I'm wondering if the defensive value of armour stacks directly, or if it's more complicated than that.


As an example...

If it took one joule to get a deadly shock through mail, with a warhammer--and, if it takes ten joules to get the same shock through a padded gambeson: Would it take 11 joules to get the same amount of shock through mail with a a padded gambeson?

It is far more complicated than that. If it takes 10 joule to get through the gambeson that force is applied in one spot, the spot you hit with your weapon. But the armour spreads the force over a large area of the Gambeson and makes the hit thereby survivable.

In The Knight and the Blast Furnace, Alan Williams treats the protection provided by padding as additive. Throughout the book, Williams repeats that padding increases the energy required to defeat any given armor by 50 J.

Williams contradicts this himself (pp 943-945)

In one test he measures the amount of energy required for his blade to cut through a certain number of layers of linen in a padded target: 100J – 5th layer; 120J – 9th layer; 140J – 16th layer; 160J – 23rd layer; 180J – 26th layer.
So it took 100J to cut through 5 layers of a 26-layer target

In another test he tells us that it only required 80 J to cut all the way through a 16 layer target.

It seems that the very act of placing the target over additional padding helps to stop penetration of the primary target. I wouldn't be surprised if this difference is noticable no matter what type of armour is tested. It is yet another reason to dismiss all the tests we have seen that don't put some kind of padding behind the armour. Even if the padding itself provides no additional protection (which it obviously does), it seems to improve the defensive capacity of the primary armour as well.

Yes. The armour being able to move with the impact is the key, as discussed in http://www.myArmoury.com/talk/viewtopic.php?p...ht=#254513

Padding is a good way to get this movement, but there are others. Freely hanging pieces might be even better (e.g., necks, skirts), but that that isn't so useful for torso armour.

Have they created a math formula to simulate this effect? Or, is it still debated...?

A simple treatment of free-to-move armour is easy enough, for penetration by arrows:

Start with the initial energy or speed of the arrow (either will do; they're related). Find the momentum. When the arrow hits the armour, momentum is conserved. That is, the final momentum of the arrow + armour after the impact is the same as the momentum of the arrow before the impact. Calculate the final kinetic energy of the armour - this energy isn't available to penetrate the armour. So, energy available for penetration = initial energy of arrow - final energy of armour.

If it was just a piece of armour floating in empty space, the kinetic energy of the arrow after this impact wouldn't be available either. But the armour will presumably be stopped by padding/backing/laces, so we will keep this energy available.

Example:

100g arrow, 50m/s, kinetic energy E = (1/2) m v^2 = 125J, momentum p = mv = 5Ns.

Hitting a movable 10cm by 20cm iron plate, 1.3 mm thick:

Mass of armour plate = 200g.
Total mass (armour + arrow) = 300g.
Final speed = momentum/mass = 5/0.3 = 16.7m/s.
Energy of armour = (1/2) m v^2 = 28J

So, only 97J is available for penetration, instead of the original 125J. Every bit helps!

More stuff can be included in the model, but this is a reasonable and simple beginning.

the effect is described in physics as impulse

http://en.wikipedia.org/wiki/Impulse_(physics)

its how airbags work. When you're in a car crash the change in momentum you undergo will always be a constant but if you lengthen the time under which your momentum changes (in this case going from a high speed to a stop), you'll be accelerated more slowly. It's pretty intuitive, if one car goes from 0 to 60 in 20 seconds that person will feel a lot more force than someone who accelerates from 0 to 60 in a minute. If you were struck wearing a plate with padding underneath, the padding would crush, lengthening the time that it was under acceleration, lowering the force. That's thinking of it in terms of conservation of momentum.

Thought of another way, in terms of conservation of energy, if you dangle a piece of armor, and strike it with an arrow, a lot of energy would be expended accelerating the armor backwards. If you held it rigid all the energy goes into deforming the material around the area of impact, breaking the bonds with sufficient energy. If there's padding behind armor that gives under an impact, its going to absorb that energy and "waste" it. If the person is accelerated back with the impact that absorbs even more energy. Of course, if the person is moving into the impact then he's adding energy to the collision. Also a really spongy material with a lot of give will make light blows more comfortable, but will flatten almost immediately when struck hard and be virtually useless. This is called bottoming out. A relatively stiff and rigid material that deforms more slowly under high impacts will protect better in those situations, but when hit with a small force they're practically rigid and useless.

the figures Dan Howard quoted make sense to me. It took less energy to cut through more layers of less padding. When you add more padding to change time t in the impulse equation, the force of the blow on the top layers is lowered, and less layers can be cut through even when he applied more energy.

There's a huge amount of research put into this in terms of armor, for military applications as well as for sports such as cycling and football. Modern helmets generally use a combination of soft "low-impact" padding and hard "high-impact" foam. Suspension systems were previously widely used and operate on the same principles but are inferior to modern padding materials.

http://www.bhsi.org/general.htm
this site has a lot of useful information for helmets in general that's easy to understand.

http://www.pas.rochester.edu/~blackman/ur10helmets.pdf
this power point is pretty technical but might have some useful information. A lot of what they model could be applicable to a pollaxe striking a helmeted man in the head.

http://designscience.umich.edu/pdf%20files/APD-2003-05.pdf
This study addresses foam thickness in a helmet. Since the experimental helmet is a steel shell with some amount of foam underneath it, it should be applicable to steel armor with padded backing.

G=C/t where G is peak acceleration during the impact, C is some constant that he calculated using a lot of parameters in his experiment but is specific to whatever helmet and foam he was using, and t is foam thickness.

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so according to his research this is how increasing thickness of padding works. Force on y and padding thickness on x. Seems like you get a huge decrease to force with just a small amount of padding, and thicker and thicker padding continues to help but gives diminishing protective returns.

I know that was a long post, hope some of this was useful or relevant to your question, the base line is its really complicated. Researching this stuff is what people do for a living because its still extremely applicable to war, sports, and general safety.

As a side note, I think your example in the original post would actually work out, but only because maille doesn't really do anything in terms of impact protection. It was worn to stop weapons and projectiles from penetrating or cutting the body. In that sense, the only thing protecting from impact would be the padded gambeson.

a steal plate worn with out padding will stop blows and disperse the energy, but will not absorb the energy. Thus a would be penetrating trauma such as from a sword against sufficient plate will be in effect a blunt trauma. So more armor of any type provides greater protection. It is all about energy, which is why major advancements in weaponry have in many cases dealt with producing and containing greater energy.