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I have a $2\text{ mm}$ sheet of HDPE at $280-320^{\circ}\text{ F}$. I want to pierce the skin of this sheet with a $2\text{ mm}$-high pin with a surface area of $3.33\text{ mm}^2$. The pin is also made of HDPE and has cooled completely. With the Young's modulus of HDPE being $0.8\text{ GPa}$, how much force do I need to apply to have the cooled HDPE pin reach the molten layer of the HDPE sheet at $280-320\text{ F}$?

UPDATE: I've been told that the pin is supposed completely weld to the layer of HDPE. Based on the melt temp of the HDPE and the starting temp of the pin, I just don't think its possible in the 2 minutes before the material is quenched. I've attempted to figure out the rate of heat transfer, but can't find the thermal contact conductance coefficient, or the thermal contact resistance. Searching online turned up a tool "CoCoE.exe" but was having trouble running the program on my computer.

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After starting to redo the analysis, I've come to realize this is a fairly complicated process, and instead I'll outline what I would do to solve this if I had more time.

First, some assumptions I'd make:

  • perfectly plastic behavior of the HDPE (i.e., the stress-strain curve is flat after yielding)

  • thermal convection from the flat surface is irrelevant (though it is taken into account for the pin)

  • the pin has an adiabatic tip

  • the metal is at a constant temperature

  • the pin pierces the surface when the energy absorbed by a cylindrical portion of the material directly beneath it is exceeds the toughness of the material multiplied by the volume of cylindrical portion of the material (I can explain this differently if desired.)

  • the contact resistance decreases as the HDPE melts (not sure how)

  • some material properties can come from this paper and others from this data sheet, and this book (this data may or may not be valid for the HDPE you have)

  • I'm not sure what value of the convective heat transfer coefficient for the pin is valid and would have to think more about this.

  • heat of fusion comes from this data sheet

Given all of this information, you can write 1D heat transfer solver for the Stefan problem to compute the temperature and thickness of the solid skin. From there, you can use the temperature of the skin to find the yield stress. My simple model suggests that $F = \sigma_\text{y} A_\text{contact}$.

Old answer:

The answer is likely zero force or nearly so, because the melting temperature of HDPE is about 266 F (it's reported a little higher here at 279 F), so the HDPE probably has already melted at the specified temperature. You'll meet the molten layer immediately upon touching the surface (at least if the temperature is uniform enough).

If the temperature range is wrong, I think I can develop an estimate assuming that HDPE is a perfectly plastic material, that the impacter is flat, and some other things, and then change my answer. Let me know if you'd like this.

Also, if you want an estimate of the force required to continue penetrating the medium, I can find an old terminal ballistics book I read that has some models for such things, but I don't know how accurate they are.

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  • $\begingroup$ The material I am using has a melt point of 260F, but I've been told the sheet is at 280-320F when the pin makes contact with the sheet. The material is being blow molded, and prior to blown open, my part is inserted. When the mold closes it clamps shut on my part and the HDPE sheet, supposedly creating a weld. There should be a skin that forms on the sheet as it cools, before my part touches it. I'm not sure how thick that skin would be. $\endgroup$
    – William B.
    Feb 10, 2015 at 22:57
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    $\begingroup$ Thanks for the reply. Can you edit your question to have a drawing of some sort? I am having trouble visualizing what you are describing and believe a simple drawing would clarify things. Also, describe how the sheet is heated, e.g., is it heated until it reaches the right temperature and then allowed to cool? If so, how long is it allowed to cool? Is it heated continuously and then the pin hits it? How long is it heated for? Etc. $\endgroup$
    – Ben Trettel
    Feb 10, 2015 at 23:08
  • $\begingroup$ The skin thickness could probably be estimated from a 1-D heat transfer model if the heat transfer coefficients and thermal conductivity are known (or measurable). As Ben Trettel noted, some clarification of the question would be helpful. $\endgroup$
    – wwarriner
    Feb 11, 2015 at 1:13
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    $\begingroup$ @starrise: Yes, I was going to try the same thing. Outline of my intended approach: Estimate thickness of material based on 1D heat transfer model (complicated by the phase change). Then, assuming the liquid phase does not factor into the computation, calculate the toughness of the material assuming it is perfectly plastic, and use that to get an estimate of the energy required to penetrate it. Suggestions are welcome. The answer you'll get this way surely will be very approximate, but hopefully accurate enough. $\endgroup$
    – Ben Trettel
    Feb 11, 2015 at 2:37
  • $\begingroup$ I realized last night that the force, not energy is asked for. I think the energy is more relevant, but in the model I intended to use (perfectly plastic, assuming only a cylindrical cutout of the material is being compressed) the force would be $F = \sigma_\text{y} A$ where $\sigma_\text{y}$ is the yield stress and $A$ is the contact area. There are a large number of assumptions and approximations here (not all of which will fit in the comment), but under this approximation the main issue is finding the yield stress, which may not be known for the temperature of the solid film. $\endgroup$
    – Ben Trettel
    Feb 11, 2015 at 16:12

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