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, V_{50}, 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_{T}U² + 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 (V_{50}) 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.

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_{T}U² + Y_{T}) · √(K_{T}- ρ_{T}α²U²)] = Y_{T}· [1 + (ρ_{T}α²U² ÷ 2G_{T}) · √(K_{T}- ρ_{T}U²)].

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_{T}U²)]

—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_{P}V²) ÷ (2R_{T}· (1 + μ))]

where μ = √(ρ_{T}÷ ρ_{P})

—as well as the ballistic limit velocity (V_{50}) for any projectile/plate pair—

V_{50}= √[(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_{P}V_{50}^{2}) ÷ (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 (10^{9}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''—

*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 toBehindBlueI'sfor his indulgence in permitting me to exercise my ''inner OCD-demons'' by allowing me to correct my prior attempt at posting this topic.