Hi Lydia,

Isn't that just the problem we all have ! There is no great solution I'm afraid, dogs plus shade are very tough on the lawn. I don't even try with any grass personally, it never holds out . I have had some luck with plants like wild ginger, nepeta, upright sedum (it will grow in semi shade) chives. 

The trick is to put down some hard scape in, I'm talking about using some type of pavers, stones, bricks, wood pieces etc. Placing them so they are not touching ,just in a Hop Scotch sort of way. This breaks the dogs claws from ripping everything out while allowing the the roots of plants to "catch" and get stronger. I know it may not be the most attractive ( see it as an aesthetic challenge), it is very effective.

Here is a link that has lots of great suggestions. You may have to just try some and see which ones work.

https://conservancy.umn.edu/handle/11299/51549

Native sedges can grow quite well in the shade ;

https://www.prairienursery.com/plants-seeds/native-plants/native-sedges.html?light=318&filterApplied=1

Perennials tend to handle shade and foot traffic better than grass. 

https://extension.umn.edu/yard-and-garden-news/ground-covers-gardens

Keep trying,

Sally Granath

St. Louis County MG