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.