Got a Real-World Weapon or Armor Question? Mk. IX

[fusilier]

Kinetic energy is useful to measure what it takes to move something or as a go to step for heat, friction and similar. There's a reason there's an entire field in mechanics that deals with momentum, impacts, impulses and such. It's really quite complex.
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Thank you very much for your post.

We somehow get into this conversation almost any time somebody mentions armor penetration, and all of this has to be rehashed again and again.

Kinetic energy is just too easy to calculate, and too easy for people to fall back upon. As a result many people think it's all that matters.

The physics behind how a human body moves, and is injured, is amazingly complex. Like how the bones can easily support our entire weight and then some from above but if the same force is exerted from the side, while doing this they, break. But if they don't have to support our weight they can usually take it.

I think this, the first part, touches upon Partysan's original question (well follow up question):

Or does a thrusting motion aid the penetration as opposed to a swinging motion (which generally has more power behind it)?

I think there may be something about a typical sword thrust that is very "rigid" (?). The body and sword are aligned to the same line, and if the thrust is square, it probably helps keep the point on a very narrow target. Not sure about it though.

[fusilier]

We are talking about the use of actual weapons though, and the vast majority of swung weapons do meaningfully extend the arm.

Yes: in significantly varying amounts. I could swing an eight-foot halberd at your head, or a 1.5 foot long bronze age short sword! So starting from the question of "is a human arm swing more or less powerful than a human arm thrust" is a totally valid place to start a line of inquiry. Otherwise, we can't speak in generalities we have to speak about each weapon specifically in it's own terms. Only if we set a basis can we then proceed to how weapon design changes the outcome.

Types of thrusts and types of swings should also be considered, as well as how weapon design effects what kinds of swings and thrusts can be used.

As mentioned earlier, there is a claim that a dagger swung in an overhand fashion generates more energy than one thrust underhand. [Aux-ash, I'm guessing that energy is safe to use here, as the weapon is the same and the attack is with the "point"? The difference would then become how and what part of the body/armor is hit?] So, swinging, may be an inherently better way of delivering a more forceful blow. If this is correct, we do not need to establish some minimum amount of extra length where a swing would be as powerful as a thrust. Longer swung weapons should increase the amount of force. So something like a pick, should do even more damage. But! is there something about the rigid stance of something like a sword thrust, that gives an unexpected advantage -- perhaps something not easily captured by simple force or energy calculations? Does the rigidity of the stance (arm, sword, and body all aligned) prevent or reduce any potential "rebound" that might reduce actual penetration?

[gkathellar]

I think there may be something about a typical sword thrust that is very "rigid" (?). The body and sword are aligned to the same line, and if the thrust is square, it probably helps keep the point on a very narrow target. Not sure about it though.

This is part of what I've been trying to say. Thank you for putting it into clearer terms, although "rigidity" isn't necessarily the word I would use. Rather, because the muscles and bones can "push" a thrust forward, they can perform more work than they would with a swing. It's less a question of rigidity and more one of physiological reinforcement.

Does the rigidity of the stance (arm, sword, and body all aligned) prevent or reduce any potential "rebound" that might reduce actual penetration?

Somewhat. Because the body is physically behind a thrust, it can absorb the actual impact better. With any kind of large swing, the arms and shoulders basically have to do this by themselves. (Smaller, more contained chops and slashes executed with a square body can capture some of the best characteristics of a thrust in terms of mechanics, but they lose out on momentum.)

We are talking about the use of actual weapons though, and the vast majority of swung weapons do meaningfully extend the arm.

Part of the problem with this line of argument is the ambiguity of what you mean by "extend the arm." If you mean "the elbow tends to straighten," then yes, obviously. But if you're talking about the momentum of a swing hyperextending the shoulder, then what you're actually talking about is losing control of your strike. Too much hyperextension can disconnect the bones and muscles from the swing, which results in an ultimately weaker blow.

[Knaight]

Part of the problem with this line of argument is the ambiguity of what you mean by "extend the arm." If you mean "the elbow tends to straighten," then yes, obviously. But if you're talking about the momentum of a swing hyperextending the shoulder, then what you're actually talking about is losing control of your strike. Too much hyperextension can disconnect the bones and muscles from the swing, which results in an ultimately weaker blow.

I mean when comparing something to an unarmed swing, you have to take into account that there is something in the hand effectively making the arm longer. How much longer varies - it might be a mere sword or axe, and on the shorter range, it might be some variety of swung polearm, but the arm and the weapon are longer. Its effectively artificial arm extension.

[gkathellar]

So all you're saying is that big things are heavier, which gives them more momentum, which makes them hit harder.

This is clearly true, but it doesn't really address the question Partysan originally asked — on whether or not the physiological action of swinging a weapon as opposed to thrusting it conveyed any advantages with regards to armor penetration.

[Spiryt]

Somewhat. Because the body is physically behind a thrust, it can absorb the actual impact better. With any kind of large swing, the arms and shoulders basically have to do this by themselves. (Smaller, more contained chops and slashes executed with a square body can capture some of the best characteristics of a thrust in terms of mechanics, but they lose out on momentum.)

Pretty much this.

Absorbing impact can often allow for more 'straightforward" heavy strikes - while smashing stuff with an axe often ends with wrists in pain, because all in all energy won't ever perform work only on target....

[Knaight]

So all you're saying is that big things are heavier, which gives them more momentum, which makes them hit harder.

This is clearly true, but it doesn't really address the question Partysan originally asked — on whether or not the physiological action of swinging a weapon as opposed to thrusting it conveyed any advantages with regards to armor penetration.

Its not a matter of mass, at least not entirely. Basically, over a period of time the object moves a certain distance, and if it moves further over that time it has more energy. Getting hit by faster objects tends to hurt more, all other things equal. Now, with a swung object, a wider arc comes from more length, which means more distance moved, which means a faster object that contacts, which isn't the case with thrusting motions, which aren't affected by the length of what is being thrusted with nearly as significantly - if anything, longer objects are likely to be slower. Hence, even if thrusting has more energy unarmed, increasing the swung arm length by, say, 50% translates to a 125% gain in energy, which correlates to force to some extent (though there is the huge matter of pressure, which is a different matter entirely). As such, one has to look at the whole "weapons as arm lengtheners" aspect.

[Spiryt]

Its not a matter of mass, at least not entirely. Basically, over a period of time the object moves a certain distance, and if it moves further over that time it has more energy. Getting hit by faster objects tends to hurt more, all other things equal. Now, with a swung object, a wider arc comes from more length, which means more distance moved, which means a faster object that contacts, which isn't the case with thrusting motions, which aren't affected by the length of what is being thrusted with nearly as significantly - if anything, longer objects are likely to be slower. Hence, even if thrusting has more energy unarmed, increasing the swung arm length by, say, 50% translates to a 125% gain in energy, which correlates to force to some extent (though there is the huge matter of pressure, which is a different matter entirely). As such, one has to look at the whole "weapons as arm lengtheners" aspect.

All energy has to come from somewhere however, and in result with large lever, energy of body momentum still goes into the blow, but not directly - instead it speeds up the business end of the weapon, doesn't it?

If it actually increases the total amount of energy received by target, depends on very many things, but longer weapon arm won't just mean harder strike necessarily.

[Aux-Ash]

[Aux-ash, I'm guessing that energy is safe to use here, as the weapon is the same and the attack is with the "point"? The difference would then become how and what part of the body/armor is hit?

Euh... good question. We're talking a overhand stab versus a thrusting stab, right?

Spontaneously I'd say that in a thrust you're going for a quick impulse, delivering a lot of force over very little time and thus doing very little Work (energy). So high force, low energy.
Whereas in a overhand attack (or a cut/slice) you're after a longer point of connection so that you can exert greater Force over time. You want as long a connection as possible. Moderate force, high energy.

The former would then primarily be about destabilising, and perhaps penetrating through merit of being too quick for the force to spread evenly. Whereas the latter is all about pushing through.

Mind that's this is a guess. I could be very wrong about this.

Also. I use a vastly different set of muscles in a overhand strike than in a thrust. This will matter for a lot. Perhaps even more than enything else.

One thing worth keeping in mind though, is that while a overhand grip might "deliver" more energy, the body is also much better at recieving work from above. We're made to be able to add mass (by lifting stuff) and both bones (fun fact: bones are springs on a molecular level) and tissues can handle that. And if all else fails, I can minimise the work you can use to injure me by allowing my knees to bend.


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As for the other discussion. Like some of you have touched upon, the longer the lever the more the Force. The more the Force the greater acceleration. The more acceleration the higher speed.
F = m*a. F is force, m is mass and a is acceleration.

And a weapon is just that; A lever.

But this isn't the end all be all of the mechanics behind it all. Be mindful of that.

[gkathellar]

Also, are we talking about thrusts/chops from a stationary position, or incorporating a step (or even just a shifting of the weight from back foot to front foot)? The two would look very different, and probably generate force very differently.

[razark]

I have a question: Do thrusting motions (as opposed to swinging motions) have an inherent advantage at piercing armor, or is the prevalence of thrusts to be used against amor solely a consequence of weapon design (piercing points generally sitting at the long end of weapons)?

It depends on how the weapon is designed. For each weapon, a swing or a thrust is going to be more effective.

Take the spear. It's great for thrusting. Swinging it is not going to get through armor (in general). The pick, however, would need to be swung to develop enough force. Thrusting with a pick would be rather pointless.

So yes, weapon design is much more important than motion. Once you settle on a particular design, the question of motion becomes meaningful.

[The Reverend]

If I remember my medieval weaponry class 225 this thrust versus slash was why the Egyptians designed the kopesh was designed how it was designed, as a compromise weapon system. Effective stabbing and a nice large surface area on a curved cutting plane to maximize the physics of the slice. This leads to a discussion between the main stay swords of two nible warrior groups.

The longsword of the European knight and the katana. They both were superb designs trying to overcome opponent defense and maximize their principle attack:thrust and slash. The longsword was a two handed weapon wherein the principle attack was a stab/thrust trying to pierce armor or its chinks or in wide swing to connect the last 3inches on a vital spot to provide a crushing blow. The katana of course relied on cutting motions that ran the entire edge of the blade with a stab a definite reserve possibility.

But I feel the main difference was not in the armor that was trying to overcome but in how the weapons were held, how they were drawn, and how the edge was built. The longsword was handled more like a quarter staff than a sword: swung from the blade end, used to choke, pommel strikes, trips and myriad of other tactics. Some long swords even hand an extra length of handle partway up the blade, while katana was never handled by the blade only by the handle and tactics of pommel striking and varieties in blow types were less common. The drawing of the weapon was a culturally and tactically important factor: knights often Drew blade before the battle had really started as positioning counted for more than the rocket launcher tag game that was katana duels. The draw was the first strike in a perfect katana duel and also the killing blow, hence the long dramatic stare off. Finally the cutting edge makes different tactics more important than others. The katana has an ideal cutting edge and is mastercrafted weapon with an edge like a combination \/ or() making the blade part the material while European blade edges are formed within an inward curve, a feathered point )( that causes a gathering of material on the blade. This means that the cutting element of the katana is even more emphasized.

[Spiryt]

Your rant is being troubled by some 'details', like the fact that there were literally hundreds of different longsword design combinations, and the same was true even with katanas, although they were relatively homogeneous trough the centuries.....

In second half of 14th century, for quick example, you could encounter sword like this one:

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Reconstruction of roughly similar sword

And sword like that one:


Linky


Completely different blade profiles, cross sections, balance, harmonics, in results - handling and performances. Both obviously pure longswords.

last 3inches on a vital spot to provide a crushing blow.

Last 3 inches would be pretty much never very good for "crushing blow" because that part of the sword would be pretty universally too fine - too narrow and/or too thin and the very end.

[gkathellar]

Some long swords even hand an extra length of handle partway up the blade, while katana was never handled by the blade only by the handle and tactics of pommel striking and varieties in blow types were less common.

Emphasis mine. This is just manifestly false. While inferior gauntlet technology ensured the katana could not be half-sworded, placing one hand against the back of the blade about four-eight inches from the hilt in order to reinforce a block is a documented technique still practiced in modern koryu schools.

The drawing of the weapon was a culturally and tactically important factor: knights often Drew blade before the battle had really started as positioning counted for more than the rocket launcher tag game that was katana duels. The draw was the first strike in a perfect katana duel and also the killing blow, hence the long dramatic stare off.

This is also false. The battojutsu duel, insofar as it ever existed and was not simply an invention of cinema, was a creation of the Edo period, when samurai were almost universally expected to be functional as both courtiers and warriors. If you examine battojutsu, a great deal of traditional technique is executed from a sitting position, implying either a surprise attack during a polite meeting or a quick response to such an attack. In fact, when going into war during the Sengoku period, the sword was often worn differently — if it was worn at all, since it would have been stupid and ridiculous to go into battle with a sheathed sword.

Finally the cutting edge makes different tactics more important than others. The katana has an ideal cutting edge and is mastercrafted weapon with an edge like a combination \/ or() making the blade part the material while European blade edges are formed within an inward curve, a feathered point )( that causes a gathering of material on the blade. This means that the cutting element of the katana is even more emphasized.

Cutting, yes, but not chopping. The katana's razor edge was poorly suited to hitting armored targets, which is why traditional koryu goes after openings in the armor of its day. Moreover, the blade would dull quickly specifically because it was so sharp (a feature shared among many east-Asian swords, and the reason why many have a designated blunt or blunt-ish area within the first 4-8 inches of the guard meant exclusively for blocking).

Of course, all of this applies mostly to later Japanese swords, which were (theoretically) exclusively the province of nobility and were actually regulated in size, weight and shape to a limited degree. Earlier battlefield swords had far more variation (Sengoku-period swords are where you mostly encounter very large examples), and were by no means all "mastercrafted" weapons.

[The Reverend]

Excuse mw last 6 inches typing on phone and didn't proof read. A lot variation exists in all weapon design, heck I once saw a broadsword from the 15th century with an internal ball bearing system to make swings more powerful. I have also heard though never researched it that the reason we have a lot long swords hanging around still is that the practical workaday killing implement was something more like a falchion or scimitar and since they were popular they were used till the broke and that's why fewer medieval falchions exist.

Always favored spear and shortsword myself though.

[The Reverend]

Thank god facts! I was hoping to get you guys off that physics debate minus the math necessary to describe it.

[Spiryt]

I agree about drawing sword stuff,

If you examine battojutsu, a great deal of traditional technique is executed from a sitting position, implying either a surprise attack during a polite meeting or a quick response to such an attack. In fact, when going into war during the Sengoku period, the sword was often worn differently — if it was worn at all, since it would have been stupid and ridiculous to go into battle with a sheathed sword.

Pretty much. Drawing and slashing techniques obviously and naturally makes sense in self defense, and other occasions when one is surpised etc.

No sense in battle or duel.

Anyway:

While inferior gauntlet technology ensured the katana could not be half-sworded,

Gauntlet technology doesn't really have much to do with half swording?

Half swording, as proven both by manuals etc. and modern practice can be easily done with bare hand - which AFAIU were common even among solidly armored men in Japan.

Lack of half swording had probably more to do with general shape and profile of the blade - it just wasn't comfortable and effective to handle it that way.

The katana's razor edge was poorly suited to hitting armored targets,

Is this at least positive that they generally had 'razor' edge? AFAIU, there would be great variance here too, depending on particular swordsmith tradition, geometry etc.

Like pointed out, making really razor sharp sword doesn't really make sense. It will effectively cut certain amount of clothing etc. just as well if duller, but will be only get ruined quick if sharper.

[gkathellar]

Pretty much. Drawing and slashing techniques obviously and naturally makes sense in self defense, and other occasions when one is surpised etc.

Or for assassination.

Half swording, as proven both by manuals etc. and modern practice can be easily done with bare hand - which AFAIU were common even among solidly armored men in Japan.

Lack of half swording had probably more to do with general shape and profile of the blade - it just wasn't comfortable and effective to handle it that way.

Huh. I've never actually done half-swording, so that shows what I know. There's really no risk of getting your fingers cut off?

Anyway, yeah, they did have very limited half-swording as I said.

Is this at least positive that they generally had 'razor' edge? AFAIU, there would be great variance here too, depending on particular swordsmith tradition, geometry etc.

Like pointed out, making really razor sharp sword doesn't really make sense. It will effectively cut certain amount of clothing etc. just as well if duller, but will be only get ruined quick if sharper.

Yeah, there's a lot of variation, but you see it much more in Sengoku-era battlefield swords. Edo-period swords were usually commissions for the nobility, and were intended chiefly for civilian use. I can't speak to this as a universal phenomenon, but that usually only requires cloth and flesh penetration. (They were sometimes test-cut on corpses and criminals to determine this sharpness.)

[The Reverend]

I remember reading reports from Admiral Perry's " Trade Fleet", the Japanese had guns that were almost exact duplicates of what the Portuguese had 200 years previously, buuuut they were of an incredibly high level of manufacture . They assessed the steel to be of a higher grade than their own rifles and built with a masterful level of skill and decoration.

[Spiryt]

Huh. I've never actually done half-swording, so that shows what I know. There's really no risk of getting your fingers cut off?

Anyway, yeah, they did have very limited half-swording as I sai

There's quite simply no way to cut your bones off by pressure, and even slight movement.

Risk of getting your skin and flesh screwed up was probably there, especially if sword was sharp and thin, but still better to bleed from hand than from gut.

And, obviously:

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For quick examples.

Here arma has video - thrust on maille - show some bare hands halfswording.

Have no experience either, but would simply guess that firm grip was simply vital - you can squeeze even very sharp knife hard without damage - as soon as you pull or push your hand along the blade though, you obviously will get bitten.

[Mike_G]

I remember reading reports from Admiral Perry's " Trade Fleet", the Japanese had guns that were almost exact duplicates of what the Portuguese had 200 years previously, buuuut they were of an incredibly high level of manufacture . They assessed the steel to be of a higher grade than their own rifles and built with a masterful level of skill and decoration.

I'm having a tough time believing that.

By the 1850's, western metalurgy would be pretty well developed, and Japan was always short of good steel. They bought a lot of scrap iron from us up until WWII. They probably wouldn't have been able to build a Pacific fleet if we hadn't supplied them.

[Incanur]

We somehow get into this conversation almost any time somebody mentions armor penetration, and all of this has to be rehashed again and again.

That's because it's a fundamental disagreement. As I've said before, none that complexity negates the importance of kinetic energy within the context in question. Folks who test armor use joules for a reason, and it's not because they're idiots.

[The Reverend]

I'm having a tough time believing that.

By the 1850's, western metalurgy would be pretty well developed, and Japan was always short of good steel. They bought a lot of scrap iron from us
up until WWII. They probably wouldn't have been able to build a Pacific fleet if we hadn't supplied them.

They also had very few of them.




on another note. I'm playing a 4th edition monk/avenger hybrid in a Dark Sun game and I'm fluffing the characters divine power coming from divine heritage and a bargin wrought into his peoples blood, the chosen people. So he basically is Jewish covered in qabbalistic tattoos to activate his divine powers, i thought it would fit in a desert world.

anyway he uses a fullblade: giant two handed sword that has a high crit value as it properties. I was going to fluff it as a huge ceramic Kopesh, but was wondering if anyone had other ideas of what a Jewish warrior might use sword wise. I've decided on the fullblade because of its ingame stats, but i am open to how it should look.

I've done a brief look in google at ancient israeli and seen their short swords and few other things but nothing but a giant kopesh has caught my eye.


ps if you comment something like "it wouldn't be made of ceramic because..." know this, Dark Sun has VERY LITTLE METAL. He ain't that awesome yet, yes it does have breakage rules for weapons.

thanks

[fusilier]

That's because it's a fundamental disagreement. As I've said before, none that complexity negates the importance of kinetic energy within the context in question. Folks who test armor use joules for a reason, and it's not because they're idiots.

I didn't mean to imply that they are, and I apologize if that's the way it sounded.

From the studies of armor penetration that I've seen, typically, they list the amount of joules a particular weapon or projectile needs to penetrate a particular armor.

So an arrow needs X joules, and crossbow bolt Y joules. When dealing with the same projectile, I believe joules become a useful comparison. Most studies I've seen, often just deal with a single projectile, being fired at different velocities. So kinetic energy is really the only variable.

But when you start to change projectiles things change. A good example of this would be too look at rifle caliber armor-piercing bullets. Often times they weigh the same as the standard "ball" cartridges, are fired at the same velocity, and have the same exterior shape. So kinetic energy between the two would be the same -- but one will pierce armor much better than the other. That has to do with the materials used, and how those materials react in the collision.

Likewise, I remember seeing a study posted in one of the previous threads, that tested lead and steel musket balls, and the steel musket ball needed less kinetic energy to puncture whatever armor they were using.

[fusilier]

Also, are we talking about thrusts/chops from a stationary position, or incorporating a step (or even just a shifting of the weight from back foot to front foot)? The two would look very different, and probably generate force very differently.

I was thinking about this too. For example a pike is thrust with the legs, not the arms (i.e. you attack by stepping at your opponent).

[gkathellar]

I was going to fluff it as a huge ceramic Kopesh, but was wondering if anyone had other ideas of what a Jewish warrior might use sword wise.

I'm no expert, but I'm given to understand that there's more historical relationship between the Ancient Israelites and the Ancient Greeks than is widely known. Look into the Greek Kopis and Xiphos.

[Deadmeat.GW]

Assuming liberally, that 400 paces is ~ 250 meters here, this is the distance from which your average shooter will have problems to target face from modern carbine, with it's accuracy, and ability to aim trough muzzle etc.

Targeting the face, let alone specific parts, with something that swings above you eyes in the moment of launch, is pretty much abstraction....

...

You said they were inaccurate, that is what annoys me.

Even historical texts did not claim that they would be that accurate at the range they would be effective at, but they were accurate enough to hit the head of an enemy at about 100 yards.

The reason it was a 'super' weapon was just like certain weapons could completely change the dynamics of a battle or war.

That 'super' was something I added in because they did change battles and wars dramatically.

Several archeologists have been on the books as being able to hit an archery target at about 20 meters after practising a couple of hours most of the time.
The problem is that that is not good enough for a war or a battle, there you need to be able to hit from far further and hit far harder.
And that takes a lot more training, especially you need training to speed up your shots without loosing accuracy.

I can use a sling to hit a tin can at 20 meters one shot out of 4, the misses mostly would be within half a yard of said can.
I do not practise with slings daily, heck I have done sling practise for maybe a couple of weeks with about 3 or 4 hours a day with a group of re-enactors.

In the hands of someone who has no experience sure a sling is inaccurate, heck, even dangerous to the wielder (I have clubbeb myself in the head a few times with the sling in the beginning because the release did not make a nice perfect arc) if you are not carefull.
But then in the hands of a rank amateur any ranged weapon is inaccurate and in some cases even outright dangerous to people not anywhere near the target.

Now if someone who does not practise daily can hit a tin can, which keep in mind is smaller then the face, then I fail to see how someone who is a professional with a sling would not be able to do better.
Heck, if you claim that a swing from above your face cannot be accurate check out some of the ways people pitch for baseball or cricket.
A fair few use overarm throws, where you do not line your eyes up with your arm.
It is no different then learning how to shoot a remote drone weapon.
On top of this there are dozens of variations on how to release a sling, overhead helicopter style is just one of them.

[Traab]

Circling back to the question, is a thrust or a hack better at piercing armor? The answer is, it depends. Every weapon is designed differently, each one has its own answer. There is no single motion that will universally make it easier to pierce armor with any weapon. The question is far too broad. A better one would be, "Is a hacking or thrusting motion more effective at piercing armor, when using a broadsword?" Or, a kopesh, or a jian, or a halberd. Pick a weapon, or at the least a weapon TYPE to narrow it down.

[Aux-Ash]

That's because it's a fundamental disagreement. As I've said before, none that complexity negates the importance of kinetic energy within the context in question. Folks who test armor use joules for a reason, and it's not because they're idiots.

Without knowing how the tests are set up and the hypothesis they went for (or the background of the people doing it), I can't really speculate in why they chose joules. Maybe there's a crucial bit of information in the algebra that would make it all work, I don't know.

Out of curiosity. Is there scientific articles treating this subject? Not books, articles. In serious scientific journals?

It just looks iffy due to he huge difference between what it takes for a knight to move forward at a gallop (29 kJ) and the numbers suggested bein needed for penetration. Not to mention that I could hit 110 joules by simply pushing an arrow straight at the armour... no velocity involved (granted, it'll take a few seconds).

However... be mindful that even if work isn't the primary factor it's still a contributing one. There will be a work exerted and it will affect the chain of events. I'm just arguing it's the wrong thing to look at.

[fusilier]

Without knowing how the tests are set up and the hypothesis they went for (or the background of the people doing it), I can't really speculate in why they chose joules. Maybe there's a crucial bit of information in the algebra that would make it all work, I don't know.

Out of curiosity. Is there scientific articles treating this subject? Not books, articles. In serious scientific journals?

There are, I'm trying to track one down that I might have access to . . .

However, I did revisit "For Show or Safety" by Sylvia Leever, an article in which they performed destructive testing on a pair of 17th century breastplates. While there are some wrongheaded conclusions in the article, they do present some equations. They don't go too much into why they would use those equations however.

At one point they calculate sheer stress needed to puncture a bullet sized hole, based upon force applied to an area. From that they develop an equation based on bullet radius, some constant which has to do with the nature of the material, and the thickness of the material, to determine how much energy is needed to puncture the armor. Note, that this energy is dependent upon bullet radius, so a different sized bullet will mean a different amount of energy is needed. They also admit that this is a simplified approach.