REF] Xu | Designing conformal cooling channels for tooling[END_REF] need to be adapted to this new material and its behaviour. The objectives of the present paper are, using numerical simulations on a disc-shaped mould, to show :

• the advantage of changing the cooling circuit geometry,

• the influence of the mould making process (massive mould machining or lamination),

• the influence of the bonding technique (glue or brazing) on the heat transfer within a simple discshaped mould.

INFLUENCE OF THE MOULD GEOMETRY

Constitutive equations

Steady state computation in the mould is used to compare the mould capacity to evacuate heat provided by the polymer to the cooling fluid. Steady state heat transfer in the mould yields equation (1). The boundary conditions associated with the problem are (see fig. 1):

• A heat flux density representing heat provided by the polymer piece during the injection cycle is imposed on the mould cavity surface according to equation [START_REF] Gley | From design to rapid tooling an application of Stratoconception : rapid tooling for foam casting[END_REF].

Φ = ∇ - Ω ∂ ∈ ∀ n T M M P . λ (2)
where λ M is the mould material thermal conductivity. Heat provided by the polymer is due to cooling and crystallization according to equations ( 3) and ( 4) where ρ P is the polymer density, V P is the polymer volume, C P is the polymer specific heat, ∆T is the difference between the polymer injection temperature and the mean ejection temperature, χ is the crystalline rate and H the massive latent heat of crystallization.

T C V Q P P P cooling ∆ = ρ (3) H V Q P P crystal χ ρ = (4)
As the objective is to compare mould geometries, only a rough estimation of the heat flux is necessary and therefore, heat provided by crystallization is neglected. The heat flux density Φ is supposed to be uniform on the mould cavity surface and is then estimated by dividing heat provided Q cooling by the mould cavity surface A and the estimated cooling time t [START_REF] Menges | How to make injection molds[END_REF].

• A heat convection conditions is imposed on the cooling channel boundary [START_REF] Xu | Designing conformal cooling channels for tooling[END_REF]. h is the convective heat transfer coefficient and T is the fluid temperature.

∞ ) ( . ∞ - = ∇ - Ω ∂ ∈ ∀ T T h n T M M CC λ (5)
h is estimated from fluid flow properties :

D Nu h F λ = ( 6
)
where D is the channel diameter, λ F is the fluid conductivity and Nu is the Nusselt number.

The flow rate is chosen to get a turbulent flow (Re>2300). In this case, Nu can be estimated using equations ( 7) and (8). where Re is the Reynolds number, Pr is the Prandtl number, Q is the fluid flow rate, C F is the fluid specific heat, ρ F is the fluid density, µ is the fluid viscosity and λ F is the fluid thermal conductivity.

• The outer surface of the mould is isolated as heat exchange with the ambient air and the press is neglected (less than 5% of the mould heat loss [START_REF] Himasekhar | CAE mold cooling injection molding using three dimensional numerical simulation[END_REF]).

0 . = ∇ - Ω ∂ ∈ ∀ n T M M ES λ (9)

Numerical computation

Equation ( 1) is solved by a finite element method using ABAQUS TM . Different configurations of the cooling circuit are tested to evaluate the influence of the number, size and position of the channels. Injection of a polypropylene disc in a steel mould cooled with water is considered (Tables 1 and2).

Results

Temperature distribution is shown inside the mould in fig. 2 and on the mould cavity surface in fig. 3. Tendencies known by mould makers are similar to computations: a high number of cooling channels should be preferred to get a low homogenised temperature. Channels close to the mould cavity should be preferred for a low temperature but with a loss of the temperature homogeneity at the mould cavity surface. These results show the advantage of creating a cooling system surrounding the polymer piece.

EFFECT OF THE MOULD LAMINATION

Introduction of the lamination in the computation model

Stratification introduces discontinuities and material changes influencing the heat transfer in the mould. Among the different possibilities, it is taken here into account in the computation by introducing a thermal contact resistance on the interface between laminates.

λ (W.m -1 .K -1 ) 1.8 410 R M/M (m 2 .K.W -1 )
1.6 10 -4 7.3 10 -8

Results

Distance

The effect of the stratification is different according to the bonding method (fig. 5 and6). When brazing is used, no differences are found compared with the original mould. Thermal contact resistance is too low to influence the heat transfer between the mould cavity surface and the cooling channel. When glue is used, the contact thermal resistance becomes high enough compared to the mould conductivity to influence heat transfer leading to a higher temperature on the mould cavity surface.

Constitutive equations

The transient heat transfer equation ( 10) is applied to the mould and the polymer simultaneously. Several studies have been carried out to determine the value of the thermal contact resistance [START_REF] Delaunay | Nature of convection between polymer and mold in injection molding. Part 1 : Influence of a nonperfect thermal contact[END_REF][START_REF] Quilliet | Transferts thermiques à l'interface polymèremétal dans le procédé d'injection des thermoplastiques[END_REF]. This resistance is dependant on polymer pressure and shrinkage during the injection cycle. In order to avoid a coupled thermo-mechanical computation, the resistance is supposed to be constant in the model and equal to 10 -3 m 2 .K.W -1 . In our case, the initial temperature in the polymer is uniform and equal to the injection temperature Tinj = 250°C.

T t T C M i i i P M i i 2 ) , ( ∇ = ∂ ∂ Ω ∈ ∀ = λ ρ (10)
inj P T t T M = = Ω ∈ ∀ ) 0 ( (12)
The initial temperature in the mould is homogenous and equal to the coolant temperature at the beginning of the first cycle. At the beginning of the following cycle, the initial temperature is equal to the final temperature of the previous cycle (13).

   = = = = Ω ∈ ∀ - ∞ cycle N T t T cycle T t T M th N st M 1 ) 0 ( 1 ) 0 ( (13)

Results

The results obtained here confirm the previous obtained using the steady-state model. At the cooling channel surface and at the cavity surface (fig. 8), the temperature obtained using the brazed mould is equal to the temperature obtained using the massive mould whereas the glued mould leads to a higher temperature. In the polymer, the temperature obtained at the end of the cycle (for the same cooling time) is higher with a glued mould. It confirms that heat transfer in the mould is less efficient with the glued assembly. Steady state and transient models of the mould thermal regulation are detailed here and used to compare cooling channels geometry, mould making technique and bonding method. Results show the advantage of the brazed laminates making method that allow the realisation of complex shapes for the cooling system, without a modification of the heat conduction performance of the mould.

Fig. 1 .

 1 Fig. 1. Massive mould geometry and domain definition

Fig. 2 .Fig. 3 .

 23 Fig. 2. Steady-state temperature distribution in the mould

Fig. 4 .

 4 Fig. 4. Laminated mould geometry and domain definition This resistance is evaluated by dividing the thickness e of the joint by its thermal conductivity λ.

Distance

Fig. 5 .Fig. 6 .

 56 Fig. 5. Temperature distribution on the mould cavity surface according to laminate assembly technique

Fig. 7 .

 7 Fig. 7. Transient model domain definition Boundary conditions on the mould channels and mould surface are the same as the steady-state model (5, 9). For points located on the polymer surface or on the mould cavity surface, a new contact condition is introduced (11) where ∆ T is the temperature difference between the polymer surface and the mould surface, R M/P is the thermal contact resistance at the mould/polymer interface and Φ is the heat flux crossing the interface. Φ = ∆ Σ ∈ ∀ P M P M R T M / /

Fig. 8 .

 8 Fig. 8. Evolution of the mould temperature at the cooling channel surface and at the cavity mould surface

Table 1 .

 1 Mould and polymer thermal properties

		Mould (Steel)	Polymer (PP)
		λ (W.m -1 .K -1 )	36.5	0.15
		ρ (kg.m -3 )	7820	980
		C (J.kg -1 .K -1 )	460	1700
	Table 2. Fluid properties and processing conditions
	λ F	0.6 W.m -1 .K -1	∞ T	20 °C
	C F	4.18 J.kg -1 .K -1	µ	10 -2 P
	ρ F	1000 kg.m -3	∆Τ	250-100=150 °C

Table 2

 2

		Glue	Brazing
	e (mm)	0.3	0.03

. Interface properties according to assembly technique (RTG, Loctite, CEA)


Sylvia Andrieu

Fabrice Schmidt

Yannick Le Maoult

Experimental and numerical infrared heating of thermoplastic sheet during thermoforming process

Thermoforming includes a heating stage of thermoplastic sheets by infrared lamps. The temperature distribution on the surface and through the thickness of the sheet conditions the distribution thicknesses of the thermoformed part. Our objective is to understand the heat transfers between the infrared heaters and the thermoplastic sheets. For that we developed an infrared heating device which makes it possible to carry out temperature measurements of a sheet during the heating with an infrared oven. An AGEMA 880 LW infrared camera allows measuring the back surface (non-directly exposed to the radiation) temperature distribution of the sheet. An indication of temperature discrepancy through the thickness is given by a RAYTEK Thermalert TX pyrometer. Two infrared ovens (halogen and ceramics lamps) were studied characteristics of infrared ranges (short and long wavelengths respectively) on a polymer representative of the thermoforming market (white PS 3mm-thickness sheets). These experimental measurements allow validating the numerical models. A method of radiative transfer based on ray tracing is used to calculate the irradiance with lamp reflectors. The Rosseland approximation is used to take into account the optical properties of the polymers that were measured using infrared spectrometry. The transient heat balance equation (including 3D radiative diffusion) is solved.

Introduction

Thermoforming is an industrial process that involves a heating step before forming [START_REF] Gruenwald | Thermoforming, a plastics processing guide[END_REF]. During this stage, the thermoplastic sheet is warmed up to forming temperature. The range of forming temperatures varies from polymer to polymer [START_REF] Throne | Technology of thermoforming[END_REF]. Basically, the range is large with amorphous polymers like polystyrene and narrow with crystalline polymers like polypropylene. The thermoforming machines are generally equipped with ceramic heaters that function in the long-wavelength infrared (? max ~ 3 µm). In our study, we use also halogen lamps that are short-wavelength infrared (? max ~ 1.2 µm). The aim of our study is to understand the radiative transfers between infrared heaters and thermoplastic sheets. We want to measure sheet temperature in order to obtain a better temperature distribution over the sheet surface and through the thickness. We want to study thin-gage and heavy-gage thermoplastic sheet. So it is important to obtain an indication of the temperature gradient throughout the sheet. A numerical method of radiative transfer based on ray tracing has been developed to compute the view factors between the oven and a thermoplastic sheet. The Rosseland approximation is used to take into account the optical properties of the polymers which were measured by infrared spectrometry. The transient heat balance equation is solved using PLASTIRAD software based on a control volume method. Comparisons between experimental and numerical results allow validation of numerical model.

In this paper, we will present the experimental infrared heating device, the characterization of the infrared ovens, the thermal and optical characterizations of the polymer, the heating measurements and the results of the numerical simulations.

Experimental infrared heating set-up

The infrared heating set-up (fig. 1) has been developed in order to study the interaction between the infrared radiation and polymers during the heating conditioning step of the thermoforming process. Thermoforming machines do not allow measurement of the sheet temperature during the heating except in specific location using a pyrometer. The infrared device described below makes these measurements possible. In addition, this device could be used as an optimisation tool of temperature distribution: each lamp is power-regulated.

The infrared heating device consists of an aluminium structure (1) that supports an infrared oven (2) power-regulated by an electrical power box [START_REF] Spruiell | [END_REF]. A metallic frame (4) clamps the thermoplastic sheet [START_REF] Cai | phD of Ecole des Mines de Paris[END_REF]. The aluminium structure has been designed in order to change the distance between the sheet and the oven. For the experiment, in this paper, the distance is fixed to 15 cm. A polished aluminium mirror ( 6) is adjusted at 45° under the sheet to measure the sheet temperature on the surface that is not directly exposed to the infrared radiation ("back surface") using an infrared camera (7) (880 LW AGEMA). A pyrometer [START_REF] Lentes | [END_REF] has been placed to measure the local sheet temperature on the "front surface". Local back and front temperature measurements allow to measure temperature gradients through the thermoplastic sheet thickness.

Characterization of infrared ovens

A halogen oven is constituted of twelve 500W-halogen lamps (13169Z CLEAR Philips 500W-230V). A ceramic oven is constituted of ten 600W-ceramic lamps (Elstein HTS/1 600W-230V). The interval between halogen lamps is 80 mm; it has been optimised using numerical simulations performed with the PLASTIRAD software. As sketched in figure 2, the lamps positions in front of the 350 mm x 350 mm thermoplastic sheet are given. In addition, the location of the oven hole (in order to measure locally the temperature using a pyrometer) is noticed. The oven irradiance cartography has been measured using a XY bench. This experimental set-up permits to measure the irradiance of a heat source using a thermopile MEDTHERM 64 that functions from 0.6 up to 15 µm. The measures are performed at 15 cm of the oven. In the following irradiance cartography (W/m 2 ), 1681 points have been measured in 800 mm x 800 mm surface around the oven. The values of irradiances are very similar in both cases. Our goal is to model the interaction between the infrared radiation and the halogen lamps. For that purpose, we have to measure the thermal and optical properties of the polymer.

Thermal and optical properties of PS sheets a. Thermal properties

In order to obtain reproducible results, we have to extrude thermoplastic sheets. The extrusion machine used is an OMV D120. The prescribed flow rate to extrude is 150 kg/h because of the small quantities of polymer. The HIPS used is a mix between EMPERA 251 and EMPERA 622 with 2% of white pigments in weight. The specific mass is 1040 kg/m 3 and the thermal conductivity is 0.18 W/m/K. We used 3mm-thickness thermoplastic sheets. The forming temperature range is between 127°C and 182°C.

The heat capacity C p is an important parameter for solving the heat balance equation. For that, DSC measurements have been performed using a classic 10°C/mn heating speed. The heat capacity versus temperature is plotted in the figure below: We can notice that the glass transition temperature is close to 100°C. The glass transition is characteristic of an important increase of the heat capacity Cp. The numerical simulations have to take into account C P variation for the accuracy of the model.

b. Optical properties

An infrared spectrometer Perkin-Elmer (FT-IR) that functions in the 1.3-25 µm wavelengths range has been used to measure optical properties. Two modes have been employed the transmission and specular reflection methods [START_REF] Spruiell | [END_REF]. The measurements have been performed at ambient temperature. The mean reflection coefficient r(T) is calculated using the data versus wavelength.

Tg ∫ ∫ = Äë (T)dë 0 ë L Äë (T)dë 0 ë L ë r r(T) (1)
L 0 λ is the spectral intensity of a black body. The mean reflection coefficient is low about 3% in the 1.3-25 µm wavelengths range. According to the Wien law displacement, we have to proceed to measurements in lower wavelengths. We used a near infrared spectrometer functioning in the 0.4 to 1.1 µm wavelengths range to complete the previous measures. In this wavelengths range mean reflection coefficient is about 80%. In the range from 1.1 to 1.3 µm, we consider a linear interpolation. According to this, the HIPS mean absorption coefficient is 27% for the halogen lamp and 3% for the ceramic lamp.

The transmission method gives the spectral transmission coefficient T λ at ambient temperature. The mean transmission coefficient T is calculated using previous method. The value obtained (0.1 %) is characteristic of an optical thick polymer.

The thermoplastic sheets have been cut using a microtome (HM 355 MICROM). The thickness sample is between 10 and 100 µm. The relation between the spectral transmission coefficient T λ and the transmissivity τ λ could be simplified to:

λ λ λ τ λ r r T + - = 1 1 (2)
If we assume the transmissivity follows the Beer-Lambert law, then:

d e λ κ λ τ - = (3)
And then we obtain:

λ λ κ λ C d T + = -ln (4)
For each wavelength, the data are well-fit by a straight line and the spectral absorption coefficient could be deduced simply using the slope of each straight line. Using this methodology, the absorption coefficient κ λ becomes independent of the thickness sample.

The polymer spectral emissivity is obtained using the energy balance as follow:

1 = + + λ α λ τ λ r (5)
At wavelength thermal equilibrium, the emissivity is equal to the absorptivity

λ α λ ε =
. In the 8-12 µm range corresponding to the infrared camera and pyrometer the emissivity of the polymer is ε=0.96. This value is used to calibrate the infrared camera during experimental measurements [START_REF]Techniques de l'ingénieur, R 2740[END_REF].

Infrared heating

The two infrared ovens have been used to heat 3mm-thickness PS sheets. When the temperature measured by pyrometer (front surface) is 160°C, the heating is stopped. Using ceramics lamps it is impossible to reach the forming temperature due to the rapid increase of the front surface temperature. But forming state is possible using halogen lamps.

The difference between the two kinds of heaters is due to the reflectivity variation. The reflectivity value is greater for the wavelengths range characteristics of halogen lamps (27%) in comparison with ceramic lamps (3%) Figure 5-Experimental temperature mesurements . These measurements will permit to validate the numerical model described in following section.

Numerical simulations a. Ray tracing method

We developed a ray tracing method to take into account the effect of lamp reflectors. This method is very accurate to model the radiation [START_REF] Cai | phD of Ecole des Mines de Paris[END_REF]. In this section, we present the irradiance computed using ray tracing method in the case of halogen oven.

The lamp power [START_REF]La transmission de la chaleur[END_REF] is calculated using the classical Stefan-Boltzmann law

w S w T w q 4 σ ε =
where w ε is the average emissivity, T W the temperature and S W the surface of the tungsten wire, σ is the Stefan-Boltzmann constant. We obtain 473W for a lamp of electrical power of 500W. The difference is due to loss by so-called "Joule effect". The cylindrical surface of the lamp is meshed. The number of nodes per lamp is 4800. 3481 rays are emitted from each node like lambertian source. A diffuse reflection (3481 ray per node) has been taken into account for the plane reflector with a reflection coefficient of 80% given by the supplier. The mean and the maximum irradiance computed by ray tracing method are close to experimental data. But the relative error is maximum for surface edges because the oven edges are not taken into account.

b. Rosseland approximation

The Rosseland approximation [START_REF] Modest | Radiative heat transfert[END_REF][START_REF] Lentes | [END_REF] is used to take into account the optical properties inside the polymer. This method avoids long calculations time like those obtained using ray tracing method. This approximation permits to solve heat balance equation in a conduction approach for an optically thick medium. The radiative conductivity (or Rosseland conductivity) is:

3 ) ( 3 16 T T Ross K Ross k σ = (6)
The Rosseland absorption coefficient is obtained versus temperature by integration over the frequency ( The Rosseland coefficient and the conductivity are computed using the absorption coefficient obtained by infrared spectral measurements. For the PS, the Rosseland conductivity range is between 0.0006 up to 0.0018 for 20°C to 170°C. This conductivity is low in comparison to the thermal conductivity (0.18 W/m/K). This is characteristic of an optically thick polymer. The Rosseland approximation is included into the PLASTIRAD software [9,10] based on a control volume method to solve the 3D heat balance equation.

λ ν / c = ): ∫ ∞ × ∫ ∞ = 0 ) , ( 1 0 ) , ( ) ( 1 )

c. Control volume method

The source terme computed by ray tracing is used as a boundary condition. A convection coefficient of 8W/m 2 /K is applied at the sheet surfaces. We used the heat capacity measured by DSC. The reflection coefficient used is 27% according to the spectral measurements in the near and average infrared. The maximum difference between experimental and numerical result is about 10°C. But we obtain an accurate forming point (same pyrometer and camera temperature measurements). The experimental data give 131°C for 174 s and the numerical simulations 132°C for 177 s.

Conclusion

The experimental infrared heating set-up has been developed in order to study and model the heating during the thermoforming process. Surface temperature measurements have been successfully performed using an infrared camera and a pyrometer.

A ray tracing method has been developed to compute the irradiance of an infrared oven. This method allows taking into account the lamp reflectors. The agreement is fair between the computation and the irradiance measurements performed by mean of a XY bench and a thermopile. The Rosseland approximation is valid because of the optically thick property of the white HIPS. This approximation has been included in PLASTIRAD. This sofware, based on a control volume method, compute the temperature for 3D geometry. The forming point is in agreement with data measurements performed by the experimental infrared heating set-up.

A sensibility analysis demonstrates that the reflectivity parameter is preponderant. Measurements in the 1 to 2 µm wavelengths range that is poor technically will be very interesting.

The thermal procedure developed in this work will be used in a global step of optimisation and notably in the case of heating under constraint.

Figure 1 -

 1 Figure 1-Infrared heating set-up

  mm x 350 mm surface around the oven corresponding to the sheet dimensions, the cartography irradiance for the two ovens is given in the

Figure 4 -

 4 Figure 4-PS heat capacity

Figure 6 -

 6 Figure 6 -Comparison experimental / numerical data

table below :

 below

	Oven	Ceramic	Halogen
	Electrical power	6000	6000
	Power receive by the	1221 (20.3%)	1246 (20.8%)
	35cm*35cm surface (W)
	Mean irradiance (W/m 2 )	9421	9612

Table 1 -

 1 Mean irradiances

Table 2 -

 2 Computed intensity versus experimental data

	The mean, minimum and maximum irradiance computed in a 350 mm x 350 mm
	thermoplastic sheet are summarized in the table below:

Acknowledgements

The authors would like to thank our industrial partners Philips Lighting and EDF R&D Companies for technical support and funding. They thank also their EMAC collaborators Georges Nepsinsky, Ludivine Moga, Guilhem Bocquet and Jean-Michel Mouys. The DSC measurements have been performed at ENIT school engineers (Tarbes France). Thanks to Michel Boutoleau.

This document was created with Win2PDF available at http://www.daneprairie.com. The unregistered version of Win2PDF is for evaluation or non-commercial use only.


C Champin
email: cedric.champin@ensmp.fr

M Bellet

F M Schmidt

J.-F Agassant

Y Le Maoult

G Denis

3D FINITE ELEMENT MODELING OF THE STRETCH BLOW MOLDING PROCESS Paper n° 148

The stretch blow molding process of PET bottles is a two-step process. First, a cold tubeshape preform is heated using an infrared oven above PET glass transition temperature (about 80°C) in order to reach the forming temperature. The softened preform is then simultaneously stretched and inflated with a rod and air pressure. The final wall thickness of the bottle is both related to heating parameters as well as stretch blow molding ones. It leads to a complex thermo-mechanical problem for which specific numerical models must be developed. In this work, a complete 3D finite element modeling of the stretch blow molding process has been developed including both infrared heating and forming steps.

The energy transfer between the infrared oven and the irradiative surface of the preform is modeled using a ray tracing method. In the same time, the amount of radiation intensity absorbed by the polymer is approximated with a Rosseland model. Owing to that, the radiation heat transfer results in a pure conductive heat transfer. All the thermal computations will be compared to the so-called PLASTIRAD control volume software [MON2001] and to a temperature analytical model.

Considering the deformation step, a Mooney-Rivlin hyperelastic model has been implemented in Forge3® software in order to account for the PET rheological behavior. The numerical model is developed using a velocity pressure formulation and P1+/P1 tetrahedral finite elements. In order to validate the hyperelastic behavior, computations are compared to a Mooney-Rivlin analytical model of a free inflation tube. This model enables to obtain the tube internal radius versus a given pressure on the internal surface.

I-Introduction

The injection stretch-blow molding process is the most widely used in the P.E.T bottle production [ROS1989]. This process is segmented in two main stages. First, a cold preform is heated in an infrared oven above glass transition temperature (80°C for P.E.T.). Then, the softened preform is inflated and stretched with assistance of a rod in a mold which has the shape of the desired bottle (cf. figure 1). The performance of P.E.T. bottles so-produced depends on many variables including the initial preform shape and several process parameters. It is well-known that these having major impact on mechanical properties are the initial preform temperature and the bi-axial elongation rates.

Since last decade, the reduction of weight became one of the main demands for the design of packaging. Therefore, process optimizing tends more and more to produce bottles of minimum thickness having the best mechanical properties. But mechanical properties of bottles also depend in turn on final thickness distribution and P.E.T. strain hardening resulting from stretching. In consequence, optimizing the process becomes a huge task when only based on experimental studies. For that reason, several numerical simulations for stretch-blow molding process optimization have been developed to save time and provide more understanding of the process. However, in order to be accurate enough, the simulation necessitates a 3D numerical model taking into account both the heating and inflation stages.

As illustrated in figure 2 a typical infrared oven is constituted of a row of halogen tube lamps which allows a rapid heating (about 30 seconds), two kinds of reflectors (back and front of preform) and a cooling fan. The cooling fan is necessary to insure that the external preform surface does not burn but also to prevent surface temperature preform exceeding P.E.T. thermal crystallization temperature (about 135°C ). The preforms are transported across the oven while they rotate continuously. The rotation allows to obtain an homogeneous temperature distribution in the angular direction. The threaded portion of the preform is protected by a heat shield in order to avoid heating. Once warmed, the preform exits from the oven and is allowed to thermally equilibrate before being placed in the stretch-blow mold. Thus, there are several processing parameters to set (heating time, equilibrium time, rotation speed, heat transfer coefficient,...) in order to heat conditioning preforms. A few models of the heat transfer inside an infrared oven have been developed. In 1992, Lebaudy [LEB1992] has developed a two-dimensional finite difference model in which the radiation flux is assumed to be uniform along the preform height. The value of the heat transfer coefficient between the air and the polymer is assumed to be constant to account for the air forced convection due to the cooling fan. Shelby [SHE1991] employed a simple numerical approach to determine the effect of infrared lamp temperature on reheat rate of PET. Radiative heat transfer was assumed. The methodology predicted the transient onedimensional temperature distribution. DiRaddo et al [DIR1993] proposed a two-dimensional approach using the finite element method where the fraction of energy leaving the heaters surface arriving at the preform surface (i.e. the view factor) is estimated analytically. Hartwig [HAR1996] in his model includes the back and front reflectors and the complex displacement of the preform (translation and rotation). Venkateswaran et al [VEN1997] use an analytical approach to calculate the view factors too.

The blowing step has been equally the object of a lot of research this last two decades. 2D viscoelastic model [SCH1992] has allowed to understand the complex thermo mechanical problem implemented. Rapidly, some studies have been specifically focused on the PET behavior heated up to its glass transition temperature. Thus, the Institute of Physical and Chemical of Research (RIKEN) programmed a 3D shell finite element software using a viscoplastic behavior law [WAN1998]. But major part of the works have shown that this behavior was more hyperelastic than viscoelastic, and as a result, numerical model must be developed using Mooney-Rivlin or neo-Hookean potentials. Thus, the Santa Clara University (SCUDC) developed a software using membrane elements with a Lagrangian formulation and a Mooney-Rivlin potential. The Queen's University of Belfast programmed a special Abaqus version dedicated to the blow molding simulation. It used Mooney-Rivlin, Ogden and more interesting, Buckley hyperelastic law [MEN2000]. Recently, Gorlier [GOR2001] or Marco [MAR2003] noted correlations between the Edwards Vilgis potential [EDW1986] and the deformation of PET subjected to multi-axial loads. Moreover, for a numerical point of view, the thin shell models proved their limits for the mechanical as for the thermal problem, regarding the more and more complex bottle shapes. As a consequence, the two steps of the stretch blow molding process will be 3D modeled in this work, with a finite element method implemented in the Forge3® commercial software. First, a modeling approach capable of predicting the three-dimensional transient temperature profile in the preform during the conditioning station (reheating and equilibrating) is presented. Conduction and radiation heat transfer modes have to be taken into account. The mechanical problem is considered using a hyperelastic Mooney-Rivlin law implemented with a velocity/pressure formulation and remeshing phases during the calculations, if necessary.

II-Infrared heating model

In its classical form, the heat balance equation can be written :

( ) ( ) r p q T k dt dT c ⋅ ∇ - ∇ ⋅ ∇ = ρ [1]
where r q is the radiation heat flux (cf. nomenclature) Using the following boundary conditions :

( ) 0 , T O x T = on Ω , initial temperature imp r n q φ = . on flux Ω ∂ , imp φ an imposed heat flux ( ) ( ) ext conv T T h n T k - = ∇ . on conv Ω ∂ , for heat convection
Rosseland has developed a diffusion approximation of the radiation heat transfer in the case of large optical thickness [MOD1993]. This kind of approximation has been widely used for glass forming processes [LIN2002]. Numerical simulation using this method speeds up, because the radiation heat transfer is solved using a conductive heat transfer form. Thus, the radiation heat flux reduces to :

( ) ( ) ( )       ∇         ⋅ -∇ = ⋅ ∇ T T T k q ROSS SB r 3 3 16σ where ( ) ( ) ( ) ∫ ∫ ∞ ∞ = 0 0 , , 1 1 ν ν ν ν κ ν d dT T dB d dT T dB T k ROSS [2]
We have implemented this method in Forge3® in order to test the accuracy of this approximation. This software is based on the Galerkin weak formulation. As a consequence, the heat balance equation can be reformulated in a matrix form :

[ ] [ ] [ ] Q T K dt dT C = + [3]
which is solved by a diagonal preconditioned conjugate gradient method. For more details, one can refer to [MAR1997]. The validation case is a semi infinite sheet in P.E.T. heated with a constant heat flux imp φ of 20 000 W/m² (about three ovens with for each seven lamps) on its front face (fig. 4). Taking into account the thermal conductivity and radiation absorption using Beer Lambert law [4], an analytical expression for temperature [5] can be calculated [MON2001] : In figure 6, the temperature evolution for the analytical case and numerical one are plotted. The temperature computed on the front face is greater than the analytical one (maximum of 40%). This difference is due to the fact that the analytical model takes into account the radiative absorption (Beer-Lambert law) in the temperature calculation, whereas in the same time, the numerical model (i.e. the Rosseland approximation) assumes an equivalent conductivity representing only 1.5% of the thermal conductivity of the P.E.T. Consequently, the heat flux imposed on the front face is not diffused in the sheet thickness, but in a very thin layer ( ROSS k , and so the optical thickness is about 150 µm). This comparison demonstrates the necessity to develop a new method in order to solve radiation heat transfer. Thus, a ray tracing method has been implemented to improve the numerical model. As a preliminary result (fig. 7), this technique allows us to compute the view factors between a preform and a punctual source.

y imp e κ φ φ - = [4] ( ) [ ]                 - +                 + +                 + - = =

III-Mechanical model

As Gorlier [GOR2001] showed in his PhD, the PET is assumed to have a hyperelastic rheological behavior above its glass transition temperature. This is characterized by the existence of a deformation energy W function of the transformation gradient tensor F . The classical relationship for the Cauchy stress tensor versus W if we consider PET as an incompressible material is [GER1973] :

1 2 1 ' 2 2 - ∂ ∂ - ∂ ∂ + - = B I W B I W I p σ [6]
In a first approach to our study, the Mooney-Rivlin potential has been chosen for its simplicity and its capacity to model the PET behavior :

( ) ( ) [ ] 3 3 2 1 - + - = I I C W α [7]
Finally, the stress tensor is obtained by :

' ' 1 ' 2 2 σ α σ + - = - + - = - I p B C B C I p [8]
This rheological model has been integrated into Forge3® using a velocity/pressure updated Lagrangian formulation together with tetrahedral elements. The Galerkin method could be written, neglecting gravity and inertia contributions :

= - ⋅ ∫ ∫ Ω Ω ∂ dV dS v S * . * : ε σ [9] with * v a virtual velocity field.
This formulation is discretized in space using P1+ / P1 tetrahedral finite elements. This element allows to calculate the velocity and the pressure in each node additional velocity degrees of freedom ("bubble" type degrees of freedom that are interpolated linearly in the four sub tetrahedral (fig. 8)) in the middle of the element. In this case, the Galerkin method gives a second equation in the case of the bubble and finally, a third one for the incompressibility assumption ( )

0 * ' = + ∫ Ω dV b v div p [10]
with v the node velocity and b the bubble one.

The discretization of [9] results in a non linear set of equations, the unknown of which are the nodal velocity field and the nodal pressure field. This set of equations is solved by a Newton-Raphson method, requiring an analytical form of the tangent matrix.

( ) ( )

        ∂ ∂ + ∂ ∂ ∆ - + ∂ ∂ ∆ = ∂ ∂ ∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆ + lq s m t t
x N F F x N F F t C F F x N t C v v δ δ ε ε ε α δ δ σ 0 0 2 0 ' 2 [12]
F is then computed using the following relationship :

n i k n m oi k m ij X N X N F ξ ξ ∂ ∂         ∂ ∂ = -1 [13]
This formulation will be the basis of all the future development to solve the system.

For the validation, an analytical solution has been developed in order to assess the numerical model. An axi-symmetrical thick tube is inflated using an imposed internal pressure (fig. 9). The lower and upper pressure transverse sections of the tube are supposed to remain in fixed horizontal planes (in other words, the height of the tube section is fixed). The polymeric tube is assumed to have a Mooney-Rivlin behavior. The following relationship exists between the radial coordinate R at current time t and o R the initial one t = 0 [SCH2000] :

2 2 2 2 2 2 2 R R r r R r R r o o o o + - = ⇒ - = - [14]
Deriving [14], we calculate the gradient tensor F [15] and then the Cauchy stress tensor σ . ), -an inflation pressure is applied on the internal surface of the tube with a linear variation from 0 MPa at 0 second to 0.6 MPa at 1 second, -the initial mesh has 42 nodes and 101 elements (figure 10), -the initial internal radius is 5mm and the initial thickness is 2mm. In figure 11, both numerical and analytical internal radius versus pressure are plotted as well as error. The agreement is good (maximum error 4%).

IV-Blow molding computations

In a preliminary computation, an isothermal free inflation simulation of a preform has been performed using a neo-Hookean behavior. The parameters used for this calculations have been referenced in Table 1 and the mesh preform in figure 12. As a first result (fig. 13), this calculations took a long time to be achieved but allowed finally to obtain a good result. Moreover, the relaxation phase due to the hyperelastic behavior could be checked numerically. Indeed, if the pressure applied inside the preform decreases to zero, then, the preform relaxes to its initial shape. The shape of the preform during free inflation will be soon compared with experimental measured realized using a fast CCD camera. In a second example, a confined blow molding simulation has been lead. In order to simplify as much as possible and to estimate the numerical parameters bound to the simulation of the process, the constitutive behavior is assumed as a Newtonian law during the blow molding simulation. Indeed, it allowed to reduce efficiently the CPU time.

Neo-Hookean parameters

The mould used is a prototype one developed in the CROMeP (Table 2). It produces 50cl water bottle. As for the tube inflation in part III, just an eighth of the preform has been mesh Figure 14 : Geometries of the preform and the mold.

The followings results (fig. 15) showed that Forge3® is able to model the blow molding process. Indeed, the calculations allowed to estimate with precision the evolution of the shape of the preform during the inflation. Moreover, the software defined perfectly the contact between the mold and the material at any moment of the calculations.

The future works will be to take into account a Mooney-Rivlin law for the P.E.T. material in order to obtain a numerical thickness distribution at the end of the process. This one could be then compared with experimental data. Moreover, the stretch step will be soon added to the model. The final purpose will be to find with the simulation of the stretch blow molding process, the influence on the thickness distribution of parameters like the speed of the rod or the value of the blowing pressure.

Figure 1 :

 1 Figure 1 : Scheme of the injection blow molding process.

Figure 2 :

 2 Figure 2 : Infrared heating experimental set-up.

Figure 3 :

 3 Figure 3 : Infrared heating of the perform.

Figure 4 :

 4 Figure 4 : Semi infinite sheet heated by a constant heat flux.

Figure 5 :

 5 Figure 5 : Mesh used for the temperature calculation (°C).

Figure 6 :

 6 Figure 6 : Evolution of temperature in polyethylene sheet.

Figure 7 :

 7 Figure 7 : Distribution of the rays emitted by a punctual source.

Figure 8 :

 8 Figure 8 : P1+/P1 element.

Figure 9 :

 9 Figure 9 : Hyperelastic tube inflation.

  the external radius at current time t. This analytical model enables to obtain the internal radius versus the pressure.The numerical computations have been processed using the following boundary conditions:-neo-Hookean hyperelastic behavior (i.e.

Figure 10 :

 10 Figure 10 : Section of inflation tube mesh.

Figure 11 :

 11 Figure 11 : Internal radius variation versus pressure.

Figure 13 :

 13 Figure 13 : Intermediate preform free inflation (CPU time = 6 hours).

Table 1 :

 1 Calculations parameters.Figure 12 : Mesh of the preform (44 403 nodes, 223 155 elements).

Acknowledgements

Nestlé Waters supports this work. Special thanks go to L.G. for the English correction.

V-Conclusion

A numerical thermo-mechanical model has been developed using Forge3® velocity/pressure formulation and tetrahedral elements. From a thermal point of view, the calculation lead proved that the Rosseland approximation is not adapted to estimate the amount of energy absorbed inside the P.E.T. plastic. Indeed, as the Rosseland conductivity represents less than 2% of the thermal conductivity, the sheet is too much heated on the face submitted to the heat flux. As a consequence, the thermal gradient in the thickness of the material is different from the analytical one. To solve this problem, a complete ray tracing method is being implemented to calculate the irradiative flux value used for the heat balance equation. Both the exchange of energy between the infrared oven and the irradiative surface of the preform, and the amount of radiation energy absorbed inside the material will be thus accurately calculated. From a mechanical point of view, a Mooney-Rivlin hyperelastic law behavior has been implemented with a complete finite element analytical formulation. In the same way, the model of a plan strain blowing of a tube has been developed in order to validate the calculations. It allowed to validate the implementation did. Future works will be to implement the more complex potential of Edwards-Vilgis, which is more representative of the PET behavior heated up to its glass transition temperature. Those next developments will enable us to model accurately the complete PET bottles fabrication process, namely, the infrared exchange in the oven and the stretching and blowing step.