Background

In Part 1, I presented experimental data on sliding of E&C syntactic foam on 0.120-inch black general-purpose polystyrene sheet at a temperature substantially above its glass transition temperature. But I did not address the question: What is sliding against what? In this part, I try to identify the conditions under which the plastic sheet slides against the plug. To do this, I propose a series of experiments using a natural rubber sheet and a plug having various surface conditions.

What is Sliding on What?

Consider a hemispherical plug pushing into a plastic sheet, Figure 9:

Hemispherical Plug Entering Sheet Surface

Figure 9. Hemispherical Plug Entering Sheet Surface

It is apparent that the sheet is being stretched between the rim and the plug surface. The question is whether the sheet on the plug is being stretched to the same degree as the sheet free of the plug or whether it is being stretched at all. Consider the first scenario. If the sheet is stretching uniformly whether in contact with the plug or free of the plug, the sheet must be sliding freely on the plug. When the sheet is stretching only where it is free of the plug, it must be firmly “attached to” or “bonded with” the plug. If we relate this to frictional resistance, when the sheet is sliding, the frictional coefficient is zero. And when it is not sliding, the relative frictional coefficient is unity.

So, to test the theory I reconstructed a device I used many years ago when I was studying the theory of plane strain deformation. The device consists of a scissor jack, a kilogram scale, and a fixture that holds the sheet, Figure 10:

Experimental Setup

Figure 10. Experimental Setup

As shown on the right side of Figure 10, the plug is centered below the membrane. As the jackscrew advances, the plug pushes into the membrane. The penetration height and membrane resistance to penetration are measured. For these experiments, I used a 0.015-inch thick natural rubber membrane and wooden plugs. I used hemispherical plugs of two diameters – ¾-inch and 3-inch. The membrane was 6¾-inch in diameter.

Some Experiments on Plugs

The force-penetration curves for these two plugs are shown in Figure 11.

Force-Penetration Curves for Two Diameter Hemispheres

Figure 11. Force-Penetration Curves for Two Diameter Hemispheres

Because the ¾-inch diameter plug contact area was so small, I considered that the contact area of the sheet on the plug was very small when compared to the membrane area, as seen in Figure 12.

This allowed me to calculate the equivalent modulus, E, for the rubber from the following equation: F = 2πδET0/ln(a/b).

Where F is the force, δ is the penetration depth, t0 is initial sheet thickness, a is plug diameter, b is sheet diameter. In addition to measuring the force as a function of penetration, I also measured the contact area between the plug and the sheet.

Surface Contact for Two Diameters of Hemispheres

Figure 12. Surface Contact for Two Diameters of Hemispheres

Figure 13 compares the calculated and measured force-penetration curves for the two plugs. As expected, the experimental values for the small-diameter plug agree quite well with the calculated values. The experimental values for the large-diameter plug are quite a bit lower than the calculated values.

Calculated and Measured Force-Penetration Values

Figure 13. Calculated and Measured Force-Penetration Values

How Does the Plug Surface Affect Force-Penetration?

Dry Plug Surface

In both cases, the plugs were as-received wooden spheres. I then began to modify the surface of the large plug. In the first set of modifications, I simply smoothed the plug surface, according to the following schedule:

  • As-is (coarse, exterior PT pine)
  • Sanded w/180 grit paper, blown free
  • Sanded w/320 grit paper, blown free, waxed with furniture wax
  • Sanded, waxed, polished on rouge wheel
  • Sanded, waxed, polished, then coated with talc

The results of these experiments are shown in Figure 14.

Effect of Surface Texture on Force-Penetration

Figure 14: Effect of Surface Texture on Force-Penetration

There is relatively little difference in the force-penetration curves, as seen in the expanded scale, Figure 15.

Expanded Scale of Figure 14.

Figure 15. Expanded Scale of Figure 14.

To carry this one step further, I measured the increase in surface area of the free portion of the sheet. This was done by inscribing a 2.5 cm circle on the membrane prior to stretching. When the membrane is stretched, the circle is stretched into an ellipse. The increase in area is inversely proportional to the reduction in thickness:

Raoriginal circle = π r² , where r is the radius

Raellipse = π a b, where a is the major axis of the ellipse and b is the minor axis

The relative increase in surface area is:

Arearelative = ab/r²

Since the volume is constant, the relative decrease in thickness of the membrane that is free of the plug is:

t/t0 = 1/Arearelative = r²/ab

If the plug had zero radius, the entire sheet thickness would be reduced by this amount. As noted, the contact area of the ¾-inch diameter plug is very small, relative to that of the membrane. As a result, we would expect the free membrane thickness to be nearly identical to that of a zero radius plug. This is the case, as seen in Figure 16.

Now consider the larger ball. If the sheet easily slides on the large ball surface, the thickness of the stretched sheet should be essentially the same as that for the smaller ball. This is shown in Figure 16. If the sheet adheres in any way to the surface of the larger plug, we might expect the free membrane thickness to be reduced as the plug penetrates the membrane. Simply because a substantial portion of the sheet is in contact with the larger ball. This is also shown in Figure 16.

Calculated Reduced Thickness v. Penetration Depth

Figure 16. Calculated Reduced Thickness v. Penetration Depth

Here’s what the measurements show:

Penetration No Frict Max Frict Exptl*
2.0 in 0.832 0.813 0.805
2.4 in 0.778 0.738 0.744
* Average of 9-10 experiments

In other words, it appears that the sheet is in fact sticking to the surface of the plug.

Lubricated Plug Surface

As seen above, plug surface condition does not seem to lead to substantial alterations in the force-penetration curves. As a result, I added lubrication to the plug surface:

  • Molybdenum white grease – oil-based
  • Glycerin – water-based

It is apparent in Figure 17 that water-based lubrication leads to substantial reduction in the amount of force required to stretch the membrane. What is surprising is that the white grease did not appear to affect the stretching resistance. It could be that the rubber membrane is adhered to the grease but slides quite well on the glycerin.

Okay, What About Thickness?

Thicker rubber sheet was not available. So I chose to double and triple the 0.015-inch membrane to simulate thicker sheet. Because of the limitations of the scale, I did not increase the thickness beyond 0.045-inch. According to the plane strain force-penetration formula given earlier, the amount of force required to stretch the membrane a given amount is proportional to the initial membrane thickness. As is seen in Figure 18, the experimental values mirror the theory quite well. In other words, at least this part of the theory works.

Effect of Lubrication Between Sheet and Plug

Figure 17. Effect of Lubrication Between Sheet and Plug

Comparison of Calculated and Experimental Measurements for Thicker Sheet

Figure 18. Comparison of Calculated and Experimental Measurements for Thicker Sheet

What Can We Conclude?

We can conclude several things.

So long as the plug surface is dry-

  • The surface quality of the plug does not dramatically affect the force-penetration curves for thin membranes.
  • It appears that the sheet tends to lay on the plug surface without additional stretching, rather than sliding over the plug surface.

When the plug surface isn’t dry –

  • Plug surface lubricity affects the force-penetration curve but apparently the nature, oil v. water, of the lubricant is important.

Okay, how does this compare with the work of others? The work of References 8 and 9 represent the most comprehensive efforts to date to understand the interaction between the plug and the sheet. The focus in Ref. 8 was the measurement of the plug force-penetration of a syntactic foam plug [Formplast 2000] with either flat or hemispherical geometry. The researchers tested only dry plugs. The hemispherical plug was used either rough or polished. Their peak load data are shown in Fig 19. Their results indicate that sheet resistance is higher for smooth plugs than rough plugs, but that the differences in resistance values are not as substantial as the differences in plug and sheet temperatures. Our results with room temperature rubber sheet appear to corroborate these results.

Literature Results on Plug Force v. Sheet Temperature

Figure 19. Literature Results on Plug Force v. Sheet Temperature

So, Are We There Yet?

Not yet. Knowing what we know now, we need to reconsider the original question: What is sliding on what? This is the focus of Part 3.

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