A projectile striking a taut woven fiber panel does not deform the whole sheet at once. Two independent mechanical waves race outward from the impact point along the fibers, and this 2D cross-section renders both of them separately and simultaneously from their own closed-form speeds (Smith & Roylance ballistic-fabric theory):
Longitudinal (extensional/stress) wave — fast, along the fiber axis:
c_L = √(E / ρ) radius r_L(t) = min(c_L·t, R)
Transverse (deflection) wave — slower, sets the "tent" cone radius:
η = v₀ / (2·c_L)
U = c_L·√η radius r_T(t) = min(U·t, R)
Cone geometry (straight fiber segments, apex depth d):
fiber strain ε(t) = √(1 + (d/r_T)²) − 1
Post-engagement phase (transverse front pinned at panel edge R):
effective spring k = n_layers · E · A_eff / R
ω = √(k / m_proj), d(t) = d₀cos(ωt') + (v₀/ω)sin(ωt')
elastic energy absorbed PE(t) = ½·k·d(t)²
projectile kinetic energy KE₀ = ½·m_proj·v₀²
- Longitudinal wave: since c_L ≫ U for every material here, the stress signal reaches the clamped edge almost instantly — visible as a fast bright ring flashing outward at fire.
- Transverse wave: the actual "tent" deformation cone grows much more slowly at U while the impact point sinks in at roughly v₀, exactly as small-deflection ballistic-fabric theory predicts.
- The race that decides the outcome: every frame the sim compares the fiber strain ε(t) at the impact point against the material's elongation-at-break, and separately compares the elastic energy PE(t) already absorbed by the engaged fiber network against the projectile's starting kinetic energy KE₀. Whichever threshold is crossed first wins — strain hits the break point first → PENETRATED; absorbed energy reaches KE₀ first (the projectile is brought to a momentary stop) → ARRESTED. Nothing is scripted — it is read off the live strain and energy curves each frame.
- Why a lighter, "weaker" fiber can win: c_L scales with √(E/ρ), so a low-density fiber (Dyneema) can have a higher wave speed than a stiffer but denser one (glass) — a higher c_L spreads the impact strain over more fiber length, lowering the peak strain at the front for the same v₀.
- Simplification: the post-engagement spring constant uses a fixed effective engaged cross-section per layer — it captures how material and layer count change the outcome, but is not a substitute for a full finite-element ballistic model.