So instead of calling it Expertise, Skill, Martial Damage Dice, etc..
Let's call it power dice.
If a fighting class has the Parry maneuver, he rolls power-die+str mod when reducing damage with his strength based weapon.
Or the fighter can spend his 1 die on Deadly Strike and do power die+str mod extra damage.
If a caster uses an abjuration spell or a movement spell such as "Levitate" as a defensive reaction, she can reduce her damage by her power die+int mod.
If a cleric is healing an ally as a reaction, he can heal power die+wis mod.