Unfortunately, there are no easy solutions to control armadillo damage. Repellents have not been shown to be effective on the species, and at this time no products have obtained federal registration.
Essentially, you have two options, exclude them with fencing or trapping.
Use sturdy fencing that extends 2 inches into the ground and then bends out at a 90 degree angle for an additional 18 inches from the area needing protection. With this underground skirting, the armadillo will be unable to dig under the fence.
Where fencing is impractical, armadillos can be trapped using cage traps of raccoon size or larger. Before trapping, be sure to learn and follow all state and local laws.
Unfortunately, a highly effective bait for armadillos has not been discovered. Some animal damage controllers have found success with mealy worms, the kind used in fishing, but the results are mixed.
The key with all successful trapping is location, location, location. Place wings on your trap to help funnel/guide the armadillo toward the trap. A picture can be found in the link below.
Of course, the more traps you use, the more likely you are to be successful. Always, always check traps on a daily basis (including weekends/holidays). Be prepared to catch a skunk! Learn about how to handle a skunk BEFORE you catch one. Visit Handling a trapped skunk.