Shooting Holes in Wounding Theories:The Mechanics of Terminal Ballistics |
V. Analytical Modeling of Terminal BallisticsV. A. IntroductionI love analysis, and ever since I have been interested in firearms I have been actively searching for theoretical models for interior, exterior and terminal ballistics. I soon discovered that accurate models for both interior and exterior ballistics have been available for some time. The Powley computer calculates optimal loads based on the IMR series of powders for specific cartridge capacities and pressure constraints and gives you a predicted velocity. Several exterior ballistics models are available in software packages compatible with PCs and I was able to write a very satisfactory one for myself based upon the information provided at the rear of my Sierra reloading manuals and published drag data. However, no one had a generally accepted model for terminal ballistics. The chief reason for this dearth of analytical joy is that there is no consensus on how to quantify terminal performance - or even on the phenomena contributing to wounding potential! This is surprising because such things have been available in the military sector for many years and are used to calculate the penetration of shaped charge jets and long rod penetrators through different armors. This, in spite of the fact that the mechanics of lethality, as distinguished from penetration mechanics, is debated in the military sector just as in the firearms community. If the firearms community could agree that it is the hole which counts (at least as a first order approximation), then something of this nature could be made applicable. The drawback to these penetration models, of course, is that they are extremely complicated and require great computational power to run. Something more readily available and understandable to the average shooter is what is needed. |
V. B. Fidelity in ModelingExperiments described in the preceding section on methods of experiment demonstrate several things about the terminal performance of bullets:
Any model which attempts to predict bullet penetration and cavitation must account for these observations. An understanding of what is taking place is a necessary starting point for developing a realistic model. In non-hydrodynamic (non-deforming) penetration the depth of penetration and cavity diameter are a linearly increasing function of velocity, since bullet diameter and mass are constant (note that the tissue in the target is in hydrodynamic flow until penetration stops). For deforming bullets, there is an initial phase of penetration in which no deformation occurs. This is usually termed the upset distance. Deformation occurs suddenly due to pressures exceding the flow stress of the bullet. The penetration of the upset distance is a reflection of the resistive forces (molecular friction) within the bullet. The hydrodynamic (deforming) penetration phase is very brief and at the conclusion of this phase the bullet will have traveled only a short distance. The stronger the bullet, the longer a distance it will travel while it deforms. Correspondingly, at higher velocities a bullet will travel a shorter distance as it deforms due to more rapid deformation. The final phase of penetration is non-hydrodynamic, starting at the threshold critical velocity, and accounts for most of the penetration and cavitation of a bullet. This explains why a deforming bullet has a nearly constant penetration over a wide range of impact velocities and frequently a diminishing penetration with increasing velocity. Penetration terminates when the velocity reaches a point where the flow stress of the tissue exceeds the stagnation pressure of the penetration. Elastic penetration may occur at this stage, wherein the bullet travels forward some distance before the tissue returns to its original position. This is similar behavior to temporary cavitation but occurs in the axial direction and is most commonly recognized in a bullet being found "lodged under the skin" on the far side of an animal. I am attempting to translate these behaviors into a model that will predict the performance of a bullet of prescribed characteristics. The most promising means appears to be some form of the Tate equations of penetration, but the difficult part of the development is in the expression which will describe the process of bullet deformation because this drives the whole event. My initial efforts have resulted in some encouraging results, but the ideal model of bullet expansion continues to elude me (at least one which will run quickly on a PC!). There are a lot of things going on, including strain hardening and failure, which simple analytical expressions do not address well. I will keep the reader apprised of any breakthroughs. What follows is my best effort so far. |
V. C. A Simple Analytical Model of Bullet PenetrationAt this location you will find the documentation for the analytical model of terminal ballistics described in the following figures. The discussion is highly technical and I won't repeat it here. While it may seem daunting, anyone with a decent grounding in science and high school calculus could write a simple numerical integration of these basic equations, as I did, to get similar results. There are some tricks of Fortran code development that I used (yeah, yeah, I'm a dinosaur - I never learned C or C++) and if I can get my aging compiler installed on my new computer I will post an executable for anyone to download who doesn't want to reinvent the wheel. As alluded to in the previous section, the real secret to making any analytical model really useful is using inputs that allow you to truly compare specific bullets of a particular design and construction and manufacture. What real use is it to compare hypothetical bullets? So, unfortunately, any analytical model needs empirical data, that is, test data. Now, as it turns out a few manufacturers have occasionally provided some such data, at least in gross terms. I gathered some information from the published photos of the Swift Scirocco and also from the North Fork Technologies bullets. Both give you expanded diameters and eroded lengths as a function of impact velocity. What is missing for the Scirocco is the retained weight, but you can sometimes get that kind of information from other sources. To be used in a model like this one, you also need the metallurgical data and in that regard, I had to make some assumptions, particularly related to work hardening. A fair amount of trial and error is required, but you can adjust the parameters until you see the model reflect the test results. We have to do this sort of calibration with any analytical model. Figure 1 below is an example of a plot of expanded diameter as a function of deformed length (which is a primary variable in the model and itself a function of impact velocity), made from data for a bonded core bullet. Never mind what brand, weight, style of bullet - this is for illustrative purposes. I did this with Microsoft Excel and you can see the third order polynomial function that was found by the regression algorithm. This is a very good fit, sometimes it not nearly so good and looks more like a roller coaster. In such cases you may need to artificially smooth the data or get yourself a better regression solution or even try a piecewise curve fit. When it comes to model development there are many ways to skin a cat. I like this approach because I can create an input deck and just read coefficients A, B, C, D of a polynomial of the form Ax^{3} + Bx^{2} + Cx + D. Note that the curve reaches a maximum and then decreases. You observe this kind of behavior in many bullets as the impact velocity increases. Figure 1: An Example Plot of the Function for Bullet ExpansionFigure 2 shows a similar plot for the function of retained weight as a function of deformed length. Here is a curve fit like a roller coaster. If you look at the data points it doesn't behave like that, but this was the closest fit that the Excel regression algorithm could achieve. This one isn't too awful. Its usable, but you're starting to introduce some error. An inflection point (or two) like this can introduce instability in a model, preventing convergence. That's where Fortran tricks come into play to keep the thing from going open-loop. Figure 2: An Example Plot of the Function for Bullet Retained WeightFigure 3 below is an example data table for the mechanical properties of the bullet from the tip to the base. This is where you capture things like the changing diameter of the bullet (the Tate equations assume a rod of constant diameter), variations in jacket thickness and variations in jacket hardness. This is important in calculating the expressions for composite density and yield strength as a function of deformed length. These would end up being two more plots with curve fits, which I have not shown. Figure 3: An Example Table of the Input Data for Mechanical PropertiesFigure 4: An Example Plot of the Wound Profile for a Conventional BulletFigure 5 shows a similar plot, this time for a bullet with a heavier jacket, but otherwise identical design features. It achieves slightly more penetration at typical velocities, but significantly greater penetration and appreciably reduced maximum wound diameter as the velocity approaches the minimum upset velocity. This is the sort of traditional soft copper and lead design that performs so well in classic cartridges of modest velocity. You can most clearly see that penetration steadily decreases with increasing impact velocity in this plot. Figure 5: An Example Plot of the Wound Profile for a Conventional BulletThis plot (Figure 6) is representative of the sort of behavior that you would see with a more modern jacketed bullet of complex design and metallurgy. It has a thin, tapered jacket at the tip for rapid expansion, but rapidly thickens to limit expansion. Additionally, it can only deform to a fixed length and the expanded diameter is well supported, so that it maintains a larger wound profile at depth in the target. Note how closely the higher velocities match in total penetration (582 mm and 562 mm). The lowest velocity is not far behind with 465 mm. By tuning the jacket thickness and metallugy one can move the depth at which the maximum expansion occurs and also control the diameter of that expansion. Although it is obscured, in this data the penetration is deepest in the middle range of impact velocity, consistent with most tests of controlled expansion bullets. This is the behavior of a bullet of nearly constant expanded diameter. Figure 6: An Example Plot of the Wound Profile for a Bonded Core BulletFinally, in Figure 7, we have the classic behavior of a monolithic style of bullet (eg, the Barnes X-Bullet). It achieves the greatest depth of penetration and exhibits the narrowest, most nearly uniformly tapering wound profile. This class of bullet reminds me very much of the most advanced form of shaped charge now in existence in its economy of penetration. Here the penetration depth is greater at the lower and higher impact velocities, as we see in testing. This is the behavior of a bullet of modest expanded frontal area that varies with impact velocity. It is also the sort of trend you see with a premium bullet like the Nosler Partition in smaller calibers and the Speer Grand Slam because the expanded mushroom is severely swept back at higher impact velocities. Figure 7: An Example Plot of the Wound Profile for a Monolithic BulletWhile none of these plots absolutely represents any particular bullet (even though I used real data where possible), they serve to illustrate that the model gives us the distinct behaviors of different design features and metallurgical properties. In short, the model replicates reality to a reasonable degree. There are obvious criticisms that may be leveled against the model I have presented here. I went to some length to show that a simplistic linear momentum based approach was not accurate in my discussion of the physics of terminal ballistics and some may think that I am now being found a hypocrite by promoting this set of equations. However, the careful observer will note that the momentum equation is only used in the hydrodynamic phase of penetration, where the simplification of quasi-fluid behavior can be used with reasonable confidence. Penetration past the point where the penetration velocity and the projectile velocity are the same, I have used an energy function based on a failure criterion and classical fluid drag parameters. More significantly, at every step of the event, I have been forced to rely on a separate analytical model for permanent hole dimensions. The linear momentum approach is quite incapable of providing this insight (without a dynamic finite element stress analysis) and it was a necessary input to the energy method that was employed in the rigid body phase of penetration. So, from my standpoint, the argument against simplistic linear momentum analysis has not been diminished. Stronger arguments might be raised against the analytical relations used (ie, the Tate equations themselves), but all I will say to this is that these approaches have been used with reasonable accuracy in the defense sector for some time. Are they the final word, the true equation of the governing mechanics? Probably not, but they work. Similarly, I have applied no constitutive models for the mechanical properties of the materials in the bullet or target, but there is nothing inherent in the method to prevent this, if one has a good model (such things are also strongly empirical in derivation and if you want strain hardening behavior, then the Hopkinson bar test data used needs to be in the same strain rate regime). The approach that I used is far simpler for the average person to implement because you can get your own data from wetpack tests. This model also fails to describe the upset distance because I don't have any relationship for the inertial effects of plastic strain. If anyone reading this is disgusted with my notion of a "simple" model of bullet penetration, understand that much of my motivation for developing and publishing this model was to illustrate just how difficult it is to accurately estimate real performance. This is why the kinds of shortcut calculations we've all seen fall flat when you get to specifics. I hope that anyone who has persevered to this extent can now appreciate why I have emphasized that simplistic answers are not meaningful, and tend only to reinforce generalizations which we knew already. |
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