Having already done this once, I decided to go with a lighter material for the core, and a smaller haft for the axe. Still PVC with an Oak dowel core, but 1" versus 1.5".
You'll still need some form of material for the core, that big rubber sheet in this case. A few bolts, a haft and padding are the rest of what you'll need.
Step 1: Cutting and Attaching the Core
So, first thing is to come up with a vague design for the axe head. Mine's going to be a stocky bearded axe, so, I cut out that part of the shape. Tagged in the first picture.
To give it a little more rigidity, I added enough for a fold over the top. In pictures 3 and 4, you can see how it was folded, then drilled and bolted through.
Since this was a little more finicky than the last one, I had to clamp down the rubber and the handle, then use a hand drill to get it in roughly the correct spot. It did not want to stay on a drill press table long enough to clamp properly.
Four bolts and corresponding washers should keep the core together with little movement or tearing.
Step 2: Cutting the Foam
Since the large tube is a little long for the width of the axe head, I cut off the front portion just beyond the core, and cut off a small portion of the lower back, so it looks more like an axe than a wad of foam attached to a stick.
Waste not, want not. I used the cut off portions for reinforcement. The long strip cut from the front will be reattached as more padding for the "blade," and the portion that was cut from the back of the axe will add more padding over the top of the core.
Step 3: Firmly Attaching the Foam.
The key points to make sure are secured are around the front and across the top. The back isn't as much of an issue because it will be secured to everything else. I tied one end of the laces to the haft and started wrapping. Two laces were enough to make sure it wouldn't move.
One below the "blade" and "top" layers and just over the layer covering the core. Then some gaff tape to secure the next layers of foam while the second lace is tied over.
After that, more gaff tape. Your best bet is to reinforce all the laces already attached by making sure they won't move using gaff tape. It's strong, it's durable, and, if the laces are tied securely, it'll just make it all the stronger and less likely to fall apart.
Step 4: The Beatings Will Continue Until Morale Improves.
I'll be sparring with it shortly and probably coming up with a design for a short and long sword soon. So, await parts 4 and 5!