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It's a common trope in the visual design of dragons, giant bats and other fictional flying beings to have wings that appear to be severely torn, cut, pierced, and otherwise worn down. It's also quite likely that the wing of a flying creature would be a prime target for its enemies in combat, so it might take on that appearance unwillingly.

My concern is that this would obviously affect the flying beast's ability to fly, but I'm not keen on the specifics of that. I can make a wild guess that this:undamaged wing would be solid enough to support flight, and that this:heavily damaged wing would let entirely too much air though to have sufficient lift remaining.

What I want to know is, at what point do holes in the wing become too much of a hindrance for the animal to achieve and sustain flight?

My interest is especially focused on combat with dragons. Would a single lucky sword-stroke, bite, or bullet be enough to ground the beast or would it take dozens of hits until half the surface area of the membrane is gone?

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Aerodynamics is hard stuff. This is more of a physics problem that heavily depends on the specifics of the creatures. Things like the mass and volume of the Dragon, the strength of the wing muscles and bones, the strength of the actual skin/membrane, properties of the air they are flying in will all determine the answer to this.

Dragons have a problem with reality, in that their wings almost never are large enough to provide adequate lift / thrust to counteract their immense weight. So figuring out how much damage they can take depends first upon figuring out how the hell they are flying in the first place.

If they rely on utterly massive wings to keep their hefty bodies aloft, then small holes won't bring them down. Adding more holes will gradually reduce their thrust until they fall.

Additionally, if their skin has a "realistic" material composition, enough holes will eventually cause larger portions to fail under stress and tear/flap away.

Arrow holes in the middle of the membrane sections would cause less damage than the tears shown next to the support "fingers" in your image. The attachment points to the "fingers" would be under a ton of stress when flying, so having it frayed like that would probably cause larger tears and failure of the wing.

The real answer for how much damage a dragon wing can take is: How ever much you need them to. Anyone accepting that Dragons exist in your world, and that they can fly, can also accept that after "enough" damage, they'll fall from the sky.

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    $\begingroup$ For some idea of just how tricky of a physics/biology problem this could be, check out this paper and see how complex bat flight even is in the first place! This one also has details on flying with torn wings specifically, and this one attempts to figure out how common such injuries actually are in the real world, which may give some idea about how much damage can be tolerated before a bat is unable to feed itself. $\endgroup$ – realityChemist Dec 19 '19 at 22:06
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    $\begingroup$ @realityChemist Your comment seems like the beginning of a really cool answer to this question. $\endgroup$ – Harabeck Dec 19 '19 at 22:16
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    $\begingroup$ @Harabeck I could give it a shot, but for the reason that abestrange pointed out I doubt I could come to a better conclusion: dragons are not really aerodynamically feasible as they're usually depicted, so there's got to be something else going on. Still, I'll look into expanding my comment as an answer and see if I can come up with anything useful. $\endgroup$ – realityChemist Dec 19 '19 at 22:23
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    $\begingroup$ OK, I don't think this is worth expanding into its own answer, but here are a few interesting points. First, at least some species of bats are able to continue flying with up to a 21% asymmetric reduction in wing area (although they may be somewhat impaired). Second, only about 3% of captured bats in one study showed any damage at all. Finally, the most common injuries were loss of membrane from an edge (wing/tail/finger) and the least common were holes. This could either be because hole injuries are bad enough to often be lethal, or because they're infrequent and heal well. $\endgroup$ – realityChemist Dec 19 '19 at 22:42

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