Liriope and creeping phlox might be good choices for you.
Steve gives you a couple of good options, Myra. You might also try blue star creeper (Isotoma fluviatillis), but for any of these choices it is crucial you keep your dog off until the plants are very well established. Good luck!
Myra, you may want to see a very similar thread asking the same question. http://www.hometalk.com/activity/148889
We have found that the only thing that holds up to dogs is Artificial Turf. We have used it on many back yards for dog areas and all of our customers swear by it. It is a 5-6 year investment compared to sod but well worth it. You will always have beautiful green lawn that the dogs can run freely on with no damage or dirt tracking back into the house. Here is the link to the turf we use the most. They can even help you out with a contractor
to install it if you need one. www.ecowaterlessgrass.com
My friend trained her dog to only go in a certain space way in back of yard with mostly mulch. Training is the biggie! The acidic urine will kill grass wherever you let the pet willynilly pee anywhere. Glad I don't have that problem any more and I had an untrained pet. I want to know how to keep neighbors cat from pooping in our flower beds!! They need to have control of their pets since laws protect us from others dogs and cats in our yards but people with pets think we all LOVE cleaning up poop!