Edging a lawn by hand is something I just tried. Instead of using the typical power tools one usually sees used for this task - like a string trimmer or a bladed edger - I used a half-moon edger and some manual labor.
I stood on the grass to do this, not the concrete. I placed the blade of the edger where the concrete and grass meet and stepped down gently on the edger to make sure I got it in the right spot. I didn't force it all the way down. Just enough to cut through the overgrowth that was coming on to the concrete. I then lowered the handle of the edger to the floor, bending my knees to do so while scraping the blade against the inside edge of the concrete to pry out a little slit of grass, soil, and debris. It makes a scraping noise, but the blade is strong. I then shift over a bit and repeat.
I took this photo to document the result.
I use the half moon edger the usual way you would when cutting the edge of a lawn right into the soil. Place it, step down on it, remove it, shift it one blade length to the left or right and repeat until you have covered the entire section. The only differences are, I didn't go as deep, and I bent the tool down to pry up the material I had cut instead of lifting the blade straight out of the ground.
How deep did I go? Only as deep as I had to.
You will likely lift up the edge of the grass because you have literally pried it away from the concrete under ground. Just step on it to pat it back down.
In the image below you can see towards the top is where I have edged. It creates almost a dark outline around the lawn as edging a lawn by hand in this way creates a narrow slit between the concrete and the lawn.
Is it slower than using a power tool? Sure. But it is something you could do early in the morning which you can't say for a string trimmer because of the noise that makes.
You can then use a broom or a blower to get rid of the debris and you'll have a nice neat and clean edge to your lawn. Because I liked the result, as of right now, this is the way I am going to edge the lawn around concrete for this coming season. I'll post more results from edging this way in the future.