An axe will literally inflict blunt damage in addition or even in absence of its cutting damage. You can have the edge taken off an axe entirely and it still remains a devastating weapon, and hitting heavy armour will be equivalent to a flanged mace or similar.
I should know -- for years I carried around a camp-axe with me while backpacking that was never sharpened and blunt as anything. It still chopped wood.
So an axe would do chop damage,
unless it cuts through a hardened wood helmet.
And maybe a flanged mace would do semi-chop...
and a falchion chould do 50/50 shopcut.
More severe now :
I think the weapons should do damage based on their energy,
and based on their shape would try to turn this into cutdamage, or not.
Otherwise the damage would remain blunt.
A mace and an axe of equal proportions would do the same amount of damage.
But the axe would try to turn this into cut.
At the price of balance, the cut-advantage would be traded off for handling.
A sword of the same length would be lighter,
it would also try to cut, would have the best handling but
it would deliver less damage than axes&maces.
A sword of the same weight of a mace would loose the handling-advantage,
but this gets traded off for reach/length.
So why should one try to get cut-damage ?
Because it would be higher than blunt-damage.
This would work well with locational damage.
The highest advantage for cut above blunt would be at limbs and neck.
An armor would subtract a certain amount of damage,
what gets through would be turned into cut.
The subtracted damage doesn't vanish, it just stayes blunt.
Now a certain % from this blunt would get substracted, depending on cushioning...
and the final calculated damage would be blunt&cut combined.
( but the healing rates could be different )
And thrusting attacks could turn cut-damage into pierce-damage.
A thrusting attack would do little base-damage, because the weapons doesn't get swung.
Therefore cut-damage could get turned into pierce-damage,
wich would be especially effective at the chest/torso and the head.
The multiplier would be higher than for cut ( but less at limbs and neck ).
The more pointy a tip is, the higher the portion of pierce damage.
More pointy weapons would also decrease the effectiveness of armor.