H Louche
email: herve.louche@univ-savoie.fr

F Piette-Coudol

R Arrieux

J Issartel

An experimental and modeling study of the thermomechanical behavior of an ABS polymer structural component during an impact test

Keywords: ABS polymer, impact test, kinematics and thermal fields, thermomechanical couplings, finite element model

An experimental and modeling study of the thermomechanical behavior of an ABS polymer structural component during an impact test is presented. The structural component was a heel of a woman's shoe made of ABS polymer material reinforced or not by a pin. Kinematics and thermal full field measurement techniques were used to observe the material and structural component during preliminary experimental tensile and impact tests. With the kinematic fields it was possible to identify the stress-strain response, which takes the necking localization into account. Positive volume variations were also observed during these tensile tests, which were associated with the crazing damage mechanisms in this type of polymer. The thermal fields measured during these tests showed high temperature variations (a few K to 25 K) in the zone where strain was localized.

The associated thermal softening, estimated by the stress strain responses at various imposed room temperatures, was taken into account in the Johnson Cook material model. This model was selected because it can easily include the strain, strain rates and temperature effects. A thermomechanical simulation was built. This weak coupled model is based on the assumption of both the adiabaticity of the problem and a fixed ratio (90%) of the local volume plastic power of the heat sources.

The finite element model was restricted to the first impact test. A specific instrumentation of the machine was performed to validate the model. Displacements, linear and angular velocities and impact load were measured. In addition to these local data, kinematics and thermal full field measurements were proposed and provided a rich spatiotemporal database to compare the experimental and numerical results. A qualitative agreement for the strain distribution and an underestimation of about 20% of the impact load was finally obtained by the model.

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INTRODUCTION

Polymers are often used in applications where impact resistance is required. In this context, specific polymers such as ABS (acrylonitrile butadiene styrene) have been created with rubber particles in order to increase the toughness. The particles act as craze stabilizers, thereby delaying crack initiation and increasing the strain to failure [START_REF] Socrate | A micromechanical model for multiple crazing in high impact polystyrene[END_REF][START_REF] Ramsteiner | Influence of void formation on impact toughness in rubber modified styrenic-polymers[END_REF][START_REF] Seelig | Localized plastic deformation in terny polymer blends[END_REF]. The deformation of a structural component or a tensile sample during characterization tests can be accompanied by intense strain localization and more or less temperature variations, depending mainly on the material and strain rates [START_REF] Rittel | On the conversion of plastic work to heat during high strain rate deformation of glassy polymers[END_REF][START_REF] Fond | Localisation des déformations et mécanismes d'endommagements dans les polymères multiphasés[END_REF][START_REF] Sell | Experimental characterization of deformation damage in solid polymers under tension, and its interrelation with necking[END_REF][START_REF] Wattrisse | Thermomechanical effects accompanying the localized necking of semi-crystalline polymers[END_REF][START_REF] Billon | Effet de couplage thermomécanique dans la caractérisation du comportement de polymères solides[END_REF].

However, these characteristics are not systematically taken into account when the material mechanical models are designed [START_REF] Billon | Effet de couplage thermomécanique dans la caractérisation du comportement de polymères solides[END_REF]. The main reason is that competition between the strain rate hardening and the temperature is hard to characterize in experiments. The tests are supposed to be homogeneous in order to simplify identification. This hypothesis is open to criticism in certain situations. For example, [START_REF] Billon | Effet de couplage thermomécanique dans la caractérisation du comportement de polymères solides[END_REF] considers that the effect of intrinsic dissipation is significant for polymers as of 10 -2 s -1 . Indeed, the thermomechanical couplings in polymers due to the irreversibilities (viscosity and/or plasticity and/or damage, etc.) generate heat sources (intrinsic dissipation), leading to local rises in temperature. These temperature variations, whose intensity depends on the heat sources and heat conduction problem, can, according to their intensity and the thermal softening of the material, modify its flow and strain rate. In this context of heterogeneous tests, full field measurements (kinematics and thermal), while bringing additional local thermomechanical information (displacements, strains, strain rates, temperatures, dissipation, etc.) must be exploited to improve the identification of material models in sample tests and the validation in structural component tests. Moreover, few thermomechanical modeling studies have been conducted during impact test loading [START_REF] Cros | Experimental study and numerical simulations of the impact behaviour of polycarbonate and polyurethane multilayer[END_REF][START_REF] Duan | A uniform phenomenological constitutive model for glassy and semicrystalline polymers[END_REF][START_REF] Duan | Analysis of multiaxial impact behaviour of polymers[END_REF][START_REF] Duan | Impact behavior and modeling of engineering polymers[END_REF][START_REF] Dean | Modelling the behaviour of plastics for design under impact[END_REF][START_REF] Dean | An evaluation of the use of finite element analysis for predicting the deformation of plastics under impact loading[END_REF]. Among these, the most advanced was the DSGZ model proposed by 10]. Their phenomenological termoelastoviscoplastic model, defined by eight parameters, took into account the loading mode, through a ponderation of the hydrostatic pressure. Implemented with a Vumat program in the Abaqus/Explicit software, the resolution was conducted using a weak coupled adiabatic thermomechanical problem, including a fixed ratio (90%) of plastic power converted into heat.

The model was then applied to a multiaxial impact behavior on a glassy ABS polymer structural component [START_REF] Duan | Analysis of multiaxial impact behaviour of polymers[END_REF][START_REF] Duan | Impact behavior and modeling of engineering polymers[END_REF].

The objectives of this paper are: 1) to present the different steps of a thermomechanical study of an industrial impact test on an ABS polymer structural component; 2) to show how thermomechanical field measurements improve the modeling of this study.

The structural component is a heel of a woman's shoe. This kind of product and its commercial success closely depend on shoe fashion. Industrial stakeholders have to propose competitive products at the right time. Numerical tools, like structural component finite element simulation, can help this industry to optimize their products, mainly through geometry and material choices, and reduce the design time. A practical impact test is currently used to validate the design choices. This test involves using a pendulum. The base of the heel is fixed, impacts are applied at the end of the

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3 heel in the normal direction along the principal axis. To be validated, the heel has to withstand a number N of impacts, typically N=300. This practical test can be critical for some heel solutions and, in these cases, can substantially increase the design time. A numerical model of the real impact test is thus required in order to optimize the design choices (mainly the geometry and the material) of the heels and to reduce the design time. The present study relates only to a slanted heel type, without or with a reinforcement pin, and made by plastic injection with an ABS material.

The model used in this study was restricted to the "first impact". This choice was based on: (i) the complexity of the deformation mechanisms, with damage due to crazing during dynamic loading; (ii) high dispersion in the number of cycles at failure. Even if the rupture of the structural component happens at the end of some impacts (classically a few tens), the first impact is often very critical for certain geometrical designs. The model of the first impact is thus essential for continuation of the design project. The prediction, beyond this first impact, of the structural behavior is complex and would require the development of oligocyclic fatigue models.

The paper is organized as follows: in Section 2 we describe the thermomechanical tensile test experiments used to identify the material constitutive model described in Section 3. A specific instrumentation of the pendulum developed to validate the model is presented in Section 4. We then present the finite element model in Section 5 and the main results in Section 6.

Complementary information on this study is available in [START_REF] Piette-Coudol | Modélisation et simulation de la tenue aux chocs des talons de chaussure femme[END_REF].

TENSILE TEST EXPERIMENTS

The tested ABS was a commercial polymer (Novodur P2M Grade). The specimens used for identification were flat, with a reduced gauge length of L*l*e=80*10*4 mm 3 . Injected at one injection point, they were made under process conditions close to those used to make the structural component (via injection).

During the impact, the structural component was primarily subjected to bending, thus inducing loading of traction and compression. The ideal would thus be to carry out compression and tensile tests at the same time. However, compression tests are difficult to perform. They require cylindrical test specimens and a restrictive hypothesis to estimate the stress. Moreover, only tensile tests were conducted because the traction state was more severe for crazing development. In order to study ABS under the same conditions (strain rates and temperatures) as during the impact test, tensile tests at various strain rates and temperatures were thus carried out. A large strain rate range was used, from quasi-static (10 -3 s -1 ) to higher values (2 s -1 ), which can be reached at the time of the impact. Tests with imposed temperature were carried out for temperature values ranging from 20°C (room temperature) to 60°C.

Global measurements

Global load displacement curves, during tensile tests, are plotted in Figure 1. In each situation (strain rate or imposed temperature) three samples were tested, the results presented in this figure are representative of these tests.

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An environmental chamber was used for tests at a "fixed" imposed temperature. Figure 1a shows the high sensitivity rate of the material, with an increase in the yield stress and plastic flow with the strain rate. An increase of the Young modulus with an increase of the strain rate can also be noticed. This phenomenon can be associated to viscous effects in this apparent "elastic" loading. Such dependence was not taken into account in the following of this work. Conversely, thermal softening is observed in Figure 1b for three temperature values, at a constant imposed strain rate. The strain rate hardening and thermal softening effects for the yield stress (engineering stress) is plotted in Figure 1c. This yield stress pattern, associated with the beginning of the macroscopic irreversibility of the material, will be used later for identification of the material constitutive model.

a) b)

Figure 1: Global mechanical response for: a) various strain rates

(V 1 =2 s -1 , V 2 =1 s -1 , V 3 =0.2 s -1 , V 4 =3.10 -2 s -1 , V 5 =1.10 -2
s -1 , V 6 =1.10 -3 s -1 ), b) various imposed ambient temperatures (T 1 =40°C, T 2 =50°C, T 3 =60°C), at 0.1 s -1 . c) Yield stress vs.

strain rates, for various imposed room temperatures (T 1 = 27°C, T 2 =40°C, T 3 =50°C, T 4 =60°C).

Kinematical fields

The displacement fields u=(u 1 (x 1 ,x 2 ), u 2 (x 1 ,x 2 )), where x 1 and x 2 are respectively the axial and transverse position, were obtained using a digital visible camera (Hamamatsu, 1280 × 1024 pixels, maximum sampling rate 9 Hz, ambient lighting) and a digital image correlation (DIC) processing (7D ® software, see [START_REF] Vacher | Bidimensional deformation measurement using digital images[END_REF][START_REF] Coudert | Reconstruction tridimensionnelle du volume intérieur d'une chaussure[END_REF]). This correlation approach can

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be used to measure strain fields, deduced from the displacement fields, from 0.01% to 300%. The displacement accuracy of the DIC approach was typically about 0.01 pixels for deformations under 5%, but this value depended on many parameters, e.g. image quality, strain gradient and the order of the transformation describing the local deformation. The grey level interpolation chosen in the software was bicubic and the strain fields were computed on the basis of bilinear expression of the spatial displacement distribution. The optimisation method used to find the minimum correlation function was a gradient method. With the above characteristics, and a correlation pattern of 12x12 pixels, errors under 0.01 pixels were obtained in the displacement fields for: a) rigid body motions imposed on synthetic images, and b) for homogeneous deformation imposed on synthetic images [START_REF] Coudert | Reconstruction tridimensionnelle du volume intérieur d'une chaussure[END_REF]. All the 7D ® strain results were obtained using the Green-Lagrange strain tensor E. In the following, the axial component E 11 of this tensor or the Logarithmic axial strain ε 11 will be used, as determined from the local axial displacement u 11 (expressed in the initial configuration).

For each test, it was then possible to study the strain fields and to specifically analyze localization phenomena and volume variations.

Strain localization detection

The measured kinematic fields enabled detection of the position and intensity of the strain localization (Figure 2). For the ABS polymer tensile tests, the localization was associated with necking, which was located in a fixed position at all times (here close to point 1 plot in Figure 2a). Knowledge of the kinematics fields at all acquisition times allowed us to take the localization into account [START_REF] Wattrisse | Thermomechanical effects accompanying the localized necking of semi-crystalline polymers[END_REF][START_REF] Tabourot | Numerical determination of strain localization during finite element simulation of deep drawing operations[END_REF]. It was obtained while focusing on the local stress strain response of measurement points centered on cross-sections distributed along the axial direction of the sample (Figure 2-a). The stress was the "true stress" obtained with the local axial strain measured at these points. The local mechanical responses showed that the sample behaved like a structural component. The deformation was only elastic in certain parts of the sample remote from the necking zone. On the other hand, those located in the necking zone underwent very high unrecoverable deformations of up to 65% (Figure 2-c). The incompressibility assumption used to calculate the "true stress" will be discussed hereafter. Moreover, it was supposed, which is no longer true when necking is strongly localized, that the stress state remained uniaxial and homogeneous on the cross section. In the following, the stressstrain response used to derive the material behavior will be identified using the local strain, measured in the necking area where higher strain values were obtained.

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Volume variation study

Due to crazing phenomena, deformation of the ABS polymer was usually accompanied by a volume variation. The main mechanism associated with this damage phenomena was cavitation near rubber particles, which generates volume variation [START_REF] Fond | Localisation des déformations et mécanismes d'endommagements dans les polymères multiphasés[END_REF][START_REF] Sell | Experimental characterization of deformation damage in solid polymers under tension, and its interrelation with necking[END_REF]. This could be estimated by the formula

33 22 11 0 ε ε ε Δ ε + + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = V V ln V
where ii ε is the logarithmic strain in the principal direction i, i =1 is the loading axis. Using the measured kinematic fields and assuming strain homogeneity in the transverse direction ) ( 33 22 ε ε =

, it was then possible to plot the volume variation patterns (see Figure 3a). The volume variation increased from the very start of the plastic range, which was not the case for all the polymers (see [START_REF] Sell | Experimental characterization of deformation damage in solid polymers under tension, and its interrelation with necking[END_REF] for example).

In our study strains in 22-and 33-directions were equalled. As shown in Fig2a, the observed plane by the camera is [START_REF] Socrate | A micromechanical model for multiple crazing in high impact polystyrene[END_REF][START_REF] Ramsteiner | Influence of void formation on impact toughness in rubber modified styrenic-polymers[END_REF], so it's was not possible to measure the ε 33 strain field. This should be possible with a second camera oriented in the 3 direction or using a mirror oriented with an angle of 45° in order to reflect the image of the "thickness side" of the sample in the unique camera. These volume variation observations must be taken into account to calculate stresses. Indeed, the passage of the total response (load, displacement) to the local responses (stress, strain) can be obtained by correcting or not the cross section of the sample. Three assumptions are classically used: The mechanical responses were very different (Figure 3b). The elastic part was the same, then a difference appeared after the yield point. The hypothesis of deformation with incompressibility implied a higher flow stress value. The response obtained with hypothesis 3 was located between the two other responses. This hypothesis was based on the cross section variations obtained only by external observations: strains ε 22 and ε 33 measured on the external surfaces.

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However, crazing damage due to porosity development around rubber particles may decrease the effective cross section area. Such a mechanism was compatible with the volume variation measurement given in the previous part. A better stress estimation could be obtained with a damage model, and the mechanical response would be located between the responses given by hypothesis 2 and 3.

In this work, a thermoelastoviscoplastic model, assuming volume conservation, was used to simulate the mechanical behavior of the impacted structural component.

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Thermal fields

The temperature field T recorded during a test was obtained with a fast multidetector infrared camera (CEDIP Jade III MW, 145 Hz maximum frame rate). This camera had a noise level of less than 20 mK, and a spatial resolution of with intense dissipation in the necking zone during failure. In the two other points, the temperature increase is mainly due to thermoelasticity linked to elastic spring back.

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MATERIAL CONSTITUTIVE MODELING

Johnson Cook model

In this study, the material constitutive model was developed on the basis of tensile tests. Other tests, like compression tests, which are harder to interpret, could ultimately supplement this identification.

The use of finite element software for this impact test study on a structural component determined the choice of material constitutive model. Among the possible models available in this software, none could simultaneously take the damage, strain rate sensitivity and temperature into account. We thus decided to use elastoviscoplastic models of behavior that can take thermal softening into account.

In this context, the Johnson Cook model was selected:

( m pl pl T ln . C B A n - ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = 1 1 0 ε ε ε σ & & ) , ( 1 a
T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS 10 ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = transition melting transition pl T T T T B A m n ln 1 ln ε σ (2)
The identification of m was then achieved at the yield stress, at different temperatures (40°C, 50°C and 60°C) in the various tests, and a mean value of m was finally chosen.

The identified values of the eight model parameters are indicated in the Table 1 below.

The strain rate sensitivity parameter C was identified in the strain rates range [V4=10 -3 s -1 , V6=3.10 -2 s -1 ]. This range was limited to 3.10 -2 s -1 because it was not possible to measure the strain fields above 3.10 -2 s -1 , the visible images were too fuzzy. An example of the identified stress strain curves for the two strain rates V4 and V6 are plotted in Figure 5.

A B C

Validation in a tensile test

Experimental and numerical simulations of a tensile test were carried out on a notched sample (Figure 6a) in order to locate necking at the center of the specimen. Thermal variations in the sample were calculated using a thermomechanical model based on the adiabaticity hypothesis, compatible with the thermal diffusion of the material and the

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test duration. The heat sources used in the heat diffusion equation were fixed, i.e. 90% of the local plastic power. The specific heat of ABS, provided by the manufacturer, was 1500 JKg -1 K -1 , the volume mass ρ=1050 Kg m -3 and the thermal conductivity k=0.17 W m -1 K -1 .

Only half of the geometry was chosen for the numerical model (Figure 6b). Solid elements (bricks) were used, with a higher density of elements in the sample medium (Figure 6c). One side of the sample was embedded, the other was connected to a mobile part in order to impose the displacement. This one was gradually carried out to avoid elastic impact at the beginning. The global load displacement response is plotted in Figure 6d. Simulated and experimental responses were equal until the yield stress. Then, since damage was not taken into account in the model, the apparent experimental softening behavior was not well reproduced, i.e. the calculated loads were higher than the measured loads. The local calculated and observed results were compared just before necking (Figure 7), at a global displacement of about 3.18 mm and an imposed strain rate of 2.10 -2 s -1 .

Measurement of field kinematics carried out with the 7D ® software showed a strain localization in the center of the sample (Figures 7a,b). The logarithmic strains reached 30% in the no remained weak deformed. The same distribution was obtained for the calculated strain fields, with a maximum of 32% of the logarithmic strain at the center of the sample (Figure 7c). For the same tests, temperature fields were simultaneously recorded with the infrared camera. An example of the measured temperature field is given in Figure 8a, with an axial and transverse profile showing the marked increase of temperature in the reduced section (Figure 8b). This field was recorded just before failure. The maximum temperatu reached was about 43°C. This value represented a temperature variation of +17°C compared to the room temperature (26°C). However, it co

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Machine

An uniaxial impact test was conducted with a pendulum machine (Figure 10a). A mass of 1 kg, placed 0.5 m from the rotation axis, imposed (by gravity) rotation of the clapper and displacement of the striker. This latter struck the polymer structural component so that the impact occurred on the vertical plane of the clapper and in the normal direction with respect to the main axis of the structural component (Figure 10b). The machine was equipped with a driving system and a clutch to enable repetition of the impacts. The frequency usually used was 30 impacts per minute, i.e. a frequency of 0.5 Hz (Figure 11a). The base of the structural component was embedded in an aluminum alloy at low melting point (Figure 11b).

Measurements

The modeling of this impact test complied with the characteristics of the machine in order to obtain results close to the real situation. Instrumentation of the testing machine was thus carried out in order to obtain experimental data to validate the model.

Local measurements

Generally, the displacement and load were measured, and this information was obtained through various sensors. In this study, we chose a method based on three types of sensor: a piezoelectric load sensor (PCB 208C04), a piezoelectric accelerometer (Kistler, 500 g) and a gyroscope. We thus measured, respectively, the impact load, the acceleration of the striker and the angular velocity of the clapper (Figures 12a,b). Finally, with the two last sensors it was possible, by time integration, to measure the displacement of the striker and the angular position of the clapper, respectively. An example of a load signal during an impact, with an horizontal initial position of the clapper (angle 90° from the vertical position), is plotted in Figure 12c. Other techniques to measure the load are available, [START_REF] Seidler | Application of the Instrumented Impact Test to the Toughness Characterization of High Impact Thermoplastics[END_REF]21] used strain gauges stuck on the striker.

The linear velocity during the impact was obtained by integration of the horizontal acceleration γ:

∫ + = t dt t V t V 0 0 ). ( ) ( γ . ( 3
)
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The constant V 0 , corresponding to the linear velocity just before the impact, was estimated with the angular velocity given by the gyroscope. After various tests, the angular velocity before the impact was estimated at 2.1 rad.s -1 , which corresponded to V 0 =1.46 m.s -1 .

The displacement of the structural component in the impact zone during the impact was obtained by a double time integration of the acceleration. Other experimental techniques can be used to measure displacements at the time of the impact tests, e.g. techniques based on optical lasers are available [0] but more expensive to implement.

A value of 0.73 kg.m 2 for the inertia of the clapper was estimated using CAD software to model the geometry. It was possible to estimate the energy (kinetics) of the impact (1.61 J) with this value and the angular velocity. Due to the friction in the kinematics chain of the clapper, we noted that the value of the impact energy was markedly inferior to the energy (potential) of a perfect machine (5 J). Close-up of the gyroscope, fixed in the clapper, used to measure its angular velocity. c) Impact load response measured during an impact test at 90° on a slanted heel with a pin.

The permanent displacement of the end of the structural component during repetition of the impact was also used to validate the model. This displacement was measured with the 7D ® DIC software using sticky labels of random gray levels (Figure 13). Pairs of recorded images, at the initial stage and after an impact, were analyzed by the software to estimate the displacement after impact. Only the horizontal displacement component at the center point of the upper

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17 sticky label is reported here. Figure 13b also shows a damage zone (called "swiveling zone"), close to the support, where crazing phenomena took place. For this type of structural component, this zone was the future site of failure after repeated impacts.

Images were taken after the first impact and then after every ten impacts, when the clapper was up and oscillations of the heel damped. The tests were repeated and allowed us to compile a database on horizontal permanent displacement of the structural component during the impact test. An example of the displacement patterns is plotted (Figure 14a) for various tests carried out on 10 of the same type of heels (slanted and reinforced by a pin). Displacements increased very quickly as of the first 10 impacts, and then the variations leveled off and decreased. The dispersion of the results was weak at the beginning of the test, but progressively increased with the repetition of the impacts, with variations of up to 7 mm for the same number of impacts. We noted that only three heels, among the 10 tested, reached the required 300 impacts.

Due to the difficulty in modeling the repetition of impacts, as well as the substantial dispersion of the experimental results, we were limited to studying and simulating only one impact at a time. In order to be able to compare the simulation results with the experimental results, we observed permanent displacement associated with the first impact.

We noted a mean value of 1.1 mm and a standard deviation of 0.19 mm (Figure 14b).

Full field thermal measurements

In order to gain insight into the extent thermomechanical couplings in the studied polymer (ABS), we monitored temperature changes during an impact test on various heels. Using an infrared camera, surface temperature variations of the structural component and temperature patterns during impacts in the heating zone were studied. These observations revealed critical zones in the structural component, which corresponded to the zones of failure in the heel. Such zones differed according to the heel geometry. Figure 15 shows a typical result of thermal fields recorded during repetition of impacts on a slanted heel. The heel temperature at the beginning of the test corresponded to the room temperature, i.e. 22.7°C. The temperature increased sharply at the beginning of the test because of the important irreversibility mechanisms involved in the impact and because of the weak heat conduction of the polymer. Then, due to the heat losses, the variations became less marked and a plateau appeared for many impacts. Temperature fields were recorded on the heel surface, so it is likely that heating was greater at the core of the structural component. For this type of heel, two heating zones were observed during repetition of the impacts. In the first one, i.e. the impact zone, marked temperature variations (about 18°C after 208 impacts) were mainly due to friction, viscoplasticity and damage mechanisms. The second one, which was critical for the structural component because it turned out to be the site of failure, was the "swiveling zone", which was located at the bottom of the heel. Temperature variations here were about 12°C after 208 impacts.

The impact test is a critical experimental evaluation for the heel design. The objective of this study was to replace visual observations in real tests by a numerical model analysis in order to accelerate the design cycles and reduce costs.

Instrumentation of the machine was necessary to monitor and understand the test. We thus collected information during the impacts on: the force, velocities, global displacements, displacement and thermal fields. This experimental information was necessary to build and then validate the numerical model.

The thermal fields showed substantial temperature variations (more than 10°C at the surface) in the future failure zone.

Previous experimental results obtained during tensile tests demonstrated that such temperature variations were not

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negligible for the material. A test conducted at 40°C, compared to a lower one conducted at 27°C, led to a 4 MPa decrease for the yield stress σ y . This indicated that thermal softening of the material should be taken into account. In order to approach a more realistic temperature distribution in the structural component, we also simultaneously attempted to solve the thermal problem in adiabatic conditions.

FINITE ELEMENT MODEL

Geometry

A presumably rigid clapper with a known angular velocity applied an impact on the heel model. The impact energy is a function of the inertia and angular velocity.

The two parts of the model, i.e. the clapper and heel, were obtained by CAD. The geometry of the clapper was simplified as compared to the real one. The striker was modeled by a perfect cylinder. In the case of the slanted heel, the geometry of the heel base was truncated to facilitate mesh generation and to avoid excessive distortion of the elements in this zone. Static elastic simulations have previously shown that stresses were weak in this zone. The geometry of the heels did not take into account the part embedded in the alloy. The heels were thus shortened a 10 mm height.

Mesh

Rigid shell elements were used for the external surfaces of the set clapper-striker (Figure 16). Solid elements, brick type with 8 nodes and reduced Gauss integration, were chosen for the heel. To optimize the time for computing, the density of the elements changed along the axis of the heel. There was a high density at the point of impact and in the "swiveling zone", a coarser mesh was applied in zones with a weak stress gradient. For the striker, the cylindrical end was the critical part and thus contained more elements than the remainder of the clapper.

The pins used to reinforce the slanted heels were made of steel (C60). They were developed starting from sheets which were folded to obtain the cylindrical form (Figure 17a), then quenched and tempered at 440 °C. On the pin surface, we noted some marks which increased the adhesion with the injected ABS polymer. Moreover, the pin was not a perfect cylinder, it had a split throughout its length. However, static elastic finite element simulation of a pin (split and not split), embedded at one extremity and subjected to a radial load at the other end, showed that the presence of the split was negligible for the end deflection (about 2% variation). Thereafter, in order to simplify the mesh, the pin was thus modeled by a cylinder without a split. A study to choose the type of elements (shell or solid) was carried out. Solid elements were selected (Figure 17b). An optimum of two elements in the thickness direction was obtained, which was a good trade-off between the accuracy and the computing time.

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Initial and boundary conditions

In order to decrease the computing times, the impact was carried out in only one stage, i.e. the approach phase was not simulated. At t=0 s, the striker was positioned just before the impact, with an angle of 0.1° of the vertical position. Its angular velocity was determined before and controlled by a reference node with a nonzero value for its rotation degree of freedom and zero values for the others. The heel was embedded at its base. The duration of the simulated impact was 0.04 s.

RESULTS AND DISCUSSION

In this part, various simulations are presented: slanted heel without a reinforcement pin then with a pin. For each simulation, the results are discussed and compared with the experimental data.

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Slanted heel without pin

The slanted heel did not support even one impact without a reinforcement pin. We thus simulated an impact at weaker energy, with a starting angle of 45°, with a speed and an energy identified according to the instrumentation of the machine. The heel was meshed with 8510 elements, the computing time for this simulation was about 2 h 20 min. Von

Mises equivalent stresses are first presented (Figure 18a). Since they changed during the impact, they were plotted at the time when the force and displacement were maximal (time 0.018 s). Two zones of stress concentration appeared, one on the impact zone and the other on the "swiveling zone". Higher stresses were observed in the load application zone. The stress values reached in this zone were strongly dependent on the mesh density (52 MPa for the mesh presented). However, the behavior in this zone seemed to have little effect on the impact resistance of the structural component. The stresses remained low at the base of the heel. In the "swiveling zone", they reached 48 MPa. The yield stress of ABS, increased by the strain rate effect, was apparently not exceeded.

This was checked by looking at the displacements obtained by the numerical model, where no permanent displacement was observed, as in the experiment (see Figure 19a).

The logarithmic strain according to the heel axis estimated by the model are plotted in Figure 18b. When submitted to bending loading, one side was loaded in traction and the other in compression, with maximum strain values of about 2%.

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Monitoring displacement of the heel extremity in the impact direction enabled us to determine if there was permanent strain. This simulated value of the extremity displacement was compared with the measured value (Figure 19a).

The displacement obtained by the simulation corresponded to the horizontal displacement of the heel, for a node located in the impact zone (5 mm from the heel tip). The curves highlighted a difference in the simulated and experimental impact duration: the time of simulated impact was shorter by more than 0.005 s. This 18% variation could be related to the uncertainty of the inertia determination obtained by a simplified CAD model of the clapper and/or to the fact that the damping of the base support of the heel was not taken into account. However, maximal displacements were well estimated. The boundary conditions related to the heel was a perfect embedding without damping, this choice could explain the oscillations observed on the figure after the impact. However, the average oscillation value was centered on the zero ordinate axis, which indicated that there were no permanent deformations of the material. Using the load sensor, it was also possible to compare the forces brought into play during the impact with those obtained by the simulation (Figure 19b). Simulated and experimental variations in the forces were close, with a difference between the modeling and experimental results of just 5%. The duration of the impact obtained by the simulation was slightly shorter than that of the real impact. Such a difference was already observed for the displacement.

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Slanted heel with pin

The Johnson Cook model described in section 3 was used for the heel structural component. The material model used for the pin was an elastoplastic law presented in the form of table in the finite element software. It was identified according to a tensile test which required a modified experimental setup. The yield stress of the pin material identified was 1000 MPa, with an uncertainty of approximately 50 MPa. During injection, the pin was completely embedded in the heel. Numerically, this characteristic was modeled by a solid-type embedded interaction.

Simulations were carried out for an impact with an initial orientation of 90° for the clapper. The maximum stress level was reached at time t = 0.0082 s, the results are presented for both the pin and heel (Figures 20a and20b). The stress localization zones were located in the same places as the temperature localization zones. The stresses in the pin were high compared to those in the heel. The presence of the pin involved a shift in the "swiveling zone", which was located in a lower position than that of an impact on a heel without a pin. The pin seemed to support the entire impact. The yield stress of the latter was exceeded and there was unrecoverable deformation during the first impact, as observed in the experiments. The maximum stresses were reached in the same "swiveling zone" on the side loaded in traction.

Because of the risk of creating and opening cracks during the repeated impacts, this zone was more critical than the compression zone. For the heel, the stress values reached 56 MPa, i.e. close to the yield stress at this strain rate. To determine whether the yield stress was reached, the levels of equivalent plastic strains available in the software were plotted (Figure 20c). These deformations were observed only on the pin, and no unrecoverable deformation was observed in the heel. Simulated and measured displacements at the end of the heel are plotted on Figure 21a. The various experimental tests showed dispersion of the permanent displacement after the first impact. The average measured permanent displacement was 1.1 mm, with values ranging from 0.81 mm to 1.75 mm. The numerical simulation gave an average value of 1 mm, which closely agreed with the preceding experimental values. However, the maximum displacement was not correctly restored, higher displacement was obtained during the experiment. Figure 21b shows temporal variations in the impact load. Differences were observed between the load obtained by simulation and that obtained experimentally. The impact load found by simulation was weaker (variation of 100 N). This variation is in agreement with the observed displacements. The higher values for the measured impact load involved larger displacements. As for the simulation of the heel without pin, the duration of the simulated impact was shorter than in reality.

CONCLUSION

The different development steps of a numerical model for an impact test on a slanted heel structural component are presented in this paper. This work was associated with a global project of time and cost reduction for heel shoe design studies. Preliminary experimental tensile and impact tests were conducted to observe the material and structural behavior. In these two situations, kinematics and thermal full field measurement techniques were used. With the kinematic fields, it was possible to identify the stress strain response, which takes the strain localization (necking) into account. Positive volume variations were also observed during these tensile tests, which were associated with the damage mechanisms of crazing in this type of polymer. The thermal fields measured during these tests showed marked temperature variations (a few K to 25 K) in the zone where the strain localization appeared. At the same time, the temperature variation fields recorded during a repeated impact test showed that marked temperature variations (18 K after 208 impacts) appeared in the "swiveling zone", which was the future zone of failure for the heel structural component. Temperature variations were due to possible irreversible mechanisms: viscosity, and/or plasticity, and/or damage. The associated thermal softening, estimated by the stress-strain response observations for various imposed room temperatures, was high and had to be taken into account in the material model. A Johnson Cook material model was selected because it can easily include the strain, strain rates and temperature effects. Its coefficients were identified using tensile tests at various strain rates and temperatures. In order to predict a more realistic decrease in yield stress due to thermal softening, a thermomechanical simulation was built. This weak coupled model assumed both the A finite element model of the impact test was then built using a commercial explicit finite element software. To validate the model, a specific instrumentation of the machine was carried out. It was thus possible to measure the displacements, linear and angular velocities and impact load. In addition to these local data, kinematics and thermal full field measurements were proposed and gave a rich spatiotemporal database to compare the experimental and numerical results. They supplied, at the surface of the structural component, displacement estimations and temperature variation fields.

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Preliminary simulations on a notched sample were performed and compared with the experimental results to test the model in the simple tensile deformation mode. Global load-displacement responses and strain fields were in line with the experimental results. However, the calculated temperature variations were underestimated. This difference could be attributed to the absence of damage, which was strongly dissipative, in the material model. Simulations, on a slanted heel without a pin, of a weak energy impact test with an initial clapper angle position of 45°, were then carried out. The results closely agreed with the experimental data. Then simulations on a heel reinforced by a pin submitted to an impact test at the nominal measured energy of 1.61 J (corresponding to an initial angle position of 90°) were performed. An underestimation about 20% of the impact load was obtained by the model. A study of sensitivity of the heel and pin geometry, of the elastic properties of the ABS and the steel of the pin, and of the friction coefficient between the striker and the heel showed that the influence of these parameters on the results was less than 5%. Other causes could explain this difference. The first one could be a difference in the ABS material between the heel structural component and the tensile samples used for material identification, which was due to different process parameters during manufacturing by injection. The others could be (i) the aging of the material, (ii) the identification of the material based only on uniaxial tensile tests and the (iii) Mises yield surface used in the Johnson Cook Model. Future experiments, including compression and/or shear tests, on this material could improve the material model. In particular, it could be interesting to use (or implement) other types of models, such as damage models or Drucker-Prager model well adapted to pressure sensitivity of ABS materials [START_REF] Dean | An evaluation of the use of finite element analysis for predicting the deformation of plastics under impact loading[END_REF].

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V 1 =2 s -1 , V 2 =1 s -1 , V 3 =0.2 s -1 , V 4 =3.10 -2 s -1 , V 5 =1.10 -2
s -1 , V 6 =1.10 -3 s -1 ), b) various imposed ambient temperatures (T 1 =40°C, T 2 =50°C, T 3 =60°C ), at 0.1 s -1 . c) Yield stress vs. strain rates, for various imposed room temperatures (T 1 = 27°C, T 2 =40°C, T 3 =50°C, T 4 =60°C). given by the extensometer is 0.09, instead of the local value of 0.65 given by the strain field analysis.

6

 6

Figure 2 :

 2 Figure 2: a) Photograph of the sample with the virtual grid, the position of the three local points and the virtual extensometer, b) logarithmic axial strain field ε 11 observed for a global displacement of 9 mm, at a strain rate of 8.10 -4 s -1 , c) Local response (true strain and true stress) obtained in the three different points of Figure a). The mean axial strain given by the extensometer is 0.09, instead of the local value of 0.65 given by the strain field analysis.

Figure 3 :

 3 Figure 3: a) Volume variation measured during a tensile test at 8.10 -4 s -1 , b) Stress strain responses, for the three hypotheses, at a strain rate of 8.10 -3 s -1 .

•••.

  Hyp. 1: engineering stress, without volume correction: Hyp. 2: true stress, corrected section with the incompressibility hypothesis: Hyp. 3: true stress, corrected section with volume variation [6This formula is based on the assumption that the strain in the transverse ε 22 and thickness ε 33 directions were equal.

Figure 4 :

 4 Figure 4: a) Thermal field, at a fixed time (t=2s, see Figure b), observed during a tensile test at a strain rate of 4.5.10 -2 s -1 , showing a localization at the bottom of the sample (point 1). b) Temporal variations in the temperature at the three spatial points of the previous figure. The high temperature jump (+26°C) observed at point 1, at time 4.3 s, is associated

T

  where σ is the equivalent stress, pl ε is the equivalent plastic strain. The first term of the model describes the plastic behavior of the material, parameter A represents the yield stress, B and n are linked to the plastic flow. The second term reflects the strain rate sensitivity. The last term corresponds to the thermal softening of the material. Below the transition temperature ( ), the yield stress was unchanged. Thermal softening appeared at temperatures ranging between the transition temperature and the melting temperature ( ). The yield stress was null beyond this temperature. Finally, m characterizes the extent of thermal softening. The model parameters were identified according to the experimental results of the tensile tests at various strain rates and temperatures. The hardening parameters A, B and n of the Johnson Cook model were identified on the basis of various tests at room temperature. The temperature sensitivity parameter m was identified during tensile tests at imposed temperatures between and , while the strain rate of transition T melting T 0 ε& , m was obtained by the equation:

Figure 5 :

 5 Figure 5: Identification of the Jonhson Cook model. Comparison of the stress-strain response given by the model and by the experimental results for two imposed strain rates: V4=10 -3 s -1 and V6=3.10 -2 s -1 .

Figure 6 :

 6 Figure 6: a) Geometry of the notched specimen. b) Global view of the mesh. c) Close-up of the mesh in the notched part. d) Comparison of the model and experimental results, under imposed strain rate (4.10 -2 s -1 ) and room temperature (27°C) conditions.

  while the remainder of the sample igure 7: a) Digital image of the notched zone, with the virtual grid used in the 7D ® DIC software. b) Experimental and re rresponded to the heating on the sample surface, whereas the heating within the sample was certainly higher. Heat diffusion along the sample axis was very weak in the ABS material, it covered only a length of 20 pixels (about 2 mm).

  fields of the axial logarithmic strain in the notched zone.

Figure 8 :

 8 Figure 8: Tensile test at 2.10 -2 s -1 . a) Temperature field measured in the notched zone, just before failure. b) Axial and transverse temperature profile (central column and central line of the previous figure)

Figure 9 :

 9 Figure 9: Temperature variations (K) calculated by the Johnson Cook model during a tensile test on a notched specimen.

14 4 .

 4 INSTRUMENTATION OF THE PENDULUM

Figure 10 :Figure 11 :

 1011 Figure 10: a) Pendulum impact test machine, b) Scheme of the support and load conditions applied to the heel structural component.

Figure 12 :

 12 Figure 12: Instrumentation of the machine. a) close-up showing the load sensor used to measure the impact force. b)

Figure 13 :Figure 14 :

 1314 Figure 13: Digital images used by the DIC software to estimate displacements at the end of the structural component: a) initial state, b) after 140 impacts, with an indication of the measured displacement.

Figure 15 :

 15 Figure 15: Temperature field measurement in a slanted heel. a) Temperature field after 208 impacts. b) Temperature variations in the "swiveling zone" (point A) and in a point located at the core of the heel (point B).

Figure 16 :Figure 17 :

 1617 Figure 16: a) Clapper and striker mesh, b) slanted heel mesh (5000 elements)

Figure 18 :

 18 Figure 18: a) Von Mises equivalent stress (Pa) field given by the model. b) Axial component (in the heel direction) of the logarithmic strain.

  The simulation of the heel without a pin gave results that agreed with the experimental findings. The impact carried out with such an angle was purely elastic, as observed experimentally. The displacement and force values corresponded to the experimental values. Nevertheless, there was still some variation between the simulated and experimental impact durations.

Figure 19 :

 19 Figure 19: Model and experimental results for a 45° impact, on a slanted heel without a reinforcement pin. a) Variations in the displacement of a point located at the heel extremity. b) Variations in load during the impact.

T

Figure 22 :

 22 Figure22: Friction coefficient influence on the impact load for a 90° impact on a slanted heel with a reinforcement pin.

  27adiabaticity of the problem and an estimation of the heat sources equaled to a fixed ratio (90%) of the local volume plastic power.

Figure 1 :

 1 Figure 1: Global mechanical response for: a) various strain rates (V 1 =2 s -1 , V 2 =1 s -1 , V 3 =0.2 s -1 , V 4 =3.10 -2 s -1 , V 5 =1.10 -2

Figure 2 :

 2 Figure 2: a) Photograph of the sample with the virtual grid, the position of the three local points and the virtual extensometer, b) logarithmic axial strain field ε 11 observed for a global displacement of 9 mm, at a strain rate of 8.10 -4 s - 1 , c) Local response (true strain and true stress) obtained in the three different points of Figure a). The mean axial strain

Figure 3 :

 3 Figure 3: a) Volume variation measured during a tensile test at 8.10 -4 s -1 , b) Stress strain responses, for the three hypotheses, at a strain rate of 8.10 -3 s -1 .

Figure 4 :

 4 Figure 4: a) Thermal field, at a fixed time (t=2s, see Figure b), observed during a tensile test at a strain rate of 4.5.10 -2 s -1 , showing a localization at the bottom of the sample (point 1). b) Temporal variations in the temperature at the three spatial points of the previous figure. The high temperature jump (+26°C) observed at point 1, at time 4.3 s, is associated with intense dissipation in the necking zone during failure. In the two other points, the temperature increase is mainly due to thermoelasticity linked to elastic spring back.

Figure 5 :

 5 Figure 5: Identification of the Jonhson Cook model. Comparison of the stress-strain response given by the model and by the experimental results for two imposed strain rates: a) 10 -3 s -1 , b) 3.10 -2 s -1 .

Figure 6 :

 6 Figure 6: a) Geometry of the notched specimen. b) Global view of the mesh. c) Close-up of the mesh in the notched part. d) Comparison of the model and experimental results, under imposed strain rate (4.10 -2 s -1 ) and room temperature (27°C) conditions.

Figure 7 :

 7 Figure 7: a) Digital image of the notched zone, with the virtual grid used in the 7D ® DIC software. b) Experimental and c) simulated fields of the axial logarithmic strain in the notched zone.

Figure 8 :

 8 Figure 8: Tensile test at 2.10 -2 s -1 . a) Temperature field measured in the notched zone, just before failure. b) Axial and transverse temperature profile (central column and central line of the previous figure)

Figure 9 :

 9 Figure 9: Temperature variations (K) calculated by the Johnson Cook model during a tensile test on a notched specimen.

Figure 10 :

 10 Figure 10: a) Pendulum impact test machine, b) Scheme of the support and load conditions applied to the heel structural component.

Figure 11 :

 11 Figure 11: a) Scheme of temporal variations in loading. b) Slanted heel structural component embedded in a low melting point alloy.

Figure 12 :

 12 Figure 12: Instrumentation of the machine. a) close-up showing the load sensor used to measure the impact force. b)Close-up of the gyroscope, fixed in the clapper, used to measure its angular velocity. c) Impact load response measured during an impact test at 90° on a slanted heel with a pin.

Figure 13 :

 13 Figure 13: Digital images used by the DIC software to estimate displacements at the end of the structural component: a) initial state, b) after 140 impacts, with an indication of the measured displacement.

Figure 14 :

 14 Figure 14: a) Variations, with the number of repeated impacts, in the permanent displacement measured at the end of the structural component; the 10 curves are associated with the 10 heels tested. b) Permanent displacement after the first impact, for the different tests.

Figure 15 :

 15 Figure 15: Temperature field measurement in a slanted heel. a) Temperature field after 208 impacts. b) Temperature variations in the "swiveling zone" (point A) and in a point located at the core of the heel (point B).

Figure 16 :

 16 Figure 16: a) Clapper and striker mesh, b) slanted heel mesh (5000 elements)

Figure 17 :Figure 18 :

 1718 Figure 17: a) Radial and axial view of a pin. b) Mesh of the pin with a different number of elements in the thickness direction.

Figure 19 :

 19 Figure 19: Model and experimental results for a 45° impact, on a slanted heel without a reinforcement pin. a) Variations in the displacement of a point located at the heel extremity. b) Variations in load during the impact.

Figure 20 :

 20 Figure 20: Model results for a 90° impact on a slanted heel with reinforcement pin. Equivalent Von Mises stresses: a) in the pin, b) in the heel. c) Axial logarithmic strains in the heel.

Figure 21 :

 21 Figure 21: Model and experimental results for a 90° impact, on a slanted heel with reinforcement pin. a) Variations in the displacement of a point located at the extremity of the heel. b) Load variations during the impact.

Figure 22 :

 22 Figure22: Friction coefficient influence on the impact load for a 90° impact on a slanted heel with a reinforcement pin.

Table 1 -

 1 Identified values of the Johnson Cook model parameters

				m	n	transition T	melting T	0 ε&
	39 MPa	48 MPa	0.053	0.879	1.5	300 K	513 K	8.10 -4 s -1

Table 1 -

 1 Identified values of the Johnson Cook model parameters

	Tables

× 240 pixels. The spatial resolution (pixel size), which depends on the focal distance adjustment, was estimated to be close to 0.25 mm for the carried out tests. Prior to testing, the surface of the tube was coated with a highly emissive paint in order to obtain blackbody properties compatible with the calibration law of the camera obtained in the same conditions. Moreover, temperature results were always note relative to an equilibrium room temperature, which led to a temperature variation accuracy of less than 0.1°C.The temperature variation was measured locally on one face of the sample (Figure4a). The temporal temperature variation patterns at three different spatial points showed the thermal history of different zones of the sample used for material identification (Figure4b). This test, which was performed at high strain rate (mean value of 4,5 10 -2 s -1 ), revealed surface temperature variations of up to 26°C in the necking zone. For zones far from the necking localization, the test could be considered as isothermal (temperature variations were less than 1°C).On materials with low thermal diffusion, temperature variations during a tensile test can locally reach a few degrees and consequently modify the behavior of the material 4, 8]. As previously noted for the kinematic measurements, it would be necessary to take the temperature heterogeneity into account in the identification of the material constitutive model.In addition to the irreversibilities detection and the quantification of the temperature variations, the thermal fields also make it possible to establish energy balances[START_REF] Chrysochoos | An Infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF]. Such assessments, which are not presented here, could highlight the fraction of mechanical energy which was converted into heat, i.e. a parameter fixed constant at the time of the simulations presented hereafter.

Acknowledgement

The support of the LMOPS laboratory (Université de Savoie -France), particularly Patrice Mélé, for carrying out the high-speed tensile test is gratefully acknowledged.

Influence of friction

Temporal variations in the impact load given by the numerical model showed considerable noise. A study of sensitivity to the model parameters was carried out to determine the origin of these perturbations. The friction coefficient between the striker and the heel seemed to influence these fluctuations. The results, for an impact test with an initial clapper position at 90° on a slanted heel with a pin, are plotted in Figure 22. The curve oscillations increased with the friction coefficient. This study showed that the impact load values were not very influenced by the friction coefficient.

Hereafter, an arbitrary coefficient of 0.1 will be used.


David Dubois

Harald Zellmer

Eric Markiewicz

Experimental and numerical analysis of seat belt bunching phenomena

Keywords: Seat Belt, Bunching, Finite Element, Correlation

In current cars, loops are commonly used to redirect the webbing which reels out from the retractor to the passenger's shoulder. Some types of pillar loops, also called Drings, lead to a non-systematic instability. The webbing, which should scroll without hindrance through the D-ring, laterally shifts, bunches and produces the overturning of the ring.

In this paper, this so-called seat belt bunching phenomenon is parsed during a first step with sled test campaigns data. The results of designs of experiments are analysed and discussed.

To expertise this instability issue, an innovative fixture is exploited during a second step to reproduce the phenomenon in a fully controlled manner for dynamic and quasi-static loadings. To assess these sub-system tests, a Digital Images Correlation system is employed to evaluate the strain distribution of seat belt webbing during the bunching phase. Based on these local measurements, a correlation of a Finite Element model of seat belt bunching is achieved using a new shell element for webbing fabric, before proposing an explanation of the phenomenon.

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Introduction

Automotive manufacturers and suppliers commonly conduct crash tests in order to assess the performance of safety systems on new vehicles. Crash scenarios are reproduced using dynamic tests. For instance, the EuroNCAP protocol; a 64 km/h frontal impact in a 40% offset deformable barrier is used to simulate a car to car impact.

An assessment protocol is then applied to achieve a rating for each body region. The head acceleration, the chest deflection, the femurs loading, compression / extension / flexion of the neck are measured on crash test dummies and a star rating is given. The safest cars obtain a 5 star performance.

During the vehicle deceleration, the dummies are subjected to the collision forces and in the case of a frontal impact, they move forward. To prevent severe contacts between the passenger and the car interior (dashboard, steering wheel), a three-point seat belt restrains the passenger's motion. In parallel, the driver and the passenger airbags reduce the acceleration peak applied to the occupant and distribute the restraint loads on the upper part of the body. To take benefit of their coupled actions, these restraint systems are developed in interaction. Nevertheless, although an airbag has a spectacular action during a collision, the belt, part of the passive restraint system, is the only protection against the ejection of the passenger away from his vehicle.

A safety belt webbing is a fabric material using polyester threads woven on Jacquard weaving looms. The webbing of a width of 48 mm of the belt restraint system has to resist to dynamic loads up to 14kN. Depending on the type of vehicle, about 3.5m of webbing are used in the seat belt. The interactions between the warp and weft threads of the fabric control the behaviour of the seat belt during the crash event.

Seat belt systems are usually combined with a load limiter retractor and also pretensionner integrated in the retractor, fixed at the buckle or at the lower part of the B

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pillar (the pillar on which the rear doors are fixed). A D-ring is used on the upper part of this pillar in order to redirect the seat belt, which reels out from the retractor to the passenger shoulder. It is designed to adapt its angular position according to the passenger's motion. Numerous different kinds of D-rings (also called webbing guides) are used in current vehicles: most of them are made of a metal insert on which polymer part in nylon or in acetyl is moulded (see Fig. 1.a). In some cases, the ring frame and the guidance surface are made of one piece of metal (see Fig. 1.b). In the latest vehicles, the complete webbing guide assemblies are integrated into the trim panel, and only the diagonal belt portion appears (see Fig. 1.c).

For some belt geometries (the 3D position of the anchorage points of the seat belt on the vehicle), the use of certain D-rings has led to a non-systematic instability, which is very disadvantageous. The webbing, which should scroll without hindrance through the webbing guide, laterally shifts, bunches and produces the overturning of the ring as shown on

Initial position

Lateral shifting Bunching of pillar loop Fig. 2, where the D-ring has been masked for confidentiality reasons. This phenomenon is an issue because it might prevent the restraint systems connected to the webbing from working in a normal manner (pre-tensioner, load limiter). In the face of such a problem, various studies [1], [START_REF] Pedrazzi | Simulation of belt movement in the D-ring during a crash CAD-FEM, User's meeting Internationale FEM-Technologietage[END_REF], [START_REF] Pedrazzi | Aspects of Seat belt Material Simulation, Third European LS-DYNA Conference[END_REF], [START_REF] Dubois | Analysis of Seat Belt Behaviour Under Dynamic loading[END_REF] have been carried out to control this phenomenon and needed to be pursued.

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Analysis of the phenomenon based on sled tests

To develop restraint systems, numerical simulations are conducted and to confirm the airbag and the seat belt performances, dynamic sled tests are performed.

A sled is a mechanical welded assembly developed to simulate the car interior.

The anchorage points of the seat belt are positioned in accordance with the car project specifications. The dashboard geometry is reproduced using foams and seat prototypes are used.

The bunching phenomenon was occasionally observed during these nondestructive sled tests. Previous studies [1] shown that during this type of tests, the plastic deformations of the crash frame are not reproduced and the corresponding absorbed energy is transferred to the webbing (no deformation of the B pillar on which the D-ring is fixed, no deformation of the car floor on which the seat is fixed, no dynamic pitch of the car body). The load applied to the seat belt webbing (measured locally by specific load cells) is thus often higher than the load measured during the corresponding complete vehicle crash tests. Sled tests are known to overload the webbing and to maximise the seat belt bunching likelihood.

To expertise belt behaviours, frontal sled tests campaigns were performed. Based on this experimental approach, designs of experiments have permitted to point out that one of the major factors which influence the phenomenon is the seat belt geometry [1].

Several risky seat belt geometries have been then defined but no generic assessment has been developed to explain the seat belt bunching phenomenon.

To pursue the research, a critical analysis of the design of experiments results was proposed. It appears that two consecutive and supposed identical tests may result in two different local behaviours of the ring (bunching for the first one and no bunching for the second one). For instance, sled tests performed with two seats on the same platform (and thus the same pulse, the same dummies, the same belt geometries) gave different

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ARTICLE IN PRESS

results: bunching on one side and no bunching on the other side (see Fig. 3). This status questioned the initial condition of the seat belt webbing and D-ring interactions. An analysis of the initial lateral position of the webbing on the D-ring has been then conducted (see

Average webbing position

Front webbing position Rear webbing position Fig. 4, where the D-ring has been masked for confidentiality reasons). This critical analysis underlined an unexpected factor which influences the bunching likelihood, the initial position of the webbing according to the D-ring slot corners. A tiny variation of the webbing position can modify or delay the bunching phenomenon.

The initial positioning of the webbing on the D-ring was not known at first as being an influential factor on the crash test results because it is usually not a factor under control. Several points can explain this insufficiency:

• The initial position of the webbing on the D-ring is a consequence of the belt geometry. Moreover, the size of the dummy (Hybrid III 05, 50 or 95% ile ) and the seat position (slide position, height adjustment position) also vary the webbing/ Dring boundary conditions.

• It is also controlled by component specification, the friction level between the Dring and the webbing, the friction value between the webbing and the dummy, the spring strength inside the retractor and also the webbing slack resulting from the manual buckling of the seat belt on the dummy's body.

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• The dynamic ring technology has also an influence on the webbing positioning on the D-ring. In a case of a direct crash test (sled decelerated using deformable bumpers), the initial webbing position is often modified by the tiny sways of the dummy during the acceleration phase. These perturbations can potentially be reduced by the use on an inversed crash sled (sled accelerated using hydraulic valves), but as the seat belt bunching phenomenon is an instability problem, small variations of the sled pulse can also interfere on the webbing behaviour.

The critical analysis of the results of designs of experiments shows that experimental studies based on sled tests are helpful to investigate the phenomenon but do not lead to a final solution to the instability of D-rings. They enable to list several important influential factors and to show tendencies. Detailed analysis proved that sled dynamic tests are not adapted to the study of the webbing/D-ring interactions, because this test method provokes local dispersions which have an effect on the final result of the experiments.

Numerous sled tests would be necessary to evaluate the trends according to the initial position of the webbing on the D-ring, but the high cost of these sled tests does not enable to pursue in that way.

Analysis of the phenomenon based on global sub-system tests

To complete the analysis of the seat belt bunching phenomenon, a global subsystem test, called Webbing Guide Drop Test (WGDT) has been used [START_REF] Dubois | Innovative test method for seat belt D-ring[END_REF]. The principle of this bench is: a half cylinder drop weight, fixed to the sled of a drop test, impacts a horizontal webbing portion between a set of rods (see Fig. 5). The horizontal webbing portion is then pulled and the shoulder belt reels out. The D-ring is screwed on a vertical column. Its column is oriented at 45° to facilitate the shoulder belt orientation. The total energy applied to the system is controlled by the mass and the speed of the impactor.

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Experimental campaigns showed that the WGDT is a consistent method to test Dring in dynamic loading conditions. With this device, the D-ring's instability can be easily reproduced (see 32ms 38ms 46ms Fig. 6) with loading levels and scrolling speeds equivalent to those observed during R16 sled tests [START_REF]UNECE agreement, Addendum 15 Regulation 16, Safety-belts and restraint systems for occupants of power-driven vehicles[END_REF]. The advantage of this test method compared to sled test is that results are completely repeatable and the cost per experiment is low compared to dynamic sled tests.

The use of the WGDT to investigate the seat belt bunching phenomenon enables to complete the global understanding of the webbing behaviour. It confirms that the initial position of the webbing influences the global stability of the ring. Nevertheless to assess and to solve the ring's instability; a local analysis of the webbing has been investigated.

Analysis of the phenomenon based on local sub-system tests

Based on sled test results and the WGDT results, the need of a local sub system test has emerged [START_REF] Dubois | Prevention and Prediction of the Seat Belt Bunching Phenomena[END_REF].

Sub-System Test

To reduce the variance of designs of experiment based on sled test and to focus the study on the behaviour of D-rings, a sub-system test method has been used. Thus, an innovative experimental bench was developed by Autoliv North Germany [START_REF] Markvardsén | Expermentelle Untersuchung des Verhaltens von Sicherheitsgurtumlenkern in Frontalcrashverlauf[END_REF], [START_REF] Markvardsén | Untersuchung einer neuentwickelten Prüfvorrichtung zur Simulation von Sicherheitsgurtumlenkern unter Lastbedingungen[END_REF].

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This innovative experimental bench called WGRAT (Webbing Guide Rotative Arm Test), consists in two main parts (see Fig. 7) fixed on a quasi-static tensile test machine:

• Part 1: a support structure with a rotating arm on which the D-ring is fixed.

• Part 2: a fixture on which the retractor and a load cell are mounted.

This mechanism has a multi-angle rotative arm, which allows to reproduce all the belt orientations seen in an automotive vehicle. It is usually fixed to the lower jaw of a tensile test machine. For each test, a retractor is mounted on the device, the D-ring is fixed on its axle and the webbing is set through the ring. Then, the free end of the webbing is fixed to the upper jaw of the tensile test machine. The experimental fixture (see Fig. 8) enables to load the D-ring in similar loading conditions to a sled test.

The use of two devices of angle variation allows re-creating the seat belt curvature through the D-ring. Compared to a real car set up, the loading conditions are varied as following:

• The position of the retractor and the position of the loading webbing end are reversed.

• The loading direction is constant (during a sled test, the dummy is moving forward and the loading direction is varying).

• The speed of loading is limited by the pulling device specification (it can reach up to several meters per second during a sled test).

• No deceleration pulse is applied to the D-ring and to the webbing.

Experimental analysis based on WGRAT

The local assessment on the seat belt bunching phenomena based on experimental tests has been pursed in two steps:

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• For dynamic loadings, using an hydraulic jack and a high speed video camera.

• For quasi-static loadings, using a tensile test machine and a Digital Images Correlation for strain distribution measurement.

Dynamic WGRAT Test Set Up

The WGRAT device was adapted to dynamic loadings to obtain the same outlet speed of the webbing through the webbing guide than the speed measured during dynamic sled tests (see Fig. 9). Thus, the fixture was mounted on a high-speed hydraulic jack at ONERA Lab.

Center of Lille (The French Aerospace Lab.). The frame is made of two vertical columns fixed on ground (2.5 x 2.5 m²). This ground is fixed to a 40 tons seismic mass suspended and damped. The columns are connected by a horizontal crosspiece to two degrees of freedom, on the middle of which the hydraulic jack is mounted.

Its loading capacity is 50kN in dynamic. Its maximal speed is close to 10m/s. The cylinder travel exceeds 250mm. Its guidance is obtained by hydrostatic bearings and allows radial loadings.

Dynamic WGRAT Test Result

For a 200mm jaw displacement at 3.5m/s and a 45°/45° angle between the vertical belt and the diagonal belt, the webbing initially in a central position, slides laterally and bunches in the left corner of the ring at 22 ms (see These dynamic tests enable to visualize the deformation of the vertical belt portion. Initially flat, the webbing deforms and several weaves appear on the surface close to the ring. In parallel, the analyses of horizontal lines drawn on the belt show discontinue local perturbations which grow until bunching.

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Quasi-static WGRAT test set up

During a second step, the WGRAT device has been used for quasi-static loadings to evaluate the strain distribution through the webbing (see Fig. 11). To complete the assessment of the seat belt bunching phenomena, an analysis of the strain distribution through the webbing during the instability issue has been investigated.

In order to study the strain field distribution over the seat belt webbing during the bunching phenomena, tensile tests, performed on an INSTRON TM tensile test machine, were undertaken and post-analysed by ''GOM/ARAMIS TM '' Digital Images Correlation system [START_REF] Aramis | version 4.6.5[END_REF].

To enable the strain measurement, the specimens needed to be prepared. For this purpose, a very thin layer of white flexible silicon was spread on the surface of the vertical portion of webbing. A black colour spray was next applied to obtain an irregular contrast pattern results.

During these quasi-static tests, the following set up was used:

• Retractor without load limiter (to reduce the reel out distance of the webbing through D-ring and to focus the analysis area).

• 45°/45° belt geometry.

• The webbing was positioned in the middle of the D-ring.

• The loading speed was fixed at 100 mm/min.

• The camera frame was fixed to one frame per second.

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• A 1240*1240 pixels resolution digital video camera was used.

The sample was installed in the tensile testing machine and the ARAMIS system was calibrated and positioned in front of the sample (see Fig. 12). During the tensile tests, a camera recorded the 2D displacements of the splash pattern. During the same time, the displacement of the jaw, the tensile load and the retractor load were recorded.

A Digital Images Correlation of the movie pictures was performed to measure the longitudinal and transversal strain distributions. This method has enabled to quantitatively capture the development of an unbalanced strain field of the webbing before and during the bunching phenomenon.

Quasi-static WGRAT Test Results

Three repeatable tensile tests were performed. During these tests, left bunching phenomena occurred. The two main directions of the strain tensor were calculated and they showed different behaviours. The measurement field, size about 10 x 4 cm² was located in the center of the specimen where the largest deformation was expected.

Finally, the deformation behaviour in the whole measuring area was observed in realtime until bunching.

By eliminating the jaw movement of the specimen, the local deformation of the specimen surface can be made visible. By a mathematical derivation of these images, the strains were determined and displayed as a whole field distribution (isolines of equal strains). The distribution epsilon X and epsilon Y was calculated and graphically shown by plotting strain curves variation exported in a specific ASCII-format. Thus, the behaviour of the sample during the test shall be evaluated and displayed graphically.

Transversal strain

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The analysis of the deformation of the seat belt webbing (in the weft direction)

shows a homogenous and a close to zero strain distribution of the webbing before the bunching phenomena. As soon as the webbing begins to bunch, the out of plane deformation of the samples disturbs the measurement.

Longitudinal strain

The analysis of the longitudinal strain of webbing shows that its behaviour varies in several stages: From 0 to 34mm, a homogenous distribution of the longitudinal strain can be seen on the webbing sample (see Fig. 13).

• At 34mm, a dissymmetry of the longitudinal strain appears and this dissymmetry increases from 34 to 67mm. At that stage, the difference between the strain values calculated on the right edge and the ones calculated on the left edge of the webbing sample reaches 30%. (see Fig. 14).

This phenomenon was computed along several sections of the webbing (see sections 1 to 5, Fig. 14). An increasing difference reaching more than 2% of longitudinal strain value (a variation of 30% between each edge, see Fig. 15) was measured between the two edges of the webbing before the overturning of the D-ring.

• At 68mm, the bunching phenomenon begins, and 15mm later the bunching phenomenon is completely achieved (see Fig. 16).

• After bunching, at 83mm, the out of plane deformation of the sample disturbs the measurement in the center of the specimen but the edges deformation can be assessed. For each edge, the longitudinal strain reaches the same average value: 9% (see Fig. 17).

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Judging from the results obtained by the Digital Image Correlation method, the phenomenology of the seat belt bunching phenomenon can be discussed: At the beginning of the test, the belt is uniformly loaded, thus the system is in equilibrium. A disequilibrium appears, the longitudinal strain distribution becomes inhomogeneous along the width of the webbing. That unbalanced strain distribution grows up to reach a 30% variation between each edge of the webbing and leads to the instability of the system by the overturning of the D-ring. The system has reached a new stable state with a symmetric strain distribution.

At that stage, the dissymmetry of the strain distribution and the D-ring rotation appear to be synchronized. To complete the results obtained with the Digital Image Correlation method and to pursue the assessment of the bunching phenomena, a FE numerical model of D-ring / seat belt interactions has been developed. The use of this numerical tool aims at organizing into hierarchy the factors influencing on the stability of D-rings.

Numerical analysis based on WGRAT tests

To complete the experimental analysis of the seat belt bunching phenomena based on the WGRAT dynamic and quasi-static test results, a numerical analysis is proposed.

A specific FE model of seat belt webbing is developed. Based on the numerical innovation, the experimental tests described in the previous chapter are modelled.

FE model -new shell finite element model for webbing

To improve the finite element model of seat belt webbing fabric, an innovative shell finite element is developed using Pam-Crash TM packages [START_REF] Safe | Solver Notes Manual[END_REF]. This shell element takes into account the low but effective bending stiffness of the seat belt webbing by the use of a plate bending element with a decoupled and orthotropic bending stiffness.

According to the results of tests on seat belt webbing, a method to simulate its

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behaviour was studied. To evaluate its stability, it is applied to realistic belt geometries and loadings.

The global behaviour of seat belt webbings corresponds to the behaviour of a fabric. Fabrics are usually modelled with membrane elements, which do not take the bending stiffness into account. This method has proved to be satisfactory for the simulation of thin fabrics, such as fabrics used for airbags, but cannot be extrapolated to the case of seat belt webbing because the bending stiffness cannot be neglected during the bunching phenomenon.

Thus a new simulation method was used to model the behaviour of seat belt webbings. It consists in the superposition of two layers of shell finite elements (see Fig.

18.

). The two layers are defined with the same nodes and the density of each layer corresponds to half the density of seat belt webbings.

The first layer is a membrane element usually used for fabric element; it defines the membrane behaviour. The first part of this model corresponds to the membrane stiffness. Its main use in the field of dynamics is the simulation of airbag fabric. This material corresponds to a linear elastic membrane material which consists in two families of fibres, running at given angles in two directions, embedded in an isotropic matrix. The following general relationship between membrane stresses and strains describes the general isotropic case. fabric = isotropic matrix + layer 1 at 0° + layer 2 at 90°

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - - = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡
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Where Em is the Young's Modulus of the isotropic matrix, E 1 and E 2 are Young's tension/compression moduli of the threads, G 1 and G 2 are shear moduli and ν is the Poisson's coefficient. This element is formulated in a total Lagrangian approach, in which the Green-Lagrange strains (ε GL =1/2 (L 2 -L 0 2 ) / L 0 2 ) and the second Piola-Kirchhoff stresses are used (σ PK = (L/L 0 ) E ε GL ). This element is completely integrated by four Gaussian points.

The second part of the model corresponds to a bending plate stiffness based on a Mindlin shell element [START_REF] Mindlin | Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic[END_REF]. That plate bending element [START_REF] Safe | Solver Notes Manual[END_REF] is usually used in the field of non-linear dynamic loadings to simulate the behaviour of composite ply made of two components, the fibres and a matrix.

Then, the behaviour of uni-directional continuous fibres reinforced composites is modelled with this shell using the superposition of plies composed of two phases: a unidirectional fibres phase and an orthotropic matrix phase (see Fig. 19). The original aspect of this numerical model is the distinction of these two phases.

Each phase of each ply is defined independently in tension and in compression.

For each ply, the stiffness is calculated by superposing the effects of the orthotropic elastic matrix (matrix minus fibres) and of the one-dimensional elastic fibres (see Fig. 20).

Within the context of the seat belt webbing, only the matrix phase has been used (no fibres phase has been used). To transform it into a bending plate stiffness, this shell element was degenerated by removing from the code the calculated membranes forces transmitted to the nodes (see Fig. 21).

Once superposed, the resulting layered element is thus a multi-layered quadrangle element (Q4) with a decoupled and orthotropic bending stiffness. The orthotropic

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material phase is defined with the following strain-stress relation (expressed in the directions of orthotropy). The parameters of the stiffness matrix were identified with rheological tests performed on seat belt webbing fabric [START_REF] Dubois | analysis of seat belt bunching phenomena[END_REF]. Several experimental tests were undertaken:

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 23
-Picture frame tests to measure the real in plane shear modulus -Cantilever tests to evaluate the bending stiffness -Tensile tests for Young's modulus and the Poisson's ratio This new shell finite element was used to simulate the behaviour of seat belt webbing and to investigate the webbing kinematics observed experimentally.

Numerical model of Dynamic WGRAT test

A numerical model has been achieved to reproduce the seat belt bunching cases which had been experimentally obtained. Special care has been devoted to model the tribology of the phenomenon in a most correct way.

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Model development

A model is achieved to reproduce numerically the seat belt bunching cases experimentally obtained. The belt orientation (45°/45°) used during the experimental test campaign was reproduced using pre-simulations. At one end of the webbing, the Pam-Safe TM model of retractor is used to simulate the spool-out effect of the webbing on its spindle and to take into account the load limiting device existing in the retractors used during the tests set. At the other end, the loading method used for this simulation is an imposed displacement applied to the nodes of the free edge of the webbing (see Fig. 22). The ''displacement / time'' law used corresponds to the one measured at the pressure cylinder during the dynamic WGRAT tests.

The original aspect of this test is that the retractor is near the D-ring. In order to obtain a realistic simulation, a fixed cylinder (called ''rigid wall'') is used to reproduce the scrolling conditions of the webbing out from the retractor.

Friction Model

As the friction between belt webbing and D-ring rod may play a significant role in webbing instability, an in-depth analysis of the friction has been performed. Safety belts are now generally woven out of Polyester. To improve the properties in slip and the abrasion resistance, a new generation of webbing is produced with polyester threads without torsion. The use of threads without torsion reduces the friction resistance because the filaments are strongly directed in the longitudinal direction.

The safety belts are woven according to a cross weave (2x2) in which, each wire passes above two consecutive weft threads then above two following weft threads. From a macroscopic point of view as the Jacquard weaving mode being directed, the surface quality in the weft and warp directions differs (see Fig. 23). The longitudinal and transversal tribologies are then different.

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D-rings are often made of a metal insert on which polymer part (Hostaformor TM , Delrin TM , etc.) is moulded. As polymeric materials badly hold the heat, the scrolling of the webbing on the D-ring dissolves the sliding surface (the melting point is lower than 200°C) and modifies the tribology.

The analysis of the webbing after loading shows that the polymer melts after some centimeter of scrolling (see Fig. 24). A molten polymer film is formed between the webbing and the ring. That phenomenon reduces the friction coefficients and homogenizes its value in the warp and weft directions.

Several D-ring types have been experimentally tested (see Fig. 25). D-rings with sheeted lines on the sliding surface or with a PTFE coating (teflon TM ) on metal ring have shown the same seat belt bunching behaviour.

The smooth contact algorithm (type 46) from Pam-Crash TM data base [START_REF] Safe | Solver Notes Manual[END_REF] has been used to manage the sliding of the webbing on two smooth surfaces extrapolated from the surface nodes to avoid perturbations through the belt forces. This algorithm ensures the contact between the webbing and the D ring, and the self-contact of the webbing when bunching occurs (see Fig. 26).

A contact thickness defines a volume around the meshes of the webbing and the D ring. As soon as a node is entering this zone, a contact force is generated. A penalty method is used, for which the penetrations are allowed but penalized. The penalty force is linearly proportional to the depth of penetration. The interpenetration check frequency has been fixed to 1000 cycles to stabilize the contact.

Fcontact= k . penetration

where k = 0.1x kstable and kstable is the stable stiffness to integrate with the given time step.

For a simplified 1 degree of freedom (see Fig. 27) kstable= M1.M2/(M1+M2). (1/Dt2)

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To model the friction, several friction models have been alternatively tested.

Isotropic Standard Coulomb Friction, orthotropic, pressure and/or velocity dependent friction models have been numerically evaluated.

Due to the molten polymer film between the webbing and the D-ring, a homogenized isotropic friction has been first chosen.

Pressure and velocity dependent friction models haven't proved significant improvement in the correlation process. As the modeled kinematics is an energetic and quick process, the melted state of the polymer is rapidly established and the dynamic friction value remains low during the whole loading.

Starting from the academic value of PTFE friction coefficient [START_REF] Gieck | Formulaire technique[END_REF] equals to a 0.05, a parametric study has been performed to analyze the effect of the friction value variation (between 0.0 and 0.1) on the global kinematics of bunching. In that range, the overturning of the rings was equivalent. Preliminary results of a design of experiment, in which geometrical and physical parameters were varied, have also confirmed its low influence.

Model results

Based on the previous analysis, the friction between the webbing and the D-ring has been modelled using a coulomb friction fixed at a usual value of 0.05. 16ms 18ms 20ms 22ms Fig. 29 shows several steps of a corresponding numerical seat belt bunching phenomenon obtained for a 45°/45° belt orientation.

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The validation of the model is obtained by the comparison of the load levels through the webbing versus the webbing's end displacement measured experimentally and calculated by simulation (see Fig. 28).

The same global behaviour experimentally observed has been numerically reproduced (see 16ms 18ms 20ms 22ms Fig. 10). Weaves on the webbing can be seen before bunching. A fine post-processing has been carried out to compare the strain distribution through the FE webbing to the one visualized experimentally using the previous ''GOM/ARAMIS TM '' Digital Images Correlation method.

The numerical simulation is conducted as an adiabatic process; then, there is no heat transfer between the webbing and the D-ring. As the thermodynamics effects are not modelled, there is no modification of the sliding friction coefficient during Finite Elements runs. The effect of the inertial forces is also neglected between quasi-static and dynamic tests. Despite these strong hypotheses, the local phenomena are studied.

The comparison of the strain distribution fields obtained for two different velocities of loadings (experimentally in quasi-static loading see Fig. 14, numerically in dynamic loading see Fig. 30) shows the same strain dissymmetry.

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Thus, the results of FE model show that the strain distribution is equivalent to the experimental measurement. At 17ms, the range of strain deformation between the two edges of the webbing extends from 6 to 8% (a variation of about 30%) and the iso-strain areas are localized in the same zones. At 22ms, after belt bunching, the overturning of the D-ring interrupts the disequilibrium. A new homogenous distribution of longitudinal strains (about 9%) can be observed on the vertical belt portion.

A design of experiment will be performed based on this correlated numerical model. That numerical study will take into account:

• The friction coefficient between the webbing and the D-ring.

• The bending stiffness of the webbing.

• The behaviour of the revolute joint of the loop.

The aim of this analysis will be an improvement of the stability of the D-ring.

Numerical simulation data enable to complete experimental tests results and to assess the chronology of the phenomena. Up to the round off load value of 4kN, the webbing/D-ring couple is stable. The webbing reels out normally and the angular D-ring position is constant. Beyond that load level, the diagonal portion of the webbing shifts laterally on its guide. That shifting causes a dissymmetry of the D-ring loading which generates an angular rotation. That rotation imbalances the strain distribution of the webbing. In order to reduce this perturbation, the webbing's shifting grows. The two phenomena become coupled and quicken until the complete overturning of the ring.

Conclusion

The objective of this study was to assess the Seat Belt Bunching Phenomenon and to develop a numerical model of the instability issue. In this paper, the webbing / Dring behaviour is first investigated using several experiments. An analysis of dynamic An efficient Finite Element model for seat belt webbing is developed using an original two layers shell quadrangle. The first layer is a membrane element usually used for fabric element. The second layer corresponds to a degenerated bending plate stiffness. Once superposed, the resulting layered element is a multi-layered quadrangle element with a decoupled and orthotropic bending stiffness. FE models of the sub-system tests are developed using the layered shell element.

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The correlation of these finite element models is proved to be good. A fine post processing of the local subsystem test results is used to assess the behaviour of seat belt webbing. The internal strain distribution of the webbing measured by Digital Image Correlation is very well reproduced by the FE model and a numerical bunching phenomenon has been detected for an equivalent loading. Based on experimental and numerical data, an assessment of the phenomenon is proposed. To complete this research study, further investigations are planned. This validated model will be used for a numerical Design of Experiment in order to analyze the stability of the D-ring and to fully control the bunching phenomenon.

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  11 the modulus in direction 1 E 22 the Young's modulus in direction 2 G 12 the in-plane shear modulus G 13, G 23 the transversal shear moduli ν n,m the Poisson's ratio in n,m directions This plate-bending element (12 d.o.f.) defines two different bending stiffnesses, in the warp and in the weft directions, independently of the membrane stiffness.

  sled tests results is proposed. Innovative global and local sub-system tests are developed and are used to investigate the overturning of the ring.

Fig. 1 .Fig. 3 .Fig. 4 Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 13 .Fig. 14 .Fig. 15 .Fig. 17 .FigFig. 20 .Fig

 134678910111314151720 Fig. 1. Webbing Guides (courtesy Autoliv)

Fig. 30

 30 Fig. 30 Strain distributions through the vertical webbing portion at 17-22ms

Acknowledgements

The authors would like to thank Jacky Fabis from ONERA Lab Centre of Lille for his valuable contribution during the WGRAT tests.