Mulch alone is not an effective smother for a vine grass. The first layer needs to be solid. Strong, corrugated cardboard is what I use. Newspaper will work but it needs to be a lot of layers, at least 10 pages. Any holes or seams need to be covered, generously overlapping, at least 6". A foot is better. Then cover with organic matter.
Bark is not a viable option because it is not solid or heavy enough, there are too many gaps. But actual mulch or other, more solid organic matter will work. It should be at least 4" to block all of the light and be heavy enough to keep the cardboard in contact with the ground. As long as the smother effectively prevents the grass from putting a blade in the sun, it will die. It needs to be heavy enough so the grass can't push it up enough to creep to the edge.
It's much more tricky when the patch one is smothering is surrounded on all sides by grass. In that application, it's necessary to go around the edge with a shovel to sever all ties connecting the un-smothered grass from that under the smother. Otherwise, the grass under the smother will stay alive because it's connected to other parts of itself that are growing just fine. The next "rule" is to leave it alone for a long time to make sure the grass is dead. I usually wait at least 6 months.
If covered as described, plastic could work, but will also wreck the microbiological activity by starving the area of moisture and oxygen. And then, when you are sure the grass is dead, there is additional work in removing the plastic. The cardboard or paper method requires no additional work, provides improvement to the soil, just dig in and use the spot once you are sure the grass is dead.
I don't see any soil in the pictured area, just dirt. Soil is alive.
