There is not much that wll stand up to two big, robust and playful dogs like that! Even turf grasses, which are generally tolerant of more foot traffic than any other kind of groundcover, would be toast in no time, even if in full sun and with no trees around.

Local dog parks use wood/playground chips to cover pathways and open areas. If an adequately thick layer, it keeps mud and dust to a minimum, easy to clean up and soft on paws. And lots of times, you can get wood chips for free from arborists or utility companies.

I'd design some dedicated 'dog areas' that are carpeted in the chips and a few other, maybe perimeter areas that have groundcovers or some durable plantings that will stand up to the occasional romp. You can have a groomed landscape OR a dog playground but it is not an easy process to combine the two unless the dogs are very well trained.

As someone who has both an extensive, shady garden and who has had a number of dogs over the years - and frequently dog-sits other dogs, I believe the two are not incompatible if the dogs' needs are addressed when planning and maintaining the garden. Of course, I'm in a considerably different planting zone than you so that would make a difference to the specifics of what would work. Looking at your pictures, several thoughts come to mind:

- Very nice dog-house :-)

- There appears to be more grass surviving in the central area (probably gets the most sun - in mid-day when the sun is high overhead. If you're going to grow grass (or lay sod) put it in that area.

- There appears to be a 'path' at the fence-line - dogs NEED to run the fence-line to patrol their territory. If you don't give them paths there, they will make them! It looks like yours have made them....

For me, the starting point of making a dog-compatible garden is looking at their traffic patterns and making sure their needs are considered - patrolling territory; an elevated observation point so they can survey their territory (here that is the back porch....); play space (which can be rough-and-tumble play or chasing play - lots of paths are good for that (IME dogs tend to take 'the path of least resistance' so having a network of paths gives them the ability to run and chase each other without running through planted areas). Once you figure out in detail through close observation how they are currently using the space - i.e. what their current detailed traffic patterns are - plan a bed and grass layout that accommodates that. As gg48 suggests, using some sort of wood material (mulch or chips) in the vigorous play space makes sense, and paths can be mulched with the same material for continuity of appearance - and feel for the dogs' feet. Select tough plants (perennials, shrubs, trees) because the dogs will still occasionally go where they aren't supposed to! Water any pee spots on grass as soon as possible after the pee incident to dilute the dog-applied high-nitrogen fertilizer that burns the grass. Over-seed grass in late summer or fall to help it stay strong but re-sodding may eventually be needed - our shady and dog-trafficked lawn was just re-sodded in early June - but it had lasted fairly well with just periodic over-seeding for 15 years.

So, IMO/IME, it is possible to have both a garden and dogs. It takes some planning and effort - but it's worth it for both the people and dogs!

Some pictures:

Dogs in surveillance mode:

Racing collies on path (in early May before most plants - including the grass! - have filled out....):

Good luck - it can be done (at least in this zone!) with work and planning.