Arrows vs. plate armor

1. Feb 12, 2016

Warbow

This question is about the English Warbow in use during the War of Scottish Independence, Hundred Years War and War of the Roses. In 1545, way past the heyday of the Warbow, Mary Rose sank in the English channel. In 1981 they salvaged her. What they found was extraordinary. She carried 250 longbows and 172 have been recovered. Reproductions of these bows show us an average draw weight of the heavy bows at 150-160 lbf and one even reached 185 lbf. The lowest draw weight was around 100 lbf. That's beyond what a lot of people think is possible to draw. However, men like Simon Stanley, Mark Stretton and Joe Gibbs shoot bows like these on a regular basis.

This whole post is dedicated to the understanding of the physics of these bows in terms of hitting power, or more specific, plate armor penetration.

Calculating the kinetic energy is fairly easy (1/2mv^2) even for me, but this seems to be an inadequate quantification of the armor penetration ability.

I will give some examples to illustrate the problem. Here are some numbers from the book: The Great Warbow. A 150 lbf bow will shoot a 57.8 gram arrow at 62.25 m/s. This will give us 112 joule.

In comparison, Mark Stretton, in the book The Secret of the English Warbow, talks about his test of a 144 lbs bow shooting a 102 gram arrow at 47,23 m/s generating 113.76 joule.

According to Alan Williams a 2.4 mm plate with a fracture toughness of 120-150 kJ/m^2 (wrought iron) requires 112,5 joule if you want to penetrate sufficiently enough to hurt the person wearing the plate armor.

These are the only examples I could find with the kinetic energy approximately the same while the momentum changes. The 57.8 gram arrow have a momentum of 3.59805 N s. The 102 gram arrow have a momentum of 4.81746 N s. That's 1.21941 N s more than the lighter arrow. When you shoot these heavy arrows they always penetrate deeper than the lighter arrows, by far. How are these two related to each other. I mean, is it possible that an arrow can substitute some of its kinetic energy with momentum and penetrate just as deep?

A lot of people argue for kinetic energy to be the major determining factor when it comes to penetration, and they bring up a bullet, say a 7.9 gram bullet fired from an AK-47 traveling at 715 m/s generating 2019 joules of energy as an argument. And I have to agree, in this scenario the kinetic energy is the major determining factor. The momentum of the bullet is 5.6485 N s. In comparison to a 160 lbs bow shooting a 110.5 gram arrow at 54.864 m/s yielding 166 joule and a momentum of 6.062472 N s, the momentum is probably irrelevant to the bullet. But when it comes down to arrows striking plate armor, only 10 joules will make the difference between penetration or not (look at the chart below).

Is it possible that an arrow of 63.7 gram shot from a 175 lbf bow at 64 m/s, generating a momentum of 4.0768 N s and 130.5 joule is equal to an arrow with more momentum and less kinetic energy? Something like a 102 gram arrow at 48.5 m/s giving us 4.947 N s and 120 joule in comparison. (That's just an example.)

Also, Mark Stretton penetrated a 1.6-1.8 mm modern plate with around 0.5 % carbon with a 124.74 gram arrow at 40.843 m/s,104.08 joule and 5.09475582 N s. According to Alan Williams it takes between 127 - 139 joule to do that with a 64 gram arrow striking mild steel with 0.35-4 % carbon and a fracture toughness of 235 kJ/m^2 ( look at the chart below).

What is going on here? Is momentum a determining factor at work here?

To put things in perspective here. The strength of these heavy warbows are their ability to shoot heavy arrows in the 95-125 gram range without a significant drop in velocity in comparison to lighter arrows. My examples demonstrate this quite well.
Here are some data I have collected.

175 lbf shooting a 63.7 gram arrow at 64 m/s.
160 lbf shooting a 110.5 gram arrow at 54.864 m/s.

Now, how fast will the 175 lbf bow shoot a heavy arrow around 120 gram? 54 m/s? If that's the case a 175 lbf bow have a potential of 175 joule and 6.48 N s point blank. The 110 lbf bow will only reach around 54 m/s with 64 gram arrows. That's 93 joules and 3.456 N s.

Alan Williams' test of different plate qualities in the book: The Knight and the Blast Furnace.
J = joule not impulse.

An arrow of 64 gram will penetrate a plate with a fracture toughness of 120-150 kJ/m2, 3-4% slag and minimal carbon. Iron munitions armour such as the plate made in Koln.
1mm = 27.5 J, 1.1mm = 33.5 J, 1.2mm = 39 J, 1.3mm = 45 J, 1.4mm = 52 J, 1.5mm = 57.5 J, 1.6mm = 63 J, 1.7mm = 68.5 J, 1.8mm = 75 J, 1.9mm = 82 J, 2mm = 87.5 J, 2.1mm = 93 J, 2.2mm = 100 J, 2.3mm = 106 J, 2.4mm = 112,5 J, J 2.5mm = 118,5 J, 2.6mm = 125 J, 2.7mm = 131.5 J, 2.8mm = 137.5 J, 2.9mm = 144 J, 3mm = 150 J, 3.1mm = 156 J, 3.2mm = 162.5 J

An arrow of 64 gram will penetrate a plate with a fracture toughness of 180-210 kJ/m2, 1% slag and 0.3% carbon. Low-carbon steel armor such as the plate made in Nurnberg.
1mm = 41.25 J, 1.1mm = 50 J, 1.2mm = 59 J, 1.3mm = 68 J, 1.4mm = 77.5 J, 1.5mm = 86.5 J, 1.6mm = 95 J, 1.7mm = 104 J, 1.8mm = 113.5 J, 1.9mm = 123 J, 2mm = 131.5 J, 2.1mm = 141.1 J, 2.2mm = 150 J, 2.3mm = 160 J, 2.4mm = 169 J, 2.5mm = 178.5 J, 2.6mm = 188,5 J, 2.7mm = 197.5 J, 2.8mm = 207 J, 2.9mm = 216 J, 3mm = 225 J

An arrow of 64 gram will penetrate a plate with a fracture toughness of 235 kJ/m2, 0% slag and probably between 0.35-4% carbon. Modern mild steel plate armor.
1mm = 55 J, 1.1mm = 67 J, 1.2mm = 78 J, 1.3mm = 91 J, 1.4mm = 102.5 J, 1.5mm = 115 J, 1.6mm = 127 J, 1.7mm = 139 J, 1.8mm = 151 J, 1.9mm = 163 J, 2mm = 175 J, 2.1 mm = 182 J, 2.2 mm = 198 J

An arrow of 64 gram will penetrate a plate with a fracture toughness of 240-260 kJ/m2. <1% slag and 0.6% carbon. Medium steel armor such as the 15th-16th century plate made in Milan or Greenwich before 1530.
1mm = 60.5 J, 1.1mm = 74 J, 1,2mm = 87.5 J, 1.3mm = 100 J, 1.4mm = 113 J, 1.5mm = 127 J, 1.6mm = 140 J, 1.7mm = 153 J, 1.8mm = 167 J, 1.9mm = 180 J, 2mm = 192.5 J, 2.1 mm = 206.5 J

An arrow of 64 gram will penetrate a plate with a fracture toughness of >300 kJ/m2, <1% slag and 0.6% carbon but the carbides are present as tempered martensite rather than pearlite. Medium carbon hardened steel armor such as the plate made in Innsbruck, Augsberg, Landshut, or Greenwich (after 1530).
1mm = 82,5 J, 1.1mm = 100.5 J, 1.2mm = 118.3 J, 1.3mm = 136.2 J, 1.4mm = 154.2 J, 1.5mm = 172 J, 1.6mm = 190 J.

2. Feb 12, 2016

Staff: Mentor

Welcome to PF!

What is your interest in the physics of longbows?

3. Feb 12, 2016

Warbow

Thank you.

I'm interested because of military history and experimental archaeology.

My interest is how heavy arrows effect armor penetration and the relationship between kinetic energy and momentum. Or whatever else is at work.

I want to make a chart like the one I posted for every arrow weight (64 g, 66 g, 68 g, ... 130 g) and every draw weight (105 lbf, 110 lbf, 115 lbf ... 200lbf) and test different qualities of plate (probably 5 or 6 qualities) to find out what it takes to penetrate armor of different thickness (0.6mm, 0.8mm, 1mm, 1.2mm, 1.4mm, .... 4mm). Probably a unrealistic goal...

4. Feb 12, 2016

Staff: Mentor

I didn't read your whole post but did they do anything special with the arrowheads to make them penetrate armor better? For example, armor piercing shells use several tricks to do a better job...

5. Feb 12, 2016

Warbow

The slower arrow had a rhombic or lozenge cross section, significantly bigger than the lighter arrow. The lighter arrow had a relatively small square head. The Vickers plate hardness was the same. The reason for a lozenge cross section is two sharp edges to slice through the arming doublet underneath. The heavy arrow was at a disadvantage to the lighter arrow.

http://2.bp.blogspot.com/-iljsXxA3j4w/VqqfI00aQlI/AAAAAAAAEik/TWwfA4LDOX4/s1600/Test+19.jpg

6. Feb 12, 2016

Staff: Mentor

Your post reminded me of the Michael Crichton book about traveling back in time to medieval France. He went through a lot of detail concerning the armor and how much weight the knight had to carry. He also invented night arrows i.e. Flaming arrows shot at night to help with targeting. The novel was called Timeline.

https://en.m.wikipedia.org/wiki/Timeline_(novel)

The other thing it reminded of was the development of crossbows during the Qin dynasty which allowed ordinary farmers to become lethal archers with bows that could penetrate traditional Asian armor. The documentary went into arrow design using two metals, arrowhead shape, penetration and the firing mechanism and mass production of the crossbows. Truly amazing.

http://www.pbs.org/wgbh/nova/ancient/emperors-ghost-army.html

7. Feb 12, 2016

Bystander

Google "Sectional density," or "X-sectional density," or "Cross-sectional density."

8. Feb 12, 2016

Johnny Reb

With firearms, cmapring different bullet weights in the same caliber, the heavier will penetrate more, even though it is moving slower. For example, a 158gr(that's grain, not sure what it is in grams) will travel at approximately 1400fps will a 125 gr slug will travel at 1600 fps. The 158 will penetrate far deeper than the 125. Again, even in hunting, you use heavier bullets when you need penetration