COMMUNITY - FORUMS - GENERAL DISCUSSION
The physics behind arms and armour

To start things of, lets clear up two terms I will use multiple times throughout the thread ... maybe.

Ultimate Tensile Strength measures how much force a material is able to withstand before breaking. With it we can calculate how much energy is required to cut or pierce a piece of armor.

Once the Yield Point is reached materials deform permanently, instead of elastically.

So ... lets get started.

The formula to see if something fractures (or pierces) the armor is pretty easy:

Thickness of the material in mm x UTS (or 3 x YS in the case of mild steel) in MPa=Joule/cm² you need to apply to fracture (or pierce) the armor.

For a 2 mm plate of mild steel this would be:

0.2 x 840 = 168 or 0.2 x (3 x 250) = 150

For other types of armor this is a bit more difficult to calculate and you will have to estimate (mail for example). Since the UTS is given for cm³ you'll need to figure out how many cm³ of material you will have to go through for each cm² of area of impact.

Finding the UTS and YP for the material you want can be a bit tricky, as there are many modern alloys, and degrees of hardening etc. so I will always mention the UTS or YS of the material I am talking about, and would ask you to do the same. If you think I got the wrong numbers please tell me.


The second important part is about how we apply/generate energy with weapons.

We can calculate with how much energy the weapon impacts by calculating it's kinetic energy. (Momentum will get it's own thread at some point in the future.)

Once again we will start with a simple example ... an arrow:

0.5 x the arrows weight in kg x (the arrows velocity in m/s)²=kinetic energy in Joules

In the case of an 100g heavy arrow usable in war, moving at around 60 m/s:

0.5 x 0.1 x (60)² = 180

Depending on the tip used, the arrow could have very well pierced the steel plate we used as an example before. (The bodkin points shown above, would most likely have pierced.)

... And that's already it. You now understand the basics you need to know, to understand how armour works, and how weapons work.


When you apply this knowledge, be aware of some things though:

  • The angle at which you hit the armor matters

  • For melee weapons the weight isn't the weight of the whole weapon

  • You don't necessarily have to pierce or break the armour to injure the wearer

  • Not all the energy of an impact gets transfered to the armour.

  • Some types of armour deform even before the yield point is reached (mail comes to mind)

  • When you want to calculate how much you deform an armour, keep in mind, that you deform more, than just the cross section of the impact

  • Projetiles lose velocity as they move, (and in turn KE)

There are definitely more things to keep in mind, but this should suffice for now.

I hope this cleared some stuff up.


Edit: for fracture toughness:

While fracture toughness might be a more accurate representation for the stress resistance I'd like to calculate all of this with, it is sadly a quite inconsistent value. Even the same material can differ by as much as one order of magnitude depending on many factors, for example method of processing. It is a lot more convenient to use the UTS to see how much energy is required to damage the armour, even though it might not yet be enough to actually injure the person wearing it. @Huntsmaster might elaborate on fracture toughness at some time in the future though, for those interested.

Back to the collection thread


The truth is born in argument

8/17/2017 11:41:19 PM #1

Nice.


\[T]/

8/18/2017 3:02:02 AM #2

I'll pile onto this one!

EDIT: I have rewritten this post and the following post to break some thoughts apart and make everything a bit more clear.

Posted By Luminios at

So ... lets get started.
The formula to see if something pierces the armor is pretty easy:
Thickness of the material in mm x UTS in MPa=Joule/cm² you need to apply to pierce the armor.

[snip]

The second important part is about how we apply/generate energy with weapons.
We can calculate with how much energy the weapon impacts by calculating it's kinetic energy. (Momentum will get it's own thread at some point in the future.)

Once again we will start with a simple example ... an arrow:
0.5 x the arrows weight in kg x (the arrows velocity in m/s)²=kinetic energy in Joules

[snip]

Depending on the tip used, the arrow could have very well pierced the steel plate we used as an example before. (The bodkin points shown above, would most likely have pierced.)

I agree with the calculation of the kinetic energy and the approximation of penetration depth of an arrow (or similar low speed projectile). I do have some concerns about the use of UTS to determine the ability of an arrow to pierce, which I will discuss in the following post.

However, since this thread is "The Physics of Arms and Armor", let’s step through the physics, shall we? (Warning: math ahead!)

Taking an Arrow to the Knee

(or, The Work-Energy Theorem, as applied to adventurers)


For penetration by a projectile such as an arrow to actually occur, the pressure applied by the projectile must be greater than or equal to the strength of the material. Following the analysis in the OP, we will use UTS to represent the strength (for now):

  • P = F/A ≥ UTS

where F is the applied force of the projectile, A is its area, UTS is the strength. The SI unit of pressure is the Pascal (N/mm2). Rearranging the pressure equation gives:

  • F ≥ A * UTS

As penetration depth increases, the projectile energy decreases by the work-energy theorem:

  • dE = - F * dL

where dE is the change in (kinetic) energy, F is the force required for deformation, and dL is the change in penetration depth. Substituting our equation for force into the energy equation gives

  • dE ≥ - A * UTS * dL

Integrate both sides (in the -L direction, indicating penetration) to obtain:

  • E ≥ A * UTS * L

Now, if we substitute in the kinetic energy equation (neglecting potential energy), we can calculate the minimum thickness of armor required to resist penetration by a projectile with known mass, velocity, and cross-sectional area:

  • ½ m * v^2 ≥ A * UTS * L

This is the physics that connects your analysis in the OP of an arrow’s energy and an armor’s toughness. Physically, it is the minimum amount of energy required to push a volume of armor equal to the arrow's penetration. We can take all the OP values for arrow mass and velocity, and plate thickness and strength to estimate whether the plate was penetrated. The missing parameter is the diameter of the arrow, so the quantity we will find is the maximum arrow area that could penetrate given the assumptions of energy and strength:

  • (m x v^2)/(2 x UTS x L) ≥ A
  • (100g x (60 m/s)²)/(2 x 840 MPax 2 mm) ≥ A = 107 mm^2

A blunt arrow of reasonable diameter (1/4”-3/8”) would probably not puncture a 2 mm steel plate. However, the effective diameter of those pyramidal bodkin point arrows is very small at the tip. This serves to focus their energy on a very small area, increasing the effective pressure and making them better at piercing armor. This can also be calculated, but the details get messy.

8/18/2017 3:13:43 AM #3

Ok, here’s the rewrite of my earlier comments on penetration, fracture toughness, and armor.

Posted By Luminios at 1:57 PM - Thu Aug 17 2017

To start things of, lets clear up two terms I will use multiple times throughout the thread ... maybe.

Ultimate Tensile Strength measures how much force a material is able to withstand before breaking. With it we can calculate how much energy is required to cut or pierce a piece of armor.

Once the Yield Point is reached materials deform permanently, instead of elastically.

You linked my old post in the other thread, but here it is again for reference. The key points I want to bring over from that thread are these:

Ultimate Tensile Strength


UTS is a measure of the applied force at start of fracture (which can be ductile or brittle). This quantity is measured in tension, by grabbing hold of the ends of a sample piece and pulling it apart. This is a different stress state than what you have in penetration by arrows, in which the primary stress state is compression (all materials are at least as strong in compression as in tension, and brittle materials are always stronger in compression). It’s an ok approximation to use this as a measure of durability as you have, but know that it is only an approximation because of how this number is tested.

Yield Strength


YS is a measure of the applied force at which the material begins to deform permanently, again in tension. It is always lower than the UTS; how much lower depends on the material. Generally speaking, the more ductile (deformable) the material, the larger the ratio UTS/YS. This ratio points towards an interesting and important phenomenon known as strain hardening, which explains how metals become harder to work as you work them. Anyone who has ever bent a paper clip back and forth, or tried to straighten a bent wire, has experienced the effects of strain hardening.

Failure

(Or why it’s not always good to be strong)


Imgur
Mechanisms of Plate Perforation – “Impact: The Theory and Physical Behavior of Colliding Solids”, Werner Goldsmith

For thin plate and low projectile velocities such as plate armor and arrowheads, panels B and C in the above figure represent the typical failure modes. For these two, plastic deformation of the metal dominates. Panel A shows the fracture “plug failure”, and D shows the fragmentation mode more often associated with brittle materials.

For B and C, both forms of ductile deformation failure, we can look at classic contact mechanics for insight into the damage mechanics. Per K.L. Johnson’s ”Contact mechanics”, the yield criterion for quasi-static penetration (velocity < 100 m/s) is:

  • P ≥ 1.6 Ys

where P is the impact pressure, and Ys is the lower of the tensile yield strengths of the projectile and the target. This accounts for the elastic deformation in the ring surrounding the impact site. For full penetration by plastic deformation as depicted in C, the experimental-confirmed yield criterion is:

  • P ≥ 3 Ys

One interesting implication here, for "low speed" penetration such as arrows, is that penetration is strongly dependent on deforming the metal (everything ties back to yield strength). An armor constructed of plates of bone or stone might actually be preferable to a metal armor, at least when facing steel tipped arrows. If a plate were struck with an arrow, it would break, reducing the arrow's energy and its higher hardness would deform the arrowhead. Back it with a thinner layer of leather or cloth to catch the fragments, and you basically have the principle of some modern body armor.


For both A and D, the principles of fracture toughness apply. This is, as Luminous has indicated in the OP, an area of materials research fraught with uncertainty. This is true of all metals, from steels to advanced aerospace titanium alloys. One of the latter shows great variation in facture toughness, even at the same yield strength:

Imgur
A classic work displays the variability of fracture toughness (vertical axis) at similar yield stress (horizontal axis). Reference in figure.

So what’s going on here? What’s a “microstructural difference”? Well, atoms in a metal are lined up in well-ordered rows. There are local areas of order called “grains”, where all the atoms are aligned in the same direction, this orientation is somewhat random, and adjacent grains are not aligned in the same directions (if they were, they would merge to form a larger grain). As it turns out, the average grain size is very important for both yield strength and fracture toughness. The qualitative effect is shown here:

Imgur

Grain size is affected primarily by three things: the amount of mechanical work in the material (simply: the time spent hammering), and the time and temperature of subsequent heat treatment. There is room here for the master smith to maximize both yield strength and fracture toughness in his armor. Chemical variation can also have a large effect, as shown here by the effect of small amounts of sulfur in a pressure vessel steel:
Imgur

This is why most traditional smiths used charcoal instead of coal. Un-coked coal can contain high levels of sulfur, which will inevitably burn off during the forging process and deposit in the worked metal. Again, there is room here for the master smith: either to do the hard work of charcoaling lumber for the forge, or discovering the technology needed to eliminate sulfur from mined coal.

8/18/2017 9:39:52 AM #4

Firstly I'd like to apologize as I'm completely confused about what's being talked about here but I'm thinking it's about how strong armor is and how well weapons can break armor right?

I'm also kinda confused on how to apply this knowledge in game.

I mean i don't know if gaming technology is advanced enough to determine if the angle of approach for an attack will penetrate/deform armor, then keep track of the durability of the armor for the location it was hit, and also keep track of the durability for the armor of thousands of other players and NPCs, but it might be in 2019 and we could always hope for the best.

I mean we are all expecting a level of realism with this game but this might be too much to comprehend for me right now.


UDL

8/18/2017 10:06:57 AM #5

Posted By Yoruninja808 at 05:39 AM - Fri Aug 18 2017

I mean i don't know if gaming technology is advanced enough to determine if the angle of approach for an attack will penetrate/deform armor, then keep track of the durability of the armor for the location it was hit, and also keep track of the durability for the armor of thousands of other players and NPCs, but it might be in 2019 and we could always hope for the best.

Well I don't know about keeping track of durability in particular locations, but calculating the angle of the hit is not too hard to do. World of Tanks does it and I'd wager other similar games have such a mechanic. I don't know how efficient it would be if the check had to be done for hundreds of players though, so that might be a valid point.

At any rate, the knowledge is useful and interesting even if the game doesn't literally use the same equations. The core ideas could inform the design even if the math is simplified.


8/18/2017 10:25:50 AM #6

The way I've seen the design of the game to this point, is that things work the way you'd expect them to work in the real world.

So the idea behind this post was to give you guys the knowledge necessary to estimate how different weapons will interact with different armour. It isn't meant to be super accurate, the bullet points at the end are just there to remind you that we are just getting an approximate image of what is going to happen, once the weapon impacts the armour.

I'll edit the OP some time later this day though, after talking to @Huntsmaster about fracture toughness. (Done!)

What I hope this achieves is, that with a little effort you can estimate whether the weapon you are using will be effective against your opponents armor, or not, based on how it would work in the real world.


The truth is born in argument

8/18/2017 11:56:10 AM #7

First off I have enjoyed this post a good amount... particularly the little Tid bit at the end regarding angle of impact. I know some armies were designed with more curvature than other armies with the intention of altering the angle of impact and lowering the chance of piercing/penetration. Well that's my tidbit - good posts.


8/18/2017 12:53:01 PM #8

This is fantastic!


Imgur

8/18/2017 10:31:15 PM #9

I rather like this post. It's full of science (math and physics) and also serves as a warning to groups of bandits who might have considered trying to rob an individual or a few individuals armed with bows. Makes me recall every scene of Legolas vs a horde of orcs.


8/18/2017 11:03:59 PM #10

Posted By MReal at 05:06 AM - Fri Aug 18 2017

Posted By Yoruninja808 at 05:39 AM - Fri Aug 18 2017

I mean i don't know if gaming technology is advanced enough to determine if the angle of approach for an attack will penetrate/deform armor, then keep track of the durability of the armor for the location it was hit, and also keep track of the durability for the armor of thousands of other players and NPCs, but it might be in 2019 and we could always hope for the best.

Well I don't know about keeping track of durability in particular locations, but calculating the angle of the hit is not too hard to do. World of Tanks does it and I'd wager other similar games have such a mechanic. I don't know how efficient it would be if the check had to be done for hundreds of players though, so that might be a valid point.

At any rate, the knowledge is useful and interesting even if the game doesn't literally use the same equations. The core ideas could inform the design even if the math is simplified.

I would say world of tanks does it poorly, but that is completely besides the point.

It technically would be possible to have SpatialOS track the location and "damage" to the armor done by an arrow in this case. If SpatialOS is already planned to make dents in armor from sword strikes, then there is the potential to do such an algorithm like those stated above.

The question then becomes is it worth it? Especially since that will likely increase the total overhead. Personally I could take or leave such a thing.


8/19/2017 12:50:38 AM #11

I love the work the two of you are doing but you make my brain cry at equations I never wanted to need to do again. I have seen this done with arrows, and more often bullets, on many sights, some good, some bad.

Now lets see some melee weapons, lets say an arming sword, a bar mace, an axe, a quarter staff, and a two handed sword (you can use zweinhander or even claymore). Here the Yield factor and deformation size is going to be more indicative of the damage done, but the math is beyond me.


8/19/2017 7:50:16 AM #12

Melee weapons will be the topic of sunday ... today it's going to be arrow weight and momentum.

Yesterday we had potential energy and bows, if you are interested ...


The truth is born in argument

8/21/2017 10:58:45 PM #13

I've re-written my posts on penetration and fracture toughness above. I'll admit to an error on my part, corrected after refreshing my memory with the K.L. Johnson "Contact Mechanics" referenced above: fracture toughness does not scale by 3x(yield strength), but penetration by quasi-static plastic deformation, such as that induced by arrows, does scale by 3x(yield strength). I apologize for the improper syntax!

8/22/2017 12:14:23 AM #14

did I just see the mention of softbody physics from swords on armor? Please tell me its true.


If you have items or assets you no longer have use for feel free to send them my way.

8/22/2017 8:14:33 AM #15

Quick release to drop armor weight would be pretty wicked. Even equipment being stuck on because of arrows.