Hi, 

Unfortunately there isn't a ground cover that withstands heavy foot traffic or pet urine and as you can see even turf grass struggles with this. If you can aerate and overseed yearly in the early fall this will help reestablish and maintain the grass, as it is the best option for pet traffic and play. You will need to fence off the area probably for a couple of months to the get the grass growing and strong enough for pets again. Some people immediately hose off the turf after pets urinate on it to dilute the high nitrogen content that is burning the foliage. This is often not that practical though. 

Alternatively you could put in a path leading out of the house with thick mulch, woodchips, or pea gravel and install plant material to direct the dogs further out into the yard. Perhaps this will spread out the heavy traffic paths and urine concentration. If you install plants you will also probably need to fence them off for a while so they can establish and grow large enough that they don't break if getting trampled. Native ornamental grasses could be an option. 

Some people choose to install synthetic turf for the dogs. There are pros and cons to that as well but maintenance seems to be less than turf grass. Many dog parks will have this, mulch or pea gravel since the high compaction and concentration in pet waste is too much for plants to grow. 

You can review our Lawn Renovation and Overseeding page for further information. Also feel free to send in a photo(s) of the area that you are struggling with so we can see how to better advise you. 

Here is an article on dog urine and turf grass from Perdue University that may be helpful. 

Emily