I'd probably just design specific things for specific martial classes or subclasses, rather than try to have a generic 'martial use of concentration'.

Like, a class which can stack different kinds of buffs into its attacks as it lands hits, but which are lost if concentration is interrupted.
A class ability (barbarian? monk?) to buffer hitpoint damage (and at higher levels, things like poisons and status conditions) as long as concentration is sustained successfully - you take the damage/effects/etc the moment concentration is interrupted or allowed to lapse, but you can receive healing to drain the buffer or lose the status conditions.
A rogue ability to sustain sneak attack eligibility against a target you have successfully landed a sneak attack against as long as you sustain concentration.

Etc...