About the only thing that will withstand the onslaught of a dog in winter is astroturf - but that doesn't root into the soil, and so won't hold it in place, which is necessary on a steep incline. Do you have to walk over the grass to get somewhere, or is it just for looking at? If you don't have to walk over it, is it possible to plant ivy and let it grow down the bank, or is that not a suitable alternative? Without a photo, I don't know what the layout of your garden is.
The best solution is to prevent the dog accessing the grass in winter, or when its very wet.

... large size gravel ?

Wouldn't it just avalache down the steep slope Terratoonie? The only way I can see that working is to use larger bits of rock and build a drystone wall against the slope. And if the dog has a go at climbing up or down the slope the rocks will be dislodged. :-(

I suppose you could use a hard wearing groundcover, or even strawberries (we plan to do this on the driveway strip) but you'd have to keep the dog off while it was all taking root...

What about periwinkle (vinca) ?-Evergreen-grows quickly -prefers shady areas ,but in my experience grows anywhere! -and,is very hard to destroy !Though, once established, may be hard to find dog poo ! +lots of varieties to choose from .

Sorry... I was a bit tired when I put the gravel suggestion...

you could put gravel at the base of the slope to help with drainage, and possibly "terrace" the other areas, and add pavings or ground-cover plants ...

I like the terracing idea - there's someone on GoY who's done that and put up lots of pictures - will go & look for them.
Here I am, back again! It's Titchy who has a very steep back garden. Have a look at the photos -
http://www.growsonyou.com/titchy

Hi Beattie ...
That's a good link to Titchy's sloping garden..
well done ...thanks... :o)

Gerry555 ... I've added some of Titchy's photos to GoYpedia Sloping Gardens... so if you click on those words above [underneath "featured on"] that should give you more ideas for your sloping area.. I hope this helps...

mine is sloping and shaded so its mostly moss but its nice and green!! would you like some of my moss?

There are lots of small garden ideas here http://gardnersgardens.co.uk/townhouse-gardens/ they also seen to use a lot of artificial grass on their gardens. Surely if it is securely fixed it would be ideal even for a sloping site.