Key Engineering Materials Vols. 554-557 (2013) pp 1602-1610 (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/kem.554-557.1602 Numerical simulation of plug-assisted thermoforming: application to polystyrene C.A. Bernard 1, a, J.P.M. Correia 1,b, N. Bahlouli 1,c and S. Ahzi 1,d 1 University of Strasbourg/CNRS, ICUBE, Department of Mechanics, 2 rue Boussingault, 67000 Strasbourg, France a chrystelle.bernard@unistra.fr, b jpm.correia@unistra.fr, c nadia.bahlouli@unistra.fr, d ahzi@unistra.fr Keywords: Finite element simulation, Plug-assisted thermoforming, Polystyrene, Hyperelasticity. Abstract. In the present work, the thickness distribution in a plug-assisted thermoforming process is investigated using finite element (FE) simulations. Numerical simulations have been performed with the FE code ABAQUS/Explicit. The contact between sheet and tools is considered as isothermal. Moreover, the coefficient of friction between plug and sheet is assumed constant. The behavior of the material is described by three hyperelastic laws available in the FE code. The comparison between experimental and FE results highlights that the neglected thermal effects (conduction and convection) and the thermal dependence of coefficient of friction should be considered. For future work, we propose an elasto-viscoplastic model which appears to better describe the behavior of the material than a hyperelastic model. Introduction Plug-assisted thermoforming processes are used in various industries such as automotive (vehicles internal panels) or packaging applications (food, medical and pharmaceutical packaging). Moreover, plug-assisted thermoforming processes can be used for a wide range of polymers (amorphous, semi-crystalline ). In these processes, a thin polymer sheet is heated above its glass transition temperature Tg for amorphous polymers, or just below its melting temperature Tm for semi-crystalline polymers. As the material is softened (has a viscoelastic behavior), it can be easily deformed. Then, the polymeric sheet is clamped around its edge. Next the polymeric sheet is stretched using a plug. At the end of the stretching, the polymeric sheet is formed into a mold by an air pressure. With the contact of cold mold surface, the deformation of the polymeric sheet ends. During the thermoforming process, different defects can occur such as wall-thickness variations, rupture, shrinkage and warpage [1, 2]. Several process parameters with complex interactions influence the plug-assisted thermoforming process [3]. In order to ensure the quality of the product as well as the efficiency of the process, it is necessary to understand the effects of the process parameters. Classically, the choice of the process parameters for a plug-assisted thermoforming process is determined empirically by an iterative procedure involving a large number of expensive tests. Several experimental studies [4, 5] have been performed in order to study the influence of the material behavior and of contact conditions (temperature, friction, velocity, plug material and geometry, pressure, sheet material) on wall thickness distribution. Nowadays, with the increase of computational capabilities, various finite elements (FE) analyses are increasingly used to understand physical phenomena occurring during a plug-assisted thermoforming process [6, 7, 8]. In various numerical studies [9, 10], hyperelastic constitutive laws are used in the FE codes to simulate thermoforming processes. Hyperelastic models can be developed by the construction of a strain energy function for an isotropic material. A wide range of hyperelastic models are available in commercial FE codes, but their use is limited. One of the main difficulties in applying any FEM analysis is to reliably reproduce the material behavior by a proper constitutive law with consistent model parameters. The present study is focused on the modeling of a plug-assisted thermoforming process. The plug-assisted thermoforming process geometry proposed in the experimental work of Erner and Billon [11, 12] has been chosen for performing FE simulations. The polymeric sheet is a All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 130.79.50.141, Institut de Mécanique des Fluides et des Solides, Strasbourg, France-10/04/13,14:20:02)
Key Engineering Materials Vols. 554-557 1603 polystyrene blend (PS and HiPS) used in the work of Ener and Billon [11, 12]. Firstly, the mechanical behavior of the polystyrene is modeled with different hyperelastic models. The hyperelastic models are compared with the experimental tensile curves of Erner [12]. FE simulations of plug-assisted thermoforming have been carried out with ABAQUS/Explicit. The FE predictions are compared to the experimental results in the literature [12]. Then, another approach is proposed, as a future work, to model the elasto-viscoplastic behavior of the material. In order to take into account the strain rate and temperature dependence of material, the elasto-viscoplastic model proposed in the work of Richeton et al. [13] is presented. The approach proposed in [13] allows us to describe the mechanical response of amorphous polymers below and above the glass transition temperature. Hyperelastic models In most of numerical studies, the polymeric sheet behavior is modeled with an isotropic hyperelastic model. With this type of material behavior, Amiri et al. [14] have obtained fairly good wall-thickness predictions in FE simulations of a free thermoforming process of PMMA sheet. Several isotropic hyperelastic models are implemented in the commercial FE code ABAQUS/Explicit. In the present study, the Mooney-Rivlin model [15], the Yeoh model [16] and the Arruda-Boyce model [17] are chosen. The first model used is the Mooney-Rivlin model. In ABAQUS/Explicit, the Mooney-Rivlin form of the strain energy potential per unit of volume is given by: ( ) ( ) ( ) (1) where C 10, C 01 and D 1 are temperature-dependent material parameters, and are the first and the second strain invariants defined as: (2) where ratio and are the deviatoric stretches, J is the total volume ratio, J el is the elastic volume are the principal stretches. The initial shear modulus and bulk modulus are given by: ( ) (3) The Yeoh form of strain energy potential is a reduced polynomial form of order 3. In ABAQUS/Explicit, the strain energy potential per unit of volume is written as: ( ) ( ) (4) where C 10 et D i are temperature-dependent material parameters. The initial shear modulus and bulk modulus are given by: The last model used in the present work is the Arruda-Boyce model. In ABAQUS/Explicit, the Arruda-Boyce form of the strain energy potential per unit of volume is given by: (5)
1604 The Current State-of-the-Art on Material Forming ( ( ) ( ) ( ) ( ) ( )) ( ) (6) where a d are temperature-dependent material parameters. The initial shear modulus and bulk modulus are given by: ( ) (7) The material coefficients of the hyperelastic models (in Eq. 1 to Eq. 4) have been calibrated by ABAQUS/Explicit from the experimental stress-strain curve at a temperature of 130 C and at a strain rate of 1 s -1. For the Arruda-Boyce model, the fitting made by ABAQUS/Explicit was not acceptable, so the calibration has been made manually. The material constants used for the different hyperelastic models are reported in the Table 1. Table 1: Material constants of hyperelastic models. Mooney-Rivlin model D 1 = 1.71E-9 C 10 = -7770 C 01 = 124544 Yeoh model C 10 = 72305 C 20 = -4806 C 30 = 188 D 1 = 2.76E-9 D 2 = 1E-9 D 3 = 1E-9 Arruda-Boyce model μ = 200000 MPa λ m = 7 D = 1E-9 The experimental tensile curve at 130 C and 1 s -1 from the work of Erner [12] and the predicted curves from ABAQUS/Explicit are plotted in Fig. 1. Among the three hyperelastic models used, the Arruda-Boyce model does not fit well the experimental curve when the nominal strain is greater than 0.5. Figure 1: Experimental and predicted tensile curves of blend of polystyrene at 130 C and 1 s -1.
Key Engineering Materials Vols. 554-557 1605 FE Model The geometry of the thermoforming process is defined in [11] and illustrated in Fig. 2. The tooling is composed of a plug, a mold and a blank-holder. In the studied thermoforming process, the plug is assumed to exhibit a low thermal diffusivity and to have a rough surface (syntactic foam like). Furthermore, the sheet temperature is assumed to remain quite constant during all the deformation process. The blank-holder is a clamping ring. The polymeric sheet is a blend of polystyrene with an initial thickness of 1 mm and an initial diameter of 104 mm. Figure 2: Geometry of thermoforming process. Shaping of plastics by plug-assisted thermoforming consists of three main steps. In the first one, the polymeric sheet is heated above its glass transition temperature. In a second step, the heated sheet will be stretched by the plug (which can be heated or unheated). The plug goes down at controlled speed. When the plug reaches approximately 90% of depth, the plugging is stopped and the sheet is blown by air pressure from the surface of the plug into the mold. Time, pressure and blowing rate are also controlled. In the experimental study of Erner [12], the polymer sheet is heated until 133 C.The felt plug is unheated. Due to the low rate of plugging in the experiments of Erner [12], the plug temperature is assumed constant and equal to room temperature (23 C). This assumption is not true in the industrial context where the heating of the punch surface is observed due to the high rate of plugging. In the experiments performed by Erner [12], the plug velocity was fixed at 0.5 m.s -1 during 120 ms and at the end of the plugging, a blowing rate of 2.7 MPa.s -1 was applied during 56 ms. In the present study, the blowing is not considered in the FE simulations. The numerical thickness distribution is analyzed at the end of plugging and compared with experimental one [12]. The FE simulations are carried out with the dynamic explicit FE code ABAQUS/Explicit. In order to reduce the computational time, an axi-symmetric model has been developed. The tools (i.e. punch, mold and clamping ring) are modeled as analytical rigid parts and thus they are not meshed. The polymeric sheet is meshed with 7200 linear quadrilateral elements with reduced integration (denoted CAX4R in ABAQUS/Explicit documentation [18]). To take into account the evolution of sheet thickness, it is necessary to have a fine mesh. For this reason, 12 elements are used in thickness and 600 in length. The contact between the polymeric sheet and the tools has been described by the Coulomb friction model. In agreement with the experimental work of Erner and Billon [11] on friction effects in plug-assisted thermoforming, a coefficient of friction equal to 0.7 is defined between the felt plug and polymeric sheet. However, for the contact between polymeric sheet and the other tools (holder and mold), a coefficient of friction value of 0.3 is chosen. Similarly to work of Erner [12], a concentrated force is applied on the clamping ring and is kept constant during the FE analysis. The plug velocity is fixed to 0.5 m.s -1. The final plug displacement is taken equal to 60 mm, corresponding to 92% of depth. Convection effects appear on the free surfaces of the sheet. Due to rapidity of the plug-assisted thermoforming (approximately 180 ms), we assume that convection effects can be neglected in our FE simulations.
1606 The Current State-of-the-Art on Material Forming Results Before plotting the numerical results, we have ensured that the FE analyses of the forming process were quasi-static analyses by comparing the kinetic energy with the internal energy. The evolution of plug force with respect to plug displacement has been plotted in Fig. 3. The predicted plug force is underestimated for all hyperelastic models. However, the contact between tools and the sheet is assumed isothermal (no conduction effect) and the convection effects are neglected. In Fig. 4, experimental and numerical thickness distributions are compared. For all hyperelastic models, the predicted thickness distributions are not in agreement with the experimental thickness distribution. For the same temperature of tools and sheet, a high discrepancy is observed between the FE predictions (for the three hyperelastic models) and the experimental results (Fig. 4). In the FE simulations, for the same holder force applied on the clamping ring, the polymeric sheet slips down between the clamping ring and the mold. This phenomenon is observed for Arruda-Boyce model and Mooney-Rivlin model. Consequently the holder force must be chosen carefully. Figure 3: Plug force vs plug displacement curves. Figure 4: Thickness distribution along the sheet at the end of stretching. Figure 5: Von Mises stress obtained with the Yeoh form for three levels in the sheet at the end of stretching. Figure 6: Von Mises stress obtained with the Arruda-Boyce form for three levels in the sheet at the end of stretching.
Key Engineering Materials Vols. 554-557 1607 Figure 7: Von Mises stress obtained with the Mooney-Rivlin form for three levels in the sheet at the end of stretching. The stress state in the sheet is plotted for each hyperelastic model, Figs. 5-7. Three fibers in the sheet are considered: the top of the sheet in contact with the plug and with the clamping ring, the bottom of the sheet in contact with the mold, and the middle of the sheet. As illustrated in Fig. 5, Fig. 6 and in Fig. 7, the von Mises stress is homogeneous through the sheet thickness except for one specific point. For all the hyperelastic models, a maximum stress value is observed at 30 mm. This position corresponds to the end of contact between plug and sheet. Between 0 and 30 mm, the von Mises stress is almost constant for the three models. Moreover, after this critical point, the evolution of von Mises stress is inverse to the evolution of the thickness. The von Mises stress is more important in the wall for Mooney-Rivlin model than for the two other models (Fig. 7). This is probably due to the high decrease of thickness in the wall. Discussion and perspectives In Fig. 4, a discontinuity in sheet thickness distribution is observed at 30 mm for all models. This discontinuity corresponds to the end of plug-sheet contact. Another one appears at a radius of 50 mm for Yeoh model, which correspond to a point in the middle in the wall. This discontinuity is due to a mesh entanglement for Arruda-Boyce and Yeoh models. Therefore, the mesh needs to be refined on the superior fiber of the sheet in order to obtain a better prediction of thickness distribution near the end of contact between plug and sheet. Under the plug, for a radius ranging between 0 and 30 mm, the polymeric sheet is two to three times more stretched in the FE simulations carried out with Arruda-Boyce model or with Yeoh model than with the Mooney-Rivlin model. After the end of the contact between plug and sheet (Fig. 4), the thickness increases in the FE predictions with the Arruda-Boyce model and with the Yeoh model, whereas it decreases in the FE prediction with Mooney-Rivlin model. Other simulations made with lower values of coefficient of friction have been performed for the three hyperelastic laws. These simulations show that lower coefficient of friction leads to more stretching of the sheet under the punch. However, the obtained thickness distribution may not be accurate since these calculations are conducted without accounting for thermal dependence of the coefficient of friction. Although the Mooney-Rivlin model predicts, with more accuracy, the behavior of the polymeric material (Fig. 1), the thickness distribution is not in agreement with the experimental results. Thus, in the presented thermoforming process with a plug, the hyperelastic approaches seem to be not a good solution to model the behavior of polymeric sheet. The mechanical behavior of polymeric sheet should be predicted with more accuracy with an elasto-viscoplastic model where constitutive equations take into account the strain rate and temperature dependence.
1608 The Current State-of-the-Art on Material Forming In the present study, the approach of Richeton et al. [13] is proposed, as a future work, for a better description of the mechanical behavior of the polymeric material. In this constitutive model [13], presented in Fig. 8, three modules are used to describe the elasto-viscoplastic behavior of material. At small deformations, the polymer is assumed to have an elastical behavior and Hooke s law is used to predict the mechanical response. However, the Young s modulus is strain rate and temperature dependent. Then, when the yield stress is reached, the mechanical behavior of the polymer is described by a visco-plastic model. The viscoplastic model takes into account the isotropic resistance due to plastic flow and the anisotropic resistance due to molecular chain alignment. Figure 8: Schematic representation of constitutive model [13]. To take into account the strain rate effects on the Young s modulus, Richeton et al. [19] proposed a model where the Young s modulus is written as follows: ( ) ( ( ) ( )) ( ( ( ) ) ) ( ( ) ( )) ( ( ( ) ) ) ( ) ( ( ( ) ) ) (8) where a d are respectively the β-transition temperature, the glass transition temperature and the temperature corresponding to the beginning of the flow region. E i represent the instantaneous modulus at the beginning of each transition region. The last coefficients (m i ) are Weibull parameters which describe the probability of bond failure. To identify the yield stress of polymer, the model proposed by Richeton et al. [20] is used. This model assumes that yielding is due to a cooperative jump between polymer chain segments. The expression for the model proposed by Richeton et al. [20] is given by Eq. 9 for and Eq. 10 for : ( ) ( ) ( ) ( ) (9) ( ) ( ( ) ( ( ) )) (10) where k is the Boltzmann constant, V is the activation volume and n describes the cooperative character of the yield stress.
Key Engineering Materials Vols. 554-557 1609 When the polymer is stressed to overcome the intermolecular resistance, polymer chains begin to orient themselves and align in the direction of maximum stretch. To describe this orientational hardening of the polymer chain segment, the 8-chain model, developed by Arruda and Boyce [17], has been used. The principal components of the network stress tensor are given by: ( ) ( ) ( ( ) ) (11) where C R (T) is the rubbery modulus, N(T) is the number of rigid links between entanglements, λ i are the principal components of the plastic stretch tensor and is the inverse Langevin function. Conclusion Numerical simulations have been performed to simulate the plug-assisted thermoforming of a blend of polystyrene. The mechanical behavior of the polymer has been modeled with three hyperelastic laws. For all hyperelastic models, the predicted thickness distributions at the end of stretching have been compared with experimental data [12]. The discrepancy between the predicted results and experimental data allows us to say that conduction effects and the temperature dependence of coefficient of friction should not be neglected. The convection effects neglected for the plugging should also be taken into account for the blowing. For the moment, we can not assess if the convective effects have a strong influence on the thickness distribution in the wall during the plugging step. The hyperelastic models are not suitable to simulate the presented thermoforming process. The influence of the behavior laws is more significant in the thickness distribution than stress distribution along the sheet. Therefore, the behavior of material needs to be approached with precision in order to improve thickness distribution prediction. Experimentally, the contact between the cold punch on the heat sheet induces a local cooling to the polymeric material. This contributes to stiffening of the sheet since the material is sensitive to temperature. In order to increase the accuracy of FE simulations, we suggest to take into account a thermal flux at the surface of the plug and also to consider in constitutive equations accounting for the temperature and strain rate dependence of the polymer. References [1] H. Hosseini, B. V. Berdyshev, A solution for the rupture of polymeric sheets in plug-assist thermoforming, J. of Polymer Research 13 (2006) 329-224. [2] H. Hosseini, B. V. Berdyshev, A. Mehrabani-Zeinabad, A solution for warpage in polymeric product by plug-assisted thermoforming, European polymer Journal 42 (2006) 1836-1843. [3] A. Aroujalian, M. O. Ngadi, J. -P. Emond, Wall thickness distribution in plug-assist vacuum formed strawberry containers, Polymer Engineering and Science 37 (1997) 178-182. [4] R. McCool, P. J. Martin, The role of process parameters in determining wall thickness distribution in plug-assisted thermoforming, Polymer Engineering and Science (2010) 1923-1934. [5] P. J. Martin, P. Duncan, The role of plug design in determining wall thickness distribution in thermoforming, Polymer Engineering and Science (2007) 804-813. [6] P. Gilormini, L. Chevalier, G. Régnier, Thermoforming of a PMMA transparency near glass transition temperature, Polymer Engineering and Science (2010) 2004-2012. [7] S. aus der Wiesche, Industrial thermoforming simulation of automotive fuel tanks, Applied Thermal Engineering 24 (2004) 2391-2409. [8] A. Makradi, S. Ahzi, S. Belouattar, D. Ruch, Thermoforming process of semicrystalline polymeric sheets: modeling and finite element simulations, Polymer Science 50 (2008) 550-557.
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