Lawn Alternative for High Traffic and Varied Exposure

I live in Plainfield, CT (zone 6) USA. I have 1.7 acres with a lot of trees (deciduous and evergreen). I would like a low growing groundcover. Not that I'm lazy or anything, it's just that I've already destroyed two lawnmowers. I'm not rich and they don't make disposable lawnmowers yet. I also have 3 children and a large dog. A ground cover that will deal with dog urine, foot traffic, a variety of light conditions, and if at all possible is evergreen. I've thought about maybe an english moss or something in the moss family. Any suggestions?

I don't think that any moss would survive the first ball game your children play on it. And it would fly off after the first "after-relieving himself" scratch your dog did on it! Moss is almost not rooted – it just sits on top of the substrate.
There is no magic answer for you. The biggest problem is the trees the varied shade on the area. I think your best bet would be creeping thyme (if artificial turf is out of the question!). Wild thyme variety can be started from seeds and thus is the most economical of the thymes to grow as a lawn alternative. But does best in full sun and light shade and under the biggest trees it probably won't do well.
Check our website for other suggestions for ground covers especially for shady areas where the thyme is not going to thrive. Go to the "Q&A" section and type in "ground cover" in the search tool box.