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

[the Schwartz]

Simple, closed-form extensions of the Alekseevskii-Tate (AT) equation* for quasi-static rigid penetrators in semi-infinite ductile targets

Mandatory Excedrin Warning!

If the math required to balance your check book gives you a splitting headache, this is a really good point at which to stop reading. You've been warned. 😉


In the past, I’ve often relied upon the Alekseevskii-Tate (AT) pressure interface model, which is really nothing more than a modified Bernoulli pressure equation that describes the hydro-dynamic process of penetration and deceleration of projectiles impacting ductile target materials, to explain certain terminal ballistic phenomena. But, there are other applications for which the AT equation may be used. Although the underlying math is not terribly complicated, I thought that for those few who might be interested in availing themselves of the extended utility of the AT equation, I would take the time to offer some simple closed-form extensions of the AT model that will allow the determination of the ballistic limit velocity, V50, and maximum thickness of a ductile metal plate (for example, cold-rolled sheet steel panel typically found in automotive construction) that a bullet can defeat among other things.

The Alekseevskii-Tate pressure interface equation is—

½ρP(V-U)² + YP = ½ρTU² + RT

YP = projectile yield strength (Hugoniot Elastic Limit)
RT = target resistive strength
ρP = projectile density
ρT = target density


Mathematically manipulating the AT equation allows it to be used to determine the ballistic limit velocity (V50) for any projectile/plate pair, if the corresponding value of RT can be accurately derived. The greatest difficulty in such modeling is producing an accurate mathematical model of the target plate’s resistive strength, or RT, as it exists during hydro-dynamic flow conditions. The expansion of the cavity in the target material from a zero-radius results from the pressure/stress field created by the projectile’s impact. RT is strongly dependent upon the velocity of the projectile and determines the boundary (shown in red in the diagram below) of the resultant plastic/elastic flow field in the target that forms ahead of the projectile.

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With diminishing projectile impact velocity that approaches zero, RT increases asymptotically to a maximum value that is determined by a target flow field in which cavity expansion governed by a von-Mises yield surface is assumed. A critical part of any such analysis is the determination of α, the ratio of the target’s plastic zone to the radius of expanding cavity produced by the passage of the rigid-plastic projectile through the target. The calculation of α is non-trivial because it is intimately coupled with the constitutive model assumed for the target.

The following transcendental equation, which expresses the relationship of α to the density, shear (GT) and bulk (KT) moduli of the target material is—

[1 + (ρTU² + YT) · √(KT - ρTα²U²)] = YT · [1 + (ρTα²U² ÷ 2GT) · √(KT - ρTU²)].

Assuming a steady-state flow stress field in the target ahead of the projectile and setting 'm' as the slope of the intact yield strength/pressure curve (where m = ¾) of the target material, solution of the transcendental equation for α is—

α = √[(2√3GT) ÷ (2σyT + ½mρTU²)]

—where α must then be utilized in the computation of RT —

RT = (7 ÷ 3) · LN(α) · σyT

σyT = yield strength of the target material.

Because the one-dimensional AT extensions are extremely sensitive to effective projectile length (LEFF), it is important to determine that dimension correctly. Such an equation is easily constructed for that purpose. All one needs to supply is the density (ρ) of the projectile and the number of caliber radius lengths (n) contained within the nose of the projectile ogive. Computation is straight-forward.

LEFF = LOGIVE + LCYL = (n·r) + [(MTOTAL - (⅔πr²nrρP)) ÷ (πr²ρP)]

With the constitutive value of RT having been determined, the maximum thickness (T) of an armor plate that a projectile can defeat is computed as—

T = LEFF · [(ρPV²) ÷ (2RT · (1 + μ))]

where μ = √(ρT ÷ ρP)

—as well as the ballistic limit velocity (V50) for any projectile/plate pair—

V50 = √[(T · 2RT · (1 + μ)) ÷ (LEFF · ρP)]

The velocity at which the projectile-target pressure interface advances through the target is—

U = V ÷ [1 + √(ρT ÷ ρP)]

Another way of thinking of this behavior is imagining it as the velocity at which the bottom of the penetration cavity advances as the penetrator moves through the target material.

Of course, if RT > YP, then the projectile will not penetrate the target. The critical velocity (VCRIT) for penetration of the target is—

VCRIT = √[2 x (RT - YP) ÷ ρP]

—and erosion of the projectile nose occurs when—

VEROSION = √[2 x |YP - RT| ÷ ρT]

The transition velocity (UPF) for plastic flow within the target material ahead of the projectile occurs when—

UPF = √[HELT ÷ ρT]

The HEL (Hugoniot Elastic Limit) of the target is—

HELT = σyT [(1 - qT) ÷ (1 - 2qT)]

q = Poisson's ratio of the target material

Otherwise, the stress field ahead of the projectile remains within the elastic regime and the displaced target material rebounds after the passage of the projectile.


The final diameter of the penetration cavity, ØCAVITY, produced by the passage of the projectile through the target is—

ØCAVITY = ØPROJECTILE · √[YP ÷ RT) + ((2ρP(V-U)²) ÷ RT)]

After the penetration process is complete, the final length of the projectile, if it has eroded (that's the ''quasi-static'' part), is—

Lf = LEFF · e-x

where x = [(ρPV502) ÷ (2YP)]

and e is Euler's number ≈2.718281828459045




All units for these computations are in SI units of kilograms, meters, seconds.
Target material yield strengths, pressures, bulk and shear moduli are expressed in GPa (109 x Nm-2).
Target and projectile material densities are expressed in kgm-3.
Velocities are expressed in ms-1.


For those who are not so mathematically-inclined, here is some ANSYS™ ''goodness''—


https://www.youtube.com/watch?v=Vz3Oin5AU6o&feature=emb_logo




*A theory for the deceleration of long rods after impact, Tate, A.; Journal of the Mechanics and Physics of Solids, 1967




Note: A special ''Thank you!'' goes to BehindBlueI's for his indulgence in permitting me to exercise my ''inner OCD-demons'' by allowing me to correct my prior attempt at posting this topic.

[5pins]

https://4.bp.blogspot.com/-BiF9x4fXzj8/XDrRYcGMI5I/AAAAAAAAA4w/qqaSTZFs2jcoPyvtP_nYLLPaQ3y_T5c5QCEwYBhgL/s1600/79.%2BMath%2BMeme.jpg

[luckyman]

tS, how far away is a modern controlled expand bullet from the assumption of “quasi-static rigid penetrator”? Or are you only using this for non-expanding projectiles?

[the Schwartz]

tS, how far away is a modern controlled expand bullet from the assumption of “quasi-static rigid penetrator”? Or are you only using this for non-expanding projectiles?

Good question.

To your first question, the ability to penetrate ductile metallic targets exhibited by low-L/D aspect projectiles like those fired from pistols and rifles, whose length is usually no more than 2 - 5 times the projectile's diameter (DP), at typical ordnance velocities (300 - 1,100 ms-1) is limited. They do not penetrate more than a few bullet diameters into semi-infinite ductile targets due to entrance phase effects. Entrance phase effects dominate when the projectile begins to open the expansion cavity from a ''zero-radius'' at the surface of the target and dominate the penetration process to a depth of about 6DP. So, the answer to your first question, ''How far away is a modern controlled expand bullet from the assumption of “quasi-static rigid penetrator?” is, ''Not far at all.''

As for your second question, ''Or are you only using this for non-expanding projectiles?''...

Also a good question.

The AT pressure interface equation can be used for expanding projectiles.

Because deceleration during the entrance phase is not constant, there needs to be a quantitative assessment for the influence of the entrance phase, so we must first define an effective (average) resisting stress (REFF),—and an effective (average) deceleration—during the entrance phase that accounts for both the resistive effect of the target material and the lateral forces constraining the expansion of the projectile during the process of penetration.

In order to do so, it is necessary to determine the radial constraint for expansion of the projectile's nose by the target material using this equation:

ØCAVITY = DPROJECTILE · √[YP ÷ RT) + ((2ρP(V-U)²) ÷ RT)] = DP'

This is the ''new'' DP' that would be used in the quasi-static equations above and that follow.

Once that computation is made, the corresponding ''control volume''—"O" in the diagram—of the expanded/expanding projectile nose must be assumed to fill the lateral dimension of the expanding interface cavity as illustrated here—

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For hemispherical- and ogival-nose projectiles where b = 0.50 and 0.15 respectively, the equations for REFF and normalized penetration for hemispherical- and ogival-nosed projectiles are then—

REFF = bρTV2

Hemispherical

P/DP' = 1.2V4/3·[(ρPLEFF)÷(REFFDP')]2/3

Ogival

P/DP' = 1.77V·[(ρPLEFF)÷(REFFDP')]1/2

[luckyman]

Good question.

To your first question, the ability to penetrate ductile metallic targets exhibited by low-L/D aspect projectiles like those fired from pistols and rifles, whose length is usually no more than 2 - 5 times the projectile's diameter (DP), at typical ordnance velocities (300 - 1,100 ms-1) is limited. They do not penetrate more than a few bullet diameters into semi-infinite ductile targets due to entrance phase effects. Entrance phase effects dominate when the projectile begins to open the expansion cavity from a ''zero-radius'' at the surface of the target and dominate the penetration process to a depth of about 6DP. So, the answer to your first question, ''How far away is a modern controlled expand bullet from the assumption of “quasi-static rigid penetrator?” is, ''Not far at all.''

As for your second question, ''Or are you only using this for non-expanding projectiles?''...

Also a good question.

The AT pressure interface equation can be used for expanding projectiles.

Because deceleration during the entrance phase is not constant, there needs to be a quantitative assessment for the influence of the entrance phase, so we must first define an effective (average) resisting stress (REFF),—and an effective (average) deceleration—during the entrance phase that accounts for both the resistive effect of the target material and the lateral forces constraining the expansion of the projectile during the process of penetration.

In order to do so, it is necessary to determine the radial constraint for expansion of the projectile's nose by the target material using this equation:

ØCAVITY = DPROJECTILE · √[YP ÷ RT) + ((2ρP(V-U)²) ÷ RT)] = DP'

This is the ''new'' DP' that would be used in the quasi-static equations above and that follow.

Once that computation is made, the corresponding ''control volume''—"O" in the diagram—of the expanded/expanding projectile nose must be assumed to fill the lateral dimension of the expanding interface cavity as illustrated here—

65641

For hemispherical- and ogival-nose projectiles where b = 0.50 and 0.15 respectively, the equations for REFF and normalized penetration for hemispherical- and ogival-nosed projectiles are then—

REFF = bρTV2

Hemispherical

P/DP' = 1.2V4/3·[(ρPLEFF)÷(REFFDP')]2/3

Ogival

P/DP' = 1.77V·[(ρPLEFF)÷(REFFDP')]1/2

Thanks for the detailed reply.

Perfect result...my brain expanded but not to the point of exploding [emoji1]

[the Schwartz]

Thanks for the detailed reply.

Perfect result...my brain expanded but not to the point of exploding [emoji1]

Glad to hear it. The P-F sanitation engineers (Remember when they used to be called 'janitors'?) are busy enough as it is. ;)

[the Schwartz]

These videos illustrate the FEA simulation of a regime in which the projectile remains rigid through-out the penetration event—

https://www.youtube.com/watch?v=Tj5phpdWLMI

https://www.youtube.com/watch?v=LbyLoVjTR9o

[awp_101]

These videos illustrate the FEA simulation of a regime in which the projectile remains rigid through-out the penetration event—
[Beavis and Butthead]heheheheheh[/Beavis and Butthead]

[the Schwartz]

[Beavis and Butthead]heheheheheh[/Beavis and Butthead]

I should have known better. ;)

[4given]

Reminded me of this video:


https://www.youtube.com/watch?v=QfDoQwIAaXg

[the Schwartz]

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

[DueSpada]

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.

[the Schwartz]

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

[peterb]

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 :-)

[the Schwartz]

“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".

[the Schwartz]

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—

94910

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—

94911

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.