Material engineering, projectile deformation

In summary, the conversation discusses the difficulties of modeling ballistics, specifically in the area of terminal ballistics. The focus is on the deformation of projectiles when penetrating a medium, with algorithms and constants being used to approximate the deformation and retained mass. The conversation also addresses the complexity of modeling jacketed bullets and the importance of considering the shape of the projectile when determining stress distribution. The main question posed is how to calculate the deformed length of a projectile, particularly in the case of a solid rod.
  • #1
tuoni
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[EDIT] Would this maybe be more appropriate in "Engineering > Materials & Chemical Engineering"?

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; modeled 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 modeled 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 penetrating backwards over 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 modeled 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 possesses 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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  • #2


Thank you for sharing your work and seeking help from the scientific community. Your project sounds very interesting and it is great to see someone with a passion for ballistics and a determination to understand the science behind it.

Regarding your question about whether this topic would be more appropriate in "Engineering > Materials & Chemical Engineering", I believe it would be a good fit for that category. Material engineering is a crucial aspect of ballistics and I believe you would find more experts in that field who could provide you with valuable insights and suggestions.

As for your algorithms and calculations, I am not a material engineer and therefore cannot provide you with specific advice. However, I would recommend reaching out to material engineering forums or contacting experts in the field for further guidance. Additionally, you could also consider consulting with a professional material engineer who could help you with your model and calculations.

I wish you the best of luck with your project and hope you are able to find the answers and support you are looking for. Keep up the great work!
 
  • #3


As a scientist with a background in material engineering, I can offer some insights into this topic. Projectile deformation is a complex phenomenon that involves a combination of material properties, fluid dynamics, and structural mechanics. It is not surprising that you have encountered some difficulties in trying to model this process.

Firstly, I would like to commend you for your efforts in developing a ballistics model from scratch. It takes a lot of dedication and perseverance to learn about a new field and apply it to a practical problem. However, I would like to caution you that without a solid understanding of the underlying principles and theories, it may be challenging to develop an accurate and reliable model.

In terms of material engineering, there are a few key concepts that are relevant to projectile deformation. These include material strength, stress, and strain. Material strength refers to the ability of a material to withstand external forces without breaking or deforming. In your model, you have used the yield strength of the projectile and the medium to calculate the hydrodynamic velocity limit. This is a good start, but it is important to note that the yield strength is not the only factor that affects projectile deformation. The material's modulus of elasticity, which measures the stiffness of the material, also plays a crucial role. It determines how much the material will deform under a given amount of stress.

Additionally, it is important to consider the stress distribution within the projectile. As you mentioned, different nose shapes and jacket thicknesses can lead to different stress distributions, which in turn affect the deformation behavior. This is where your algorithm for calculating the yield force and drag force may need to be refined, as it assumes a uniform stress distribution. In reality, the stress distribution may vary along the length and diameter of the projectile, depending on its shape and composition.

Finally, I would like to address your question about calculating linear deformation. This is a challenging problem, as it involves both elastic and plastic deformation. For elastic deformation, the material will return to its original shape once the external forces are removed. However, for plastic deformation, the material will retain its deformed shape. In your model, you have assumed that the projectile will deform symmetrically, but in reality, the deformation may be more complex. The material's Poisson's ratio, which measures how the material changes in shape when subjected to stress, can also affect the deformation behavior.

In conclusion, material engineering plays a crucial role in understanding projectile deformation. It involves a combination of material properties, stress analysis,
 

1. What is material engineering?

Material engineering is a field of science and engineering that focuses on understanding the properties of materials and how they can be manipulated to create new and improved materials for various applications.

2. How does projectile deformation occur?

Projectile deformation occurs when a projectile, such as a bullet or a missile, impacts a target. The impact causes the projectile to change shape, often resulting in damage or destruction of the target.

3. What factors affect projectile deformation?

There are several factors that can affect projectile deformation, including the material properties of the projectile and the target, the speed and angle of impact, and the shape and design of the projectile.

4. How do material engineers study projectile deformation?

Material engineers study projectile deformation through experimental testing and computer simulations. They use specialized equipment and techniques to measure and analyze the deformation of projectiles and their effects on different materials.

5. What applications does material engineering have in relation to projectile deformation?

Material engineering has many applications in relation to projectile deformation, such as designing more effective armor materials, improving the performance of ammunition, and creating safer and more durable structures that can withstand impacts from projectiles.

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