We are not able to get a good grass coverage under the tree canopy...

You got this!! Check it out https://www.gardeningknowhow.com/ornamental/trees/tgen/tips-for-growing-grass-under-a-tree.htm

Shade caused a thinning in my St Augustine grass. The company that fertilizes my lawn puts potash out to take care of the problem. There are a number of suggestions on the web. Potash does work for us.

A few problems. The shade of the trees. The tree roots rob the water and nutrients from the grass. Some trees drop fine sap that can cause nothing to grow.

Most trees, including oaks do not like having things planted directly underneath them. Some trees will actually sterilize the soil inside their drip line so that nutrients and water won't be stolen by other plants. The drip line is the widest part of the canopy diameter, and then visualize straight vertical lines down to the dirt, which will be all of the dirt directly under the tree's widest part. Oaks especially do not like having 'wet feet', which is when a high-water user like lawn or perennials get frequently watered in a fairly shallow manner. Mature trees like to root deeply for water, or get an occasional deep watering by those spikes with holes that one can press into the ground and attach to a slow hose. A good shade tree could have some light, bright chairs underneath it and some mulch or chips and stepping stones...?

i did the wood chip mulch....the next spring there was grass appearing.

I live in a mobile home park near Sacramento CA & we have many oak trees (which are "protected"), meaning they are NOT to b cut down. Grass was always planted around them. Oaks can't have ANYTHING that needs constant watering planted under them. You are killing your tree. The park had to put gravel in an area as large as the shade span & the tress rebounded. Nature will take care of the tree. If u need color, perhaps potted plantings. Nature is nature & will NOT capitulate to MAN. Enjoy your oaks