One reason a person might stay in a abusive relationship is

Lack of self worth. Meaning that the person who is in the abusive relationship feels as if they don't deserve any better than that. Or perhaps that they need the abuser because they aren't anyone without them.

Denial, maybe they feel like it isn't happening or that it is their fault for what the other person is doing.

The illusion of the abusive person making them happy. They may say "But when is really happy, our relationship is amazing." In reality, the only happiness that is present creates a high because most of the time it is negative. The emotions in the relationship are either extremely low, or extremely heightened.

The person in the abusive relationship most likely did not go into the relationship getting abused from the get-go. Usually signs are shown overtime and things start going out of control when it is almost too deep to back out. I know that people usually hold onto the person that the abuser could be. "They were nice once, they can change" It's often caused by holding onto the false image. Or the hope that they could still be person they were.

Finally, a more obvious one would be that they are either dependent on the person or scared of leaving due to their safety.

I don't know if this is what you were looking for, but I hope i helped.

This is coming from someone who has been all three sides. The friend, the abuser, and the abused.

clean the wound to make sure of no infections.