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Introduction

[Please take time to read through the entire post, especially if you know material engineering.]

Material engineering is NOT one of my "specialities" and I actually know quite little. Hopefully someone will be able to help me, or at least provide a few pointers, suggestions, or thoughts.

I have already posted a few posts here dealing with various things related to ballistics. For a few years I have been working on a ballistics model, basically starting from scratch, knowing hardly anything at all about ballistics and firearms (aside from what I knew from games).

For internal ballistics I have the "Advanced Coppock Internal Ballistics Model" that I'm still trying to understand (in the solved numerical form it works, but I'm trying to work out the integrations so I can make more use of it), and external ballistics is pretty much complete (by far the simplest of the three).

Terminal ballistics is the tricky one, have not studied material engineering, and I have no decent material to learn/work from. I have been able to come up with something that looks pretty good (and realistic), but with its own problems (difficulty in determining some of the variables).

To simplify things, deformation is assumed to be symmetric (projectiles are 2-d shapes rotated around their longitudinal axis), so it should be possible to treat it as a "simple" 2-dimensional problem. That is how my model works at the moment: when penetrating a medium, an overall stress is exerted on the projectile, which results in a deformation in length, which in turn results in a deformation in diameter and loss of mass.

Thus projectiles are treated as point-mass-bodies (rather than just point-masses; but still not full-blown simulations), I wasn't satisfied with treating all bullets as non-deforming projectiles, or using some arbitrary system to model the "wounding potential" of deforming projectiles.

Algorithms

Using the following simple algorithm:

[tex](1) \;\; \sigma = \frac{F}{A}[/tex]

which can also be expressed as:

[tex](2) \;\; \sigma = \frac{\rho v^{2}}{2}[/tex]

the following approximations are found:

[tex](3) \;\; v_{limit(h)} = \sqrt{\frac{2(\sigma_{p(TY)} - \sigma_{m(SY)})}{\rho_{m}}}[/tex]

v_limit(h)— hydrodynamic velocity limit (m/s); velocity at which the projectile begins to deform; projectile always deforms if medium strength is equal to or greater than the projectile strength.

σ_p(TY)— tensile yield strength of projectile (Pa, N/mm^2); projectile is assumed to deform primary in tensile erosion.

σ_m(SY)— shear yield strength of medium (Pa, N/mm^2); a solid or semisolid medium has not only fluid strength but static strength as well, thus the overall stress is the combination of medium shear strength (σ_p(TY) - σ_m(SY)) and fluid stress (dynamic pressure). [a1] For a true fluid σ_m(SY) is always 0.

ρ_m— mass density of medium (kg/m^3).

[tex](4) \;\; v_{limit(e)} = \sqrt{\frac{2\sigma_{m(SY)}}{\rho_{m}}}[/tex]

v_limit(e)— elastic velocity limit (m/s); velocity at which the elastic properties of a solid or semi-solid medium is able to withstand the stress of a penetrating projectile, thus ceasing penetration; in an elastic medium the projectile may travel further relative to an external frame of reference (medium is stretched), however, it will not penetrate any further into the medium (projectile bounces back to its position at the elastic limit).

[tex](5) \;\; F_{yield} = \frac{\sigma_{m(SY)}\pi d_{h}^{2}}{4}[/tex]

F_yield— yield force (N); force required to induced yielding in a solid or semisolid medium; the model does not include material behaviour beyond the yield point.

d_h— diameter of yielded (crushed) mass (m); may be greater than the actual projectile diameter in tissue and other elastic media.

[tex](6) \;\; F_{drag} = \frac{k_{drag}\rho_{m}v_{s}^{2}\pi d_{x}^{2}}{8}[/tex]

F_drag— drag force (N); force required to overcome fluid forces of fluid or yielded medium.

k_drag— drag constant (n/1); dimensionless constant determined by shape and Mach ratio; expresses the efficiency of a body to move through a fluid medium.

v_s— velocity of projectile front (m/s); as a projectile deforms, there arises a velocity gradient between the rear and front of the projectile; the determinant velocity is the actual penetration velocity, i.e. that of the projectile front. [a2]

d_x— deformed diameter (m); the actual deformed diameter of the projectile; modelled primarily as a function of deformed length (L_x).

Additional algorithms

[tex](a1) \;\; \sigma_{1} = \sigma_{m(SY)} + \frac{\rho_{m}v_{s}^{2}}{2}}[/tex]

[tex](a2) \;\; v_{s} = \frac{st - s_{0}t}{t}[/tex]

s— length of current step (m)

s_0— length from previous step (m)

t— time step of integration (RK4)

[tex](a3) \;\; a_{1} = \frac{F_{yield} + F_{drag}}{m_{x}}[/tex]

a_1— overall deceleration (m/s^2)

m_x— retained mass (kg)

Deformation constants

Like the drag constant, constants for deformed length (k_L(x)), deformed diameter (k_d(x)), and retained mass (k_m(x)) are expressed in the generic form:

[tex](7) \;\; k = ax^{b}[/tex]

and calculated over the range 0.0 (no deformation, or fragmentation (all mass lost)) to 1.0 (deformation or fragmentation (overdeformation)).

To find the final value, simply multiply it by the corresponding constant. E.g. if a projectile with diameter 9 mm, and at a deformed length of 0.3 has k_d(x)=1.25 and k_m(x)=1.00, the final deformed diameter will be 11.25 mm and it will have retained all its mass.

The problem

The following illustration depicts 8 bullets, the left column being monometallic solid bullets, and the right column jacketed bullets, and there are 4 different nose designs.

http://wb.bob.fi/domains/flonne.org/temp/material.png

For the monometallic bullets, deformation is fairly simple, with the shape of the bullet being the only variable of concern. With constant stress, the nose will have the quickest deformation (change in the rate of deformation depends on bullet shape) due to reduced area. From deformed length it should then be possible to calculate an approximate deformation of diameter as a function of shape, especially the nose. The retained mass should also be possible to approximate from deformed length, deformed diameter, and possible one or two other variables.

Since I am not very good at material engineering, I am unsure exactly how to calculate the deformed length, even with such a simple case as a solid rod. Do I need to have something like a tensile modulus, or can I somehow simply use the tensile strength?

For the jacketed bullets, deformation becomes more complex. I think that now shape is even more important, to determine how the stress is distributed over the nose: sharp noses probably very close to the tip, round noses equidistant from the point, and flat at the sides; in a hollow nose/point the stress would be forced to the sides producing the distinctive "flowering" seen in many "Hollow Point" bullets. Jacket thickness and type (entire nose covered, or exposed, leading to lessened strength of jacket) would also be very important to how the stress would be distributed.

Conclusion

The primary problems are how to calculate linear deformation (the problem is simplified to a 2-dimensional problem, and with each deformation modelled separately, rather than creating a more complicated (and exhaustive) 3-dimensional model with a very accurate modelling), given stress over time, how much would a bullet (simply a solid rod with varying diameter) deform. From deformation of length and how stress is distributed (as a function of shape; and varying as shape changes).

Now that I think about it, deformation of diameter (deflection of mass to the sides) may actually add to the deformed length...

Would it maybe be possible to make a simple approximation by first approximating the strength of the jacket alone (material and shape), and then calculate how this strength increases (?) as the empty middle is filled with a particular material (e.g. lead)?

Some possible variables:

1. Jacket alone

- jacket thickness
- jacket material
- nose shape (flat, cone, ellipse, tangent, secant, power, parabola, Haack)
- truncation (flat, cone, sphere, ovoid)
- fineness ratio (nose length/nose radius)
- integral jacket? or "compromised" at nose or base?

results in "structural" tensile strength over deformed length (Pa, N/mm^2)

2. Filling (for jacketed bullet), or Solid (for monometallic solid bullet)

- material
- bonding? (does any jacket sliding occur)
- shape, cavities

results in addition to "structural" tensile strength, or overall tensile strength for monometallic solid bullets (should probably be the same as σ_p(TY)) (Pa, N/mm^2)

Furthermore there is also a third problem, inert cores (e.g. steel, tungsten), however, I think this is a little simpler...I think that the inert core does not contribute to "structural" strength in the same manner. As a bullet is penetrating a medium, and consequently deforming, the inert core would probably contribute by the softer filling penetratingbackwardsover the very strong core. The bullet penetrates forward through a medium, the softer filling deformes backwards, and the harder core penetrates forward through the filling that is moving backwards over the core. Or something like that.

3. Inert core

- material
- diameter, nose shape
- drag constant

should probably be modelled as a separate penetration

The end result would be diagrams something like this (not usable values, just what it would look like):

http://wb.bob.fi/domains/flonne.org/temp/7.62x39%20mm%20Elizarov-Sjomin,%20M43,%20PS.png

http://wb.bob.fi/domains/flonne.org/temp/diag_k_drag.png

http://wb.bob.fi/domains/flonne.org/temp/diag_Lx.png

http://wb.bob.fi/domains/flonne.org/temp/diag_dx.png

http://wb.bob.fi/domains/flonne.org/temp/diag_mx.png

Is anyone able to help me, provide some pointers, or add anything else to this discussion/plea for help? It has become quite frustrating that I have finally come this far, and now I have run into a wall and been unable to progress any further for a few months.

I can often get pretty technical and delve pretty deep, before I simplify the algorithms and find the final solution ^_^ This just makes it all the more realistic. What makes it worse is that I am only an amateur and do not possess the amount of skill and knowledge an educated professional would have, I also don't have the possibility of using such advanced things like CFD or other simulation software (also wouldn't work with my utlimate goal of creating a simple yet efficient and realistic approximation, but within limits, like F=ma: great for everyday things, but not always enough for more advanced things, or things of different mangitudes). A "commoner" should be able to painlessly use it, and I'm also planning on using it in a game of mine, which is also why I first began, I became hooked a few months later ^.^

Please, please, please! If you have any clues, or any hunches, let me know. Seems most of this stuff is primarily insider material T_T Damn scientists/engineers need to learn to share! RIP Carl Sagan.

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# Material engineering, projectile deformation

Can you offer guidance or do you also need help?

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