Physics & Methodology

Five penetration models, one flight model, and a set of shared post-impact equations. Which penetration model runs is decided by the target material, not by the projectile: brittle solids, metals, wood, fibres and granular media fail in different ways, and a single closed form cannot describe all of them. Every formula below is the one the engine executes, and every source is one this project has read rather than cited second-hand.

Which model runs against which material

The engine holds 18 materials. Each one carries a model tag; the tag decides the branch. The model name in the third column is the string the calculator prints with every result, so a report can be traced back to this table.

Material	Tag	Model as reported
Red Brick	PONCELET	Poncelet + Confinement
Concrete	PONCELET	Poncelet + Confinement
Tempered Glass	PONCELET	Poncelet + Confinement
Ballistic Gel 10%	PONCELET	Poncelet + Confinement
Drywall (Gypsum)	PONCELET	Poncelet + Confinement
Cinder Block	PONCELET	Poncelet + Confinement
Alumina Ceramic	PONCELET	Poncelet + Confinement
Water	PONCELET	Poncelet + Confinement
Steel A36	JC	Johnson-Cook + Poncelet
Steel AR500	JC	Johnson-Cook + Poncelet
Aluminum 6061	JC	Johnson-Cook + Poncelet
Polycarbonate	JC	Johnson-Cook + Poncelet
Pine Wood	ORTHO	Orthotropic + Poncelet (⊥ grain)
Oak Wood	ORTHO	Orthotropic + Poncelet (⊥ grain)
Kevlar NIJ III	KEVLAR	Fiber Deformation (Thermal-Aware)
Dyneema UHMWPE	KEVLAR	Fiber Deformation (Thermal-Aware)
Packed Sand	MC	Mohr-Coulomb + Dynamic Locking
Compacted Earth	MC	Mohr-Coulomb + Dynamic Locking

A sixth mechanism, Tate–Alekseevskii erosion, is not a material branch. It is a ceiling applied on top of the rigid-body result whenever the target can consume the projectile — see below.

1. Flight — velocity at the target

Before anything strikes anything, the round has to get there. Drag is integrated with a G1 point-mass model; air density falls with altitude and rises as temperature drops.

v_target = v₀ · exp( −(ρ_air · Cd · A · dist) / (2·m) )
ρ_air = ρ₀ · exp(−h/8500) · (273/(273+T))

Impact angle is applied to the arriving velocity, not to the geometry of the target:

v_eff = v_target · cos(θ)

Used for: every calculation, every material. Source: standard G1 drag reference; the trajectory module is checked against published drop tables for five loads out to 800 yd (worst deviation 0.16%).

2. Poncelet + Confinement — brittle solids

Fig. 1 — Poncelet branch. The projectile pushes against a constant strength term and an inertial term that rises with the square of velocity; confinement multiplies both, and applies to brittle targets only. Schematic — not to scale.

The Poncelet law resolves the retarding force into a strength term and an inertial term. Integrating it gives a closed form for depth:

P = (m / (Cp·ρ_c·A)) · ln( 1 + (Cp·ρ_c·v_eff²) / (2·σ_c) )

σ_c is the dynamic compressive strength, not the static one; ρ_c is target density; Cp is a nose-shape drag coefficient carried per bullet type. A confinement factor (1.2–1.5× for brittle solids) raises the effective strength, because a brittle target restrained on all sides resists more than an unconfined coupon.

Used for: brick, concrete, tempered glass, cinder block, alumina ceramic, drywall, ballistic gel, water. Source: Rosenberg & Dekel for the modern treatment; Forrestal et al. (1994) for the concrete comparison; Gaylord et al. (2013) for retarding force in gelatin.

3. Johnson-Cook + Poncelet — metals

Fig. 2 — Johnson-Cook branch. Metals do not comminute; both bodies deform. The flow stress rises with strain and strain rate and falls as the impact heats the material. Schematic — not to scale.

Metals do not have a single compressive strength. Their flow stress depends on strain, strain rate and temperature, and at impact all three are far from laboratory conditions. Johnson-Cook supplies the dynamic flow stress, which then enters the Poncelet integral in place of a static σ_c:

σ_flow = ( A + B·εⁿ ) · ( 1 + C·ln(ε̇/ε̇₀) ) · ( 1 − T*ᵐ )
T* = (T − T_room) / (T_melt − T_room) · ε̇ ≈ v_eff / d

The strain rate is estimated from impact velocity over projectile diameter, and T* is clamped to 0.9 so the term cannot drive the flow stress to zero. The result is floored at the material's quoted σ_c.

Used for: mild steel A36, AR500 armour steel, aluminium 6061-T6, polycarbonate. Source: Johnson & Cook (1983) for the constitutive model; AUTODYN material library for the A, B, n, C, m constants.

One departure worth stating. Polycarbonate is tagged JC and its σ_c floor is raised well above the ~70 MPa a real sheet yields at. This is phenomenological: monolithic Poncelet under-resists against laminated UL752 panels, so the floor is calibrated to stop 9 mm at UL752 Level 1. It reproduces the standard, but it is a fit, not a derivation.

4. Orthotropic + Poncelet (⊥ grain) — wood

Wood is not isotropic. Strength across the grain and along it differ by a factor of several, so the model carries both and uses the perpendicular value for a shot into a face, with the grain ratio reported alongside the result.

σ_eff = σ_⊥ · ratio = σ_∥ / σ_⊥
P = Poncelet(σ_eff), then flattened above ~420 m/s

Above roughly 420 m/s the rigid-body form over-predicts, because rifle rounds begin to yaw, deflect and fragment in wood instead of driving straight. The curve is flattened in that regime only. Handgun rounds are stable in wood and are deliberately left untouched — an earlier rifle-tuned threshold was suppressing them wrongly.

Used for: pine, oak. Source: Koene & Broekhuis (2017), the measurement set this branch is checked against.

5. Fiber Deformation (Thermal-Aware) — aramid and UHMWPE

Fig. 3 — Fiber branch. Load spreads into a cone behind the impact; a wider cone engages more material. Dyneema is the stiffer fiber but melts at 145 °C against Kevlar’s 450 °C, so it alone carries a friction-melting penalty at rifle speeds. Values from the engine’s material table. Schematic — not to scale.

Soft armour does not resist by compressive strength. Energy leaves the projectile through a transverse wave spreading down the fibres and into a cone of deforming material behind the impact. There is no Poncelet term here at all — the mechanism is different, so the equation is different.

σ_fib = σ_c · ( 1 + 0.35·(v/v_ref)^0.6 ) · ( 1 − T_env − T_friction )
cone_factor = 1 + tan(θ)²

Two thermal terms matter and are usually left out elsewhere. Ambient temperature degrades the fibre relative to its melting point, and above 800 m/s friction heating degrades low-melting polymers further. Dyneema melts near 145 °C and Kevlar near 450 °C, so the same impact treats them very differently. The combined weakening is clamped so it can never reduce the fibre below 5% of its rated strength.

Used for: Kevlar NIJ Level III, Dyneema UHMWPE. Source: NATO STANAG 4569 for the protection-level framing; fibre parameters from the AUTODYN library and published V50 data.

6. Mohr-Coulomb + Dynamic Locking — granular media

Fig. 4 — Mohr-Coulomb branch. A grain of sand has almost no strength; a metre of sand does, because resistance is built from overburden, friction and lateral confinement. Above roughly 50 m/s the pores collapse and the medium behaves like weak rock. Schematic — not to scale.

Sand and soil have almost no cohesion. They resist by internal friction under confining pressure, which means depth depends on how deep into the medium the projectile already is.

σ_v = ρ · g · (T/2) · σ_MC = c + σ_v · tan(φ)
K₀ = 1 − sin(φ) · σ_lateral = σ_v · K₀ · 3

Three effects are stacked: static Mohr-Coulomb resistance from cohesion and friction angle, lateral confinement via the Jaky K₀ coefficient, and pore locking — at ballistic rates the pore space collapses and the medium behaves closer to a solid than to loose grains.

Used for: packed sand, compacted earth. Source: Børvik, Dey & Olovsson (2015), the granular penetration study this branch is compared against.

7. Tate–Alekseevskii erosion — the ceiling on hard targets

Fig. 5 — Tate–Alekseevskii. Applied as a ceiling on the rigid-body result, not as a material branch: when the target can consume the projectile, penetration stops being a question of how much energy arrived. Schematic — not to scale.

A projectile striking something harder than itself does not stay intact. It erodes at the nose, and its depth is bounded by the eroding-rod limit rather than by the rigid-body Poncelet depth. Without this cap the engine over-predicts badly at rifle velocity — 5.56 read about 45 mm into mild steel where field data gives 15–25 mm.

½ρ_p·(v−u)² + Y_p = ½ρ_t·u² + R_t
u = v / ( 1 + √(ρ_p/ρ_t) ) · L = m / (A · ρ_p)

u is the penetration velocity from the Tate Bernoulli balance with strength terms on both sides. Projectile yield strength Y_p is carried per bullet type — 400 MPa for FMJ, 300 for expanding designs, 4000 for AP cores — which is why an AP round behaves entirely unlike a hollow point against the same plate.

Applied to: rigid solids only — ceramics and metals. Never to fibres or granular media, where erosion is not the limiting mechanism. Source: Alekseevskii (1966); Tate (1967, 1969).

8. Shared post-impact equations

These run regardless of which penetration branch was taken.

Residual velocity — only when the round gets through (P ≥ T):

v_res² = ( v_eff² + α/β ) · exp( −2·β·T ) − α/β
α = σ_c·A/m · β = Cp·ρ_c·A/(2m)

Crater radius:

R_crater = 0.84 · d · ( E_k / (σ_c · d³) )^1/3

Fracture zone — linear elastic fracture mechanics:

K_I = σ_c · √(π · R_k) · r_crack = (K_I/K_IC)² / π

Fragmentation — Grüneisen energy partition:

N = 8 · ( η·E_k / (G_c·A_avg) )^0.62 · η = 0.15 + 0.15·min(P/T, 1)

Assumptions & limitations

What follows is what the model does not do. A result that omits this list is not a result a report can rest on.

Geometry and incidence

Normal incidence is the default. Impact angle enters only as cos(θ) on the arriving velocity. There is no ricochet model, no angular deflection of the path inside the target, and no asymmetric loading of the nose.
Semi-infinite target in the lateral direction. Edge effects, shot placement near a boundary and plate flexure are not modelled.
The projectile is a rigid body except where Tate–Alekseevskii erosion applies. Nose deformation below the erosion threshold is not tracked.

Where narrower models are applied

Yaw is modelled on drywall only. The onset distance (0.20 m for a 9 mm round-nose FMJ) comes from Fackler and is not generalised to other materials. Wood gets a velocity-threshold flattening instead, which is a different correction with a different justification.
JHP expansion is modelled in soft tissue only — ballistic gel. A hollow point that has already passed through a hard barrier is treated as clogged, and clogged behaviour differs from fresh expansion. Expansion in any other material is not computed.
Fragmentation is a count, not a trajectory set. The engine reports how many fragments the energy partition implies. It does not track where any of them go.

Not modelled at all

Multi-hit. Every calculation is a first shot into undamaged material. Degradation from a previous impact is not carried.
Backing effects. A ceramic plate with a fibre backer behaves unlike the same ceramic alone; the engine models the layers in sequence, not the interaction between them. Behind-armour blunt trauma is not computed.
Confinement beyond a scalar. Confinement is a multiplier on strength (1.2–1.5× for brittle solids), not a boundary condition.
Spin, precession, and gyroscopic stability during penetration.

Parameter uncertainty

Material strengths are ranges, not constants, and the engine carries the range. Reported depth is accompanied by an error margin derived from the σ_c variance for that material — typically ±15%, wider for granular media. A single depth figure quoted without its margin misrepresents what the model claims.

References

Sources this project has read in the original. Where a figure was taken from a specific table, the article that used it names the table.

Penetration models

Alekseevskii, V. P. — Penetration of a rod into a target at high velocity. Combustion, Explosion and Shock Waves, 2(2), 1966.
Tate, A. — A theory for the deceleration of long rods after impact. Journal of the Mechanics and Physics of Solids, 15(6), 1967.
Tate, A. — Further results in the theory of long rod penetration. Journal of the Mechanics and Physics of Solids, 17(3), 1969.
Johnson, G. R. & Cook, W. H. — A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. Proceedings of the 7th International Symposium on Ballistics, The Hague, 1983.
Rosenberg, Z. & Dekel, E. — Terminal Ballistics. Springer.

Measurements and validation

Koene, L. & Broekhuis, M. — Bullet Penetration into Wooden Targets. 30th International Symposium on Ballistics, 2017. DOI 10.12783/ballistics2017/16976
Børvik, T., Dey, S. & Olovsson, L. — Penetration of granular materials by small-arms bullets. International Journal of Impact Engineering, 2015.
Forrestal, M. J., Altman, B. S., Cargile, J. D. & Hanchak, S. J. — An empirical equation for penetration depth of ogive-nose projectiles into concrete targets. SAND92-1948C, 1994.
Noonan, F. M. & Steves, H. — The Performance of Small Arms Ammunition When Fired Into Water. NOLTR 70-174, US Naval Ordnance Laboratory, 1969. DTIC AD0713445
Gaylord, S., Blair, J., Courtney, A. & Courtney, M. — Bullet Retarding Forces in Ballistic Gelatin by Analysis of High Speed Video. arXiv:1305.5215, 2013.
Fackler, M. L. & Malinowski, J. A. — The wound profile: a visual method for quantifying gunshot wound components. Journal of Trauma, 25(6), 1985, 522–529.
US Army — FM 3-06.11, Combined Arms Operations in Urban Terrain, Table 7-3.

Material parameters and standards

Ansys AUTODYN Material Library — constitutive constants for metals and fibres.
NATO STANAG 4569 — protection levels for occupants of armoured vehicles.

What is deliberately absent. The Poncelet law dates to the 19th century and is used here in the form given by Rosenberg & Dekel rather than cited to a specific original edition. Where a secondary source could not be checked against its original, it is not listed above.

Run it

A methodology without a tool is a paper. A tool without a methodology is a black box. The calculator applies every equation on this page, names the model it used, and prints the parameters that produced the number.

Run your own scenario →