I put down 4mil plastic that I get shipped for free from Home Depot, I get 10X100ft rolls, they have 20X100ft rolls that would be better on your 40X40 plot. It is hard to keep down and will come up like a sail when you first put it down if it isn't anchored or weighted down well. I have collected a lot of used metal T posts over the years and use them to hold the edge of the plastic down along with huge staples I make out of grape arbor wire. Even with all of that I scatter about a dozen water filled gallon plastic jugs around to make sure the wind doesn't pick up the plastic. I burn holes for things like beans and okra with a propane torch, I cut about a foot diameter hole for the tomatoes and squash with a box cutter. I put newspaper on the ground around the tomato plants to act as mulch to keep the weeds from coming up. Once your plants get established the plastic stays put.

After I get the plastic down I go over it with a turning fork and punch thousands of holes in it randomly to let the rain flow through. I usually get two years out of the plastic, sometimes three if a shady plant it planted in it
Lately I have been mixing landscape fabric with the plastic because of the scorching summers we have had lately, the plastic may get the ground so hot that the plant seeds of some varieties have trouble germinating. Tomatoes, squash and okra don't mind the heat, beans don't like it. I have found I get a bigger bean crop if I put a strip of landscape fabric under my trellises to plant my beans through. I grow all of my squash, tomatoes and cucumbers from plants instead of seeds. I start the plants in March. The plastic in the picture is 2 or 3 years old, I pull it up before planting my winter garden, spread the dirt side up to let it dry and fold it up for future use. I try to use it as long as it will stay together. If I decide to plant something different through it that doesn't match the cut holes, I fill the holes with squares of newspaper once it is on the ground and cut the new holes that I need for the different crop.

Delicate plants like spinach have to have a bed of landscape fabric, the seeds won't germinate in hot ground. There is cheap landscape fabric that I use for beans and lasts one year, there is a commercial grade fabric that never rots that I got from my sister-in-law that I use year after year for my spinach. Spinach can't compete with the rampant weeds that come up in my garden from years of manure application.
Spinach, the little fence is to keep the rabbits out, they can eat an entire spinach crop in a couple of nights.