Jim Throne, Sherwood Technologies, Inc., Dunedin Florida 34698 Copyright 2006


In Part 1, we focused on simple experiments to determine the interaction between a 3-inch hemispherical wood plug and a 0.020-inch thick natural rubber sheet. We examined the transfer of grease dots from the wood plug to the rubber sheet. In addition, we noted local differential down-plug orientation of the grease dot on both the wood plug and the rubber sheet, particularly at the 1 ½-inch position on the plug where the sheet had been stretched locally about 50%.


For the set of experiments detailed below, we are using the EZFORMTM thermoformer (Centroform, 820 Thompson Ave., Unit 5, Glendale CA 91201) described and shown in Figure 1 of Part 1. The plastic is 20-mil flexible clear PVC, referred to by Lancs Industry, Inc. as “nuclear grade double polished PVC sheeting.”

As noted in Part 1, the thermoformer heats only one side of the sheet. The thermoformer is located in an area with no crosscurrent airflow. When the sheet is being heated, the clamp frame is against the shroud of the heater with the reverse side of the sheet exposed to the environment. A Raytek PM50 infrared thermometer with emissivity = 0.95 is used to measure the free surface of the sheet. Because the sheet is thin, the free surface temperature is assumed the same as that of the surface seeing the radiant heat.

The heating times for 20-mil FPVC are shown in Figure 8 for various spacing from the heater cage. The ” ½ inch” notation is when the clamp frame is against the edge of the heater cage. When the heater is at its steady-state temperature of 600oF, it takes approximately 45 seconds for the sheet temperature to reach 275oF.

IR temperature readings of reverse side of 20-mil FPVC sheet.

Figure 8. IR temperature readings of reverse side of 20-mil FPVC sheet. Spacing refers to the distance between the heater shroud and the clamp frame.

When the sheet temperature reaches 275oF, the clamp frame containing the heated sheet is manually pulled onto the mold frame. The transfer time is less than 1 second. The vacuum is on from the time the sheet temperature reaches 275oF until the sheet is cool to the touch. The plug and vacuum box are at room temperature of approximately 68oF. It takes approximately 3-4 seconds for the sheet to be completely drawn against the plug and vacuum box.

Differential Stretch

In the first set of experiments, permanent Sharpie dots were placed on an unformed sheet ½ inch apart outward from the apex of the plug. The sheet was then formed onto the wood plug. The cool formed part was then removed from the plug and the distances between the dots were measured. The values are given in Table 5 and in graphical form in Figure 2 (in Part 1). It is apparent that there is relatively little stretch around the plug apex but substantial stretch around the 2 inch position. This confirms the measurements given in Part 1 for the natural rubber sheet.

Table 5 Measured Distance between Current Location and Previous Location On 20-mil Flexible PVC Sheet Stretched over Plug
Dot Location from Apex, in Distance, in
0.5 0.512
1.0 0.512
1.5 0.63
2.0 0.709
2.5 0.827
3.0 0.748

Grease Dots

The grease dot experiment of Part 1 was now repeated. The grease dots were placed at ½ inch intervals down the wood plug, beginning at the apex. They were then microphotographed and measured. As with the grease dots described in Part 1, the grease dots were generally domed and generally circular in shape. The dot dimensions were approximately 2-4 mm. The values are given in Table 6. A second FPVC sheet was heated to 275oF and stretched over the room temperature plug. After the sheet cooled, it was carefully stripped from the plug and everted. This allowed the transferred grease spots to be microphotographed and the distance between them measured. Table 6 lists the dimensions of the transferred grease spots and Table 7 gives the distances between them.

Table 6 Dimensions of Grease Dot Originally on Plug, and On 20-mil Flexible PVC Sheet after Contact
Dot Dim, a x b, mm Product, a x b, mm2
Position from apex: 0 in
Original dot 4.0 x 3.5 14.00
After, on FPVC 4.0 x 3.5 14.00
Position from apex: 0.5 in
Original dot 2.5 x 2.5 6.25
After, on FPVC 3.6 x 3.6 12.95
Position from apex: 1.0 in
Original Dot 3.0 x 2.25 6.75
After, on FPVC 2.75 x 3.5 9.625
Position from apex: 1.5 in
Original dot 2.0 x 2.0 4.00
After, on FPVC 3.75 x 3.5 13.125
Position from apex: 2.0 in
Original dot 2.0 x 2.0 4.00
After, on FPVC 3.5 x 3.5 12.25
Position from apex: 2.5 in
Original dot 2.0 x 2.0 4.00
After, on FPVC 3.5 x 3.5 12.25
Table 7 Measured Distance between Current Location and Previous Location For Grease Dots Transferred From Plug to Vacuum Formed 20-mil FPVC Sheet
Dot Location from Apex, in Distance, mm Distance, in
0.5 12 0.472
1.0 13 0.512
1.5 13 0.512
2.0 13 0.512
2.5 12 0.472
3.0 ~13 ~0.512

The areas of the transferred grease dots are generally larger by a factor of three or so. However, the grease dot shapes are generally circular. This indicates that the domed or conical grease dots were uniformly flattened by the hot sheet.

More importantly, the distances between the transferred grease dots are essentially 12-13 mm apart. In other words, there appears to be no stretching taking place once the sheet has contacted the plug.

There is an apparent dichotomy between the forming of FPVC sheet and the apparent differential stretching seen with the rubber sheet. It appears that some of the smearing observed on the rubber sheet may be due to elastic recovery once the vacuum had been released and not due to sliding as the sheet is drawn onto the plug.

Force Balance

Consider a simple force balance on a sheet in contact with a solid surface, Figure 9.

Frictional experiment

Figure 9. Frictional experiment. P is the applied pressure, A are the test blocks, in this case, the plug material. B is the plastic sheet. The applied horizontal force is measured as a function of the applied pressure.

Sliding Force

Assume for the moment that the sheet is inextensible. That is, it slides but does not stretch when a force is applied. The force required to move the sheet, Fs, is given as:

Fs = C N

Where [Fs] = lbf, say, C is some measure of frictional resistance between the sheet and the solid surface [C] = unitless, and N is the applied force normal to the sliding direction, [N]= lbf. The normal force can be written as:

N = P A’

Where P is the applied pressure, [P]=lb/in2, and A’ is the contact surface area resisting sliding, [A’]=in2.

For a sheet to slide against a solid surface, the force used to slide the sheet must be greater than the resistance to hold the sheet against the solid surface. As noted in our PPT presentation at 2004 SPE Thermoforming Conference (please see the PPT presentation posted on the www.foamandform.com website), there are many definitions for C, the frictional coefficient. In this paper, we need only consider C as a constant that is a function of the natures of the sheet and plug surfaces and their respective temperatures.

Stretching Force

However, the force applied to sliding must also be applied to stretching. The stretching force, Ft is given as:

  1. Ft = τ(T) A

    Where [Ft]=lbf, τ(T) is the temperature-dependent tensile strength of the polymer, [τ(T)] = lb/in2, and A is the cross-sectional area of the sheet being stretched, [A] = in2. The area can be written as:

  2. A = t W

    Where t is the sheet thickness, [t]=in, and W is the sheet width, [W]=in.

    Consider a unit width of sheet. A=t and A’=x, where x is the length of contact involved in sliding. The total force, FT, applied a unit width of hot sheet is given as:

  3. FT = Fs + Ft = C P x + τ(T) t

    Consider the sliding v. stretching aspects of the sheet. The applied force, FT, can be factored into the force along the solid surface and that normal to the solid surface. The stretching force portion of the sheet acts solely to stretch the sheet. On the other hand, the sliding portion depends on the force normal to the solid surface. The force normal to the solid surface acts to lift the sheet from the solid surface, thereby reducing the pressure effect. As a result, the sliding portion can be rewritten as:

  4. Fs,parallel = C P x cos α and Fs, normal = C P x sin α

    In other words, as α increases in value, more of the applied force goes to reducing the effective pressure, viz, P cos α, and less goes to providing sliding forces. When α = 90 degrees, the force parallel to the solid surface, providing sliding, is zero, and the effective force holding the sheet to the surface is maximized. The force in the direction of the solid surface acts to either slide the sheet along the surface or stretch it or both.

    Schematic of sheet progression as it is laid against wood plug.

    Figure 10. Schematic of sheet progression as it is laid against wood plug.

    Consider now the progression as the sheet is laid against the wood plug, Figure 10. Initially the sheet just lies onto the apex of the wood plug, without stretching. As a result, Ft=0. Because the sheet is essentially parallel to the surface of the plug, α =zero. As a result, the total force required to move the sheet is just given as:

  5. FT,apex = Fs, apex = C P x

    If the sheet has just touched the apex, x is small. In addition, if vacuum has not been applied or the plug has not been mechanically forced into the sheet, the total normal force is just the local weight of the sheet, which for thin gauge sheet, is very small. Therefore, the force required to move the sheet is also very small.

    Now consider the sheet that has formed a substantial distance down the wood plug, as shown in Figure 10. The angle between the sheet and the plug is no longer parallel to the plug surface. However, by now, the plug has been mechanically forced into the plug and/or vacuum or pressure has been applied to the sheet. As a result, P can be much greater than just the weight of the sheet against the plug. Furthermore, it is not apparent what value to use for x, the length of sheet subject to sliding. Certainly if one assumes the value to be the length of sheet from the apex to the last point of contact, the amount of force required to slide the sheet is very large. Therefore, even though only a portion of the available force can promote sliding, that force may need to be quite large.

    Moreover, the sheet is being stretched, the amount of stretching force being strongly dependent on the hot tensile strength and local thickness of the sheet. The hotter the sheet is, the lower that amount of force needs to be. In sum, then, sheet movement on the side of the plug is preferentially one of stretching than sliding. In other words:

  6. Fs,parallel = C P x cos α>> Ft = τ(T) t

    Ergo, if sliding is to occur anywhere on the surface of the plug, it will occur preferentially at the plug apex. However, our experiments with both the rubber sheet and the FPVC sheet show essentially no sliding at the apex. Moreover, with the FPVC sheet, apparently there is no sliding anywhere along the plug surface.


The original question was “What is sliding on what?” In Part 1, we changed the emphasis of the question to “Is the sheet sliding on the plug, and if so, where?”

Experiments used a nearly hemispherical wood plug on which grease dots were placed every ½ inch down the plug, beginning at its apex.

In Part 1, we used a 20-mil natural rubber sheet to determine the effectiveness of grease transfer from the plug to the sheet. We microphotographed the grease dots before and after stretching the sheet onto the plug. From these experiments, we proposed a tentative conclusion that sliding does not occur at the apex of the plug but may occur at locations down the side of the plug.

In Part 2, we used a 20-mil flexible PVC sheet, heated to 275oF. We microphotographed these grease dots before and after vacuum forming the sheet onto the plug. From these experiments, we determined that the distances between the grease dots transferred to the sheet were essentially the same as the spacing between the initial grease dots on the plug. From this we conclude that no sliding occurred during vacuum forming.

From our technical analysis, we conclude that sliding of the sheet on the plug is most likely when the sheet first touches the plug. This is when stretching is minimal and the contact surface is smallest. By the time the sheet has been formed some distance down the plug, the contact surface is large, the applied force is high, and the resistance to stretching is low. As a result, stretching is preferred over sliding.


One criticism of this work is that the wood plug surface was rough. As a result, it may be argued, both the rubber sheet and the soft, hot FPVC sheet would preferentially stick to rather than slide against the plug surface. Further work will focus on the plug surface characteristics.


We conclude from these experiments that there is no sliding between the 20-mil flexible PVC sheet and the wood plug. As a result, there is no need to determine a value for the frictional coefficient.

Leave a Reply