Why Are Thin Layers Permeable?

1. The question of a layered target

In a single uniform target, penetration depth follows from the equation in PRT-01. Protective envelopes, however, are layered: a hard face, an energy-absorbing core, a load-bearing back. The projectile loses speed in each layer and enters the next one more slowly. The question is how to compute that loss.

2. Young's layering method

Young (1997, section 6.2) proposes the following steps for concrete structures. For each layer, a depth D_n is computed as if the projectile entered a semi-infinite medium of that layer at its entry velocity. If D_n is less than the layer thickness, the projectile stops in that layer. Otherwise constant deceleration is assumed through the layer and the exit velocity follows:

The S-number of fill between two concrete layers is halved (the fill is confined between two hard faces).

3. The method contradicts its own equation

The method has an internal consistency check: if we slice a uniform concrete into thin layers and apply the layered method, we should recover the single-block result. We do not. The single-block penetration of a ~14 t, 0.8 m diameter projectile (295 m/s) into concrete with S = 0.6 is —; if the first 0.5 m of the same concrete is sliced:

The reason: in the main equation depth grows linearly with velocity (D ∝ V − V₀), whereas layering assumes constant deceleration within a layer (an energy-proportional loss). The two assumptions assign different remaining capacity to the same projectile: layering leaves a little extra penetration capacity in every slice. In a single thick block the effect is small; across many thin layers it accumulates.

4. A self-consistent alternative

A layering consistent with the main equation can also be written: each layer subtracts exactly its own thickness from the projectile's remaining penetration capacity, and the exit velocity is the one that gives that remaining capacity. This form returns the same result however a uniform medium is sliced (the single-block value in the table above). This is our alternative, not Young's method; only layered-target tests can tell which is closer to reality. Below, the two laws are shown together as an uncertainty band.

5. Cratering and spalling: the thin-layer penalty

A nose entering a concrete layer does not decelerate fully until it is completely embedded (the "cratering" phase); as it approaches the exit face, deceleration falls again (the "spalling" phase). Young accounts for this by reducing the effective thickness of the layer. If the layer is thicker than the nose length and the projectile diameter (a thick layer), the effective thickness is T − L_n/K₃; otherwise (a thin layer):

K₃ depends on whether there is air beside the layer: 2 with air on both sides, 3 with air on one side. When the nose is longer than the layer, the penalty is large: —

6. Two text issues in the report

(1) For a layer with air on neither side the report states "K₃ = K", which is not a number. Since the series decreases monotonically (two free faces → L_n/2, one → L_n/3) and cratering and spalling are free-surface phenomena, we interpret it as ∞ (no correction); this is an interpretation. (2) Equation 6.9, which should be the deceleration-free form of 6.6, contains (T − 1/K₃) instead of T(1 − 1/K₃); subtracting a dimensionless number from a length is dimensionally inconsistent. An independent check shows 6.6 is correct: at the boundary (T = L_n) the thick and thin equations —; with the form of 6.9 this continuity breaks.

7. Results for an example envelope

The example envelope is 21 m thick: a group repeated five times (B₄C 0.15 m · UHPC 0.85 m · MAX phase 0.40 m · STF 2.40 m · aerogel 0.10 m) and a 1.5 m load-bearing layer. Hard layers are treated as concrete-equivalent; STF and aerogel are outside Young's scope, so the STF resistance is swept as a parameter. Worst-case penetration for the ~14 t threat class:

The hard layers in each group (1.4 m in total) are shorter than a long nose (— at 9 CRH): the nose passes through without ever being fully embedded in a hard group.

8. Same material, different arrangement

Keeping the material amounts fixed (hard 8.5 m, STF 12 m, aerogel 0.5 m), three arrangements were compared; for each, the worst penetration over all interpretations, both laws, two noses and two hard materials:

The front-block arrangement is insensitive to every uncertainty, because the hard block is thicker than the worst single-block penetration: the projectile never reaches the soft layers. A side benefit: all soft layers lie behind the burst point. You can run your own trials in the layered-envelope mode of the Penetration Explorer.

9. Limits

10. Check yourself

1. If you slice a uniform concrete into two and apply Young's method, is the result larger or smaller than the single block? Why?
2. If a hard layer is half as thick as the nose is long, what is its effective thickness, roughly (K₃ = ∞)?
3. How would you catch the misprint in equation 6.9 by dimensional analysis?
4. Why is the front-block arrangement independent of the STF resistance? How thick must the block be?
5. Which test would you use to validate this article's results?