Jim Throne on December 4th, 2010
Dr James L. Throne

Dr James L. Throne

Jim Throne is now doing business as Sherwood Technologies, Dunedin Florida. His consultancy focuses on advanced plastics processing technologies, including thermoforming, foam processing and rotational molding.

He was a Fellow of the SPE, Fellow of IoM3 (England), and Chartered Scientist (England). He was SPE Thermoformer of the Year 2000. He received the first Lifetime Achievement Award from the SPE European Thermoforming Division in 2004 for his technical contributions to the international thermoforming industry. In 2004, he was inducted into the Plastic Pioneers Association.

He has published ten books in polymer processing, including four in thermoforming and two in thermoplastic foam processing. He holds nine US patents, including two in thermoplastic foams and a fundamental one in thermoforming CPET. He has written more than a dozen technical book chapters and has published and presented nearly 200 technical papers. He was Technical Editor of SPE Thermoforming Quarterly and Editor of SPE Rotational Molding Division Newsletter.

His BS is in Chemical Engineering from Case Institute of Technology. His MChE and PhD in Chemical Engineering are from University of Delaware.

Jim Throne on December 4th, 2010

After 21 years as founder, owner, and president of Sherwood Technologies, Inc., an S-Chapter Delaware Corporation, I have dissolved my corporation. I’m now doing business in Florida as (D/B/A) Sherwood Technologies.

What has changed? Really, nothing except the tax status. I’m still consulting in many aspects of plastics processing. And I’m still focusing on foam processing, particle processing, and thermoforming. I’m still in the Tampa Bay area and my phone number is still 001-727-734-5081.

My in-plant seminars are always tailor-made to specific company needs. My public seminars always present up-to-date information on the general aspects of the topics I’m presenting. Oh, and don’t forget to read my technical notes. They’re updated and added to on a regular basis.

So, whether your interests are in updating your current technologies or troubleshooting vexing processing problems, let’s talk.

Jim Throne

Jim Throne on August 4th, 2008

Peram Rao, President, TechnoBiz Communications, Bangkok, Thailand, invited Jim to present two two-day seminars – Thermoforming and Thermoplastic Foams – in Bangkok, Thailand in February 2007. Based on their success, he invited him to present two two-day seminars – Thermoforming and Thermoplastic Foams – in Kuala Lumpur, Malaysia, and one two-day seminar – Rotational Molding – in Bangkok, in July 2008.

TechnoBiz Communications offers seminars by such renowned technical leaders as Chris Rauwendaal in Extrusion, Kirk Cantor in Blown Film Technology, and Jim Summers in PVC Technology. TechnoBiz is planning extensive troubleshooting seminars in Extrusion and Thermoforming. Mark Strachan will be leading the Thermoforming seminars.

TechnoBiz Communcations is also organizing a two-day International Conference, Thermoforming Asia, planned for spring 2009. The conference format is based on the very successful International Plastics Extrusion Asia Conference 2008, in which more than 20 speakers from US, Colombia, Germany, Thailand, Canada, Italy, Portugal, the Netherlands, and China presented technical papers to more than 160 attendees.

Jim Throne, Sherwood Technologies, Dunedin, Florida

Introduction

Often, when a sheet of plastic is heated prior to forming it into a part, there are planes of heaters above and below the sheet. The heaters often have many segments, with each segment temperature controllable. As a result, the energy input to the sheet can be “shaped” to provide uniform heating of the sheet. The raison d’etre for this work can be found in [1].

Differential and Finite Element Interchange

As a first assumption, we consider both the sheet and the radiant heaters to be very simple finite black body parallel planes. The differential element energy interchange yields the following well-understood form:

differential element energy interchange

(1)

For parallel differential elements, the direction cosines are just z/r, yielding:

parallel differential elements

(2)

If we replace the differential elements with finite elements, we can easily generate the standard double integral form:

standard double integral form

(3)

The extended version of this equation yields a quadruple integral form where dA1=dx1dy1 and dA2=dx2dy2:

quadruple integral form

(4)

Stokes’ Theorem

Stokes showed that an areal integral can be replaced with a integral that follows the contour of the area in a counterclockwise fashion [2]. This is written as:

Stokes’ Theorem

(5)

We can replace surface integration with contour integration and carry out the integration.

replace surface integration with contour integration and carry out the integration

(6)

If all the heater and sheet temperatures are equal, the energy interchange can be written as:

energy interchange

(7)

view factor or configuration factors referred to as the view factor or configuration factor.

When the double integral is parsed in finite elements, desired geometric elements can be algebraically related back to the Stokes’ integrated form. Consider the case where there is only one heater element and one sheet element, Figure 1 [3].

one heater element and one sheet element

The contour integral for this case yields the following equation for the view factor:

view factor

view factor

(8)

Where X=x/z and Y=y/z.

Extension of the Stokesian Model to Many Elements

Although integrating the contour integrals can be done for a few parsed elements on each surface, the practical problem involves, say, m x n heater elements and x x y sheet surface elements. When the parsed elements become many, the arithmetic becomes tedious at best and very prone to “juggling” errors. There does not appear to be a Stokesian-driven algorithm that allows the sum of all the energy interchanges between, say, element xy on the sheet and the m x n heater elements, where xy is any position on the sheet surface.

The Differential Form of the Double Areal Integral

In earlier publications, the differential area elements were replaced with differential area elements of the form [4].

differential area elements of the form

(9)

To determine the sum of energies interchanged between all the m x n heaters and sheet element ab, say, the areal integral has been replaced with summation of the form:

sum of energies interchanged between all the m x n heaters and sheet element

(10)

While this equation has been used to generate the “energy dome” for equal-sized square heater and sheet elements, it is difficult to apply when heater and sheet elements are neither of equal size nor square.

The Algorithm for the Stokesian Contour Integral

While searching for an appropriate computer algorithm, we found a reference to a 1982 catalog of Radiation Configuration Factors [5,6] The coordinates associated with the algorithm are shown in Figure 2.

Radiation Configuration Factors

a and b in the arithmetic that follows. Again, following tradition, A1 is the area of the sheet element and A2 is the area of the heater element. The view factor for heater element ab and sheet element xy is given as:

view factor for heater element ab and sheet element xy

view factor for heater element ab and sheet element xy

view factor for heater element ab and sheet element xy

view factor for heater element ab and sheet element xy

(11)

Note that there is no restriction to the sizes or locations of the various elements in this equation. It is, in fact, the general solution of the parallel plate radiant energy interchange between m x n heater elements and any or all of the x x y sheet elements.

Extension to Typical Thermoforming Heat Transfer Issues

For the black body interchange between heater element mn and sheet element xy:

black body interchange between heater element mn and sheet element xy

(12)

For the black body interchange between all heater elements M x N and sheet element xy:

black body interchange between all heater elements M x N and sheet element xy

(13)

For the black body interchange between a heater element mn and all sheet elements X x Y:

black body interchange between a heater element mn and all sheet elements X x Y

(14)

And the total black body interchange between all heater elements M x N and all sheet elements X x Y:

total black body interchange between all heater elements M x N and all sheet elements X x Y

(15)

For gray-body energy interchange, where mn is the emissivity of the heater and xy is the emissivity of the sheet (assuming these values represent diffuse energy transfer and are independent of wavelength, sheet and heater temperatures), the total radiant interchange between MxN heaters and X x Y sheet is given as:

total radiant interchange between MxN heaters and X x Y sheet

(16)

Checking the Arithmetic

Case 1. In [7] is given a plot of the view factor for radiative energy transfer between identical, parallel, directly opposed rectangles. For X=Y=Z=1, the view factor F1-2 = 0.200. When the appropriate values for the differential form for the view factor equation are used, F1-2 = 0.218.

Case 2. Consider the view factor between heater element 3 x 3 and sheet element 1 x 1. From equation (11), the view factor is 0.00448. When the appropriate values for the differential form for the view factor equation are used, F1-2 = 0.00393.

At this point, this author is unsure why these two methods do not yield exactly the same values. Although the “exact” solution, equation (11) is perhaps slightly more complicated than the differential form, both involve multiple summations and both involve relatively simple arithmetic.

Case 3. In [8] is the computer solution for energy input to a sheet containing 49 elements from a heater bank containing 49 identical elements. Here x=y=Z=1. Because the energy interchange is symmetric, only one-fourth (+one) of the elements need to be used. Given in Table 1 below are the view factors for the left upper quadrant, relative to 100% at the center element.

Table 1

Local View Factor Values Relative to the Center Element for 9×9 Elements
Using Differential Equations
(Symmetric, So Only Upper Left Quadrant Shown)
Element Number
(11) (12) (13) (14)
60.9 74.0 76.8 77.3
90.8 94.4 95.1
98.4 99.2
100

The relative view factors calculated from equation (11) are given in Table 2 below:

Table 2

Local View Factor Values Relative to the Center Element for 9×9 Elements
Using Equation (11)
(Symmetric, So Only Upper Left Quadrant Shown)
Element Number
(11) (12) (13) (14)
53.6 68.9 72.3 73.0
88.9 93.3 94.1
98.1 99.0
100

The errors between these two methods are shown in Table 3 below. It is apparent that the differential method yields more generous results than that of the more exact equation (11), particularly at the two-dimensional corners. Nevertheless, the difference in the overall energy interchange between these two methods is only 1.2%. It is believed that this difference is primarily the result of the use of the centerpoint positions on the differential model and the edge positions on the method used to generate equation (11).

Table 3

Percent Difference in Energy Interchange Values
9×9 Elements
(Symmetric, So Only Upper Left Quadrant Shown)
Element Number
(11) (12) (13) (14)
-13.6% -7.4% -6.2% -5.9%
-2.1% -1.2% -1.1%
-0.3% -0.2%
0%

Conclusion

View factors are extremely important values when determining energy interchange between various heater and sheet elements. Originally, the double areal integrals, or the equivalent quadruple position integrals, were solved by replacing the differential element equation with differential elements and simply summing the view factors.

Recently, a thorough study of the very extensive literature in this area revealed that the areal double integral equation could be replaced with a contour integral equation, said contour equation then being integrated. Furthermore, it was found that there was a general mathematical form for energy interchange where the sheet element was placed anywhere in the x-y plane and the heater element was placed anywhere in the a-b plane. Furthermore, there was no restriction on the relative dimensions of either the heating element or the sheet element.

The values of view factors generated from solving this equation for simple elements and 9×9 elements were similar but not exactly the same as those determined by differential summation. Because the general Stokesian-generated equation is more versatile, we are recommending its use rather than the differential equation.

Future Work

This model will be used to analyze the following thin-gauge processes:

  • Thin-gauge heating times
  • Multiple ovens in thin-gauge heating (including oven temperature profiling)
  • Effect of edge heating of thin-gauge sheet
  • Effect of environment on edge of thin-gauge sheet just prior to exiting the oven

Then the model will be used to analyze combined convection and radiation on the heating of thick-gauge sheet.

References

  1. J.L. Throne, Technology of Thermoforming, Hanser-Gardner Publicatins, Inc., Cincinnati OH, 1996, Section 3.7, Radiant Heating, pg. 138+.
  2. S.A. Schelkunoff, Applied Mathematics for Engineers and Scientists, 2nd Ed., D.Van Nostrand Company, Inc., Princeton, NJ, 1965, pg. 131.
  3. J.R. Howell, http://www.me.utexas.edu/~howell/tablecon.html, C-11, Identical, parallel, directly opposed rectangles.
  4. J.L. Throne, Technology of Thermoforming, Hanser-Gardner Publicatins, Inc., Cincinnati OH, 1996, pp. 155-159.
  5. J.R. Howell, A Catalogue of Radiation Configuration Factors, McGraw-Hill, New York, 1982. Dr. J.R. Howell, Professor, University of Texas at Austin kindly referred us to updates of the catalog on his website, http://www.me.utexas.edu/~howell/tablecon.html. His catalog reference C-13 is entitled Rectangle to rectangle in a parallel plane; all boundaries are parallel or perpendicular to x and boundaries.
  6. J.R. Ehlert and T.F. Smith, T.F., “View Factors for Perpendicular and Parallel, Rectangular Plates,” J. Thermophys. Heat Trans., vol. 7, no. 1, pp. 173-174, 1993.
  7. M.F. Modest, Radiative Heat Transfer, McGraw-Hill, Inc., 1993, Figure D-2, page 794.
  8. J.L. Throne, Technology of Thermoforming, Hanser-Gardner Publicatins, Inc., Cincinnati OH, 1996, Figure 3.33, pg 157.
Jim Throne on November 16th, 2006
Jim Throne, Sherwood Technologies, Inc., Dunedin Florida 34698 Copyright 2006

Introduction

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%.

In Part 2, we used a 0.020-inch 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 concluded that no sliding occurred between the PVC sheet and the wood plug during vacuum forming.

In Part 3, we replace the nom. 3-inch hemispherical wood plug with a 3-inch diameter glass sphere. The objective is to determine whether the nature of the plug surface influences in any way the contact between the plug and the stretching sheet. Again, we use the same equipment and techniques described in Parts 1 and 2.

The Experimental Set-Up

For the set of experiments detailed below, we are using EZFORMTM thermoformer (Centroform, 820 Thompson Ave., Unit 5, Glendale CA 91201), having a 13-in x 18 ¾-in clamp frame is used. The thermoformer heats the sheet only from the top. The vacuum box edge is ¾-in quarter-round. The vacuum box bottom is tempered pegboard. The vacuum system is a shop vacuum pulling approximately 25 in water.

This study uses a glass plug. The plug used is a 3-in diameter solid glass sphere, euphemistically called a ‘feng shui orb.’ It sits on a glass pedestal in a 3 ¼-in hole in the center of a vacuum box. For each experiment, we cleaned the glass surface with rubbing alcohol. This was done prior to applying grease dots.

The experimental set-up is seen in Figure 11.

Experimental set-up with glass plug in place.

Figure 11. Experimental set-up with glass plug in place.

The marking system was the same as described in Part 1, viz, small dots of grease, approximately 2.5-4 mm in diameter, placed at ½ -inch intervals down the surface of the plug, beginning at the apex. This is seen in Figure 12.

Formed FPVC sheet in contact with glass plug, left, and grease dots on everted formed sheet, right.

Figure 17. Formed FPVC sheet in contact with glass plug, left, and grease dots on everted formed sheet, right.

Experiments with Rubber Sheet

Two sets of experiments have been performed using this set-up. The first is described here. In this set of experiments, a 20-mil [0.020-in] natural rubber sheet is used at room temperature. The objective of the first set of experiments whether the rubber sheet slides against the very smooth glass plug in a manner similar to that seen against the wood plug as described in Part 1.

The first set of experiments involved determining the local degree of rubber sheet stretching. The sheet was dot-marked at ½ -in intervals from the point where the sheet contacted the center of the plug to 3 inches radially outward. The sheet was then stretched and while the sheet was still under vacuum, the distance between the dots was measured. Table 8 gives these data, along with the data from Table 1 for the similar experiment with the wood plug.

Table 8 Comparison of Dots Prepositioned at ½ inch Marks On Rubber Sheet Prior to Stretching –Measurement with Vacuum On
Position from Apex, in Distances between Dots on
Glass Wood
0.5 0.62 0.5
1.0 0.62 0.625
1.5 0.70 0.6875
2.0 0.80 0.8125
2.5 0.74 1.0

Figure 13 graphically compares these data.

Comparison of dot distances on 20-mil natural rubber sheet stretched against wood and glass plugs.

Figure 13. Comparison of dot distances on 20-mil natural rubber sheet stretched against wood and glass plugs.

The 20-mil natural rubber sheet was then vacuum formed against the glass plug. The characteristics of the grease dots were then examined on both the glass plug and the rubber sheet. The 1 ½-inch grease dot on the glass plug after contact with the sheet is shown in Figure 14. The shape of the grease is characteristic of but more exaggerated than the shapes of the grease dots at the 1-inch and 2-inch positions. The importance of this will also be discussed later.

Grease dot at 1 ½-in from the glass plug apex, after contact with rubber sheet

Figure 14. Grease dot at 1 ½-in from the glass plug apex, after contact with rubber sheet

The distances between grease dots on the relaxed rubber sheet are given graphically in Figure 15 and in tabular form in Table 9, along with the similar data from Part 1. The differences are discussed later.

Table 9 Distances between Grease Dots Spaced ½ inch Apart Then Transferred to Rubber Sheet, Relaxed Distances
Position from Apex, in Distances between Dots on
Glass Wood
0.5 0.5 0.5
1.0 0.5 0.533
1.5 0.46 0.633
2.0 0.30 0.727
2.5 0.30 0.799
Comparison of distances between grease dots, wood and glass plug, rubber sheet in relaxed state

Figure 15. Comparison of distances between grease dots, wood and glass plug, rubber sheet in relaxed state

Experiments with FPVC Sheet

In Part 2, we described experiments with 30-mil [0.020 inch] flexible PVC sheet. These experiments were repeated with the glass plug replacing the wood one. As before, the sheet was heated to 275oF prior to vacuum forming against the glass plug. The transfer times and hold times were similar to those used in the experiments with the wood plug.

Again, the extent of stretching of the sheet was determined first by placing black dots at ½-inch intervals radially outward on the sheet, beginning at the point of first contact with the plug. The sheet was then vacuum formed against the plug. The plug sat in the mold base at a height that allowed the sheet to partially wrap under it during forming. As a result of this undercut, the formed sheet had to be peeled from the plug. The distances between dots on the formed sheet were then measured. The values are given in Table 10 and are compared graphically in Figure 16 with data obtained from the similar experiment with the wood plug.

Comparison of dot spacing for FPVC sheet stretched over wood and glass plugs

Figure 16. Comparison of dot spacing for FPVC sheet stretched over wood and glass plugs

Table 10 Comparison of Dots Prepositioned at ½ inch Marks On 20-mil FPVC Sheet Prior to Stretching
Position from Apex, in Distances between Dots on
Glass Wood
0.5 0.5 0.512
1.0 0.68 0.512
1.5 0.72 0.63
2.0 0.72 0.709
2.5 0.80 0.827
3.0 0.60 0.748

Grease dots were again applied to the glass plug at ½-inch spacing, as before. The FPVC sheet was again heated to 275oF and vacuum formed against the glass plug. This time, because of the undercut to the plug, the sheet was razor-cut from the plug. This hopefully minimized smearing of the grease dots on both the plug and the plastic surfaces. The formed plastic hemisphere was then everted and the distances between the grease dots were measured. All dots were essentially 0.5 inches apart, within allowable error. This agrees with the FPVC-wood data in Table 7 of Part 2. Figure 17 provides an interesting comparison of the dots on the everted plastic form with the dots on the glass plug with formed sheet still in place.

Formed FPVC sheet in contact with glass plug, left, and grease dots on everted formed sheet, right.

Figure 17. Formed FPVC sheet in contact with glass plug, left, and grease dots on everted formed sheet, right.

Observation

We observed in Part 2, “In other words, there appears to be no stretching taking place once the sheet has contacted the plug.” This observation holds for both wood and glass plugs.

In Part 1, we also proposed two mechanisms for non-sliding contact between the sheet surface and the plug surface – compression and shear. We now reexamine Figure 14, the grease dot at 1 ½ inches from the apex of the glass plug after contact with the rubber sheet. The original dot was approximately 1.4 mm in diameter. The width of the grease dot is now about the same but the length is at least 4 times the original diameter. More importantly, the shape indicates that the original dot has been spread downward and outward. This indicates that the spreading action is one of shear rather than compression. The shearing action was depicted in Figure 7B of Part 1 and is repeated here.

Schematic of sheet shearing grease dot against plug

Figure 7B (Part 1) Schematic of sheet shearing grease dot against plug

Consider now that the grease dot is not present. Instead, the gap between the sheet and the plug is air, as seen in the cropped version of Figure 10 from Part 2, viz:

Contact angle between sheet and plug

Figure 10, cropped (Part 2). Contact angle between sheet and plug

Instead of a shearing action as given in Figure 7B, the action is one of squeezing, or pushing the air from the gap between the sheet and the plug.

Plug Surface Characteristics

In Parts 1 and 2, we used a coarse-surfaced wood plug. In this study, we used a smooth glass surface that was usually cleaned with isopropyl alcohol prior to use. [See Figure 12 for exception.] As seen in Figure 16, the surface characteristic of the plug seemed to cause little difference in the local FPVC sheet stretching dimensions. One additional set of experiments was conducted to see if a dramatic change in the interface between the sheet and plug surface could change the stretching characteristics of the sheet. The thinking being that a film of oil should alter the sliding frictional coefficient between the plug and the sheet. To ensure that any effect could be observed, we opted to coat the plug and the sheet, in turn, with a thin film of very low viscosity oil.

First, the glass plug was coated with a very thin film of vegetable oil [Pam]. The FPVC sheet was pristine. The sheet was then heated to 265oF and vacuum formed against the oiled plug. The dimensions between dots are given in Table 11, along with the data graphically illustrated in Figure 18.

Table 11 Comparison of Dots Prepositioned at ½ inch Marks On 20-mil FPVC Sheet Prior to Stretching Clean Glass, Oil on Glass, Oil on Sheet
Position from Apex, in Distances between Dots
Clean Glass Oil on Sheet Oil on Glass
0.5 0.5 0.5 0.51
0.5 0.5 0.5 0.51
1.0 0.68 0.60 0.62
1.5 0.72 0.72 0.69
2.0 0.72 0.72 0.72
2.5 0.80 0.70 0.70
3.0 0.60 0.55 0.60

Then the glass plug was thoroughly cleaned with isopropyl alcohol and the portion of the sheet that would contact the plug surface was coated with vegetable oil. The oiled sheet was heated to 265oF and vacuum formed against pristine plug. The dimensions between dots are also given in Table 11. Figure 18 now shows the data of Table 11.

Comparison of dot spacing for FPVC sheet stretched over wood and glass plugs, including data for oil interface between glass and FPVC sheet

Figure 18. Comparison of dot spacing for FPVC sheet stretched over wood and glass plugs, including data for oil interface between glass and FPVC sheet

We believe it can be argued that there is little difference in the curves of Figure 18 (within experimental accuracy). These data also support our contention that the local grease dots used as markers in earlier studies did not alter the measured values by affecting the local interfacial conditions.

Plug-Sheet Interfacial Dynamics

In our presentation at the SPE Thermoforming Conference in 2004, we reviewed the various types of frictional coefficients. They include:

  • Static frictional coefficient – Initiation of sliding between plug (and mold wall) and sheet
  • Sliding frictional coefficient – Continuation of sliding between plug (and mold wall) and sheet
  • Dry v. wet sliding
    We detailed more than one type of wet sliding, including boundary lubrication – low sliding velocity, low interfacial viscosity, high loading – and hydraulic or hydrodynamic lubrication – high sliding velocity, high viscosity, low loading, as shown in Figure 19.

Characteristic types of wet sliding.

Figure 19. Characteristic types of wet sliding.

From some rudimentary experiments, we concluded that the nature of the dry plug surface did not substantially affect the amount of force needed to stretch the membrane and that it appeared that from simple measurements, the sheet adhered to rather than slid on the dry plug.

From similar experiments, with liquid layers on the plug, we concluded that when an oil layer existed between the sheet and the plug, the amount of force needed to stretch the sheet was the same as if the plug were dry. In this set of experiments, we conclude that the extent of stretching is also essentially unchanged by the nature of the interface (wet v. dry).

It appears to us that the currently accepted sheet-stretched-by-plug mechanism that relies on a coefficient of friction is wrong. If the sheet slid on a wet surface and did not on a dry surface, we would see substantial difference in the sheet dimensions in Figure 18 and in the required forces, as displayed in the 2004 Conference presentation. We do not. We believe that there is no evidence that the sheet ever slides on the plug.

Conclusions

We combine the observations of Parts 1, 2, and 3 to draw the following conclusions.

  • We used a rough-surfaced wood plug and a smooth-surfaced glass plug. We found relatively little difference in the nature of sheet stretching over the plug. Therefore, we conclude that neither the nature of the plug (wood v. glass) nor the surface roughness (rough v. smooth) influences the way in which the sheet forms over the plug.
  • We observed grease dot deformation by vacuum forming both rubber and plastic sheets. The grease dots away from the apex of the plug appeared to be deformed down the plug surface. We conclude that the shapes of the deformed grease dots indicate squeezing between the sheet and the plug rather than the sheet sliding against the plug.
  • We altered the interface between the sheet and the glass plug by applying a thin layer of oil to one surface or the other. We found no appreciable change in the way the sheet formed over the plug.
  • In essence, we were unable to devise an experiment where we observed an effect different from the other experiments. We conclude from our experiments that the sheet does not slide against the plug.

In our presentation at the SPE Thermoforming Conference 2004, we quoted the following:

The relation that the power required to move a body bears to the weight or pressure on the body is known as the coefficient of friction.

W.M. Davis, Friction and Lubrication, A Handbook For Engineers, Mechanics, Superintendents and Managers, The Lubrication Publishing Co., Pittsburgh PA, 1903.

And we asked the question:

In plug-assist thermoforming, what is sliding against what?

If the sheet does not slide against the plug, then, by definition, there cannot be a coefficient of friction, regardless of which definition is used.

Part 4 will discuss alternative mechanisms that may affect the way in which a sheet is stretched against a plug.

Jim Throne on November 6th, 2006
Jim Throne, Sherwood Technologies, Inc., Dunedin Florida 34698 Copyright 2006

Introduction

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%.

In Part 2, we used a 0.020-inch 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 concluded that no sliding occurred between the PVC sheet and the wood plug during vacuum forming.

In Part 3, we replaced the nom. 3-inch hemispherical wood plug with a 3-inch diameter glass sphere. The objective was to determine whether the nature of the plug surface influences in any way the contact between the plug and the stretching sheet. From our experiments, we concluded that no sliding occurred between the PVC sheet and the glass plug during vacuum forming. In addition, we coated first the plug, then the sheet with a thin film of oil to see if interfacial conditions could be affected this way. Neither the force required to stretch the sheet nor the local degree of stretch was affected.

For the three-part set of experiments, we conclude that no sliding occurs between the sheet and the plug, regardless of the nature of the sheet or the surface condition of the plug.

In Part 4, we discuss a possible cause for problems seen in plug-assisted thermoforming.

A New Understanding

From the data presented in this set of experiments and those given in the 2004 presentation, we believe that a new understanding of interfacial dynamics between the plug and the sheet is needed.

So, what is happening as the sheet contacts the plug? First, we believe that the air (or any other low viscosity substance) between the sheet and the plug is squeezed out as the sheet lies against the plug surface. This is demonstrated in Figures 7B and 10, given earlier. If the two surfaces are perfectly smooth, all the air would move ahead of the closing gap. But this is usually not the case with conventional plug materials. If the plug has irregularities, as the wood plug has in our experiments, we would expect that the asperities would support the sheet as it is being drawn over the plug, as shown in schematic in Figure 20.

Schematic of plug asperities that support the stretching sheet

Figure 20. Schematic of plug asperities that support the stretching sheet

Depending on the microscopic architecture of the plug surface, the regions around these asperities may trap the air between the plug and the sheet surface or the air may be squeezed from these regions through tortuous paths along the plug surface. Although one might expect that hotter sheet would minimize the volume of the air trapped between the asperities and the sheet, as envisioned in Figure 21, the softer sheet might actually trap a greater amount of air by sealing off “escape routes” as the sheet is drawn against the plug.

Interfacial resistance between sheet and plug surface, demonstrating the effect of hot, pliable sheet on air trap.

Figure 21. Interfacial resistance between sheet and plug surface, demonstrating the effect of hot, pliable sheet on air trap.

Figure 22, a close-up of Figure 11 (Part 3) shows air bubbles trapped between the sheet and the very smooth glass plug. In many plastic processes – injection molding, blow molding, thermoforming – air pockets at mold surfaces are often called “lakes.”

Air pockets formed when smooth FPVC sheet contacts smooth glass plug.

Figure 22. Air pockets formed when smooth FPVC sheet contacts smooth glass plug.

The most obvious effects of air trap are shiny spots or areas on the molded part surface. These are the result of the plastic not replicating the mold surface, or in our case, the plug surface.

Heat Transfer through Interstitial Air

In addition to causing a surface blemish, air trap may also alter the rate at which the sheet cools against the solid surface.

The heat transfer effect of an air trap was detailed in J.L. Throne, Technology of Thermoforming, Hanser Gardner, Cincinnati OH, 1996, pp. 318-321. The general heat conduction equation through the plastic sheet is:

general heat conduction equation through the plastic sheet

The general heat conduction equation through the plug is:

general heat conduction equation through the plug

The boundary condition at the sheet/air interface is:

boundary condition at the sheet/air interface

The boundary condition at the center of the plug is:

boundary condition at the center of the plug

The heat flux equation that accounts for conduction through the air layer between the sheet and the plug is:

heat flux equation that accounts for conduction through the air layer between the sheet and the plug

Where kp, ki, and km are the thermal conductivities of the polymer, interstitial air, and plug, respectively, and Δxp, Δxi, and Δxm are the differential thicknesses of the polymer, interstitial air gap, and plug, respectively.

The initial sheet temperature is:

initial sheet temperature

The initial plug temperature is:

initial plug temperature

If Ti1 is the polymer surface temperature at the polymer/air interface and Ti2 is the plug surface temperature at the air/plug interface, the interfacial temperatures are related to the interior plastic and plug temperatures, Tp and Tm, are:

polymer surface temperature

plug surface temperature

Where the following combination of thermal conductivities is noted:

kp thermal conductivity

kp thermal conductivity

km thermal conductivity

ki thermal conductivity of the interstitial air

kp thermal conductivity of the interstitial air

km thermal conductivity of the interstitial air

kim thermal conductivity of the interstitial air

Here ki is the thermal conductivity of the interstitial air and Δxi is the apparent air gap thickness.

Because the plug is usually substantially thicker than either the polymer sheet or any air gap, km‘ is usually much smaller than kp‘ or ki‘. Even though thermal conductivity of air is around 20% of that of most polymers, the polymer sheet thickness is many times that of the air layer. As a result, we would expect ki‘ to be larger than kp‘. In other words:

ki‘ > kp‘ > km

As a result:

ki ” ≤ 1 and ki,m ” ≅ 1.

The denominator in the temperature equations given above, viz,
(1 – k i ” – K i,m “), becomes very small. The interfacial temperature difference can be written as:

interfacial temperature difference

Parametrically, we can show that ΔTi becomes quite large (several degrees F) with even a modest air gap (on the order of 0.001 inch or so). In addition to the values for thermal conductivities and sheet and plug thicknesses, the actual value for ΔTi depends on the sheet and plug temperatures.

Some Thoughts on Process Variability

So far, no consideration has been given to differential pressure across the sheet or to the speed of plug penetration into the sheet, which may be a contributing factor to the differential pressure issue. Consider the following cases.

Differential Pressure across the Sheet is Zero

Mechanical stretching is the only force moving the sheet against the plug surface. In the case of the hemispherical plug of Parts 1-3, the initial contact is one of compression of the air between the sheet and the plug. As the sheet continues to stretch, the air is squeezed from between the sheet and the plug.

If the plug is not hemispherical but flat, the initial contact is still one of compression. But as the plug penetrates the sheet, the plastic at the edge of the plug may be pulled over the edge.

We discussed a similar problem in J.L. Throne, Technology of Thermoforming, Hanser Gardner, Cincinnati OH, 1996, pp. 499-503 where we compared the force required to slide draw polymer along the edge of a mold with the force required to stretch the polymer. If D is the diameter of the plug, h is the sheet thickness, β is the angle between the sheet and the flat plug surface, cf is the coefficient of friction, τ is the tensile strength of the polymer at temperature, and P is the pressure holding the sheet against the plug surface, the angle beyond which the sheet stretches rather than slides is given as:

angle beyond which the sheet stretches rather than slides

As the sheet temperature increases, its tensile strength decreases. For high temperature thick sheet, the angle at which the sheet stretches rather than slides is very small. High frictional coefficients favor sliding rather than stretching as do large diameter flat plugs. Nevertheless, for most practical cases the plug does not penetrate the sheet very far, viz, the angle is not very large, before stretching forces prevail.

Differential Pressure is Positive

As the plug penetrates the sheet, the general effect is to hold the sheet against the plug surface. The sheet should preferentially stretch rather than slide. However, aggressive differential pressure may force the sheet against the plug in a compressive manner rather than in the squeezing fashion we observed in Parts 1-3. Air trap in the form seen in Figure 22 may occur if the sheet and plug surfaces are quite smooth, Air trap in interstices may occur between the sheet and the plug surfaces if the surfaces are quite rough.

Differential Pressure is Negative

In this case, the sheet is held away from the plug as it penetrates the sheet. As a result, the sheet is not chilled by the plug and the interstitial air temperature remains essentially constant. This mode is preferred regardless of the plug shape. Unfortunately, this mode is rarely achieved. One reason for this is that only a little negative pressure is needed. Excessive negative pressure forces the sheet against the mold surface, thus obviating the role of the plug, viz, to alter the final part wall thickness.

The Usual Case

Typically, as the plug enters the sheet, it acts as a piston to force residual cavity air through vents. If the residual cavity air is not vented sufficiently rapidly, pressure will build within the cavity, forcing the sheet against the penetrating plug. There are many technical and practical reasons for forcing the plug into the sheet. And there are many technical and practical reasons for minimizing the number of vent holes in the mold cavity. As a result, in most plug-assisted thermoforming operations, differential pressure is positive. At some time during the plug movement, the sheet will usually be forced against the plug surface.

Forward Thinking

From experiments given in Parts 1-3, we find that friction is not an issue in plug design. Altering the plug surface characteristics seems to have little effect on draw-down. The factors influencing efficient plug design must be sought elsewhere. One suggestion focuses on air trap. Interstitial air can substantially alter the local heat transfer from the polymer to the plug.

To verify the role of interstitial air on the performance of plugs, we propose two sets of experiments.

The first set of experiments is very simple. Relatively deep holes of varying diameters are drilled into the surface of a plug. The plug material is not important but should not have very high thermal conductivity. Syntactic foam and wood appear to be materials of choice. Sheet is formed over the plugs and the temperatures of the sheet over the holes and next to the holes are to be measured. The depths of the holes should be at least 5 times their diameters. This will maximize the amount of air that is trapped in the hole by the forming sheet. Because careful temperature measurement is needed to verify the air trap concept, thermal imaging should be used to measure the sheet temperature so that the overall temperature is measured as the sheet cools against the plugs.

A second, more complex set of experiments employs hemispherical plugs of identical character with the exception of their microsurfaces. Both sets of plugs are to be polished smooth to the same RMS level, as recorded with a surface micrometer and microphotographs.

The first set of plugs is to have microgrooves engraved as concentric horizontal rings, beginning at the plug apex and continuing to below the equator of each plug. The depth of each microgroove is to be at least twice its width. The width between pairs of microgrooves is to be equal to the width of each microgroove. If the plug material were aluminum, it could be EDM’d, the microgroove dimensions could be as little as 1 mm wide by 5 mm deep. Aluminum plugs could be cartridge-heated for accurate temperature control.

The second set of plugs is to have microgrooves engraved as concentric vertical rings, again beginning at the plug apex. The microgroove dimensions are to be the same as those in the first set.

What is the objective of this set of experiments? With the first set of plugs, the forming sheet should trap air in the concentric horizontal rings. With the second set, the concentric vertical rings should allow the air to be squeezed out ahead of the advancing sheet. Again, temperature measurement should be made using thermal imaging so that the overall sheet temperature is measured as it cools against the plugs.

Final Thoughts

  • We believe that we have put to rest the question “What is sliding on what?” by answering “Nothing is sliding on anything.”
  • We have proposed that interfacial air may have some influence on the way in which the plug affects sheet mechanics during stretching.
  • We have proposed some experiments to isolate the role of air trap.
  • However, we have not resolved the conundrum “What plug characteristics are necessary when designing a plug to yield the best sheet stretching protocol?”
  • At this point, we leave this discussion and debate to more competent and better equipped people.
Jim Throne on November 2nd, 2006
Jim Throne, Sherwood Technologies, Inc., Dunedin Florida 34698 Copyright 2006

Introduction

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%.

Experiment

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.

Recap

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.

Caveat

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.

Conclusion

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.

Jim Throne on October 20th, 2006
Jim Throne, Sherwood Technologies, Inc., Dunedin Florida 34698 Copyright 2006

Introduction

Nearly all thermoforming processes employ mechanical assists to differentially stretch the heated sheet prior to contact with the mold surface. The mechanical assist is usually referred to as a plug. It has always been assumed that, as the plug enters the hot sheet, the sheet slides against the plug. Mathematical models such as T-Sim (trademark of Accuform, Zlin, Czech Republic) require values for the coefficient of friction between the hot sheet and any solid surface (i.e., the plug and any and all mold elements). It is well known that it is very difficult to measure frictional coefficients in general and frictional coefficients between hot plastic and solid surfaces, in particular.

At the 2004 SPE Thermoforming Conference, we presented a talk on the type and nature of frictional coefficients and asked, “What is sliding on what?” The talk has generated substantial discussion among plug material suppliers, formers, and software manufacturers. The PPT presentation is available on our website.

This paper continues our investigation into “What is sliding on what?”

The Experimental Set-Up

For the set of experiments detailed below, we are using EZFORMTM thermoformer (Centroform, 820 Thompson Ave., Unit 5, Glendale CA 91201), having a 13-in x 18 ¾-in clamp frame is used. The thermoformer heats the sheet only from the top. This study uses a 3-in diameter wood plug that sits in a 3 ¼-in hole in the center of a vacuum box. The vacuum box edge is ¾-in quarter-round. The vacuum box bottom is tempered pegboard. The vacuum system is a shop vacuum pulling approximately 25 in water. The experimental set-up is seen in Figure 1.

Vacuum former set-up, showing top heater shroud, sheet clamp, vacuum box, and plug in place.

Figure 1. Vacuum former set-up, showing top heater shroud, sheet clamp, vacuum box, and plug in place.

Two sets of experiments have been performed using this set-up. The first is described here. In this set of experiments, a 20-mil [0.020-in] natural rubber sheet is used at room temperature. The objectives of the first set of experiments are to determine

  1. whether a simple marking system can yield useful information about local stretching and orientation, and
  2. whether, in fact, the rubber sheet slides against the plug.

The first set of experiments involved determining the local degree of rubber sheet stretching. The sheet was dot-marked at ½ -in intervals from the point where the sheet contacted the center of the plug to 3-in radially outward. The sheet was then stretched and the distance between the dots was then measured. Table 1 gives these data. Figure 2 shows them in graphical form. As expected, the sheet at the very apex of the plug shows no stretching, whereas there is substantial stretching around the 2 ½-inch mark.

Table 1 – Measured Distance between Current Location and Previous Location On 20-mil Natural Rubber Sheet Stretched over Plug
Dot Location from Apex, in Distance, in
0.5 0.5
1.0 0.625
1.5 0.6875
2.0 0.8125
2.5 1.0
Location of dots on stretched 20-mil natural rubber sheet and vacuum formed 20-mil FPVC sheet.

Figure 2. Location of dots on stretched 20-mil natural rubber sheet and vacuum formed 20-mil FPVC sheet.

The Marking System

A marking system was then employed to determine how the sheet contacts the plug. After several false starts, a very simple method was employed. Small dots of grease, approximately 2.5-4 mm in diameter, were placed at ½-inch intervals down the surface of the plug. This is seen in Figure 3.

Photograph of wood plug showing location of grease dots.

Figure 3. Photograph of wood plug showing location of grease dots.

The dots were micro-photographed for dimensional reference, as seen in Figure 4 for the dots at the ½-inch and 1 ½-inch positions. It should be noted here that the dots were not planar but instead were mostly domed or conical in shape. In addition, it should be noted that the dots were somewhat irregular in shape, not circular in diameter.

Grease dots at 0.5-in and 1.5-in positions on plug before contact with sheet

Figure 4. Grease dots at 0.5-in and 1.5-in positions on plug before contact with sheet

The rubber sheet was then vacuum-drawn down against the plug. After the vacuum was released, the sheet was carefully lifted from the plug. The transfer of grease dots to the rubber sheet is seen in Figure 5.

Photograph of grease dots transferred from plug to rubber sheet

Figure 5. Photograph of grease dots transferred from plug to rubber sheet

The dots on the plug were again micro-photographed along with the dots on the rubber sheet. The raw data are presented in Table 2. A composite of the dots on the rubber sheet, along with a dimensional bar, is given in Figure 6. The numbers on the figure are reproduced in Table 2.

Table 2 – Dimensions of Grease Dot Originally on Plug, On Plug after Rubber Sheet Contact, and On Rubber Sheet after Contact
Dot Dim, a x b, mm Product, a x b, mm2
Position from apex: 0 in
Original dot Not measured -
After, on plug 4.3 x 3.6 15.48
After, on rubber 3.9 x 3.6 14.04
Position from apex: 0.5 in
Original dot 2.3 x 2.7 6.21
After, on plug 2.9 x 3.6 10.44
After, on rubber 3.1 x 3.6 11.16
Position from apex: 1.5 in
Original Dot 2.4 x 2.6 6.4
After, on plug 2.9 x 3.5 10.15
After, on rubber 3.3 x 3.7 12.21
Position from apex: 2.0 in
Original dot Not measured -
After, on plug 3.4 x 4.6 15.64
After, on sheet 0.5 x 0.6 0.3

Composite of grease dots as transferred from the wood plug to the rubber sheet. Arrows on individual dots represent the direction down from the plug apex.

Figure 6. Composite of grease dots as transferred from the wood plug to the rubber sheet. Arrows on individual dots represent the direction down from the plug apex.

Analysis of the Data

As noted above, the marking dots were not planar, but domed or conical in shape. As a result, when the sheet contacted a dot, it flattened it. The nature of the flattening is important.

  1. If the sheet simply compressed the dot, one would expect the final dot shape to be simply larger in all dimensions, without any preferential orientation. This would imply that the sheet compressed the dot without sliding on it.
  2. If the dot became distorted in one direction – particularly in the sheet draw direction – one could conclude that the dot had been differentially sheared by the sheet. In addition, one might conclude from that evidence that the sheet slid locally, not only on the dot, but also on the plug as a whole.

To determine the nature of the flattening, the ratios of the dot dimensions on the plug, before and after sheet contact, were determined. These data are given in Table 3. Note that the initial dot dimensions for dots at 0-inch, 1-inch, and 2-inch positions were not determined. In addition, the ratios of the dot dimensions on the plug before sheet contact and on the sheet were determined. These data are also given in Table 3.

Table 3 – Effect of Rubber Sheet on Grease Dot Dimensions for Dot Positions 0.5-in and 1.5-in
Across Down Ratio, D/A
Position from apex: 0.5-in
After, on plug/original 1.26 1.33 1.06
After, on rubber/original 1.35 1.33 0.99
Position from apex: 1.5 in
After, on plug/original 1.21 1.35 1.12
After, on rubber/original 1.38 1.42 1.03

Sheet Orientation Factor

At first glance, there appears to be relatively little difference between the dot dimensions on the plug and those on the sheet. However, keep in mind that the sheet is stretched locally over the dot. An average of the stretching values of Table 1 is used to determine the local degree of stretch. These values are given in Table 4. The data of Table 3 for the rubber dimensions are now corrected by multiplying by the local stretch values, as shown in Table 4.

Table 4 – Correction for Local Rubber Sheet Stretch and Recovery At Dot Positions of 0.5-in and 1.5-in
Position from apex: 0.5-in
Stretch between 0 and 0.5 in 0.500 in
Stretch between 0.5 and 1 in 0.625 in
Avg 0.559 in
Stretch ratio, 0.559/0.5 1.12
Position from apex: 1.5-in
Stretch between 1 and 1.5 in 0.6875 in
Stretch between 1.5 and 2 in 0.8125 in
Avg 0.747 in
Stretch ratio, 0.747/0.5 1.49
Across Down Ratio, D/A
Position from apex: 0.5-in
After, on plug/original 1.26 1.33 1.06
After, on rubber/original 1.35 1.33 0.99
D/A, corrected for stretch recovery 0.99 x 1.12 = 1.11
Position from apex: 1.5-in
After, on plug/original 1.21 1.35 1.12
After, on rubber/original 1.38 1.42 1.03
D/A, corrected for stretch recovery 1.03 x 1.49 = 1.53

Observations

Plug stretching is considered as plane-strain stretching. For a hemispherical plug such as that used in this study, there is relatively little circumferential or radial stretching compared with down-plug stretching. As a result, the observations focus on down-plug stretching only.

Consider the importance of local down-plug sheet stretching. It seems that at the ½-inch position, there is little down-plug differential orientation evidenced on the rubber sheet. At that position, the sheet has been locally stretched a little more than 10%.

However, there appears to be significant down-plug differential orientation at the 1 ½-inch position as evidenced on the rubber sheet. At that position, the sheet has been locally stretched about 50%. There is no equivalent down-plug differential orientation on the plug itself. This implies that the sheet first contacted the dot at the 1 ½-inch position in compression. This allowed the grease to be adsorbed by the rubber.

Because the down-plug dimension of the dot on the plug is not substantially greater than the cross-plug dimension, it appears that the sheet did not smear the grease down the plug surface. However, it appears that the sheet was stretching while it was contacting the grease. If the sheet had been stretched prior to contacting the grease, the down-plug dimension of dot on the sheet would now be substantially smaller owing simply to the sheet recovery.

Tentative Conclusion

If we reword the question “What is sliding on what?” to read “Is the sheet sliding on the plug, and if so, where?”, then the tentative answer seems to be, “It may be sliding, but not everywhere.” The experiments indicate that down to perhaps the 1 ½-inch point on the plug surface, the sheet simply lays onto the plug, as evidenced by the compression of the grease dot without apparent shear. Figure 7A is a schematic of what this effect might appear to be.

Schematic of the possible compression of grease dot by advancing sheet.

Figure 7A. Schematic of the possible compression of grease dot by advancing sheet.

Schematic of the possible shear of grease dot by advancing sheet.

Figure 7B. Schematic of the possible shear of grease dot by advancing sheet.

Beyond that point on the plug surface, there appears to be elongation of the grease dot in the down-plug direction. Because the grease dot is not substantially elongated on the plug but is on the sheet, it would indicate that the sheet is smearing the adhered grease dot in the down-plug direction. This effect might be considered a type of sliding. However, it might also be considered a shearing or squeezing effect. As the sheet contacts the dot, the initial portion of the grease dot is forced against the sheet. As the sheet continues to stretch, that portion adheres to the sheet. As grease is picked up by the stretching sheet and adheres to the sheet, it resides behind the initial portion. Perhaps the roughness of the plug or the rapid absorption of the grease prevents any portion of grease initially adhering to the sheet to be transferred back to the plug as the sheet presses against the plug. Figure 7B is a schematic sequence of what this effect might look like.

Consider a conclusion that the sheet is in fact sliding against the plug at the 1 ½-inch point. If one surface is truly sliding against another, some form of a frictional coefficient is needed. But can it be concluded that sliding is also taking place at the ½-inch point? If not, then a frictional coefficient is not warranted. More importantly, is there some property or combination of properties that varies with the angle of attack between the plug surface and the sheet?

Continuing Work

The second set of experiments, described in Part 2, employs the same set-up described above except that the rubber sheet is replaced by 20-mil [0.020 inch] transparent flexible PVC, heated to various temperatures.

An analysis of the interrelationship between sliding force and stretching force will also be considered in Part 2.

Jim Throne on October 4th, 2006

Individual Power Points

Power Point of Thermoforming Conference 09-06 Title Page

Power Point of Thermoforming Conference 09-06 Heating Page

Power Point of Thermoforming Conference 09-06 Stretching Page

Power Point of Thermoforming Conference 09-06 Trimming Page

One ZIP File

ZIP File of Four Power Points of Thermoforming Conference 09-06

Jim Throne on April 4th, 2005

Dr. Throne was the invited lecturer at Two Indian Plastics Institute Programs in India in March 2005. He presented a plenary overview of thermoforming to more than 400 technologists at the IPI meetings in Mumbai (Bombay) on 25 March and in Delhi on 28 March. Jim began his several-hour presentation by summarizing the history of thermoforming. He then outlined new developments in materials, machinery, markets, mold and part design and controls. Reliance Industries Ltd., Supreme Petrochem Ltd., Irwin Research and Development Inc., and Wonderpack Industries PVT Ltd. sponsored his presentations. During his ten-day stay in India, he discussed thermoforming technology with more than a dozen thermoformers, machinery builders, and materials suppliers, many seeking cooperative efforts with North American companies.