Numerical simulation of acrylonitrile-butadiene

Journal of Materials Processing Technology 91 (1999) 219 – 225

Numerical simulation of acrylonitrile-butadiene-styrene material’s vacuum forming process S. Wang a,*, A. Makinouchi a, T. Tosa b, K. Kidokoro b, M. Okamoto c, T. Kotaka c, T. Nakagawa d a

Materials Fabrication Laboratory, The Institute of Physical and Chemical Research, Tokyo, Japan b Denki Kagaku Kogyo Co. Ltd., Tokyo, Japan c Polymeric Materials Engineering, Toyota Technological Institute, Tokyo, Japan d Institute of Industrial Science of the Uni6ersity of Tokyo, 2 -1 Hirosawa, Wako-shi, Saitama, Japan Received 20 January 1998

Abstract In this paper a vacuum forming simulation together with the experimental results is presented. In order to establish a material model which could describe the polymers deformation behavior precisely, the authors conducted uniaxial tensile tests using the newest type of Meissner rheometer with an acrylonitrile-butadiene-styrene (ABS) material. The tests were conducted for constant strain-rates varying from 0.01 to 1 (1 s − 1), at temperatures ranging from 150 to 200°C. A new material model, based on the test data was proposed, in which the combined effects of strain-hardening, strain-rate sensitivity, and temperature variation can be taken into account. Excellent agreement with uniaxial tensile test data was obtained. The vacuum forming processes of a square cup, under different initial temperatures, were simulated employing the proposed material model and the results compared with those of experiment, the simulated final thickness distribution showing a good correspondence to the measured thickness values. Published by Elsevier Science S.A. All rights reserved. Keywords: FEM simulation; Viscoplastic; Vacuum forming; ABS material

1. Introduction In recent years, much attention has been paid to thermoforming, including vacuum forming and various assist processes. When using a thermoforming process, large sized and complex shaped products, such as a surfing board, an inner panel of a refrigerator etc. can be produced. Because of the high speed and the large deformation characteristics of the process, it is difficult to predict the final thickness distribution, which is a determinative factor of product quality. In order to provide an economical and efficient means of thickness prediction, a viscoplastic finite-element simulation code was developed, in which the quadrilateral degenerated shell element is implemented, treating the bending effects properly. Contact between the parison and the * Corresponding author. Tel.: +81-48-4679319; fax: + 81-484624657. E-mail address: [email protected] (S. Wang)

tool is formulated as node-to-mesh contact (a node on the surface of polymer contracts to a triangular mesh element on the tool surface) [1,2]. Based on experimental observations, the sticking-contact assumption is employed [3]. There have been many FEM simulation studies reported in this field [4–10]. One of the key issues in this research field is to develop an accurate material model. Most studies employed the hyperelastic material model [5–9], assuming rubber-like deformation behavior, and neglecting the strain-rate sensitivity and temperature effect. The strain-rate sensitivity and the temperature effect are, however, quite obvious in the material test presented in this paper, using an ABS material. However, there are still almost no detailed studies reported on the combined effects of strain-rate, strain-hardening, and temperature on the elongational deformation behavior of polymers at a high temperature. In order to establish a better model, the tensile tests were con-

0924-0136/99/$ - see front matter Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 8 ) 0 0 4 4 0 - 3

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S. Wang et al. / Journal of Materials Processing Technology 91 (1999) 219–225

ducted for an ABS material. A new viscoplastic material model, based on the test data, was proposed, in which the strain-hardening, strain-rate sensitivity, and temperature effects are considered properly. The material parameters for the proposed model can be determined by a least-squares fit procedure using the experimental data. The vacuum forming processes of a square cup, under different initial temperatures, were simulated employing the proposed material model and compared with the experiment. Fig. 1. The kernel part of the Meissner rheometer.

2. Uniaxial experiment

2.1. Instrument The uniaxial tensile test has been used widely to determine the constitutive equation of a material. However, the tests are usually conducted at room temperature, and the instruments employed cannot be used directly at an elevated temperature for the polymer elongational test. This is mainly due to three reasons. First, when the temperature is near to the melting point, the polymer is quite susceptible to the clamping force, and the material in the clamping zone can be squeezed or broken before any stretching occurs. Second, most conventional testing machines operate with a constant increase of the sample length with time, and the true strain-rate changes with the stretching. Therefore, it is difficult to evaluate the strain-rate effect correctly. Third, it is almost impossible to conduct the tensile test at a high strain-rate using conventional testing machines. However a high strain-rate is a very important characteristic of thermoforming processing, including blow molding and vacuum forming, etc. These problems can be overcome successfully by the use of the Meissner type of elongational rheometer equipped with rotary clamps (Fig. 1) [11]. The rotary clamps, which have a fixed distance between them, rotate in opposite directions. As a result, the material between the two rotary clamps is elongated homogeneously, and if the speed of rotation of the rollers is kept constant, the true strain-rate will be constant. Fig. 1 shows the kernel part of this Rheometer. Each clamp has upper and lower belt fixtures made of titanium. These fixtures are equipped with a conveyor belt, made of a thin metal band. The relation between the true stress and the true straincan be obtained for the constant true strain-rates ranging from 0.001 to 1 (1 s − 1) at different temperatures. The maximum true strain is 7 and the temperature can be precisely controlled up to 300°C.

Fig. 2. True stress versus true strain at 170°C.

2.2. Uniaxial experimental results Uniaxial tensile tests were conducted, for ABS material provided by Denki Kagaku Kogyo Co. Ltd., at the strain-rates ranging from 0.01 to 1 (1 s − 1) and at temperatures varying from 150 to 200°C, which are the expected deformation ranges in the actual vacuum forming process. Because the data distribution at different temperatures is similar, the results obtained at

Fig. 3. True stress versus true strain-rate for different strains at 170°C.


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where m is the strain-rate sensitivity index, given by the slope of the straight line. Fig. 4 shows the true stress–strain relation for a constant strain-rate of 0.1 (1 s − 1) at different temperatures, from which strain-hardening can be observed for all temperatures, the slope becoming smaller gradually with an increase in temperature.

3. Material modeling

Fig. 4. The true stress versus true strain for different temperatures at constant strain-rate of 0.1 (1 s − 1).

From Figs. 2 and 3 it is found that the stress versus strain relations and the stress versus strain-rate relations satisfy the following equation: log s= log k+ n% log o+ m% log o; + w log o log o;

(3)

Rewriting Eq. (3) into a more compact form we have: s= ko n%o; m% + w log o

(4)

where n% is the slope of the log–log stress–strain relation when o; = 1, and m% is the slope of the log–log

Fig. 5. Calculated true stress versus true strain.

170°C are analyzed here as an example. Fig. 2 shows the true stress–strain relation obtained at 170°C. It can be seen that the distribution of the test data under a constant strain-rate shows a strain-hardening behavior in which the true stress s increases with the increase in the true strain o. It is also seen that as the strain-rate increases the strain-hardening increases. Applying the least-squares-fit procedure to experimental data in which the strain-rate is constant, the data distribution can be approximated by an exponential relation that can be expressed by the following equation: s = ko n

Fig. 6. Calculated true stress versus true strain-rate.

(1)

This curve appears to be a straight line in a log–log plot, and where constant n, which is a strain-hardening index, becomes the slope of this line. In order to evaluate the strain-rate effect, the true stress–strain-rate relations at constant strains obtained at 170°C were plotted as shown in Fig. 3. Again, applying the least-squares method to the data under constant strains, it is found that the distribution can be expressed in the exponential form: s =k%o; m

(2)

Fig. 7. Dimensions of 1/4 of the mold (mm).

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Fig. 8. Temperature distribution at 195°C.

Fig. 9. Final thickness distribution calculated at 195°C.

stress–strain line when o =1, in which w can be given as: (( log s/( log o)o; = c −n% w= log c

n=

( log s = n% +w log o; ( log o

(6)

( log s = m% +w log o ( log o;

(7)

m= (c "1)

(5)

where (( log s/( log o)o; =c is the slope of the log–log stress–strain relation at any strain-rate except unity, which is obtained from the experimental data. From Eq. (5) it can be seen that w represents the variation of the slope. The strain index n, and strain-rate sensitivity index m in Eqs. (1) and (2), can be expressed by:

which reveals that the strain-hardening index n is affected by strain-rate, whilst strain-rate sensitivity index m is affected by strain. Using the values of the constants of n%= 0.907, m%= 0.544, w= 0.26, plotting Eq. (4) and the experimental data distribution together, Figs. 5 and 6 are obtained. Clearly, they correspond to each other quite well. The above discussions are for tests conducted at a constant temperature of 170°C. On the other hand,


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from Fig. 4 it is known that the temperature effect should also be modeled. Because the data distribution at different temperatures shows the same tendency, and can basically be described by Eq. (4), the temperature effect is approached by first obtaining the constants k, n%, m%, and w in Eq. (4) at different temperatures and then evaluating their relation to temperature. We found that within the tested-temperature and strain-rate ranges, except for k– T, which can be best fitted by an exponential relation, all of the variables can be best described by the linear relations: k = a1T b1,

n%=a2 + b2T,

m% =a3 +b3T,

w= a4 +b4T

(8)

Fig. 12. Thickness comparison along the C direction obtained from calculation for different temperatures.

where the constants are: a1 =4.2456 ×1025, a2 =3.3183, a3 = − 0.21649, a4 =0.89966, b1 = − 8.8803, b2 = − 0.014049, b3 = 0.0045101, b4 = − 0.00382.

4. Simulation and experimental verifications The material model proposed above can be generalized to the 3-D case in terms of equivalent stress s¯ ,

Fig. 13. Thickness along C direction obtained experimentally for different temperatures.

equivalent strain o¯, and equivalent strain-rate o¯; , as follows: s¯ = ko¯ n%o¯; m% + w log o¯ where s¯ =

'

3 s% s% , o¯; = 2 ij ij

'

(9) 2 o; o; and o¯ = to o¯; dt 3 ij ij

Fig. 10. Thickness comparison along the A and B direction for 195°C.

Fig. 11. Thickness comparison along the C direction for 195°C.

Fig. 14. Thickness along the A direction obtained experimentally for different temperatures.

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three pieces of sheet were tested and the results averaged.

4.2. Comparison

Fig. 15. Thickness along the B direction obtained experimentally for different temperatures.

The viscoplastic constitutive equation can be obtained by substituting Eq. (9) into the following Levy –Mises flow rule: 2s¯ s%ij = ; o; ij 3o¯

(10)

For the details of the FEM formulation refer to [1,2]. Implementing the new material model into the viscoplastic FEM code that has been developed, a vacuum forming process is now simulated.

4.1. Vacuum forming experiment The dimensions of the mold are shown in Fig. 7. The average thickness of the ABS sheet is 1.53 mm. The experiments were performed using the vacuum forming machine FK-0431-10 produced by ASANO Laboratories Ltd. The experiments were done in two cases with the initial temperature of the ABS sheet set to be 185 and 195°C separately. The temperature distribution just before the vacuum forming was examined by a Thermoviewer JTG-6300 (JEOL Ltd.). As an example, the perspective view of the temperature distribution on the sheet at 195°C is shown in Fig. 8. Due to the ability of the temperature control of the vacuum forming machine, it was observed that the temperature variation still remained at about 4 – 5°C on the sheet. The thickness is measured using an Ultrasonic thickness gage (Panametrics, Model 25DL), along three directions on the final product, namely, the extrusion direction A, the transverse direction B, and the diagonal direction C. Because the sheet is made by an extrusion process, it is anisotropic. After heating, it is observed that the sheet has shrinkage in the A direction, whilst it expands in the B direction. Therefore, it is necessary to measure the thickness in the A and B directions separately. In order to obtain the scatter range of the final thickness distribution, for each case

Because the mold has two symmetric planes, the simulation is performed for 1/4 of the sheet, which is described by 2025 elements. The part that will come into contact with the corner of the mold is defined by a finer mesh than that for the other part. Because it is observed that there is almost no slip between the sheet and the mold [3], sticking contact is assumed. Fig. 9 gives the final thickness distribution, whilst Fig. 10 gives a comparison of the thickness along the A and B directions and Fig. 11 gives that along the C direction. The predicted thickness distribution is seen to be in good agreement with the experimental results. The same calculation is repeated for the initial temperatures of 175, 185, and 195°C. The finial thicknesses along the diagonal C direction are plotted together in Fig. 1, from which it can be seen that there is almost no difference in the final thickness distribution. The final thickness distribution obtained from the experiment along the C direction, presented in Fig. 13, also shows almost no difference for 185 and 195°C. The final thickness distribution obtained along the A and B directions at 185 and 195°C, as shown in Figs. 14 and 15, are a little different, but not significantly so. These results presumably imply that uniform or nearly uniform initial temperature distributions at different temperature levels basically repeat a similar material deformation history.

5. Conclusions FEM simulation of a vacuum forming process has been conducted, based on a newly proposed material model, which can consider strain-hardening, strain-ratehardening, temperature change, and variation of the hardening index. It is clarified that the predictions of this model are in excellent agreement with the results of uniaxial tensile tests. The thickness distribution in the vacuum forming simulation agrees with the measurement quite well. Simulations for the different initial temperatures showed similar thickness distributions, which is also supported by the results of the experiment.

Acknowledgements The authors would like to express their sincere thanks to Denki Kagaku Kogyo Co. Ltd. for providing the test material, and to K. Taneichi for his great help with the vacuum forming experiments.


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