Sounds like cutworms. They're caterpillars that live in the soil or debris and crawl out at night, wrap themselves around the stalks, and chew through them for food (a couple varieties also climb high into the plants, not doing as much damage, while some other varieties feed on roots). There are a variety of methods to control them, including watering with various pesticides and placing protective rings around the base of their plants.

I've never heard of a cutworm eating a stem that thick before -- the corn stalks are thicker than my thumb.

Does the south grow super-sized cutworms?