In Part 1 of this treatise, I discussed some of the background thinking on solids sliding on polymeric surfaces. I identified friction as part of the tribological triumvirate of friction, lubrication, and wear. I reviewed the various methods for measuring frictional coefficients and concluded that for plug-polymer sheet interaction, the extent of sliding is very small. So I conducted a simple experiment using a linear motor that pulled a weighted sled very slowly across the hot PS sheet. The contact surface of the sled was syntactic foam. It showed that, at constant sheet temperature, the towing force decreased with increasing plug temperature.

In Part 2, I examined the force required to stretch a natural rubber sheet using a hemispherical plug. I changed the surface condition of the plug from very rough to very smooth and found very little difference in the force-penetration curves. I found an interesting anomaly when I deliberately coated the plug surface with a lubricant. There was little difference between the oil-coated plug force-deflection curve and the dry plug force-deflection curve. But there was measurable difference between the water-base coated plug force-deflection curve and the dry plug force-deflection curve. The possible reason for this is discussed in this part.

I still have not fully addressed the question: What is sliding on what? Or perhaps, more pertinently now: Is anything sliding on anything?

Previous Researchers’ Data

I noted in Part 2 that it appeared from my simple rubber sheet experiments, the sheet did not stretch once it contacted the plug surface. In my opinion, it appeared that the sheet simply adheres to the plug surface. In Part 1, I listed several publications of interest to this thesis. Some of the data from Ref. 8 are most interesting and are discussed here. In this work, the researchers relate the thickness of HDPE sheet at the tip of a hemispherical plug to sheet and plug temperatures and plug speed into the sheet. I have plotted their data in Fig. 19, below.

Literature Results on Sheet Thickness v. Sheet Temperature

Figure 19. Literature Results on Sheet Thickness v. Sheet Temperature

The blue line represents average sheet tip thickness at the specific sheet temperature and the red line represents the average sheet tip thickness over all sheet temperatures. In my opinion, there is no dramatic change in sheet tip thickness for the set of experiments listed on the figure legend. In essence, this seems to confirm my rubber sheet experiments. Another interesting aspect to these data is the apparent sheet tip thickness insensitivity to plug speed. I’ll discuss this very interesting aspect somewhat later.

So, can we conclude that nothing is sliding on anything? No. Not quite. Remember the water-based v. oil-based data. It is time to review the tribological thing again. We do this by reviewing the concepts of friction and frictional coefficient.

Again, Tell Me the Meaning of the Coefficient of Friction?

The best definition is that given in the 1903 Ref. 4, to wit:

The coefficient of friction is the ratio between the resistance to motion and the perpendicular pressure.

What are the various conditions that might occur between the plug and the sheet?

  • Static frictional conditions, no sliding (coefficient max)
  • Sliding frictional conditions, no static (coefficient zero)
  • Some static, some sliding
  • Slip-stick behavior

First, what are these? And then which of these – if any – are relevant when plastic stretches against plug surface?

Static frictional force is the force needed to initiate sliding. Sliding frictional force is the force needed to move one surface against another in a steady fashion. Static frictional force can often be hundreds of times greater in value than sliding frictional force. It appears from Ref. 9 experiments, that the static coefficient of friction between a syntactic foam plug material and HDPE is about two to three times greater than the sliding coefficient of friction. The ratio for syntactic foam and HIPS may be as high as five times. However, as noted in Part 1, the static frictional coefficient appears to vanish when the HIPS sheet temperature is above the polymer glass transition temperature.

As discussed in Part 1, the key to sliding focuses on the nature of the asperities at the interface between the plastic and the plug. The sliding force is proportional to the normal stress, being the load per unit contact area. The interfacial contact area may increase as the load increases. Or the sheet may creep or flow to a greater extent as the sheet temperature is increased, thus increasing the interfacial contact area. An example of this effect is seen as Figure 21. The red area represents short contact time or low temperature. The yellow is medium contact time or temperature, and the blow represents long contact time or higher temperature.

The Effect of Increased Time or Temperature on Contact Area

Figure 21. The Effect of Increased Time or Temperature on Contact Area

As a result, the sliding friction is usually independent of load.

I already discussed wear. If sliding does occur, wear probably occurs on both the plug and sheet surface. Wear would occur on the plug because it contacts sheet perhaps thousands of times. Although the plug material is brittle when compared with the sheet, after many contacts, we should expect some asperities to be worn from the plug surface. If the sheet is brittle, as it would be if it is below its melt or glass transition temperature, its asperities should also be worn away. The caveats here, however, are 1) the sheet temperature is usually substantially above its transition values when it contacts the plug, and 2) the sheet contacts the plug just one time. So wearing away is probably not a factor in polymer-to-plug siding friction.

So that leaves us with the concept of lubrication, the third of the trilogy.

Is There Lubrication Between the Sheet and the Plug?

Probably the proper question is: “Could there be lubrication?” According to Persson (5), there are two general forms of sliding friction – dry and wet. Dry sliding assumes no lubrication between the surfaces. Now we know that plastics exude small molecules, be they low molecular weight polymers, additives, processing aids, or whatever. And we would expect that these molecules would reside between the plug and the sheet. And we would suspect that these molecules could be transferred from the sheet to the plug. And if that were the case, after many contacts, the plug surface would become coated with these small molecules. If we extrapolate this thought further, we would expect that the sliding frictional coefficient of a well-used plug might be substantially lower than that of a brand-new plug or one that has just been cleaned.

Is There More Than One Type of Wet Lubrication?

Unfortunately, yes. Boundary lubrication and hydraulic or hydrodynamic lubrication. Boundary lubrication is characterized by low sliding velocity, low interfacial viscosity, and high loading levels. Hydrodynamic lubrication is characterized by high sliding velocity, high viscosity, and low loading levels. Boundary lubrication occurs when surfaces are started and stopped. Boundary lubrication sliding resistance is often much higher than hydrodynamic lubrication, as seen in Figure 22.

The Relationship Between Boundary and Hydrodynamic Lubrication

Figure 22. The Relationship Between Boundary and Hydrodynamic Lubrication

If, in fact, sliding is taking place between the plug and the sheet, we should see some effect if we add different lubricants to the interface. Lubricants such as grease and glycerin. For the results shown in Fig. 17, reproduced below, the plug speed was held constant. The force required to stretch the water-base lubed sheet was less than that for the oil-base lubed sheet. And perhaps the glycerin had a higher viscosity than the grease. If so, then the sliding resistance for the water-based lubricant may be hydrodynamic whereas that for the oil-based lubricant may be boundary.

Effect of Lubrication Between Sheet and Plug

Figure 17. Effect of Lubrication Between Sheet and Plug

So What, You Say

Well, here’s what! The dry v. wet and boundary v. hydrodynamic lubrication issues could be quite significant as a plug continues to age and as the polymer character changes. By polymer character, I mean polymer temperature and extent and nature of the additive package. As the data from References 8 and 9 demonstrate, the sliding coefficient of friction increases dramatically as the sheet heats to the forming temperature. Part of this may be due directly to improved adhesion of the sheet with the plug. But part of it may be due to increased release of small molecules that influence the interfacial conditions between the plug and the sheet.

Thickness, Revisited

In Part 2, I presented some data showing that the plane strain theory that relates the applied force directly to the sheet thickness holds for the natural rubber sheet.

But does this hold for real-world plug-assisted thermoforming? It might not. The reason for this is relatively easy to understand. Two effects might happen as the plug – or in reality, any solid object – presses into the hot, thick sheet. The first is compression, forcing the polymer away from the point of applied force and thereby thinning it locally. The second is shear. Shear is the result of differential tension on the sheet from the surface in contact with the plug to that on the free surface opposite the plug. Both effects would serve to reduce the sheet thickness at the point of contact with the plug. Certainly, with thicker sheet, the effect of shear would be more pronounced. And of course, shear implies viscoelastic deformation of the plastic.


As you probably have guessed, I have some serious reservations about the importance of the frictional coefficient in plug-assist thermoforming. Remember the original question: What is sliding on what?

I believe that for the most part, nothing is sliding on anything.

Furthermore, if something is sliding on something else, the sliding effect is limited to region where the sheet initially contacts the plug surface. Once the sheet contacts the plug, there seems to be little additional reason for it to slide. The force-penetration curves and sheet thickness on the plug show little difference between a very rough plug surface and a highly polished, waxed, and powdered surface.

If there is substantial lubrication at the initial contact area, there may be some change in the sliding characteristics. But the nature of the lubrication is apparently quite important.

Interestingly, from other research work, it appears that the speed of the plug does not dramatically alter the stretching characteristics of the sheet. If this effect can be verified with polymers other than HIPS and HDPE, it does not bode well for the hypothesis that the rate of stretching of the sheet must be included in the mathematical models currently in vogue. In simple terms, plug-assist thermoforming may be primarily an elastic process and not a viscoelastic one. At least for thin sheet.

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