Rappelling with a (slightly) damaged rope
If your rope sheath is slightly damaged, AND the core is intact, you might be able to safely rappel.
If you rappel carefully without jumping around too much, you're not gonna put much more than about half your body weight on one strand of the rope. The core of the rope is where the real strength is, so if that’s still intact, you're probably in good shape.
If the core is still in good condition, you can wrap some tape tightly around the damaged sheath. This helps hold the sheath together and keeps the core from being further damaged. Use a small amount of tape so the rope can more easily feed through your rappel device.
However, if the core itself is damaged, even a little, you might need to resort to more advanced rappel shenanigans, like tying off the damaged part of the rope in a knot (see below) using the damaged side as a pull cord, knot blocks, passing knots if it’s a single strand, etc.
Do you have a damaged rope? Here are some options.
Is the damage near one end? Consider cutting the rope and losing a few meters, provided you can still reach the next rappel anchor.
Is the damage in the middle? Consider cutting out the whacked part of the rope and tying the two ends together. (Yes, this complicates leading, but still allows you to rappel.)
If you're concerned about preventing damage to rope in the first place, consider a Unicore rope. Here's a complete article about that.
If you do have to isolate a damaged part of the rope, you can tie a bight knot with the damaged part in the loop. The standard knot for this is the butterfly, because it holds well in both directions of pull.
This is a very good time to be conservative. If you have the slightest doubt about the strength of the rope, the best plan is probably to isolate it like in the photo below, and deal with the knot on your way down.
How strong is a slightly damaged rope?
Image: HowNOT2, Youtube, “Crampon vs Rope”
It's hard to answer because there are so many variables, but I think the general answer is: stronger than you think.
Check out this video I made with my friend Ryan Jenks, founder of HowNOT2. Probably everyone who is set foot on a glacier has crampon-spiked a rope at some point. And, we've all probably wondered, does this really hurt the rope?
Even after I spiked the rope with my crampon, AND Ryan hammered multiple nails through the rope, it tested close to full strength!
I also cover this in my series of article called “How Strong Is it?”, you can read that one here. #NoGearFear