It's hard to get grass to grow with a lot of shade. I've seen St Augustine grow ok under oak trees. You need to keep the leaves raked up and put some extra fertilizer for the grass. Also, water the grass if it gets dry since the tree roots will take up a lot of moisture.

There are quite a few ground cover plants that seem to not mind the lack of water and nutrients in the soil around the base of some trees ivies, etc.

St. Augustine is pretty-much the most shade tolerant of the turf grasses. Growing grass under oak trees can be difficult for several reasons.

The heavy/thick canopy of the oaks creates a deep-shade underneath them. You can selectively trim some limbs from the canopy, without destroying the shape and beauty of the tree. This will allow more light through the canopy.

My friend grew a beautiful vegetable garden every year, under two very large beautiful Live Oaks. The vegetables got only about 3 hours of direct sunlight a day. The rest of the light was gained by thinning the canopy.

Oaks exude "tannins" that cause major problems for some plants. This is what was used for many years to "tan" animal hides with. Plants like Asian Jasmine seem to tolerate tannins just fine.

As was mentioned above, large trees of any type will use a lot of the nutrients and water available underneath themselves. You will have to provide extra feeding and watering to keep a nice lawn under oaks or any other trees.