# How do I compute the force required to pierce HDPE?

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. 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 $F = \sigma_\text{y} A_\text{contact}$.

• 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. Feb 11 '15 at 16:12