Effects of Rheological Properties and Processing Parameters on ABS Thermoforming.

JE KYUN LEE [1]

TERRY L. VIRKLER [2]

CHRIS E. SCOTT [1][*]

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 refr igerator 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.

INTRODUCTION

Thermoforming is a processing method in which a heated thermoplastic sheet is deformed and shaped over a mold geometry by mechanical, air, or vacuum pressure. There are many kinds of thermoforming techniques, ranging from simple vacuum or drape forming, in which the softened polymer sheet is simply draped over a male or female mold to more complex processes in which a pre-stretching stage of the polymer sheet is included before contacting the mold [1-3]. Pre-stretching techniques such as plug assist and snap-back forming are used to obtain more accurate and uniform thickness distribution of parts thermoformed in complex geometries. Uniform thickness distribution and optimum average thickness are very important properties for manufacturing high quality parts economically. Some of the main processing problems in thermoformed parts include uneven thickness distribution, excess thinning in certain locations of the parts, and the incidence of webs which are caused by the gathering of excess material. Heterogeneous polymeric systems and specifically rubber modified polymeric materials such as acrylonitrile-butadiene-styrene copolymers (ABS) are of particular interest since they possess excellent properties for consumer goods (toughness, durability, chemical and heat resistance to name a few) and can be produced over a wide range of processing temperature.

The primary objectives of the present study are to investigate the effects of rheological properties and processing parameters on thermoforming behavior, thus providing fundamental information for process improvement. Understanding the effects of material and processing parameters on polymer processing is very important to establish optimum processing conditions and develop better materials for high quality products. There are a number of experimental [4-13] and simulation [14-32] efforts devoted to establishing optimum processing conditions such as temperatures, pressure, and friction coefficient and to investigating the effects of material variables such as elongational viscosity behavior or melt strength on thermoforming performance. Despite these investigations, the effects of material and processing parameters on the performance of thermoforming operations are not well understood. Also, little work has been devoted to understanding the basic relation between melt failure (i.e., necking, melt or brittle rupture), strain hardening and/or softening and thermoforming operation of heterogeneous polymeric systems like ABS polymer [30, 31, 33-37]. Investigation of rheological behavior during uniaxial and biaxial stretching is necessary for prediction of material performance in a variety of commercial applications such as thermoforming and blow molding, and it is also fundamental for polymer rheology and processing.

It is well known that a good thermoforming material must have enough viscous character to provide flow under the applied stress and sufficient elastic character to avoid excessive thinning. Viscoelastic constitutive models are required to describe both viscous and elastic property of polymer. However, many previous investigators [14-25] used the hyperelastic model to describe the rheological properties, after assuming that the polymer follows elastic behavior under the conditions of relatively high strain rate in the thermoforming process. They reported that this assumption may be quite sufficient for predicting the thickness distribution. However, most of these works have concerned simple vacuum or drape forming techniques and molds of simple geometry. Several investigators [26-31] reported in their recent studies that viscoelastic constitutive models were more proper for describing rheological properties during thermoforming operations. On the other hand, Laroche and Erchiqui [32] compared the predictabili ty of both hyperelastic and viscoelastic constitutive models for vacuum thermoforming process and concluded that both models gave similar results and over-predicted the thickness distribution of thermoformed parts. Generally, the viscoelastic constitutive models are required to accurately predict the elongational behavior including elongational viscosity and strain hardening and/or softening by fitting of experimental data as a function of strain rate and time. Based on a previous study, we used the Wagner model [38], which is a K-BKZ type [39, 40] single integral nonlinear viscoelastic constitutive model, in this study. The fitting and rheological properties of ABS polymer by viscoelastic constitutive models were summarized more detail in our previous study [41].

Several works [19, 20, 26, 29-31, 33] devoted to understanding the effect of friction between the sheet and mold on thermoforming performance concluded that friction between the sheet and mold played a very important role in determining the thickness distribution of thermoformed parts, especially in pre-stretching techniques such as plug assist thermoforming. Almost all recent studies used the coulomb friction model to describe the partial slip condition between the sheet and polymer. However, they did not show quantitatively the effect of different friction coefficients on performance of the thermoforming operation. The methods of obtaining friction coefficients and their temperature dependence are not clearly explained. Tsai [33] reported that the friction between a sheet and a mold is a function of temperature in which the friction increased dramatically near the glass transition temperature of polystyrene. He pointed out that because of friction, the thickness distribution of a thermoformed part can be m anipulated by mold temperature with vacuum and/or pressure assistance. However, he did not show the effect of the friction coefficient on the thickness distribution of thermoformed parts quantitatively. It is expected that if both the sheet and mold temperatures are very high, the friction force would be extremely high in the area where contact first occurred. As a result, the thickness distribution of thermoformed parts becomes nonuniform.

The primary reason for computer simulation of thermoforming is to provide fundamental information concerning the effects of design variables (i.e., thermoforming technique, the shape and curvature of plug and/or mold), processing variables (i.e., mold speed, pressure, temperature, and friction property between sheet and mold), and material variables (i.e., sheet thickness, rheological properties, and thermal stability) on the important properties of final thermoformed parts. Because there are many parameters, which affect the performance of a thermoforming operation, it is very difficult to design the process and optimize processing conditions and material properties by trial and error without the basic information obtained from computer simulation.

in this study simulations of the thermoforming process were conducted using commercial software (T-SIM[R] T-SIM CZ Ltd.). By arbitrarily changing the parameters in the constitutive equation ([alpha] and [beta] parameters of Wagner model) we have constructed variations in the rheological behavior in terms of the nonlinear viscoelastic regime, strain hardening and/or softening, and the temperature sensitivity of elongational behavior. This allowed us to determine which processing and rheological features are most critical for the thermoforming operation.

EXPERIMENTAL

Material

We studied a commercial ABS copolymer from Bayer Corporation widely used in thermoforming of large parts. The copolymer matrix (styrene-acrylonitrile (SAN) copolymer, [M.sub.w] = 70,000) consisted of 30 wt% acrylonitrile and 70 wt% styrene. The glass transition temperature of the copolymer was Tg = 125[degrees]C. The grafted rubber phase is primarily a narrow distribution with weight average diameter of 0.2 [micro]m constituting about 13 wt% rubber in the total composition. The ABS density at 25[degrees]C thermal conductivity, and heat capacity, which were provided by Bayer Corporation, are 1050 Kg/[cm.sup.2], 0.105 W/m K, and 2093 J/kg K, respectively. The material was extruded into 2.5 or 3 mm thick sheets for thermoforming experiments and the shrinkage of sheet in extrusion direction and cross direction were 15% and -5%, respectively. Based on transmission electron micrographs of the materials, the rubber particles were well-dispersed with no observable particle agglomeration.

Rheological Measurements

Tensile stress growth coefficients in uniaxial elongation were measured on a Meissner-type extensional rheometer (Rheometrics RME) for the melt temperatures from 140-200[degrees]C and constant elongation rates [epsilon] = 0.01-1.0 [s.sup.-1] which are typical for processing operations. The basic operating principle and detailed description of Meissner's extensional rheometer can be found in (42) and Rheometrics literature.

Shear measurements were performed on the ARES mechanical spectrometer by Rheometric Scientific (Piscataway, N.J.). The shear experiments were conducted on 8 and 25 mm diameter samples cut from the 3 mm thick extruded ABS sheet using parallel plate and cone-and-plate geometries. Small strain oscillatory shear measurements were used to determine linear viscoelastic spectra of the material in the frequency range from [10.sup.-2]-[10.sup.2] [s.sup.-1] and in the temperature range from 140-200[degrees]C. The shear stress growth coefficient measurements for the start-up of steady shear were performed for the shear rates [gamma] = 0.001-10.0 [s.sup.-1] at 170[degrees]C. The samples were examined after the experiment to ensure their uniformity and the absence of the edge instability. Details of the rheological measurements can be found in (41).

Viscoelastic Constitutive Model

In this study the Wagner model, which is a K-BKZ type single integral nonlinear viscoelastic constitutive model, was fit to shear and elongational experimental data to obtain [alpha] and [beta] parameters. K-BKZ type single integral models have the following general form:

{[T.sub.[approximate]] = - p[I.sub.[approximate]] + [[tau].sub.[approximate]]

[[tau].sub.[approximate]] = [[[integral].sup.t].sub.-[infinity]] M(t-t') [[[phi].sub.1] ([I.sub.1],[I.sub.2])[[C.sup.-1].sub.[approximate]t] (t') + [[phi].sub.2]([I.sub.1],[I.sub.2]) [C.sub.[approximate]t] (t')] dt' (1)

where [[C.sup.-1].sub.[approximate]t] (t') and [C.sub.[approximate]t] (t') are the Finger strain tensor and Cauchy strain tensor respectively, M(t - t') is the memory function, [I.sub.1](t,t') and [I.sub.2](t,t') are the first two invariants of the Finger tensor (the third invariant [I.sub.3] [equivalent] 1 for incompressible fluids), and damping functions [[phi].sub.1] ([I.sub.1], [I.sub.2]), [[phi].sub.2] ([I.sub.1], [I.sub.2]) are defined as follows:

Wagner model:

[[phi].sub.1] = exp (-[beta][[[alpha][I.sub.1] + (1 - [alpha])[I.sub.2] - 3].sup.0.5]), [[phi].sub.2] = 0 (2)

Also, the first two invariants of the Finger tensor, [I.sub.1](t,t') and [I.sub.2](t,t') for uniaxial and biaxial elongational flow are as follows:

Uniaxial: {[I.sub.1] = [e.sup.2[epsilon]] + 2[e.sup.-[epsilon]]

[I.sub.2] = 2[e.sup.[epsilon]] + [e.sup.-2[epsilon]], (3)

Biaxial: {[I.sub.1] = [e.sup.-2[epsilon]] + 2[e.sup.[epsilon]]

[I.sub.2] = 2[e.sup.-[epsilon]] + [e.sup.2[epsilon]]

The usual Lodge network model (43) was used to describe the memory function:

M(t - t') = [[[sigma].sup.N].sub.i = 1] [[eta].sub.i]/[[[lambda].sup.2].sub.i] exp (-(t - t')/[[lambda].sub.i]) (4)

The following Arrhenius relation was used to describe the temperature dependence of material rheological behavior during thermoforming simulation.

[[eta].sup.0] (T) = [[[eta].sup.0].sub.o] ([T.sub.0]) exp[-A(T - [T.sub.0])] (5)

where [[[eta].sup.0].sub.o] ([T.sub.0]) is the zero shear viscosity at reference temperature [T.sub.0], [[eta].sup.0] (T) is the zero shear viscosity at temperature T, and A is a fitted constant.

Measurement of Friction Coefficients

The dynamic friction coefficients at different temperatures were obtained using the parallel plate fixtures of Rheometric Mechanical Spectrometer (RMS-800EH) with the 200 gm-cm transducer. Disc specimens of 25[sim]33 mm diameter and 2 mm thickness were compression molded after drying the polymer sheet at 65[degrees]C under vacuum for 16 hours. A disposable flat aluminum disc plate of either 25 or 50 mm diameter was used as an actual friction surface. Between each run the aluminum plate was washed with acetone to remove any residual polymer or additives. The instrument was operated in steady shear where a lower parallel plate fixture was rotated at a continuous rate of 1 rpm. The specimen assemblies were loaded and brought to 90[degrees]C without making any contact with the aluminum friction surface. The transducer was zeroed for both torque and normal force and then, the run started. Measurements were made in temperature steps of either 5 or 10[degrees]C, depending on the temperature region, and the time requ ired for temperature changes was 5[sim]10 minutes. All runs were made under [N.sub.2] atmosphere to prevent from oxidative degradation. A thermocouple inside the lower tool contacted the bottom friction surface and was used to indicate the temperature of the sliding surfaces. The dynamic friction coefficient was calculated using the relationship between torque and normal force measured as follows:

F = 3/2R ([gamma]/[F.sub.N]) (6)

where, R = sample radius, [gamma] = torque, [F.sub.N] is the normal force. The results obtained below 140[degrees]C were very reproducible. However, the accuracy of the experimental friction coefficients measured at higher temperatures over 140[degrees]C is questionable due to low viscosity of polymer (the friction force exceeded the melt strength) and sticking of the polymer to the fixture surface. Figure 1 provides a schematic illustration of the equipment as well as the experimental data. In the thermoforming process, the mold and sheet temperatures are different. Therefore, in the numerical simulation the temperature used for the friction coefficient was the average of the sheet and mold temperatures.

Thermoforming

The method of thermoforming used in this study is referred to as vacuum snap-back forming, a schematic of which is shown in Fig. 2. Immediately after completion of the heating cycle, the sheet undergoes a pre-stretching, which is implemented by moving the top vacuum box down over the sheet and pulling a vacuum. Both the bubble height and the rate at which the vacuum is pulled can be varied. Calculations have shown that maximum strains for the part studied here in the pre-stretched bubble approach 230%. Following pre-stretching is a delay time until the rising mold contacts the deformed sheet. Forming is completed by continued upward motion of the mold, and final application of vacuum to pull in the remaining non-contacting regions. The relatively complex mold of a commercial refrigerator liner was used, as detailed in Fig. 3. Experiments were run using a Drypoll thermoformer.

Thermoforming Simulations

The commercial simulation program T-SIM[R] (version 3.4, T-SIM CZ Ltd.) was used to predict the effect of material and processing parameters on the thickness distribution. This software uses K-BKZ type single integral nonlinear viscoelastic constitutive models to describe viscoelasticity and large deformation of material. Coulomb's law of friction was enforced at areas of contact between sheet and mold. The sheet was spatially divided into 3626 elements with 1913 nodes in order to provide a balance of simulation time and accuracy. It is common practice in the industry for processes with these types of molds to be run by controlling the bubble height. Thus, a constant bubble height of approximately 390 mm was used for our simulation. Five relaxation times were used to describe linear viscoelastic behavior and the time step was 0.05s. The heat transfer was calculated after the time step and the temperature field in the elements was modified on this basis. More detailed features and descriptions of T-SIM[R] simu lation software can be found in [44]. The effects of material shrinkage and sagging of the sheet on the thickness distribution were not simulated. A recent study of shrinkage and sagging effects can be found in [45] and [46], respectively. Unless otherwise noted, the simulations were run under standard conditions of [T.sub.sheet] = 170[degrees]C, [T.sub.mold] = 90[degrees]C, f = 0.25, bubble height = 390 mm, [alpha] = 0.2893, [beta] 0.25, and sheet thickness 3.05 mm.

Evaluation of Thermoforming Performance

In order to compare the thermoforming performance in terms of part thickness distribution quantitatively, we used the Coefficient of Variation (COV), which is the standard deviation normalized by average thickness (e.g., STDEV/average thickness (mm) [*] 100). The standard deviation (STDEV) measures how widely values are dispersed from the average value (the mean), using following equation:

STDEV = [square root] n[sigma][[x.sup.2].sub.i] - [([sigma][x.sub.i]).sup.2]/n(n - 1) (7)

where n is the number of points at which the thickness was measured and {[x.sub.i]} is the set of thickness values. All measurement was made along the center line illustrated in Fig. 3. Note that the average part thickness is constant because of conservation of mass, but the average thickness along the measuring line may vary as a result of movement of material in or out of that area. Management of the COV value is commonly used in the thermoforming industry for quality control. A smaller COV indicates a more uniform and thus better quality part. Also, when the effects of material and processing parameters on the average thickness along line of symmetry are significant. the values of thickness are included with COV values for better understanding.

RESULTS AND DISCUSSION

Elongational Viscosities and the Constitutive Model

The measured uniaxial elongational viscosities for the ABS at [epsilon] = 1.0 [s.sup.-1] along with the best fit by Wagner model are provided in Fig. 4. Considerable strain hardening behavior is exhibited at lower temperatures, but at higher temperature strain hardening is reduced primarily due to increasing mobility of the copolymer matrix. The experimental elongational viscosity curve has a maximum followed by a sudden drop associated melt failure due to necking or other instability. The maximum is an artifact of nonuniform deformation and melt failure which occur during the strain softening phase [41]. The apparent Hencky strain-to-neck (strain at visually observed neck) is roughly constant (2.76 [+ or -] 0.20) within experimental error and independent of temperature and elongation rate for this material.

The Wagner model parameter values obtained by data fitting at different temperatures are given in Table 1. Note that the model parameter values vary substantially with temperature. The temperature-specific model parameter values rather than the averages were used. Although the Wagner model predicts a steady-state elongational viscosity that is not observed experimentally owing to sample failure, it simulates more accurately the experimentally observed maximum in elongational viscosity [41]. Since some part of thermoforming may proceed beyond the melt failure point, it is important to have an accurate model prediction for maximum elongation viscosity. The study of a viscoelastic constitutive model to predict the melt failure behavior is separate topic, requiring further investigation.

Comparison of Simulation With Experimental Results

Comparison of the simulation and experimental results is not emphasized in this paper, but instead in a separate study. However, one example is presented here to support the validity of the simulations. Figure 5 presents the comparison for a typical set of conditions for this combination of material and process in terms of the thickness distribution along the measuring line shown in Fig. 3. The COV was 22.8 for simulation, compared with 25.5 in the experiment. Also, the average thickness along the measuring line was 0.85 mm for simulation, compared with 0.87 mm in the experiment. Considering the complexity of the mold and process in this example, agreement is excellent. In particular, the simulated thickness distribution in the bottom area is very consistent with experimental results. Several investigators (22, 24, 32) reported large differences between experimental and simulated thickness distributions when molds with relatively complex geometries were used.

The back area is where the pole of bubble contacts the mold. The bubble formation is primarily a biaxial deformation, especially near the top of bubble. Variation of the flow mode over the surface of the part will be discussed systematically in a later section. The differences between the experimental and simulation results are partially due to the use of rheological parameters obtained from shear and uniaxial elongation experiments to predict biaxial behavior. As Treloar (47) and Koziey et al. (15) pointed out, biaxial elongational behavior calculated by using model parameters obtained from uniaxial viscosity data may give erroneous results. In addition, shrinkage (45), sagging (46) and nonuniform temperature profile over the sheet (48) occurred during the heating stage in the experimental thermoforming operation. These are not considered in this study and may play important roles in the differences between the experimental and simulation results.

Effects of the Nonlinear Rheological Model Parameters

Modification of the [alpha] and [beta] parameters of the constitutive equation provides for changes in the nonlinear rheological behavior. Subsequent numerical simulation of the thermoforming behavior of these variations yields important information concerning rheologically desirable behavior. Figure 6 illustrates the dependence of the uniaxial elongational viscosity of ABS polymer on the [alpha] and [beta] parameters. As the value of either [alpha] or [beta] increases, the uniaxial elongational viscosity is decreased and the maximum is observed at earlier times. As expected from Eq 2. the effect of the [beta] parameter is stronger than that of the a parameter. Note that the best fit values of the [alpha] and [beta] parameters to the experimental measurement were [alpha] = 0.2893 and [beta] = 0.25 at 170[degrees]C (see Table 1), which are located in the middle of Fig. 6a and b.

We also predicted the biaxial elongational viscosity behavior using the Wagner model. Since the [alpha] and [beta] parameters were obtained by fitting of experimental data in startup of steady shear and uniaxial elongation, the accuracy of extrapolation to biaxial elongation is questionable. However, the simulation software uses these parameters for all types of deformation. The results are shown in Fig. 7. The dependence of biaxial elongational viscosity on the model parameter a is different from that of uniaxial elongational viscosity. As the a value increases, biaxial elongational viscosity is also increased. From comparison of Fig. 7 with Fig. 6 it is clear that the model predicts a maximum viscosity in the biaxial elongational viscosity at earlier times than in the uniaxial extension.

First we investigated the effect of rheological properties on the bubble shape and subsequently, the thermoforming performance. Formation of the bubble is primarily a biaxial deformation, particularly near the pole. The influences of [alpha] and [beta] on the bubble shapes are summarized in Fig. 8. As the value of model parameter a decreases, the bubble shapes become narrower with a sharp pole. This is due to the lower biaxial elongational viscosity and less strain hardening (see Fig. 7a). Similar changes in bubble shape occur as the [beta] parameter is increased. The effect of the [beta] parameter is more conspicuous than that of the [alpha] parameter, as we expected from the biaxial elongational viscosity behavior. In the case of [alpha] = 0.05, [beta] = 0.4, and [beta] = 0.9, the narrowness of the bubble at its base resulted in its touching the mold in the bubble blowing stage.

Note that in each case the blowing pressure profile was adjusted to obtain similar bubble heights of approximately 390 mm. We monitored the vacuum pressure required to obtain this bubble height during the inflation stage of polymer sheet at each different combination of the model parameters, [alpha] and [beta]. It is well known that the pressure applied to polymer sheet during bubble inflation stage can be related to the rheological properties of polymer, especially biaxial elongational viscosity. The dependence of the required pressure on the model parameter [alpha] and [beta] is shown in Fig. 9. As the [alpha] value increases and the [beta] value decreases, the pressure required increases, consistent with the biaxial elongational viscosities given in Fig. 7.

The nonlinear rheological parameters have a significant influence on the part thickness distribution and COV. The simulated results of the effect of [alpha] parameter on the thickness distribution and COV are provided in Fig. 10. From Fig. 10a we observe that the thickness of the back and top areas is increased as a parameter of Wagner model increases, but the thickness of the bottom side is decreased. Also, Fig. 10b shows that as [alpha] parameter increases, the thickness distribution become more uniform. Figures 6a, 7a, and 8 demonstrate that the thickness distributions on the back and top areas depend on the bubble shape but that of the bottom side depends primarily on the extension of sheet after contact with mold. Based on the biaxial elongational viscosity behavior in Fig. 7a, polymers with higher elongational viscosity, more strain hardening, and later maximum peak show better thickness distributions. Note that severe strain softening is predicted by the model after the maximum viscosity peak. Our res ults are consistent with the previous work of several investigators (30, 31, 36, 37), in which the polymer with more strain hardening behavior shows better thickness distribution than that with strain softening behavior. We may conclude that the biaxial elongational viscosity behavior is more important than uniaxial for most of the measurement line used here. However, the thickness distribution of the bottom side determined generally after contacting the mold shows a close relationship with uniaxial elongational viscosity behavior. This is consistent with Throne's study (1), in which the uniaxial elongational viscosity had a significant influence on the part thickness after the sheet contacts with the mold.

Figure 11 shows the simulated results of the effect of the [beta] parameter on the thickness distribution and COV. As [beta] increases, the thickness distribution becomes less uniform. Values of [beta] above 0.5 could not be simulated accurately due to excessive thinning of the sheet during the inflation stage. As with Fig. 10, Fig. 11 indicates very clearly that the polymer with strain hardening and higher elongational viscosity shows more uniform thickness distribution.

The Effect of Bubble Height

A previous experimental study (4) has shown that the quality of this part is sensitive to the initial bubble height and there is clearly an optimum bubble height range, bounded by extremely small average thickness for larger bubbles and poor thickness distribution for small bubbles. Figure 12 presents the simulated results for the effect of bubble height on part thickness distribution. Simulation with bubble heights smaller than 300 mm and larger than 410 mm could not be completed because of excessive deformation after contact with mold and during the inflation stage, respectively. Figure 12 shows that as the bubble height increases, the thickness of back area is decreased and the thickness of bottom and top areas are increased. A larger bubble height means more deformation and less material in back side. Figure 12b shows that generally, as the bubble height increases up to 405 mm, a more uniform thickness distribution along the symmetry line is observed.

These results show an optimum bubble height range for the uniform part thickness distribution. If the bubble height is too large, some areas become excessively thin during bubble inflation. Also, if the bubble height is too small, there exist two areas with different thickness in thermoformed parts. Since the back area primarily depends on the bubble thickness, it becomes too thick owing to less deformation of the polymer sheet during bubble growth stage. However, polymer at the back area may not easily move to the top and bottom areas after contact with the mold owing to high friction at the contacting area. Therefore, these areas become too thin owing to excess extension of a relatively small sheet by the mold, resulting in a non-uniform thickness distribution. The simulation indicated on an optimum bubble height in the range of 380 [sim] 405 mm. Based on these results, we selected the bubble height of 390 mm for further study. The experimental investigation of this system (4) yielded an optimum bubble hei ght of 380[+ or -]20 mm, in excellent agreement with the simulation results.

The Effect of Friction Coefficient

The thermoforming simulation software utilized Coulomb's law of friction for contact between the sheet and the mold. If f*[F.sub.N] is larger than [F.sub.T] then the node is fixed to the wall (nonslip), where f is the friction coefficient, [F.sub.N] is the normal force, and [F.sub.T] is the tangential force. If f*[F.sub.N] is less than [F.sub.T], the node is allowed to slide on the mold surface with [F.sub.T] appropriately reduced due to friction. As Tsai reported (33), we also observed experimentally that the friction between a sheet and a mold is a function of temperature and increases approximately exponentially at higher temperature above [T.sub.g] (see Fig. 1).

Simulations were run with the friction coefficient varying from zero (no friction, perfect slip) to much larger values than our measured values. The dependence of the thickness distribution, COV, and average thickness along a symmetry line on the friction coefficient at 170[degrees]C are provided in Fig. 13. The thickness distribution simulated with the perfect slip condition is significantly different from that with frictional character, especially in the top area and corners. The larger thickness in the top and back areas and the smaller thickness in the bottom and corners may occur because of faster movement of more materials to the top areas, resulting from the lower friction coefficient. Note that the first contact area with the sheet is the corner between the top and back areas and the final contact occurs at the corner in the bottom area (see Fig. 3). As we expected, the average thickness along the measuring line was increased when a larger value of friction coefficient was used and this result for th e average thickness is consistent with previous studies of several investigators (19, 20, 26, 29-31, 33). A more uniform thickness distribution was observed for a friction coefficient in the range of 0.05[sim]0.2. Overall, the friction properties between polymer sheet and mold exert a great influence on the average thickness and thickness distribution. It is important to recognize this when considering the effects of other process variables which may influence the friction coefficient, such as sheet temperature and mold temperature. Observation of an optimal friction coefficient has important implications for mold design. However, in practice, transfer of material from the sheet to the mold surface and oxidation of the mold surface may change the friction coefficient over time. The mold surface may be modified in order to provide a desirable friction coefficient.

The Effect of Sheet Temperature

The dependencies of the thickness distribution, COV, and the average thickness of the finished part on the sheet temperature are provided in Fig. 14. Note that since the friction coefficient was based on the average of the sheet and mold temperatures, the friction coefficient ranged from 0.15 to 0.35 as the sheet temperature changed from 140[degrees]C to 200[degrees]C. In Fig. 14a, the thickness distribution of the top and back regions shows a similar trend in which the part thickness along the measuring line decreases as the sheet temperature increases from 140[degrees]C to 170[degrees]C; however, it is increased as the sheet temperature increases from 170[degrees]C to 200[degrees]C. This is consistent with the previous discussion of the influence of biaxial elongational viscosity and bubble shape on the thickness distribution. Interestingly enough, Fig. 14b clearly shows that at 170[degrees]C, the worst thickness distribution and the thinnest average part thickness along the measuring line were obtained. T he most uniform thickness distributions were observed at 140[degrees]C and next, 200[degrees]C. The thickness distribution effects in the bottom area shown in Fig. 14 are controlled by two primary influences of increasing the sheet temperature: a decrease in uniaxial viscosity and strain-hardening character as well as an increase in the friction coefficient. If we consider the parabolic shape effect of the friction coefficient on the thickness in Fig. 13, it leads us to conclude that the higher average thickness at 185[degrees]C and 200[degrees]C compared to that of 170[degrees]C may occur as a result of the effect of higher friction coefficients. From comparison of Fig. 13b with the temperature dependence of uniaxial viscosity of Fig. 6, the effect of the temperature dependence of friction coefficient on the average thickness seems to be more significant than the effect of the temperature dependence of elongational viscosity.

The Effect of Mold Temperature

The effect of mold temperature on the thickness distribution is potentially quite complex. The mold temperature influences the friction coefficient as well as heat transfer between the mold and sheet. Note that the range of friction coefficient of 0.15[sim]0.3 was used as the mold temperature changes from 25[degrees]C to 110[degrees]C. In addition heat transfer between the sheet and the mold changes the polymer viscosity. The effect of the mold temperature on the thickness distribution is shown in Fig. 15. The effect of mold temperature on the thickness distribution is very similar to that of the friction coefficient (see Fig. 13a). There is an optimum mold temperature of approximately 90[degrees]C, which provides uniform thickness distribution, consistent with the effect of friction coefficient. Also, note that as the mold temperature increases, the average thickness along the measuring line also increases, mainly because of the higher friction coefficient. This indicates that more material is retained alon g the centerline of the part at the expense of other regions. Changes in the thermal conductivity of polymer have little effect on the thickness distribution. This reinforces the conclusion that the primary influence of the mold temperature is through the friction coefficient. Overall, the mold and sheet temperatures play less important roles in determining uniform thickness distribution than the friction coefficient itself. However, it should be mentioned from our experimental study (4) that higher sheet and mold temperatures resulted in fewer special defects such as webbing.

The Effect of Heat Transfer (Between Sheet and Air or Cooling Media)

The effect of heat transfer between sheet and air on the thickness distribution is provided in Fig. 16. When the heat transfer coefficient increases, sheet temperature will be decreased, and as a result, the rheological properties and the friction coefficient also will be changed. From Fig. 16, we clearly see that as the heat transfer coefficient increases, more uniform thickness distribution and larger average part thickness are observed. These results may be attributed to the higher uniaxial elongational viscosity and strain hardening (see Fig. 6). According to Throne (1), if forced cooling equipment is used, the heat transfer coefficient is increased to 100 W/[m.sup.2]K. Recently, it was reported that thermoforming process cooled by spray water equipment was developed (49). Our simulation results indicate that the thermoforming process with forced cooling equipment will provide more uniform thickness distribution.

CONCLUSIONS

In the vacuum snap back thermoforming process investigated here, the initial bubble shape and height were much influenced by the rheological properties of the polymer and played an important role in determining the thickness distribution of the thermoformed part. The dominant flow mode, uniaxial or biaxial elongation, varied significantly over the surface of the part. The biaxial elongational viscosity behavior plays a critical role in the back region of this particular part. In the bottom region, the uniaxial elongational viscosity becomes important in determining the thickness distribution due to stretching of the sheet after contact with the mold. Nonlinear rheological material parameters substantially influence the thickness distribution. Optimum bubble heights and friction coefficients were determined for this process. The effects of sheet temperature and mold temperature are complex because of their influence on the friction coefficient as well as the material rheology. Overall, rheological properties and the friction coefficient played a predominant role in determining the thermoforming performance compared to other processing parameters such as sheet and mold temperature.

ACKNOWLEDGMENT 

The authors are grateful to the Bayer Corporation for the support of this work. We appreciate the work of Mr. Don Williams of Solutia for his measurements of friction coefficients and the thermoforming experiments done by Mr. Mike Nikolakopoulous and Mr. Jim Deary. We also appreciate the efforts of T-SIM CZ Ltd. for their helpful and responsive assistance with software.

(1.) Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139

(2.) Polymers Division of Bayer Corporation Springfield, MA 01151

(*.) Corresponding author.

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              Temperature Dependence of Best Fit Wagner Model
              Parameters, [alpha] and [beta], Calculated From
                 Specific Temperature Relaxation Spectra.
Temperature ([degrees]C) [alpha] [beta]
          140            0.3202  0.1036
          150            0.2678  0.2012
          170            0.2893  0.2500
          185            0.2542  0.2825
          200            0.2781  0.2757
        Average          0.2819  0.2226