Towards an accurate, back-of-napkin mathematical model of the whole ceramic armor plate II: Knockdown and Uplift Adjustment to Reflect Dynamic Properties

As discussed in the previous post in this series, the baseline BE equation offers a good approximation of ceramic performance under most conditions, yet it assumes that the ceramic’s quasi-static mechanical properties carry over into the ballistic regime without drastic changes. This assumption generally holds true for common scenarios, including steel-cored AP threats impacting boron carbide, silicon carbide, or aluminum oxide at typical small-arms velocities. However, certain specific conditions can undermine that assumption.

Two key scenarios stand out for their common nature. The first involves boron carbide (B4C) and its stress-induced amorphization under very high-impact pressures, commonly encountered when facing tungsten-cored or tungsten carbide-cored penetrators at sufficiently high velocities. The second scenario involves any ceramic (not just B4C) whose hardness (H) is substantially lower than that of the incoming penetrator core. In both cases, the ceramic cannot maintain its effective hardness and compressive strength during the crucial dwell period, and may not be able to sufficiently wear the projectile’s penetrator via abrasion, resulting in deeper penetration than the simple BE equation would predict.

To handle these cases, we have to introduce a “knockdown” adjustment factor. This factor does not alter the fundamental form of the BE equation; what it does, instead, is create a template for the modification of mechanical property inputs. By adjusting hardness and compressive strength downward under certain conditions, we can then capture the real-world degradation in ballistic performance due to non-quasi-static effects.

1. B4C Amorphization

Boron carbide’s crystal structure can undergo a phase transformation (amorphization) under high shear stresses and pressures, especially when impacted by tungsten or tungsten carbide penetrators at very high velocities. This transformation dramatically reduces B4C’s local effective hardness and, consequently, its ability to erode the incoming projectile. Empirical studies demonstrate that, under these conditions, B4C’s effective compressive strength and hardness can fall significantly below their nominal quasi-static values.

We represent this empirically with a knockdown factor λ, selected based on ballistic test data. When conditions indicate that amorphization will occur (e.g., tungsten-cored threat at >900 m/s impact velocity), we reduce the affected terms accordingly:

H_amorph = H × (1 − λ)

CS_amorph = CS × (1 − λ)

For example, if testing shows a 30% drop in effective mechanical properties upon amorphization, we set λ = 0.3, resulting in H_amorph = 0.7H and CS_amorph = 0.7CS.

2. Hardness Mismatch with High-Hardness Penetrators

Even without undergoing a phase change like B4C’s amorphization, a ceramic that is substantially softer than the penetrator becomes less effective at blunting and eroding the projectile. For steel-cored AP projectiles (hardness ~750–800 HV), ceramics like SiC or B4C (≥2600 HV) are usually harder, so no knockdown is required. However, if the penetrator core hardness equals or exceeds that of the ceramic, the erosive mechanism diminishes significantly. This often occurs when Al2O3, AlN, MgAl2O4, and other oxide ceramics and glasses face tungsten carbide-cored threats.

In these scenarios, we apply a partial knockdown factor scaled to the hardness ratio Hp/H, where Hp is projectile hardness. Specifically, if Hp > H, define a hardness mismatch factor δ that increases as Hp/H grows. For instance, if Hp is 10% harder than the ceramic, we might reduce the ceramic’s effective hardness by approximately 20%. This relationship can be linear or defined using an experimentally tuned parameter. For example:

δ = α × (Hp/H − 1), where α is chosen based on empirical data.

Then we adjust:

H_adj = H / (1 + δ)

CS_adj = CS / (1 + δ)

Thus, as the penetrator overmatches the ceramic in hardness, we proportionally reduce the ceramic’s effective mechanical properties in the BE equation.

3. Strain-Rate-Enhanced Ductility in Nominally Brittle Ceramics

Not every deviation from quasi-static behavior negatively impacts performance. A subset of ceramics – including aluminum nitride (AlN), beryllium oxide (BeO), magnesia (MgO), and certain perovskites and MAX phases – exhibit measurable plasticity at ballistic strain rates once pressures exceed approximately 5–10 GPa. Under these conditions, their compressive strength notably increases due to pressure-hardening, and limited dislocation glide allows these ceramics to absorb additional work before fracture. Historical penetration tests have documented cases such as AlN surpassing both B4C and SiC performance when impact velocities exceed approximately 1.7 km/s. High-pressure mechanical data similarly indicate substantial increases in strength for BeO and AlN under dynamic loading conditions.

To capture this effect, we introduce an uplift factor ϕ that scales the quasi-static mechanical inputs upward for ceramics known to strengthen under relevant impact conditions:

CS_dyn = CS × (1 + ϕ)

H_dyn = H × (1 + ϕ)

When to apply ϕ:  (1) When impact velocity exceeds ~1.2 km/s and estimated contact pressures surpass approximately 8 GPa.  (2) The ceramic is among the known ductility-enhanced ceramics (high-purity AlN, BeO, MgO, SrTiO3, Ti3SiC2, etc.). (3) The penetrator does not significantly exceed ceramic hardness; otherwise, the hardness-mismatch knockdown factor δ remains dominant.

Typical magnitudes:  Published dynamic compression and lateral-confinement tests report dynamic-to-static strength ratios between approximately 1.3 and 2.0 for ceramics such as AlN and BeO within 10–20 GPa pressure ranges. Conservatively setting ϕ = 0.4 (40% uplift) is reasonable for AlN impacted by steel-cored AP projectiles at approximately 1.6 km/s, while values up to ϕ = 0.8 may be appropriate for BeO at higher velocities. Where direct experimental data are unavailable, extrapolate uplift estimates from logarithmic fits of dynamic strength versus strain-rate. (For e.g., split-Hopkinson data.)

Interaction with λ and δ: The uplift factor ϕ should be applied after any hardness-mismatch reduction (δ) adjustments, but in place of amorphization adjustments (λ), as ductility uplift does not apply to ceramics experiencing detrimental amorphization effects.

Example: High-Velocity AlN vs. Steel-Cored AP

Consider a sintered AlN tile (density 3.26 g/cm³, CS = 3 GPa, hardness = 1400 HV) impacted by tungsten alloy penetrators at two velocities:

At 900 m/s (moderate velocity), we set ϕ = 0:

CS_dyn = 3.0 GPa, H_dyn = 1400 HV.

At 1650 m/s (high velocity), we set ϕ = 0.4:

CS_dyn = 4.2 GPa, H_dyn = 1960 HV.

Inserting these adjusted properties into the BE equation predicts roughly a 12–15% improvement in ballistic effectiveness, aligning closely with published historical depth-of-penetration data, where AlN performance surpasses B4C at these elevated velocities.

4. Velocity and Threat Type Dependencies

Knockdown and uplift adjustments ideally scale with specific impact conditions. At lower velocities, B4C amorphization may not meaningfully occur, making λ negligible. As velocity rises and tungsten-based penetrators become relevant, both λ and δ should increase to reflect growing mismatches and the onset of amorphization. Conversely, uplift factor ϕ becomes significant only when impact pressures and velocities are high enough to activate ceramic ductility. Experimental ballistic test data must guide the selection of thresholds and scaling parameters.

Adjusted BE Calculation

After applying knockdown (λ, δ) or uplift (ϕ) adjustments to hardness and compressive strength, the BE formula remains unchanged structurally. For example, for B4C impacted by a tungsten-cored penetrator:

BE_knockdown = 1.677 × D + 5 × ((T × D)/D) + 0.003 × CS_amorph + 0.005 × H_amorph

For hardness mismatch in another ceramic, such as Al2O3:

BE_knockdown = 1.677 × D + 5 × ((T × D)/D) + 0.003 × CS_adj + 0.005 × H_adj

For strain-rate-enhanced ceramics (e.g., AlN at very high velocities):

BE_uplift = 1.677 × D + 5 × ((T × D)/D) + 0.003 × CS_dyn + 0.005 × H_dyn

These adjusted property inputs more accurately reflect real-world high-rate behaviors within the established BE framework.  Just please note that all of this is intended as a simple toy model of ceramic performance in armor systems – a model that a human without a computer can handle with ease – and that, while it’s generally very accurate, it simplifies matters quite dramatically.