depending on how long you let it sit at under 12V it may self destruct internally
just running them down that low does damage them even for a short time
they like being FULL!
if this is a car battery it is very bad to run them down
even deep-discharge batteries don;t like it
car batteries have thin plates for high current starting supply
but they can easily corrode internally when discharged
wle
A lead acid battery has sulfur crystals form on the plates as it discharges. Sulfur comes out of solution and that is why the specific gravity falls. The sulfur redissolves as the battery is recharged. That makes the specific gravity of the electrolyte go back up.
After several days sitting there in a less than 100% full charge the sulfur crystals harden; they will no longer readily redissolve when a charge current is applied. So the battery capacity is reduced.
Plus deeply discharging a lead acid battery shortens its useful life. Just like Li-ion calls there are a certain finite number of full cycles that a lead acid battery can provide. The deeper the discharge the fewer the cycles. Automotive batteries are not meant to be deep discharged. Golf cart batteries can be discharged more deeply; they have more lead in the plates. And batteries designed for off grid power use have even thicker lead plates, and can last through many more cycles.
Here’s a good FAQ on lead acid batteries, aimed at the off grid alternative power crowd.