ARROW PENETRATION

Introduction

The introduction of carbon arrows resulted in a number of problems for target archers relating to what happens when the arrow hits the target. (In general here I'm assuming the bound straw type boss). Targets quickly became 'soft' and archers started to get lots of pass throughs - no score and damaged fletchings. In response straw targets were made a lot thicker and with tighter binding to stop the arrows. As a consequence targets became a lot heavier (hernia generators). Targets became a lot harder which resulted in an increased chance of arrow damage on impact. (Arrow breakage was further exacerbated by the growing popularity of barreled shafts which are structurally weaker where it matters). It became also much harder to remove arrows from the target - a possible cause of injury to archers and not to healthy for the arrows either.

What we want is a light, cheap, easily stored target that lasts a long time and does not damage your arrows. (dream on!). This section is about the behaviour of arrows when hitting a target.

Fundamentals

There are four basic properties of the arrow which relate to how an arrow behaves in terms of penetrating into a target:-

In order to look at the different effects of the above factors let's see at what happens as an arrow penetrates a straw target. The target can be looked at as being a bunch of elastic strands all aligned the same way that the arrow has to push its way through.

The pile of the arrow has to push the the strands to the side i.e. move them and in the process stretch them. This of course takes energy and as a consequence the arrow slows down. Along the shaft of the arrow the 'stretched' strands exert a frictional force on the arrow which act to slow the arrow down. The total frictional force will depend upon several factors including how much and how many strands are stretched (i.e. the arrow diameter) and the surface area of the shaft on which the frictional force acts (i.e. the arrow diameter and how far the shaft has penetrated into the target). Once the point of the arrow is through the target then its only the frictional force on the shaft that acts to slow the arrow down. The other thing that will happen as the arrow penerates the target is that some of the strands will snap instead of stretching. A snapped strand will reduce the energy lost via the action of the pile and exert a low frictional force on the arrow shaft. When subsequent arrows are shot into the target any already broken strands encountered will require much less energy for the pile to move aside and will generate a lower frictional force on the shaft.

In reality strands of straw are not very elastic and it is the compression/displacement of the straw by the arrow which generates the various forces on the arrow but from the point of view of looking at what is going on the elastic band metaphor seems a reasonable approach.

Kinetic Energy

The arrow arrives at the target with a given amount of kinetic energy. This energy is lost when the arrow hits the target and the arrow comes to a stop. Most of the energy ends up as heat in the target and some is lost via the flexing of the arrow stuck in the target. If 'x' is the amount the arrow has penetrated into the target at any given moment then there will be a retarding force on the arrow from the pile/shaft behaviour described above at that moment F(x). The definition of Kinetic Energy is that it is the integral over distance of the force F(x). In other words it is the arrow Kinetic Energy that defines how far an arrow will penetrate into the (elastic string) target.

Momentum

The arrow arrives at the target with a given amount of momentum which the arrow loses as it comes to a halt. The definition of momentum is the integral over time of the force F(x). In other words it is the momentum that defines how long it takes for the arrow to come to a stop. Because the momentum change relates to force x time it is also a measure of the strength of the 'impact' the arrow has on the elastic strings in the target. The arrow momentum is one of the factors which will determine whether the elastic strings will stretch or snap. As how many of the elastic strings are broken affects the value of the frictional force on the arrow F(x) the arrow momentum indirectly affects how far the arrow penetrates into the target. (An extreme example would be shooting at a suit of armour - not enough momentum and the arrow would bounce off i.e. zero penetration irrespective of how much kinetic energy the arrow had).

Point Shape

The shape of the arrow point will determine how fast the strings are pushed aside and also how much stress locally there will be in the elastic strings. With a 'blunt' point the stresses generated in the elastic strings will be spread over a larger volume then with a 'sharp' point. The sharper the point therefore the higher the proportion of the elastic strings that will be snapped rather than stretched. As snapped strings retard the arrow less than stretched strings an arrow with a sharp point will penetrate more than an arrow with a blunt point. Too blunt a point (trying to stretch too many strings too fast) and the arrow might not penetrate at all and bounce off.

Arrow Diameter

The larger the arrow diameter then the higher the number of elastic strings that need to be moved out of the way and the more each string will need to be stretched increasing the pile retardation effect. As the arrow diameter increases the more the strings are streched and so the retarding frictional force/unit area on the shaft increases. The fact that the elastic strings are being stretched more is likely to increase the chance of them snapping. With a larger diameter the shaft area on which frictional force acts is larger. The larger the arrow diameter then the more stopping power the target will provide on the arrow leading to lower arrow penetration. As the 'sharpeness' of the point will effectively increase as the arrow diameter decreases then a smaller arrow diameter will indirectly increase the probability of breaking strands i.e. more target damage.

Comparison of Carbon V Aluminium Arrows

Lets compare how target penetration varies between an aluminium and a carbon arrow, both being a good arrow match for the archer's draw weight/length.

Our aluminium arrow has an all up weight of 443 grains and an outside diameter 0f 0.89 cms. Our carbon arrow has an all up weight of 265 grains and and outside diameter of 0.61 cms.

Target arrow points vary from chisel to hemispherical in shape. The design aim is to provide the maximum stopping power while having a very low probability of a 'bouncer'. The 'parabolic' shape seems to be the most popular shape for aluminium arrows. Carbon arrows vary from parabolic to semi-chisel. This is pure speculation on my part but it seems that barrelled arrows tend to have parabolic pile shape (lower stopping power) while parallel shaft arrows tend to have semi-chisel points (higher stopping power). It may be that you have to keep the point stopping ability low on a barelled shaft to avoid too many arrows breaking on impact.

If we ignore the pile effect and assume that the frictional force/unit area is proportional to arrow diameter (the number off and the amount the elastic strings are stretched) and that the total friction force 'F(x)' is the product of the force/unit area times the area of the shaft in the target then:-

F(x) = a constant times the diameter squared times the arrow penetration

Integrating F(x) dx over the penetration distance 'X' gives you the total arrow Kinetic Energy 'E' i.e.

E = a constant times the diameter squared times the square of 'X'.

If you assume that both arrows leave the bow and arrive at an infinitely thick target with the same kinetic energy then the arrow penetration becomes inversely proportional to the diameter. I.e. the carbon arrow will have 0.89/0.61 = 1.5 times the penetration of the aluminium arrow. The fact that the target has limited thickness will in practice result in the ratio being even higher.

As far as arrows damaging the target (snapping the elastic strings) is concerned the two relevant arrrow factors are the diameter and its momentum as described above (the points assumed being the same shape). The other significant factor is the design of the target itself.

As far as the arrow diameter is concerned its a bit of swings and roundabouts in the rate of target damage. It will very much depend on the nature of the target. In my opinion probably not much in it with maybe the smaller diameter carbon arrows doing more damage.

Again assuming the two arrows have the same Kinetic Energy when they hit the target then the ratio of the carbon arrow momentum to the aluminium arrow momentum will be the square root of the ratio of the carbon arrow mass to the aluminium arrow mass i.e. square root of 265/443 = 0.77. The carbon arrow will have about three quarters of the momentum of the aluminium arrow at target impact. The following graph illustrates how the momentum of the two arrows changes relatively with time as the arrows penetrate the target . The carbon arrow, unsurprisingly, takes longer to stop.

Initially the aluminium arrow has the higher momentum but because the frictional retarding force is lower for the carbon arrow it doesn't loose momentum as fast.


The real problem with target damage relates to the increased penetration of carbon arrows. When carbon arrows were introduced the existing straw bosses could not effectively stop them. One solution would be to keep the same level of binding and increase the target thickness: Pros - limited arrow breakage, Cons - very hard to pull the arrows and very heavy targets. An alternative solution would be to increase the binding tension (less stretchy elastic strings): Pros - better arrow stopping power with an acceptable increase in weight, Cons - high incidence of arrow breakage. The actual change was a compromise of a higher binding tension and increased thickness -which you could regard as a 'best of both worlds' or a 'worst of both worlds' largely depending on what bow/arrow setup you have. If you shoot parallel shaft carbon arrows with a medium poundage bow then you are OK. If you shoot high speed barelled shafts you are in trouble, there is a higher incidence of arrow breakage and because of the tighter binding each arrow causes more target damage so they quickly go soft and you get pass throughs.

Perhaps the optimum solution with current straw targets is to use two, a soft target in front of a hard target; low risk of arrow damage, stopping power and you don't need a fork lift to move the targets. If you equate the cost of a target to three broken arrows then the cost is not to bad either.

Arrow Breakage

When you shoot an arrow there are two points where the arrow shaft is highly stressed and failure is possible, when you shoot it and when it hits the target. When shoot an arrow it's going from say 0 to 250 fps over around 28 inches. High acceleration and therefore high forces. The maximum stress point in the shaft is towards the rear somewhat in front of the fletchings resulting from the Archers Paradox bending and this is the most likely failure point. Arrows (as selected from the charts) have a big safety factor so unless to go for a seriously underspined shaft or overweight pile the chance of an arrow breaking when shooting it is extremely small. When the arrow hits the target it's going say from 250 to 0 fps over around 8 inches much higher acceleration and hence forces than when its being shot. As the pile enters the target the high deceleration results in the arrow 'whiplashing' causing a high stress zone say around 5 inches from the pile. The 'harder' the target than the higher the peak stress in the shaft and the nearer the pile it is. With barelled shafts the diameter decreases towards the pile from the middle so the arrow is structurally weaker than an equivalent parallel shaft arrow.

Varying Targets and Optimising Arrow Weight

With 'target' arrows your trying to stop arrows quickly and safely. With different targets the requirements for the arrow will be different.

For hunting purposes you want your arrow to maximise damage to vital organs so you use a 'broadhead' pile which will cut through tissue over a wide area supported by high arrow momentum to assist target damage.

Arrows in war varied in design with their purpose. In order to cause damage to someone in a suit of armour you first of all needed to penetrate the armour so you had a heavy arrow with a bodkin (needle like) point.

Penetration Example

There is often discussion with bow hunters about getting the right balance between arrow mass and arrow speed. The following is a (tongue in cheek) example of the sort of analysis involved.

Suppose our archers are going to be in a battle against foot soldiers who protect themselves with shields. Our spies have managed to acquire a couple of shields and tests on them indicate that for an arrow, of the type we use, to penetrate the shield it needs a momentum of 100 (remember that penetration of a solid depends on the arrow momentum not its kinetic energy).

With the bows we use, an arrow of mass 4 has a measured speed of 20. The kinetic energy of the arrow (half the mass times the velocity squared) is therefore 800. The momentum of the arrow (mass times velocity) is 80. The arrow of mass 4 doesn't have enough momentum (need 100) so if the arrow hits a shield it will just bounce off. The arrow kinetic energy is more or less independant of the arrow mass (the bow efficiency stays much the same).

If we use instead an arrow of mass 8 then with a kinetic energy of 800 the arrow velocity will be about 14 and the arrow momentum will therefore be 8 times 14 i.e. 112. The mass 8 arrow will penetrate the shield and injure the enemy soldier.

The downside of using the heavier arrow is the loss of speed from 20 down to 14. This means that our range will be reduced and because of the less flat trajectory our archers will probably lose some accuracy as well. The ideal arrow is one that has a momentum of 100 exactly as this gives us the fastest arrow that will penetrate the shield. The ideal arrow mass is given by half the square of the required momentum divided by the arrow kinetic energy i.e. (100 x 100)/(2 x 800) which gives an arrow mass of 6.25. The associated (fastest shield penetrating) arrow velocity is 16.

Last Revision 1 July 2009