Without sun there is not enough energy for a plant to produce good growth and resist the effect of trampling paws. St. Augustine is the most shade tolerant lawn grass but it will be less dense and in fact may not be able to grow in the amount of light in that area. Stomping paws will wear it out also. The most shade tolerant grass like plant would be monkeygrass (Mondo) and there is a dwarf form too. This doesn't spread much so it can be expensive to solid plant an area if it is very large. You might consider using shredded bark mulch in that area. Ferns are another possibility.
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