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     

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.     

Thermodynamic approach of inflation process of K‐BKZ polymer sheet with respect to thermoforming

The problem of modeling and the dynamic finite element simulation of thermoforming process for viscoelastic sheet are considered. The pressure load used in modeling 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 finite element implementation. The numerical validation is performed by comparing the theoretical solution for the uniaxial and equibiaxial hencky deformation with numerical results. Moreover, the influence of the K‐BKZ constitutive model, for three linear time distribution of airflow rate loading, on the thickness and on the stress distribution in thermoforming of containers made of HDPE are analyzed. POLYM. ENG. SCI., 45:1319–1335, 2005. © 2005 Society of Plastics Engineers     

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.     

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.     

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.     

Wall thickness distribution in thermoformed food containers produced by a Benco aseptic packaging machine

Effects of process parameters such as forming temperature, forming air pressure and heating time on wall thickness distribution in plug‐assist thermoformed food containers using multilayered material were investigated. Multilayered rollstockbase material formed into containers by thermoforming process using a Benco aseptic packaging machine. Forming temperatures in the range of 131–170°C, airforming pressures of 2, 3, 3. 5 and 4 bars, and heating times of 66, 74, 84, 97 and 114 seconds were used in the thermoforming process. Analysis of wall thickness data obtained for the thermoforming parameters used in this study showed that wall thickness was significantly affected by forming temperature, pressure and heating time at 0.05 significance level. Besides the processing parameters, wall location, container side, and their interactions significantly affected wall thickness. Forming temperature was found to be the principle parameter influencing wall thickness distribution in a plug‐assist thermoforming operation. The optimum operating conditions of the packaging machine for the thermoforming process are: 146–156°C for forming temperature, 2–4 bars for air‐forming pressure and 74–97 seconds for heating time.     

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.     

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     

Study of the thermoformability of ethylene‐vinyl alcohol copolymer based barrier blends of interest in food packaging applications

The thermoforming capacity of a number of blends of an ethylene‐vinyl alcohol copolymer (EVOH‐32, with 32 mol % ethylene) with amorphous polyamide (aPA) and/or Nylon‐containing ionomer with interest in multilayer food packaging structures have been studied. These blends were vacuum‐thermoformed between 100 and 150°C onto male molds of different shapes and areal draw ratios. It was found that EVOH/aPA extruded blends did not improve the inherently poor formability of EVOH alone. In contrast, significant improvements in thermoformability were achieved by blending EVOH with a compatibilized‐ionomer. Optimum forming capacity was achieved in a ternary blend by addition of a compatibilized‐ionomer to EVOH/aPA blends in the range of 140–150°C. Analysis of wall thickness data obtained in the thermoformed parts showed that wall thickness was significantly affected by the ionomer and amorphous polyamide content in the blend. The ternary blend showed a lower thickness reduction in the critical areas, as well as a higher uniformity in the part. A finite element analysis was used to evaluate the wall thickness distribution and the modeling results were compared with the thermoforming experiments. The simulations were performed for the vacuum‐forming process employing a nonlinear elastic material model. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 96: 3851–3855, 2004     

The role of process parameters in determining wall thickness distribution in plug‐assisted thermoforming

This article describes the results of a comprehensive investigation to determine the link between process parameters and observed wall thickness output for the plug‐assisted thermoforming process. The overall objective of the work was to systematically investigate the process parameters that may be adjusted during production to control the wall thickness distribution of parts manufactured by plug‐assisted thermoforming. The parameters investigated were the sheet temperature, plug temperature, plug speed, plug displacement, plug shape, and air pressure. As well as quantifying the effects of each parameter on the wall thickness distribution, a further aim of the work was to improve the understanding of the physical mechanisms of deformation of the sheet during the different stages of the process. The process parameters shown to have the greatest effect on experimentally determined wall thickness distribution were the plug displacement, sheet temperature, plug temperature, and plug shape. It is proposed that during the plug‐assisted thermoforming of polystyrene the temperature dependent friction between the plug and sheet surface was the most important factor in determining product wall thickness distribution, whereas heat transfer was shown to play a less important role. POLYM. ENG. SCI., 50:1923–1934, 2010. © 2010 Society of Plastics Engineers     

A viscoelastic–viscoplastic modeling approach for amorphous polymers in vacuum forming simulation

In this study, the simulation of a vacuum forming process employing a micromechanical inspired viscoelastic–viscoplastic model is investigated. In the vacuum forming process, a plastic sheet is heated above the glass transition temperature and subsequently forced into a mold by applying a vacuum. The model consists of a generalized Maxwell model combined with an dissipative element in series. Each Maxwell element incorporates a hyperelastic element in series with a viscous element based on a hyperbolical law. While the generalized Maxwell model considers the relaxation due to molecular alignment, the additional viscous element is a modification based on the approach of Bergström and thus considers molecular chain reptation. The model is designed with the aim to converge to the generalized linear Maxwell model in the limit of small deformation. Furthermore, the viscous modeling is temperature activated and follows the Williams–Landel–Ferry approach in the limit of linear viscoelasticity. To simulate rheological standard experiments, a physical-network-based implementation into Simscape is presented. To validate the performance of the model in thermoforming, it is implemented into Fortran programming language for finite element simulation with Abaqus/Explicit. It can be shown that the simulation is able to predict the thickness in high correlation with experimental results.     

Improved modeling for the reheat phase in thermoforming through an uncertainty treatment of the key parameters

This work focuses on the treatment of parameter uncertainty in the simulation of the sheet reheat phase of the thermoforming process. The approach aims to improve the quality of predictions through more accurate evaluation of the input parameters. First, the modeling approach is employed to perform a sensitivity analysis on the reheat phase. Then, a series of specialized experiments with heat flux and temperature sensors are performed on a thermoforming machine. The key parameters identified through the sensitivity analysis are the subject of these experiments. The natural convective heat transfer coefficients are evaluated by two different approaches. Through treatment of the uncertainty associated with the input parameters, the prediction of sheet reheat phase is significantly improved.     

The effects of radiation cross‐linking and process parameters on the behavior of polyamide 12 in vacuum thermoforming

Compared to amorphous thermoplastics, semi‐crystalline thermoplastics usually have a smaller processing range for thermoforming, due to their narrow temperature window for the transition from viscoelastic to viscous material behavior. On the other hand, semi‐crystalline thermoplastics offer superior properties for applications like ductility or chemical resistance. Within this article, modification of semi‐crystalline polyamide 12 by radiation cross‐linking with respect to its suitability for vacuum thermoforming as well as the effects of processing parameters and sheet thickness on the resulting strain distributions in thermoformed parts are shown. Experimental thermoforming processing studies in combination with digital image correlation measurements, thermo‐mechanical and elongational rheometry were performed to characterize the behavior of cross‐linked semi‐crystalline thermoplastics in the vacuum thermoforming process. POLYM. ENG. SCI., 2011. ©2011 Society of Plastics Engineers     

Numerical and experimental studies on the effect of thickness difference between up and down gas layers on sheet polymer gas-assisted extrusion forming

In this study, to make gas-assisted extrusion (GAE) can be better applied to conventional horizontal extrusion forming process, the thickness difference between up and down gas layer was simulated for the first time by GAE of molten high-density polyethylene sheet. In order to ascertain the effects of the thickness difference of up and down gas layers on sheet extrusion molding, a method based on the model of two-phase (gas and melt) flow was put forward, and an outer-layer GAE isothermal numerical simulation made by POLYFLOW was explored and researched. The numerical results analysis indicated that when the thickness of down gas layer is increased, the velocities, pressure drop and shear rate of the melt increase. Compared with the simplified-GAE simulation, GAE simulation method can well reflect the flow characteristics, the physical field distribution, and the morphology changes of melt extrudates. The experimental results show that with the increase of the thickness of the down gas layer, the melt falling can be obviously improved during extrusion. Thus, in the actual use of GAE for sheet production, not only the thickness difference between up and down gas layers should be reasonably controlled, but also other factors, such as melt inlet volume flow rate, gas pressure and temperature should be controlled accordingly. Finally, GAE forming technology can give full play to its advantages.     

Coupled thermo‐mechanical analysis for plastic thermoforming

An FEM software ARVIP‐3D was developed to simulate the process of 3‐D plastic thermoforming. The coupled thermo‐mechanical analysis, thermal stress and warpage analysis for plastic thermoforming was carried out by means of this software. Rigid visco‐plastic formula was adopted to simulate the deforming process. During this process, the method of comparing velocity, time and area was adopted as the contact algorithm at different nodes and triangular elements. Sticking contact was assumed when the nodes become in contact with tool surface. The Arrhenius equation and the Williams equation were employed to ascertain the temperature dependence of material properties. In order to analyze the temperature field of plastic thermoforming, the Galerkin FEM code and the dynamic heat conduction boundary condition were adopted; latent heat and deformation heat were treated as dynamic internal heat sources. Based on the above, the model of coupled thermomechanical analysis was established. Assuming that the thermal deformation occurs under elastic conditions, the thermal stress and the warpage following the cooling stage were estimated. Experiments of plastic thermoforming were made for high‐density polyethylene (HDPE). An infrared thermometer was used to record the temperature field and a spiral micrometer was used to measure the thickness of the part. Results of numerical calculation for thickness distribution, temperature field and warpage were in good agreement with experimental results.     

Modeling of Deformation Processes in Vacuum Thermoforming of a Pre-stretched Sheet

Modeling of deformation processes in vacuum thermoforming for a preliminary stretched thermoplastic sheet (plug-assist vacuum thermoforming) is investigated in this paper. The model can be used for production of polymeric articles with minor wall-thickness variation. A nonlinear rheological model is implemented for developing the process model. It describes deformation process of a prestretched sheet at any phase of vacuum thermoforming process. This process is described by a set of deformation processes, each specified by appropriate boundary conditions. For model validation, a comparative analysis of the theoretical and experimental data is presented. The wall-thickness distributions obtained from modeling results corresponded well with experiments. A method for prediction and enhancement of the quality of the final products on the basis of wall-thickness distribution criterion has been established.     

Thermoforming shrinkage prediction

Most thermoforming product development processes rely on costly and time‐consuming forming trials to determine the adequacy of the mold and process. In this paper, an analytical method is developed for shrinkage predictions on the basis of a visco‐elastic constitutive material model with initial conditions from a commercial thermoforming simulation. The theoretical analysis has been developed to accommodate different sets of materials, process conditions, and mold geometry. The shrinkage model consists of a transient thermal analysis for temperature solution; a stretching phase analysis for Inflation‐induced stress estimation; a post‐contact analysis for thermal stress and relaxation; and a post‐molding strain analysis based on stress solution. The shrinkage prediction analysis has been developed and validated with a complex geometry thermoforming application. The results indicate that the shrinkage estimates provided by the analysis were within the objective tolerances of 0.1%, as measured in terms of absolute prediction error of part dimensions.     

Improved mathematical modeling for the sheet reheat phase during thermoforming

In any thermoforming process, plastic sheet heating is the most important phase as it is responsible for final part quality as well as overall process efficiency and productivity. The goal of the study reported here was to improve existing mathematical models to accurately predict the temperature profile inside a heated sheet, where the model could be used to better control the overall thermoforming process. A mathematical model with temperature dependent, variable sheet material properties including density, thermal diffusivity, specific heat, and thermal conductivity was developed and validated against experimental data. Models with constant and variable plastic sheet properties were created, simulated, and compared in Matlab. The models were validated by experiments which obtained temperature profiles at different depths within a plastic sheet by inserting thermocouples and recording temperatures. Further, the effect of sheet color on heating was investigated by considering two extreme cases: white (transparent) and black (opaque) colored sheets, and the effect of oven air temperature and velocity on sheet heating was also investigated. Results indicated that a variable properties model was needed to control sheet reheating especially with narrow forming windows, and that the heating profiles required for colored and noncolored sheets were very different. POLYM. ENG. SCI., 2012. © 2011 Society of Plastics Engineers     

Finite element analysis of blow molding and thermoforming using a dynamic explicit procedure

This paper reports on the development of a dynamic finite element procedure for the simulation of blow molding and thermoforming of thermoplastic hollow parts. The Principle of Virtual Work written herein takes inertia effects into account. The heat‐softened parison is assumed to be a nonlinear hyperelastic Mooney‐Rivlin membrane and is meshed with classical linear triangular finite elements. We adopt the explicit central differences time integration scheme with the special lumping technique. The moid is divided into triangular elements and the contact between the parison and the mold is assumed to be sticky. Therefore, contacted degrees of freedom of the parison are fixed on the solid boundary until the end of the simultion. Performances are highly improved by the use of an adaptive mesh refinement procedure based on a geometric criterion for detection and on the simple addition of a node at the mid‐side of the longest edge for subdivision. The method is illustrated through some examples of thermoformed and blow‐molded parts. Our results are compared with both experimental and numerical results from literature to validate the present theory.