My yard is full of large oak trees. IMO, don't fight it. After fighting for decades, trying to establish a shade garden I finally gave up with in the ground gardening under those trees. I would say that it is not possible to hurt the trees by plowing the roots, but it won't help anything either. Kind of like giving them a stimulative pruning!
I love gardening under my oak trees, but limit all of that activity to raised beds and containers that are not in contact with the soil. Place a clay pots with a single hole in the bottom, and the oak roots will grow right up through the hole.
All of my bedding areas which are under oak trees are lined with two layers of plastic. First I put down sheet plastic which the roots can't penetrate. Secondly, I put down heavy duty landscape fabric, which is tougher and more durable, and is UV stabilized. Next, to give a naturalized look, we sweep a nice covering of the oak leaves over the landscape fabric. If building raised beds, as was done in two areas, those are constructed before covering the area with leaves.
Since moving to this above ground strategy, gardening under the oak trees has been very rewarding. Two vegetable and fruit areas are under the east canopy where they get morning sun until about 1 p.m. We are able to grow strawberries, blue berries, raspberries, egg plants, cucumber, green beans, and more in those areas. Anything that needs to be shaded from our brutal afternoon summer sun does very well on the east side of those trees.
In two other areas, I have large container planted shade gardens. Those plants do wonderfully as well. For the years that we tried in the ground planting, the shade perennials would do well the first year after planting, and then would get smaller each year afterward, until about the third year, everything would have to be lifted and the bed would have to be renewed. It was a lots of work that gave very little satisfaction.
Here are a few photos from 'under the oak tree' beds with the above ground strategy.
This is one of two areas that are dedicated to edible plants.
This is one of two above ground shady container flower gardens:
This is the other shady container gardening area. This one gets afternoon sun.
