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Thread: Simple closed-form extensions of the Alekseevskii-Tate equation for rigid penetrators

  1. #11
    That's an interesting video, especially the portion that starts at 4:54 that shows the hydrodynamic flow phase of the JHPs deforming in the first few inches of gelatin block. Thanks for posting it. I have seen it many times and am always fascinated by it.

    The Alekseevskii-Tate model can be used to predict not only the maximum terminal penetration depth in 10% ordnance gelatin but also the size of the temporary cavity (using equations derived from the AT pressure-interface equation to model CCE* or SCE** theory), but the math involved is quite complicated making it a bit much to try to post here. What I find mesmerizing are the high frame rate captures of the lead pellets hitting the passing projectiles starting at 2:00 in the video. Those impacts are a brilliant display of how metals flow like water when subjected to the extremely high pressures that occur when metals collide at ordnance velocities. Surprising to most folks unfamiliar to the field is the fact that the viscosity of those flowing metals is only slightly greater than that of water.

    *cylindrical cavity expansion
    **spherical cavity expansion
    ''Politics is for the present, but an equation is for eternity.'' ―Albert Einstein

    Full disclosure per the Pistol-Forum CoC: I am the author of Quantitative Ammunition Selection.

  2. #12
    ½ρP(V-U)² + YP = ½ρTU² + RT
    α = √[(2√3GT) ÷ (2σyT + ½mρTU²)]
    [1 + (ρTU² + YT) · √(KT - ρTα²U²)] = YT · [1 + (ρTα²U² ÷ 2GT) · √(KT - ρTU²)]
    and e is Euler's number ≈2.718281828459045

    I was just thinking that.

  3. #13

    Update

    Although I've been using the Alekseevskii-Tate hydro-dynamic equation to model rifle projectile penetration in ductile target materials (metals, armor systems), recently, I became interested in using the AT equation to model penetration in transparent armor systems like Plexiglas (polymethylmethacrylate), Lexan (polycarbonate), and ALON (aluminum oxynitride). Almost immediately, I was rewarded with three examples confirming the AT model's validity in modeling terminal ballistic behavior in these types of materials.

    The first example is found in the book, Terminal Ballistics, by Zvi Rosenberg and Erez Dekel, 3rd Ed. (2020). On page 92, the authors describe an experiment in which they fired .50 BMG hardened steel penetrator cores having a diameter of 0.4331'' weighing 25.92 grams (400 grains) into Plexiglas blocks at 900 ms-1. The .50 BMG AP cores did not deform and penetrated to a maximum depth of 190mm (7.48 inches).

    Setting the material parameters for the Plexiglas target as:
    ρ = 1.188 g/cc
    σy = 80 MPa
    E = 3.300 GPa
    K = 5.500 GPa
    v = 0.40

    and for the 0.4331'' hardened steel AP core as:
    ρ = 7.83 g/cc
    σy = 1572 MPa
    E = 213.3 GPa
    K = 165.5GPa
    v = 0.400

    The AT model predicts a terminal penetration depth of 189.96mm as opposed to the experimental result of 190mm.

    ==========

    The second example is found in the technical paper by Dorogoy, A., Rittel, D., Brill, A., ''Experimentation and modeling of inclined ballistic impact in thick polycarbonate plates", Int. J. Impact, 32;10, pages 804 - 814, Oct. 2011. Dorogoy fired 7.62mm AP projectiles weighing 7.45 grams (115 grains) into polycarbonate cylinders that were 243mm (9.57 inches) long and 127mm (5.00 inches) in diameter at 754 ms-1. The projectiles did not deform and penetrated to a depth of 138mm (5.433 inches).

    Setting the material parameters for the polycarbonate target as:
    ρ = 1.195 g/cc
    σy = 40 MPa
    E = 2.200 GPa
    K = 2.821 GPa
    v = 0.370

    and for the 7.62 projectile as:
    ρ = 8.885 g/cc
    σy = 1572 MPa
    E = 213.3 GPa
    K = 165.5 GPa
    v = 0.285

    The AT model predicts a terminal penetration depth of 136.93mm as opposed to the experimental result of 138mm. That's pretty darned good.

    ==========

    Finally, the Surmet Corporation, in conjunction with the Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, published a technical paper, ''Recent Advances in ALONTM Optical Ceramic'' that lists the ballistic limit velocity of V50 = 4,400 fps for a transparent ALON armor system having a thickness of 1.60 inches struck by a .50 BMG AP projectile weighing 649 grains.

    In this case, the ALON target parameters are:
    ρ = 3.688 g/cc
    σy = 700 MPa
    E = 321.050 GPa
    K = 223.419 GPa
    v = 0.2605

    and for the .50 BMG AP projectile as:
    ρ = 8.885 g/cc
    σy = 1572 MPa
    E = 213.3 GPa
    K = 165.5 GPa
    v = 0.285

    The AT model predicts a V50 = 4,418.67 fps as opposed to the experimental result of V50 = 4,400 fps for a transparent ALON armor system having a thickness of 1.60 inches struck by a .50 BMG AP projectile weighing 649 grains
    ''Politics is for the present, but an equation is for eternity.'' ―Albert Einstein

    Full disclosure per the Pistol-Forum CoC: I am the author of Quantitative Ammunition Selection.

  4. #14
    Quote Originally Posted by the Schwartz View Post
    The AT model predicts a terminal penetration depth of 189.96mm as opposed to the experimental result of 190mm.

    The AT model predicts a terminal penetration depth of 136.93mm as opposed to the experimental result of 138mm. That's pretty darned good.
    “All models are wrong, but some are useful.” —George Box :-)

  5. #15
    Quote Originally Posted by peterb View Post
    “All models are wrong, but some are useful.” —George Box :-)

    That's an interesting quote. While I agree with the sentiment generally, I think that the use of the adjective "wrong" is a little too absolute in the semantic sense. Reckon I'd go with "imperfect".
    ''Politics is for the present, but an equation is for eternity.'' ―Albert Einstein

    Full disclosure per the Pistol-Forum CoC: I am the author of Quantitative Ammunition Selection.

  6. #16
    For anyone interested in a simple armor penetration model (rigid penetrator) that takes into account the inertial properties of the target material, the Forrestal-Warren model—which relies upon CCE (cylindrical cavity expansion) theory—does an excellent job of modeling such projectile/target pair interactions so long as there is minimal erosion of the penetrator—

    Name:  Forrestal-Warren Rigid Penetrator Model with Anedrson-Walker Solution for Rt and α.jpg
Views: 147
Size:  28.5 KB

    The set of equations (in black) at the top of the sheet is the Forrestal-Warren quasi-static rigid/non-deforming penetrator model. N is a nose shape factor. 'rp' is the number of projectile radii in the length of the projectile's nose.

    The set of equations (in blue) in the lower portion of the sheet covers the Anderson-Walker solution for Rt (dynamic target resistance with respect to projectile velocity) using the Anderson-Walker modification of the transcendental equation for α which is an expression of the extent of the plastic zone in the target ahead of the penetrator nose.


    For the sake of convenience, I've also included conversions for α expressed as a function of the target material's bulk modulus (Kt), elastic modulus (Et), and shear modulus (Gt) in the following detail for use in computing the velocity-dependent dynamic resistance (Rt) of the target—

    Name:  Anderson-Walker Solution for Rt and α.jpg
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Size:  22.9 KB

    Assuming steady-state one-dimensional flow, 'm' is the slope of the intact yield strength/pressure curve for the target material.

    'm' is 1.00 for most brittle target materials (e.g.: ceramics, glass, etc.) and ¾ for ductile target materials (e.g.: metals).

    'V' is the 'tail velocity' of the rigid penetrator, 'u' is the penetration velocity of the projectile/target pair interface at the bottom of the penetration cavity.

    Yt is the yield strength of the target, 'Gt' is the shear modulus of the target, 'ρp' and 'ρt' is the respective density of the penetrator and target. P/L is normalized penetration depth.
    Last edited by the Schwartz; 09-27-2022 at 12:58 PM.
    ''Politics is for the present, but an equation is for eternity.'' ―Albert Einstein

    Full disclosure per the Pistol-Forum CoC: I am the author of Quantitative Ammunition Selection.

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