A new hybrid approach using the explicit dynamic finite element method and thermodynamic law for the analysis of the thermoforming and blow molding processes for polymer materials

In this work, the problem of the modeling of thermoforming and blow molding processes for viscoelastic sheet are considered. To take account of the enclosed gas volume, responsible for inflation of the thermoplastic membrane (which contributes significantly to the strength and stiffness of a thermoplastic structure), we considered thermodynamic approach to express external work in terms of a closed volume. The pressure load is thus deduced from the thermodynamic law of ideal gases. The viscoelastic behavior of the K‐BKZ model is considered. The Lagrangian formulation together with the assumption of the membrane theory is used in the explicit dynamic finite element implementation. The numerical validation is performed by comparing the theoretical bubble free inflation with numerical results. Moreover, the influence of the K‐BKZ constitutive model on the thickness and stress distribution in the thermoforming of containers is presented. POLYM. ENG. SCI., 46:1554–1564, 2006. © 2006 Society of Plastics Engineers     

Dynamic finite element analysis of nonlinear isotropic hyperelastic and viscoelastic materials for thermoforming applications

In this work, a dynamic finite element method is used in modeling and numerical simulation of the viscoelastic and hyperelastic behavior of a thin, isotropic, and incompressible thermoplastic membrane. The viscoelastic behavior (Lodge, Christensen) and hyperelastic behavior (Ogden and Mooney‐Rivlin), are considered. The thermoforming of the sheet is performed under the action of perfect gas flows. The pressure load used in modeling is thus deduced from the thermodynamic law of perfect gases. The Lagrangian formulation together with the assumption of the membrane theory is used in the finite element implementation. The numerical validation is performed by comparing the obtained results with measured experimental data for the polymeric acrylonitrile‐butadiene‐styrene (ABS) membrane inflation. Moreover, the influence of the Lodge, the Christensen, the Mooney‐Rivlin, and the Ogden constitutive models on the thickness and on the stress distribution in the thermoforming sheet are analyzed. POLYM. ENG. SCI. 45:125–134, 2005. © 2004 Society of Plastics Engineers     

Caractérisation viscoélastique du comportement d'une membrane thermoplastique et modélisation numérique de thermoformage

In this work, we are interested, on the one hand in the characterization of circular polymeric ABS membrane under biaxial deformation using the bubble inflation technique, on the other hand in modelling and numerical simulation of the thermoforming of ABS materials using the dynamic finite element method. The viscoelastic behaviour of the Lodge model is considered. First, the governing equations for the inflation of a flat circular membrane are solved using a variable‐step‐size‐finite difference method and a modified Levenberg‐Marquardt algorithm to minimize the difference between the calculated and measured inflation pressure. This will determine the material constants embedded within the model used. For dynamic finite elements method, we consider a nonlinear load in air flow which obeys the Redlich‐Kwong equation of state of the real gases. For numerical simulation, the lagrangian formulation together with the assumption of the membrane theory is used. Moreover, the influence of the viscoelastic model on the thickness and on the stress distribution in the thermoforming sheet are analysed for ABS material.     

Analyse Expérimentale et Numérique du Comportement de Membranes Thermoplastiques en ABS et en HIPS dans le Procédé de Thermoformage

In this work, we are interested, on the one hand in the characterization of circular polymeric ABS and HIPS membrane under biaxial deformation using the bubble inflation technique, on the other hand in modelling and numerical simulation of the thermoforming of ABS and HIPS materials using the dynamic finite element method. Hyperelastic models (Mooney‐Rivlin, Ogden) are considered. First, the governing equations for the inflation of a flat circular membrane are solved using a variable‐step‐size‐finite difference method and a modified Levenberg‐Marquardt algorithm to minimize the difference between the calculated and measured inflation pressure. This will determine the material constants embedded within the models used. For numerical simulation, the lagrangian formulation together with the assumption of the membrane theory is used. Moreover, the influence of the hyperalastic model on the thickness and on the stress distribution in the thermoforming sheet are analysed for ABS and HIPS materials.     

Effects of rheological properties and processing parameters on ABS thermoforming

Understanding effects of material and processing parameters on the thermoforming process is critical to the optimization of processing conditions and the development of better materials for high quality products. In this study we investigated the influence of both rheological properties and processing parameters on the part thickness distribution of a vacuum snap‐back forming process. Rheological properties included uniaxial and biaxial elongational viscosity and strain hardening and/or softening while processing parameters included friction coefficient, heat transfer coefficient, and sheet and mold temperatures. The Wagner two parameter nonlinear viscoelastic constitutive model was used to describe rheological behavior and was fit to shear and elongational experimental data. The linear viscoelastic properties along with the Wagner model were utilized for numerical simulation of the thermoforming operation. Simulations of pre‐stretched vacuum thermoforming with a relatively complex mold for a commercial refrigerator liner were conducted. The effects of nonlinear rheological behavior were determined by arbitrarily changing model parameters. This allows determination of which rheological features (i.e., elongational mode, viscosity, and strain hardening and/or softening) are most critical to the vacuum snap‐back thermoforming operation. We found that rheological and friction properties showed a predominant role over other processing parameters for uniform thickness distribution.     

Capillary flow of low‐density polyethylene

The capillary flow of a commercial low‐density polyethylene (LDPE) melt was studied both experimentally and numerically. The excess pressure drop due to entry (Bagley correction), the compressibility, the effect of pressure on viscosity, and the possible slip effects on the capillary data analysis have been examined. Using a series of capillary dies having different diameters, D, and length‐to‐diameter L/D ratios, a full rheological characterization has been carried out, and the experimental data have been fitted both with a viscous model (Carreau‐Yasuda) and a viscoelastic one (the Kaye—Bernstein, Kearsley, Zapas/Papanastasiou, Scriven, Macosko, or K‐BKZ/PSM model). Particular emphasis has been given on the pressure‐dependence of viscosity, with a pressure‐dependent coefficient β_p. For the viscous model, the viscosity is a function of both temperature and pressure. For the viscoelastic K‐BKZ model, the time‐temperature shifting concept has been used for the non‐isothermal calculations, while the time–pressure shifting concept has been used to shift the relaxation moduli for the pressure‐dependence effect. It was found that only the viscoelastic simulations were capable of reproducing the experimental data well, while any viscous modeling always underestimates the pressures, especially at the higher apparent shear rates and L/D ratios. POLYM. ENG. SCI., 2012. © 2012 Society of Plastics Engineers     

Effect of viscoelasticity in the film‐blowing process

Numerical simulations have been undertaken for the film‐blowing process of viscoelastic fluids under different operating conditions. Viscoelasticity is described by an integral constitutive equation of the K‐BKZ type with a spectrum of relaxation times, which can fit the experimental data well for the shear and extensional viscosities and the normal stresses measured in shear flow. Nonisothermal conditions are considered by applying the Morland–Lee hypothesis, which incorporates the appropriate shift factor and pseudotime into the constitutive equation. The momentum and energy equations are expressed in the machine direction only by using a quasi‐one‐dimensional approach introduced earlier by Pearson and Petrie. The resulting system of differential equations is solved using the finite element method and the Newton‐Raphson iterative scheme. The method of solution was first checked against the Newtonian and Maxwell results for various film characteristics given earlier. The simulations are compared with available experimental data and previous simulations in terms of film shape, velocity, stresses, and temperature. The present results show that the existing modeling of force balances is inadequate for quantitative agreement with the experimental studies. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2007     

Overall numerical simulation of extrusion blow molding process

This paper focuses on the overall numerical simulation of the parison formation and inflation process of extrusion blow molding. The competing effects due to swell and drawdown in the parison formation process were analyzed by a Lagrangian Eulerian (LE) finite element method (FEM) using an automatic remeshing technique. The parison extruded through an annular die was modeled as an axisymmetric unsteady nonisothermal flow with free surfaces and its viscoelastic properties were described by a K‐BKZ integral constitutive equation. An unsteady die‐swell simulation was performed to predict the time course of the extrudate parison shape under the influence of gravity and the parison controller. In addition, an unsteady large deformation analysis of the parison inflation process was also carried out using a three‐dimensional membrane FEM for viscoelastic material. The inflation sequence for the parison molded into a complex‐shaped mold cavity was analyzed. The numerical results were verified using experimental data from each of the sub‐processes. The greatest advantage of the overall simulation is that the variation in the parison dimension caused by the swell and drawdown effect can be incorporated into the inflation analysis, and consequently, the accuracy of the numerical prediction can be enhanced. The overall simulation technique provides a rational means to assist the mold design and the determination of the optimal process conditions.     

Influence of initial sheet temperature on ABS thermoforming

Understanding the effects of material and processing parameters on the thermoforming process is critical to the optimization of processing conditions and the development of better materials for high quality products. In this study we investigated the influence of initial temperature distribution over the sheet on the part thickness distribution of a vacuum snap‐back forming process. The linear viscoelastic properties along with the Wagner two parameter nonlinear viscoelastic constitutive model were utilized for numerical simulation of the thermoforming operation. Simulations of pre‐stretched vacuum thermoforming with a relatively complex mold for a commercial refrigerator liner were conducted. THe effects of temperature distribution over the sheet on the part thickness distribution were determined to examine process sensitivity and optimization. Effects of the temperature distribution on the material rheology and polymer/mold friction coefficient are primarily responsible for the changes in the thickness distribution. We found that even small temperature differences over the sheet greatly influenced bubble shape and pole position during the bubble growth stage and played a critical role in determining the part thickness distribution. These results are discussed in terms of rheological properties of polymers such as elongational viscosity and strain hardening.     

Derivation of optimal processing parameters of polypropylene foam thermoforming by an artificial neural network

The effects of processing parameters on the thermoforming of polymeric foam sheets are highly nonlinear and fully coupled. The complex interconnection of these dominant processing parameters makes the process design a difficult task. In this study, the optimal processing parameters of polypropylene foam thermoforming are obtained by the use of an artificial neural network. Data from tests carried out on a lab‐scale thermoforming machine were used to train an artificial neural network, which serves as an inverse model of the process. The inverse model has the desired product dimensions as inputs and the corresponding processing parameters as outputs. The structure, together with the training methods, of the artificial neural network is also investigated. The feasibility of the proposed method is demonstrated by experimental manufacturing of cups with optimal geometry derived from the finite element method. Except the dimension deviation at one location, which amounts to 17.14%, deviations of the other locations are all below 3.5%. POLYM. ENG. SCI., 45:375–384, 2005. © 2005 Society of Plastics Engineers     

Thermoforming of a PMMA transparency near glass transition temperature

To simulate the thermoforming of a transparency, constitutive equations are proposed for the nonlinear viscoelastic behavior of poly(methyl methacrylate) near glass transition temperature, which include large deformations. In a first step, they are fitted on a set of uniaxial tension‐relaxation tests at various strain levels and strain rates. In a second step, their implementation in a finite element code is performed. Finally, the thermoforming of a transparency at a constant and uniform temperature is simulated and compared with experimental results. POLYM. ENG. SCI., 50:2004–2012, 2010. © 2010 Society of Plastics Engineers     

Engineering investigations on the potentiality of the thermoformability of HDPE charged by wood flours in the thermoforming part

A dynamic finite element method is used to analyze the thermoformability of composites containing wood and a thermoplastic matrix for five different proportions of wood flour. Linear viscoelastic properties can be obtained by small amplitude oscillatory shear tests and the viscoelastic behavior is characterized using the Lodge model. To account for enclosed gas volume, which inflates the thermoplastic composite membrane, a thermodynamic approach is used to express the external work in terms of a closed volume. Pressure load is deduced by thermodynamic law using the Redlich–Kwong gas equation. The Lagrangian method together with the assumption of membrane theory is used in the finite element implementation. In addition, the influence of air flow on thickness and stress and the energy required to form a thin polymeric part in the thermoforming process are analyzed for five different proportions of wood flour in the HDPE material. POLYM. ENG. SCI., 2009. © 2009 Society of Plastics Engineers     

Birefringence and interface in sequential co‐injection molding of amorphous polymers: Simulation and experiment

Modelings of the interface distribution and flow‐induced residual stresses and birefringence in the sequential co‐injection molding (CIM) of a center‐gated disk were carried out using a numerical scheme based on a hybrid finite element/finite difference/control volume method. A nonlinear viscoelastic constitutive equation and stress‐optical rule were used to model the frozen‐in flow stresses in disks. The compressibility of melts is included in modeling of the packing and cooling stages and not in the filling stage. The thermally induced residual birefringence was calculated using the linear viscoelastic and photoviscoelastic constitutive equations combined with the first‐order rate equation for volume relaxation and the master curves for the relaxation modulus and strain‐optical coefficient functions of each polymer. The influence of the processing variables including melt and mold temperatures and volume of skin melt on the birefringence and interface distribution was analyzed for multilayered PS‐PC‐PS, PS‐PMMA‐PS, and PMMA–PC–PMMA molded disks obtained by CIM. The interface distribution and residual birefringence in the molded disks were measured. The measured interface distributions and the gapwise birefringence distributions in CIM disks were found to be in a fair agreement with the predicted interface distributions and the total residual birefringence obtained by the summation of the predicted frozen‐in flow and thermal birefringence. POLYM. ENG. SCI., 55:88–106, 2015. © 2014 Society of Plastics Engineers     

Modeling of visco‐hyperelastic behavior of transversely isotropic functionally graded rubbers

In this article, visco‐hyperelastic constitutive model is developed to describe the rate‐dependent behavior of transversely isotropic functionally graded rubber‐like materials at finite deformations. Zener model that consists of Maxwell element parallel to a hyperelastic equilibrium spring is used in this article. Steady state response is described by equilibrium hyperelastic spring and rate‐dependence behavior is modeled by Maxwell element that consists of a hyperelastic intermediate spring and a nonlinear viscous damper. Modified and reinforced neo‐Hookean strain energy function is proposed for the two hyperelastic springs. The mechanical properties and material constants of strain energy function are graded along the axial direction based on exponential function. A history‐integral method has been used to develop a constitutive equation for modeling the behavior of the model. The applied history integral method is based on the Kaye‐BKZ theory. The material constant parameters appeared in the formulation have been determined with the aid of available uniaxial tensile experimental tests for a specific material and the results are compared to experimental results. It is then concluded that, the proposed constitutive equation is quite proficient in forecasting the behavior of rubber‐like materials in different deformation and wide ranges of strain rate. POLYM. ENG. SCI., 56:342–347, 2016. © 2016 Society of Plastics Engineers     

Vacuum forming of thermoplastic foam

The process of thermoforming of foam sheet is analyzed using both finite element modeling and experiments. A simple constitutive model for finite tensile deformations of closed cellular material around its glass transition temperature is proposed, starting from well-known results from Gibson and Ashby (1988). The model is implemented in a finite element code and applied in isothermal vacuum forming simulations. The distributions of thickness and in plane strains are in adequate accordance to the experimental results.     

Finite element analysis for thermoforming process of continuous fiber reinforced thermoplastic composites

An implicit finite element method is presented to model thermoforming process of thermoplastic materials reinforced with unidirectional fibers or woven fabric. The composite materials are regarded as a homogenized anisotropic viscous fluid restricted by kinematical constrains of material incompressibility and fiber inextensibility at the forming temperature. Unlike previous numerical approaches considering quasi‐static assumptions during each increment, a high order polynomial expression is used to define continuous displacement and velocity fields. The material parameters are also updated during each increment to determine the displacement and velocity fields satisfying the equilibrium equations as well as the kinematical constrains both in the rate and total deformation forms. An iterative solution method is presented to solve the finite element formulation. The numerical results are compared with analytical solutions and the results of previous implicit finite element approaches to demonstrate the computational accuracy and efficiency of the present method. POLYM. COMPOS., 2009. © 2008 Society of Plastics Engineers     

Birefringence in injection‐compression molding of amorphous polymers: Simulation and experiment

The influence of the processing variables on the residual birefringence was analyzed for polystyrene and polycarbonate disks obtained by injection‐compression molding under various processing conditions. The processing variables studied were melt and mold temperatures, compression stroke, and switchover time. The modeling of flow‐induced residual stresses and birefringence of amorphous polymers in injection‐compression molded center‐gated disks was carried out using a numerical scheme based on a hybrid finite element/finite difference/control volume method. A nonlinear viscoelastic constitutive equation and stress‐optical rule were used to model frozen‐in flow stresses in moldings. The filling, compression, packing, and cooling stages were considered. Thermally‐induced residual birefringence was calculated using the linear viscoelastic and photoviscoelastic constitutive equations combined with the first‐order rate equation for volume relaxation and the master curves for the Young's relaxation modulus and strain‐optical coefficient functions. The residual birefringence in injection‐compression moldings was measured. The effects of various processing conditions on the measured and simulated birefringence distribution Δn and average transverse birefringence <n_rr?n_θθ> were elucidated. Comparison of the birefringence in disks manufactured by the injection molding and injection‐compression molding was made. The predicted and measured birefringence is found to be in fair agreement. POLYM. ENG. SCI., 2013. © 2013 Society of Plastics Engineers     

Three‐dimensional simulation of thermoforming process and its comparison with experiments

Three‐dimensional solid element analysis and the membrane approximated analysis employing the hyperelastic material model have been developed for the simulation of the thermoforming process. For the free inflation test of a rectangular sheet, these two analyses showed the same behavior when the sheet thickness was thin, and they deviated more and more as the sheet thickness increased. In this research, we made a guideline for the accuracy range of sheet thickness for the membrane analysis to be applied. The simulations were performed for both vacuum forming and the plug‐assisted forming process. To compare the simulation results with experiments, laboratory scale thermoforming experiments were performed with acrylonitrile‐butadiene‐styrene (ABS). The material parameters of the hyperelastic model were obtained by uni‐directional hot tensile tests, and the thickness distributions obtained from experiments corresponded well with the numerical results. Non‐isothermal analysis that took into account the sheet, temperature distribution measured directly from the experiments was also performed. It was found that the non‐isothermal analysis greatly improved the predictability of the numerical simulation, and it is important to take into account the sheet temperature distribution for a more reliable simulation of the thermoforming process.     

Effects of material properties and contact conditions in modelling of plug assisted thermoforming

Abstract

Thin walled food packaging is commonly manufactured using the plug assisted thermoforming process. In this paper the development of finite element models of the process is described. Work has concentrated on understanding the effects of material properties and contact conditions on the output from these simulations. The results have shown that a viscoelastic model must be used to simulate the deformation behaviour of the plastics. Contact conditions must also be accounted for in the models by including the effects of friction and heat transfer between the sheet and tool surfaces. For improved model accuracy, it is recommended that further experimental work should be carried out to enhance the viscoelastic material models and to provide better understanding of actual contact conditions.     

Dynamic characteristics of plug‐assist thermoforming process

Plug‐assist thermoforming is a well‐known technique in polymer processing because of its interesting features. The dynamic value of driving‐force for the stretching process is determined based on equilibrium equation. This amount of force is required for applying to a plug to stretch a sheet. It is used for calculation of the required theoretical work and power of a plug‐assist thermoforming process. By using a nonlinear viscoelastic rheological model in the proposed mathematical model, its validity was examined by performing experimental tests on ABS sheets. POLYM. ENG. SCI., 2009. © 2008 Society of Plastics Engineers