What's the angle of repose for dead zombie bodies?

Question

I'm writing a story where the superhuman heroes fight hundreds of thousands of zombies. The zombies are fast and agile and completely surround the heroes. Dead zombies pile up around them until the zombies are jumping down on top of them. The heroes realize they have to continuously climb the pile of dead zombies to maintain the high ground.

At the end of the battle (with the heroes victorious on the very top), how steep is the resulting pile?

The angle of repose for a specific material is the steepest slope which the material can be heaped without collapsing.

Here's a table from Al-Hashemi and Al-Amoudi with representative values for other materials:

Answer 1

1) A vaguely-argued upper bound.

This turns out to be a surprisingly common problem for providing answers here, and I have yet to find a good value.

In The logistics of corpse disposal I gave 45 degrees, arguing that "Most stuff is lower, I know nothing that's steeper, and steeper means fewer bodies, so 45 is likely our worst case.". I feel this is a decently defensible upper bound.

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2) A spherical-cow lower bound.

While, as others have said, the exact value may be hard to calculate, we can also at least establish a lower bound, by establishing a precise value for the case of a zombie spherical cow, which we shall presume to be frictionless except against the ground, and incompressible.

This works out at 23.8° (via https://eng.libretexts.org/Bookshelves/Materials_Science/TLP_Library_I/33%3A_Granular_Materials/33.4%3A_Angle_of_Repose).

That's reasonable as a lower bound, though as Nosajimiki points out, kinetic effects of people walking on them may mess with that.

However, as I understand the angle of repose, it's the angle of stable rest. Than angle of dune sand where you can walk down a sand dune and NOT cause an avalanche of more than the sand you displace. It's the angle of the bodies where you can scramble over them and not cause their sliding about to amplify and cascade. So I'm unsure whether this would actually have a significant effect.

Every assumption we made (spherical, incompressible, frictionless), reduces the angle of repose, so we can reasonably round up and still call it a fair minimum angle.

That still leaves us with a range between 25° and 45°. We could split the difference and call it 35°, perhaps, as a reasonable guess.

However... we don't need to guess.

We can try a PRACTICAL EXPERIMENT!

You didn't expect to find that in an answer, did you? And, OK, I'm lying. But I can at least simulate it.

Most computer games have decent friction, gravity, collision, and rag-doll physics, nowadays. So what's a game where I can create a gajillion corpses and let them just... pile up?

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3) UEBS II, 1,000 modern soldiers

Ultimate Epic Battle Simulator II (UEBS II) might be able to handle this. I note that it has a checkbox for "volumetric blood" with the note "can drown units", so this is clearly expected to handle LARGE armies.

It lets me set 1 million regular zombies to attack a thousand modern soldiers. I placed the zombies in four groups of 250,000, to surround the soldiers... OK, the soldiers died fast.

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4) UEBS II, 1,000 super-soldiers holding their ground

Let's make the soldiers be ubermensch, have ten times the firepower and ten times the health, as well as a 50% chance of blocking melee attacks.

After massacring 1,000,000 zombies, which took quite a while at about 300 zombies a second, I see that the angle of repose in this game is 26°. But I'm skeptical: in this game, this feels a deliberate effect, that they've engineered. They may have just picked a value that "looked right". On the other hand, it does look right.

Also, your suspicion was correct. I set the 1,000 supersoldiers to hold their position and fire, on the side of a gentle mound. Initially, it mattered: the zombies coming over the mound could only be fired at when they crested the rise, so they died much closer to the soldiers. On the other side, there was a significant distance that could be shot, so the zombies were kept well away.

But as the bodies built up, this stopped being an issue, as the bodies themselves made a mound, which advanced towards the soldiers as the zombies fell. Eventually, they were in the dip at the top of a pyramid, surrounded by mounds on all sides. The soldiers killed them all, but took heavy casualties, with little more than six hundred of their original number remaining.

(26 degree angle of repose: this looks about right. If you placed a body on there, you wouldn't expect it to slide to the bottom, but you could imagine it doing so if you gave it a bit of a shove.)

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5) UEBS II, 1,000 super-soldiers, moving tactically

Setting the soldiers to attack instead of hold their position, they were able to scale the sides of the body-berms and shoot the zombies as they approached up the pile. This resulted in a differently shaped pyramid, with the berms becoming a ring around their original position, expanding outwards rather than constricting inwards as it had when they'd held position. This meant their lines eventually became too thin to hold back the flood, so they had to retreat once the zombies broke through, and defended just one side of the ring, making the top of their pyramid lopsided. As that side rose up, they eventually recaptured the lower side of the ring.

Rather than losing a third of their number, they lost only two out of a thousand men. This approach was clearly better, tactically.

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6) TABS, 1000 halflings vs 1 dragon

Totally Accurate Battle Simulator (TABS) is on sale in Steam as I write, and is much cheaper than UEBS II anyway. It's for much smaller-scale battles, but with a fight on a slope, I was able to get some confidence in the the angle of repose from just 1,000 halflings, though that gave me a framerate around 3 fps.

The angle of repose in this game is just 11°! But watching the fall and slide into place, I am once again skeptical. I think it has them being VERY slippery. This would be the AoR of something like wet liver. Or perhaps the ground is slippery, and the slope they are on allows the ones at the bottom to slide sideways because of the sideways pressure from those on the ramp. Maybe testing on a ramp was a bad idea, but it was the only way I could find to make it work.

(11 degree angle of repose: this just looks too shallow. If you put a body at the top, it feels like you wouldn't be able to shove it so it slid to the bottom.)

Overall, then, my idea of simulating it fell short, but did highlight some of the variables. I didn't expect friction to be such a big deal, for example.

In the end, you'll likely have to pick a number. UEBS' angle feels about right, viscerally (ahem), and I do feel anything much steeper would look weird and unstable.

I heartily recommend using a simulator like these (UEBS II in particular), to get a sense of what your battle would be like. I'm not affiliated with either of the games, but did buy them for this question (Yes, I spent $29 after tax on trying and failing to answer this question. Shut uuuup!), and it was eye-opening in various ways.

At 300 dead zombies a second, that results in rivers of bodies flowing down the outside of their corpse-pyramid. Everyone gets spattered in blood. Of course, it's simplified: the soldiers aren't harmed by their own bullets and have perfect target-acquisition skills, etc. And blood does drain out of bodies, so the "volumetric blood" option may be an eye opener, though I suspect it's considerably less than realistic.

Well worth setting up a simulation of it all, anyway.

Answer 2

They will not pile up like you think

The best examples of what you are looking for will be from photographs of mass graves or pictures taken by over zealous hunters and fishermen (I don't recommend doing this research yourself). Based on this, I'd say that the angle of repose for a randomly stacked pile of bodies appears to be about 20-30 degrees. A greater angle can be achieved with careful stacking like you might find in certain holocaust pictures, but that does not apply here. Human bodies tend to splay out wide, but they also like to slide against each other, tumble over, etc. So, I'd expect a random pile of humanoid corpses to be closer to 20 degrees

... but this is not how a battlefield works. You have a whole army of zombies and heroes stomping around on these dead bodies, climbing them, kicking them out of the way, etc. This means that at any time the bodies stack more than 2 or 3 high, that the zombies trying to pass them will not just be walking over them like a solid hill top. They will be tripping over them, getting their feet stuck, falling when a body slides out from underneath them. Eventually the bodies may build up higher, but expect a very low hill gradient.

Also, dead bodies several deep can make terrain virtually impassable; so, unless your zombies are smart enough to clear their own dead, your heroes will slowly expand their kill radius as the zombie threat approaches more slowly further flattening the gradient of dead bodies.

This leads to a few very important conclusions:

1. Your heroes should NOT try to get the high ground. The higher you go, the less stable your footing will be. They want to keep the zombies tripping over their own dead while avoiding the bodies themselves.
2. The zombies will prevent the mound from getting very high. Instead of piling up, the zombies clambering over the dead will force the dead bodies backwards; so, instead of a giant mound, you'll just get an ever expanding field of bodies that is no more than a few bodies tall.
3. Unless the heroes are completely surrounded, they will back up as the bodies pile so that they can avoid having to stand on them.
4. This churning action of the bodies on the battle field means that some live zombies may get trampled into the dead ones. This will further destabilize any piles as they try to claw their way out and makes standing on a mound of "dead" zombies especially dangerous since a live one might in the chaos reach up through the dead ones to get you.

Also, High Ground is not an advantage against zombies

First of, the high ground is a time honored advantage in many military traditions; so, unless your heroes have a lot of experience fighting zombies specifically, they may still seek the high-ground, only to realize thier disadvantages when they get there.

In the case of killing zombies, the low ground provides your heroes better LoS (Light of Sight) to be able to kill a maximum number of zombies. In a normal modern battlefield, infantry try to spread out to minimize how many soldiers can be shot down in a single burst; so, you never really need a deep firing line, but against zombies, the enemies are much more densely packed, and require aimed head-shots instead of sustained fire to take out. This means that you should ideally have a firing line several shooters thick.

The problem with the high-ground for modern firearms is that even if the person in front of you is crouching, you may or may not be able to see past them to shoot depending on the hill gradient, but if you have the low ground, everyone in your block can see and fire up on an enemy as they come over a ridgeline. This is important when fighting zombies specifically because you want a concentration of force using aimed shots fired at close range to maximize your number of head shots and minimize your amount of wasted bullets.

Melee is also at a disadvantage on the high-ground against zombies because all of the things that give a human army and advantage on a hilltop do not apply to a horde of zombies. Some of these advantages may apply to a natural hill top, but not to a mound of zombies.

• The improved field of view that normally makes it easier to spot and respond to ambushes and flanking maneuvers don't matter if you already know you are surrounded by an endless horde of zombies in every direction.
• Many hill tops have some passable and some impassable areas allowing you to funnel the enemy into a chokepoint or guard your flanks. Zombies likely won't pile steep enough for these to form.
• Charging up a hill is tiring and slows you down, but zombies (in most canons) do not tire and they do not charge making this a moot point.
• Low velocity weapons like bows and javelins suffer a major loss in effectiveness when launched uphill, but zombies don't have any ranged weapons to mitigate.
• Hills tend to be uneven terrain which breaks up organized battle formations. Since zombies have no shield walls or phalanxes to maintain, this is almost a moot point. The unevenness of corpses will trip them up here too, but even the low points will be a rough terrain of corpses in this case.

However, there are disadvantages to the high-ground that do apply to zombies:

• It leaves your lower body much harder to defend. It's harder to parry or block attacks that are coming up at you than from on-level or above.
• It shortens your reach compared to your opponent.
• Like with the gun problem, you cannot form a phalanx over a ridge line. Even though a spear wall may not kill zombies out right, it would be helpful for stopping the advance of the zombies so that your front line has more liberty to deliver the killing blows

Answer 3

You can consider sandbags as an approximation. Those are heavy soft bodies with cloth skins, similar to your clothed zombies. Now proper stacking lets you build a mound of sandbags nearly vertical, but there's guidelines for making stable mounds, which can withstand external forces pretty well. For example the reference for dikes from the North Dakota State University.

This article lists two different recommendations for stacking, one ending up with an angle of 45 degrees and another with 33 degrees. Since we're dealing with a dike design that needs to withstand an enormous force, I think the 45 degrees angle would be stable enough on its own. That's not necessarily the natural angle of repose for sandbags (or zombies) stacked randomly but I believe it wouldn't be too far off.

For another reference, consider finding imagery of stacked human corpses. There's grim WWII imagery that may suit your needs and let you estimate an angle of repose. I'm not going to look up those pictures myself because I value my night rest.

Answer 4

Unless the arms and legs fall of when the zombies die, I would imagine they could be stacked vertically as the arms and legs are lodged under other bodies preventing them from tumbling down.

Of course the given scenario where the hero has the high ground, the zombies would die before they reach the top and tumble back down the hill, eventually creating an ever expanding circle, where the hero keeps moving forward to be at the edge, and not let the zombies on top of the hill.

Answer 5

Dead zombies offer very bad foothold. If your heroes have the room, they would be better off, retreating slowly, so the zombies are slowed by the corpses, but the heroes have a clear floor to maneuver.

Of course, this is not possible, if they are surrounded. But they would still be better off, moving around and avoiding a pileup. You will always want to have stable footing, when fighting hordes of zombies.

If the zombies are forming a horde so dense, that maneuvering isn't possible, your heroes have a real problem, superhuman or not. because that also means, that the zombies can't just fall backwards after being killed. They would fall against other zombies. The following zombies would push the killed zombies against the heroes, crushing them without the heroes being able to kill the pushing zombies through the wall of pusehd dead ones.

But let's assume, they are already on a small hill of dead zombies (or just on lightly higher ground, like a literal hill). The newly killed zombies would fall down in front of the heroes, so the heroes would quickly be on lower ground. They would either have to constantly pull the killed zombies towards themselves, to stay on the high ground. They either do not have the time for that, since the next one is alread attacking, or they have enough room, the move themselves.

In short, in a relatively realistic scenario, the zombies wouldn't pile up at all (well, maybe 2-3 falling on each other, but not really).

If you want the pile of dead zombies anyway, you are relatively free to chose an angle yourself. It's fiction after all. Just look at some pictures of piles of sand sacks and imagine a zombie trying to climb them (the sand sacks, not the pictures). If you can imagine it, you have a good angle for the slope.

Answer 6

Obligatory 'Not an Engineer' - however, I think the angle could be very steep. Steeper than 45 degrees.

I have no means nor inclination to do the maths - but here's my reasons:

1: Human Skin has a high friction, Even with Headshots, there's unlikely to be enough blood to severely impede this, plus as Blood dries, it gets very sticky.

2: Mass of each body - this means that unlike sand which has high friction but low mass, you have high friction and high mass which leads to a lot of traction force between each zombie corpse.

3: Limbs. Arms and Legs can interlock in fun ways, although not as effective as if someone consciously stacked the bodies, the limbs can substantially increase the surface area and act like a mesh of sorts to further increase the angle of repose.

Finally, unlike most of the items where we have data for the angle of Repose... Humans aren't grains or particles.