For this, we'll make a double bit axe modeled somewhat after Gimli's axe from Lord of the Rings. Why? It's good, solid and heavy.
The tricky part was making a heavy axe while balancing it with sufficient padding to not cause someone damage. With this, you'll feel the hit, but it will be something like getting hit by a Jotun's memory foam mattress rather than a heavy hammer.
You will need:
A haft (PVC with poplar dowel core left over from a previous endeavor)
A core (1/2" rubber sheet)
Padding (more neoprene-like foam)
Fasteners (Hockey Skate lace, bolts and contact cement)
Step 1: Cutting the Core
For this I used a band saw after having tried an Xacto knife and a box cutter, neither of which gave great results. So, I took a fine-tooth band saw blade and slowly cut my way through it, ending with the shape below.
I then drilled a hole through rubber, and a corresponding hole through the center of the haft (not quite center, since cylinders are difficult to drill through and attached it with a pretty decent bolt and washers.
Step 2: Shaping the Core
The biggest trick is finding the correct size bolts to push all the way through, but not to tighten past the size of the nut on the other end. However, you also need them to go all the way through fairly springy rubber. A combination of elbow grease, a hex driver on a power drill and some uncouth vocabulary was able to finish this part, and you end up with the core/skeleton of the axe head.
Step 3: Aesthetics
Step 4: Foaming the Head
Since I had a very large neoprene-like foam tube to work with, I created the first sheath for the core out of one piece of that. The axe head fits snugly in and it gives the head a little more meat without a lot more weight.
To keep it from moving around, I "stitched" the foam around the head of the axe. Using something like hockey laces makes it have a lot of surface area to keep from tearing the foam. You can either drill holes in the foam to thread it through, or try and find something to use as an awl. I tried both; the awl is just slightly less messy.
I used two separate laces, one for each bit, so they laced together and tied around the head for added security.
For all tests, it did not move, including a full-swing on the shield.
Step 5: Foaming the Blade
This gives it a double-thickness, and also keeps the stitching from moving around too much.
As a little more reinforcement, adding a wrap or two of heavy duct tape or gaff tape will keep it together a bit better.
Final weight: 6.6 lbs