If I were to have concentration effects for martials, I would probably go with them concentrating to get some bonuses in combat from how they are fighting. I have given this absolutely 0 thought before now, but what if martials could use concentration to switch up their fighting style? Base fighting style is free because that's what they do normally, but if they focus on their style they can switch that to something else that may be useful in the situation?

Or perhaps concentrating on defense or offense specifically - when concentrating, you could either take a +2AC/-5 to hit or the reverse (numbers pulled from the nether), representing that you are focusing your efforts on one or the other.

Maybe an archer could concentrate on a target, and as long as they focus on that target they get advantage on attacks. Maybe you have to make a hit first to focus so it matters when concentration is broken.

I could see a monk concentrating into a state that lets them dodge pretty much everything. Concentrate and give up attacking and you get a +10 to AC or something.

Or a different path - maybe you can concentrate on detecting hidden enemies, keeping all of your senses tuned to doing so, and you prevent anyone from hiding from you, or maybe just get a +5 to perception (including passive).

Or concentrating on specific saves - if you are concentrating on dodging, you get a bonus to Dex saves, or if you are concentrating mental focus you get a bonus on Wis saves. This may help with the other thread about how higher-level martials fail saves like that all the time.

I'm going to stop there, because I like that last one quite a bit, and need to mull it over more. Probably going to go to homebrew and workshop it.