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

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—

½ρ_P(V-U)² + Y_P = ½ρ_TU² + R_T

Y_P = projectile yield strength (Hugoniot Elastic Limit)
R_T = target resistive strength
ρ_P = projectile density
ρ_T = 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 R_T can be accurately derived. The greatest difficulty in such modeling is producing an accurate mathematical model of the target plate’s resistive strength, or R_T, 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. R_T 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.

Attachment 65580

With diminishing projectile impact velocity that approaches zero, R_T 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 (G_T) and bulk (K_T) moduli of the target material is—

[1 + (ρ_TU² + Y_T) · √(K_T - ρ_Tα²U²)] = Y_T · [1 + (ρ_Tα²U² ÷ 2G_T) · √(K_T - ρ_TU²)].

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√3G_T) ÷ (2σ_yT + ½mρ_TU²)]

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

R_T = (7 ÷ 3) · LN(α) · σ_yT

σ_yT = yield strength of the target material.

Because the one-dimensional AT extensions are extremely sensitive to effective projectile length (L_EFF), 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.

L_EFF = L_OGIVE + L_CYL = (n·r) + [(M_TOTAL - (⅔πr²nrρ_P)) ÷ (πr²ρ_P)]

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

T = L_EFF · [(ρ_PV²) ÷ (2R_T · (1 + μ))]

where μ = √(ρ_T ÷ ρ_P)

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

V₅₀ = √[(T · 2R_T · (1 + μ)) ÷ (L_EFF · ρ_P)]

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

U = V ÷ [1 + √(ρ_T ÷ ρ_P)]

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 R_T > Y_P, then the projectile will not penetrate the target. The critical velocity (V_CRIT) for penetration of the target is—

V_CRIT = √[2 x (R_T - Y_P) ÷ ρ_P]

—and erosion of the projectile nose occurs when—

V_EROSION = √[2 x |Y_P - R_T| ÷ ρ_T]

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

U_PF = √[HEL_T ÷ ρ_T]

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

HEL_T = σ_yT [(1 - q_T) ÷ (1 - 2q_T)]

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, Ø_CAVITY, produced by the passage of the projectile through the target is—

Ø_CAVITY = Ø_PROJECTILE · √[Y_P ÷ R_T) + ((2ρ_P(V-U)²) ÷ R_T)]

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

L_f = L_EFF · e^-x

where x = [(ρ_PV₅₀²) ÷ (2Y_P)]

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=Vz3O...ature=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.

https://4.bp.blogspot.com/-BiF9x4fXz...ath%2BMeme.jpg

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?

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Originally Posted by 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?

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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 (D_P), 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 6D_P. 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 (R_EFF),—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:

Ø_CAVITY = D_PROJECTILE · √[Y_P ÷ R_T) + ((2ρ_P(V-U)²) ÷ R_T)] = D_P'

This is the ''new'' D_P' 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—

Attachment 65641

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

R_EFF = bρ_TV²

Hemispherical

P/D_P' = 1.2V^4/3·[(ρ_PL_EFF)÷(R_EFFD_P')]^2/3

Ogival

P/D_P' = 1.77V·[(ρ_PL_EFF)÷(R_EFFD_P')]^1/2

Quote:

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Originally Posted by the Schwartz

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 (D_P), 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 6D_P. 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 (R_EFF),—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:

Ø_CAVITY = D_PROJECTILE · √[Y_P ÷ R_T) + ((2ρ_P(V-U)²) ÷ R_T)] = D_P'

This is the ''new'' D_P' 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—

Attachment 65641

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

R_EFF = bρ_TV²

Hemispherical

P/D_P' = 1.2V^4/3·[(ρ_PL_EFF)÷(R_EFFD_P')]^2/3

Ogival

P/D_P' = 1.77V·[(ρ_PL_EFF)÷(R_EFFD_P')]^1/2

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Thanks for the detailed reply.

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

Quote:

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Originally Posted by luckyman

Thanks for the detailed reply.

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

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

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

Quote:

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Originally Posted by the Schwartz

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

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

Quote:

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Originally Posted by awp_101

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

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I should have known better. ;)

Reminded me of this video:

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