Penetration

Achieving adequate penetration of the arrow into the target is essential for a quick kill. Over the years, much has been written on arrow penetration, much derived from bullet terminal ballistics. The incidental but seminal work by Jean-Victor Poncelet in the mid 1800's lead to the theoretical understanding of terminal ballistics or penetration mechanics.

For the technically minded, Poncelet's differential equation gives the deceleration of a projectile entering an homogenous target material:

d_(mv)/d_t  =  - A_cs.C₀ - A_cs.C₁.v²

where

d_(mv)/d_t  is the deceleration at some point in time t,

A_cs  is the cross sectional area of the projectile,

C₀ and C₁  are constant terms, dependent on the target material,

v  is the velocity at some point in time t.

When solved for the boundary conditions (initial impact velocity and final zero velocity the transformation becomes:

P = ρ_s . ln[ (C₀ + C₁ . v_s²) / C₀ ] / (2 .C₁)

where

P    is the penetration distance

ρ_s   the sectional density of the projectile

v_s   impact velocity

Poncelet's equation assumes the sectional area and the mass are constant, which for modern bullets is rarely true, however for archery is nearly always true. For archery with a long slender shape, the surface drag long the shaft is not allowed for, particularly as contact area increases with penetration.

For bullets, penetration mechanics can be complicated by shock-waves in the target, something not seen in archery. For archery there is a notional speed threshold, below which the target deforms elastically and above which the target deforms plastically with more damage.

Some have modified Poncelet's equation to include a linear velocity term that helps deal with the shaft friction issue:

d_(mv)/d_t  =  - A_cs.C₀ - A_cs.C₂.v -  A_cs.C₁.v²

To match reality, these equations need to be integrated with time, ensuring that at each distance the C_n values match the target material encountered. This is really difficult or even impossible without knowing the precise arrow's path through the animal.

A notable issue is the Poncelet's equation assumes the projectile is fully immersed in the medium as with the case of bullets. For arrows, the shaft enters progressively, so the friction surface area increases with penetration.

An Empirical Approach

An alternative to the theoretical is the experimental. Ed Ashby has undertaken penetration testing on large fresh animal carcass, most notably on the Cape and Asian Buffaloes. This has involved a great deal of work that few have undertaken. Ashby's data set is large and unique for its wide range of arrow configurations, but its presentation is rambling, wordy and the associated analysis lacks scientific rigor.

None-the-less, Ashby's data is very useful in providing insight to the penetration process. Problems arise when attempting to broaden and generalize the scope to smaller game and lighter bows. FlyingSticks makes some informed estimates to widen the application of Ashby's work.

Kinetic Energy vs Momentum Wars

There has been much debate as to what is most import in achieving the best penetration - kinetic energy or momentum. It turns out that the best answer is probably somewhere in between; a bit fuzzy and leaning towards momentum! No one can be more precise that that unless they know the materials in the path of the arrow.

Ashby clearly is on the momentum side, with his data supporting the argument to some extent. His conclusion that for a particular bow, heavier arrows will always penetrate better is probably right. (Working against this is "heavier" implies slower and slower arrows are harder to deliver accurately and allow more time for target movement).

It seems kinetic energy, which is proportional to the speed squared (≈ m⋅v²), is needed to penetrate bone and to cut flesh. Momentum, which is proportional to the speed (≈ m⋅v¹), is best for quickly moving the flesh out of the space to be occupied by the arrow. If there was sufficient data, a good statistical model could be developed, but alas such data has not yet been found. The complexity of the target makeup makes the war more ideological than meaningful!

Ok, How Does FlyingSticks Do It?

The penetration issue is a complex multi-parameter problem. FlyingSticks approach is to create a parametric physics based model and trim the various coefficients until the model output matches the known empirical data. This provides a workable fit to both the Poncelet theory and the Ashby field data.

To resolve the kinetic energy vs momentum debate, the model introduces a penetration proportionality of ≈ m⋅v^1.35 that lines up with reality (most of the time), however there can be considerable variation due to the relative proportions of fir, skin, bone, mussel and soft organ tissue with every strike. These all present different characteristics, the mix of which is unknown before any shot. To ensure at least 95% of shots have a 95% chance of ethical kill, the penetration calculation need to biased towards the minimum penetration associated with a worst case path (assuming a kill-zone strike).

FlyingSticks uses the following equation for penetration:

P  =  ( K ⋅ m ⋅ v_s^e⋅  ρ_s⋅ r_t⋅ A ) ⋅ b

where

Sectional Density ( ρ_s )

Sectional density (arrow mass / point section area) is an important factor in determining the initial penetration threshold. It is a measure of the pressure applied on impact rather than the force. It is self-evident and explains why a small point penetrates better than say a golf ball delivered with same speed and mass. Maximize the sectional density by using heavier arrows with small cross-sectional area broadheads. The sectional density is:

d = m / ( D_f² . π / 4 + (C - D_f) . T_b)

where

Some small game points are an exception to the high sectional density requirement. These points aim to cause massive blunt trauma without necessary penetrating. There is a sectional density threshold below which a projectile will not penetrate, instead it will bounce off causing some bruising and surface abrasion.

Blade Thickness

From a penetration perspective the blade should be as thin as possible to maximize the sectional density. The required thickness is determined by the forces imposed in penetrating bone, the shape efficiency and the strength of the blade material.

Fine pointed tips are prone to bending on striking thick bone, causing the arrow to deflect and significantly reduced penetration.

Rib Thickness and Spacing (r_t )

The ribs are a major obstacle for an arrow. Bone requires significantly more energy than other tissue to penetrate. Each predefined game type has a default rib thickness that may be modified.

The rib width and spacing determines the probability of a rib strike. Some game such as the Asian Buffalo have overlapping ribs so the probability is close to 100%. For larger game, FlyingSticks assumes a rib strike. For smaller game where there is more than enough penetration the ribs can be ignored.

Ashby's Modifier

A hidden complication is the inter-dependence of some factors. For example the a_foc factor is a function of FoC and mechanical advantage. This is not shown in the above equations.

So, FlyingSticks takes a pragmatic parametric approach to penetration based on the limited and sometimes confused data that is available. The parametric model has some theoretical basis, but is largely empirical and is strongly influenced by Ashby's work.

The concept of efficiency in this context needs some explanation. In general, a "normal" or "commonly used" arrow is considered 100% efficient, so when a parameter is changed, it may improve or reduce penetration. There is obviously some fuzziness here!

For example, when FoC Efficiency (a_foc) is calculated, a normal FoC arrow is considered 100% efficient. The advantages to be gain with an extreme FoC and ultra-extreme FoC are therefore expressed as an efficiency of greater than 100%.

Each of the factors is discussed in more detail:

Arrow Mass (m)

In general heavy arrows appear more effective penetraters if they can be delivered with the required accuracy.

Mass Threshold  (a_m )

Ashby's work tends to indicate there is a critical mass threshold below which the probability of perpetration through rib bone is much reduced. For Cap Buffalo he determined this threshold to be about 650 grains and the average penetration decreased by about 30%. Sub threshold masses have an efficiency ramping down to 0.7  from 1.0 for over threshold masses. See the "Penetration vs Arrow Mass by Bow Draw" predefined plot to get a feel for the effect.

Just how the observed mass threshold scales for other game sizes has not been reported, but FlyingSticks assumes it is proportional the third power of rib thickness - or in other words related to the ratio of the local rib mass to the arrow mass. The importance of the threshold quick drops away for lighter game. This will be updated should relevant field data become available.

Heavier Arrows Increase Range Estimation Errors

Range estimation accuracy becomes significantly more critical with heavier arrows due to the more parabolic path. Best using a laser range finder or per-marked ranging points.

This is a critical issue, not to be over looked. See the "Group Height vs Range by Arrow Mass" predefined plot to get a feel for the effect.

Heavier Arrows Increase Flight Time

With longer ranges and heavier arrows, the flight time increased significantly. During longer flights the game is more likely to move. This is especially true when the flight time is greater than sum of the time it takes the sound of the release to arrive at the game and the game's reaction time. The speed of sound in still air is approximately  330 m/s (1100 ft/s) and animal reaction times range from 150 to 500+ ms.

Shape Efficiency

Penetration or shape efficiency is a factor related to the general shape and configuration of a point. Moving through flesh involves three distinct actions - creating the path, displacing the tissue and overcoming the friction between tissue and arrow. This is practically the same as passing through any fluid, so similar considerations apply - keep the design streamline, more like a classic jet fighter than a Tiger Moth.

Sleek Design ( a_ff )

A sleek design without sudden protrusions or steps will penetrate better than a cluttered design. The a_ff term deals with this effect. It is 1.0 for a typical welded broadhead but rises to 1.3 for Grizzly style and drops to 0.7 for typical mechanicals.

Mechanical Advantage    (a_ma )

Ashby introduces the concept of mechanical advantage of a broadhead. It expresses the acuteness of the cut angle, discounted for the number of cuts.

K_a = L_a / L_r / n

where

FlyingSticks then modifies Ashby's mechanical advantage into an efficiency factor:

a_ma = 1.0 + k (K_a  - 2.0) / K_a

The 2.0 value creates a "normal" mechanical advantage of 2.0 - anything above will improve penetration and anything less will reduce penetration. The k is a scaling constant.

Having more than two blades will reduce penetration and can only be justified for smaller game to increase bleeding and when there is already ample penetration capability.

Tip Design  ( a_t )

The tip design is largely a statistical parameter indicating the probability of bone penetration without damage. Ashby's so called Tanto tip was found the most reliable compromise between efficiency and structural reliability on heavy bone strike. A plain needle point is most effective if it does not bend, but the probability of bending is so high that performance is greatly compromised.

There has been very little scientific study on the deformation of needle like shapes striking sturdy materials. Needle points tend to curl on bone impact, resulting high drag and poor penetration.

Interestingly, Ashby provides little guidance on the dimensions of the tip modifications. Judging by the published images, the tip feature width is typically ~20% of the cut width. The important thing here is to remove the narrow fraction of the blade that could be permanent deformed with the arrows available kinetic energy. Stronger or thicker blade material would move that point forward, creating a smaller tip feature. Ashby's work focused on the tip design shape - he compared various designs of the same general size for use on large game.There is no data for scaling up or down, so we rely on the physics.

For thinner bone, the tip shape is of less importance. There is a lack of data on this, so FlyingSticks reasonably assumes a direct proportionality to rib thickness. An appropriate rule of thumb might be the tip feature width should be ~65% of the rib thickness of the largest game expected.

A secondary effect of tip design is its efficiency in penetration through soft tissue. This is effect is not mentioned by Ashby nor modeled in FlyingSticks.

Ferule and Shaft Diameters (a_f )

Ashby's work has shown that a ferule diameter less the shaft will reduce penetration (probably due to the step - the loss of sleekness!), and that a ferule greater the shaft can improve perpetration by about 10%.

Shaft Profile (a_s )

A tapered shaft can improve penetration by up to 8% probably due to reduced friction. Barrelled tapers are to be avoided as they increase drag unless the maximum diameter is less than the ferule's.

Blade Edge (a_e )

There is an ongoing debate about the design of the cutting edge - should it be curved, single bevel, double bevel and/or serrated? The credible evidence seems to support simple straight blades, with single bevel sharpening and without serration.

Single Bevel

It has been reported that the single bevel edges tend to encourage bone splitting as opposed to simple puncturing. This is due to blade rotation induced by the asymmetrical cutting action with the effect of leveraging the bone apart. Provide a split can be initiated, it is probably more energy efficient than simple hole punching. There would be considerable leveraged force available for this action, using the blade surface as a fulcrum. Also a split bone may then reduce resistance on the shaft following through.

The single bevel also creates a helical cut when passing through soft tissue. This results in a longer effective cut and hence greater bleed opportunity. However, in soft tissue the single bevel is likely to experience a slightly higher drag due to this rotational motion needing to displace tissue (unless the blades also have an appropriate offset).

An issue not mentioned in detail by Ashby is the included angle single bevel cutting edge is likely to be half that of the double bevel. Ashby used 25° single bevels (25° included angle) and 25° double bevels (50° included angle).  This factor alone may improve cutting efficiency but at the cost of the edge being more prone to damage.

There seems to be no situation in which a single bevel is out performed by a double bevel.

Razor Sharp

Ashby's work has shown without doubt that traditional grinding and stropping is best. Certainly beats filing which leaves micro serrations. A well prepared blade should cut hair and be usable for shaving!

Serrated blades tend to clog with fibrous material and as with domestic knives are a lazy substitute for a well maintained conventional edge. They may have a place for smaller game where penetration is less of an issue.

Bevel Angle

More acute angle is better but is more likely to loose its edge.

Blade Material

From the above discussion the blade material should have the highest strength and a hard surface to support a good edge. Excessive brittleness may fracture on impact, increasing friction and compromising penetration.

If the blade can be file sharpened it is likely that it will loose its edge during penetration.

FoC (Front of Center)  (a_foc )

Ashby's and other's work has shown a high FoC can significantly (55%) improve penetration over a normal FoC with medium mechanical advantage. This believed to be due to the reduced shaft bending on impact and hence reduced angle of entry relative to the flight path, and hence the reduced friction on the broadhead and shaft.

Problems with Ashby's Published Data

Ashby would probably be the first to admit there are large holes in his published data. He understands the complexities involved. His experiments were mainly aimed at establishing the optimal arrow design for larger game - for thick bone and hide.

The optimal arrow is expensive and time consuming to construct and maintain. "Normal" or "common" arrows are fine for smaller game, but what sized game is ok for these arrows? Or, how light a bow can be used?

These questions about scaling require data that is largely missing. FlyingSticks employs a parametric physics model that is able to down-scale in the absence of this data by applying physics theory to each part of the problem, then combining the parts. As more data becomes available the model is tuned to fit the data set.

An example is the tip design. The needle tip is best at penetration but only if it does not fail in the attempt - which is very likely on large game. The tanto tip is found to be the best compromise - a little poorer at penetration, but a lot less likely to fail, so is a better choice for large game. But, at what size game is the needle tip likely to be reliable?

Another valid issue is cut width. If I already have more than enough penetration, can I trade some of that penetration for cut width to increase bleeding. Or can I use a lighter arrow to reduce range estimation errors? What is the best compromise?

Answering questions like these requires a tuned physics model.

Other Issues

Mechanicals?

Mechanical boardheads have movable blades that extend on impact. The aim is to reduce air drag and improve flight stability by moving the center of pressure rearward. By necessity they are less robust than a well-executed conventional design and the trade-off balance is generally not in their favor.

Non-Hunting Points

Standard target and field points are not appropriate for hunting as they have no cutting action and hence poor mechanical efficiency. The blood channel is likely to close almost completely or be sealed by the shaft if it fails to pass through.

A Calibration Point

The FlyingSticks uses data published by Ashby as the basis for the penetration:

This is the recommended single calibration point that can be used if modifying the penetration exponent and scalar in FlyingSticks' preferences.

Conclusion

The result of these considerations is that a slim heavy arrow is without doubt better than a light fast arrow providing it can be delivered with sufficient accuracy. Additional factors favoring heavier arrows include better bow energy transfer to the arrow on launch, and better speed maintenance in flight.

The downside of a heavier arrow is its greater drop and longer flight time, both of which impact accuracy.

The choice of broadhead and shaft is important. Considerations include:

• keep number of blades to a minimum,

• single bevel razor edge, with bevel matching fletch direction for spin,

• keep design simple and strong - straight edge with acute sweep-back,

• optionally the point's tip can be less swepted for less bending of the point and consequential arrow deflection on oblique bone contact:

  
  

• avoid serrated edges,

• ensure ferrule is aerodynamically smooth for low drag in all tissue types,

• keep blade span small (there may be a legal minimum is some jurisdictions),

• avoid mechanicals and replaceable blades - the minimal benefits are out-weighed by poor reliability on bone strike,

• ensure ferrule diameter is slightly larger than arrow shaft diameter, and a smooth shaft tapering to the rear is good.

• use a high FoC - 20% to 30% recommended.

• a heavy, strong one-piece broadhead design is good.

There is much unsubstantiated opinion, mystical belief and assertion regarding the hunting arrows. In distilling the fact and fiction mixture, the message is loud and clear:

Large Game:

• Select the heaviest arrow that can be delivered accurately and select a simple, strong, razor sharp, two-bladed broadhead with an acute sweep-back.

• An arrow of 650 grains with a 20% FoC is a good starting point.

Small Game.

• Lighter arrow with less efficient broadhead is ok.

• Focus on accuracy and short flight time.

Broadheads with more than two blades should only be considered when there is already excess penetration and the additional blades are likely to significantly increase the blood flow. Designs with closely space parallel blades would seem to fail dismally on the effectiveness score.

One reason for opting for a lighter arrow is the desirability of not having to tune the bow and sights for different arrow sets. Target shooting usually favors fast light arrows so some compromise arrow mass in the 450 to 550 grain range may be desirable. However, it would be better to adopt a practice of using two different arrow masses - say 350 and 700 grain - where the bow is turned (nock point, rest, tiller, brace and draw) for both arrows and dynamic spine is matched by arrow parameter adjustment (static spine, point weight, shaft length). The only down side of this approach is that two sight tapes are required.

The Ed Ashby Tips for Large Game

Key Ashby penetration findings for large game are:

• Structural Integrity and anti-bend tip design,

• Good arrow flight: a well tuned setup,

• High FoC: greater than 20%

• Efficient broadhead with tanto (an Ashby term) tip,

• Single blade,

• Single bevel cutting edges,

• High mass arrow: greater than 650 grains

• Shaft Diameter less than Ferule Diameter,

• Tapered Shaft.

Obviously for medium and small game, these guidelines an be relaxed, or importantly be applied for lighter bows on medium sized game.