ooh...melting is bad, very bad.
But: Convincing any Apple employee that it wasn't abuse is going to be very hard. Just winding the cord up with a kink at that spot, that could do it. You can try, but remember to keep your cool and be polite throughout.
In any case, the cord is shorting out, if you're having to position it just so to get it to work. You should replace it immediately, if not sooner, because it might damage the laptop.