The science
Impact speed. The hammer falls with m dv/dt = m g − ½ ρₐ C_d A v², v = √(2 g h) in vacuum The sim integrates this with RK4 (0.5 ms) and compares with the closed form v = v_t √(1 − e^(−2gh/v_t²)), v_t = √(2mg/ρₐ C_d A)
Stress wave. A body hit at speed v sees an initial compressive wave of stress σ = ρ c v, c = √(E/ρ) We use a peak tensile stress σ_t = κ ρ c v (κ = 0.5, shell/rim geometry factor) It depends on speed only, not on hammer mass.
Weibull flaws. Brittle strength is set by the worst flaw, so P_f = 1 − exp[ −(V/V₀)(σ/σ₀)^m ] Large m means a narrow, reproducible threshold; the volume term gives the size effect: bigger objects are weaker, σ ∝ V^(−1/m). On a Weibull plot ln(−ln(1−P)) vs ln h the slope is m/2 because σ ∝ √h.
Ductile metals. They do not have a speed threshold; the hammer's energy is absorbed as plastic work, ½ m v² = σ_y V ε, ε = ΔL / L₀ Volume is conserved, so the billet bulges: r = r₀ /√(1 − ε). It tears when ε exceeds a ductility limit ε_f (10 % scatter).
Fragmentation. Surplus energy E − U (U = σ²V/2E stored at failure) is split: a fraction η_s = 4 % makes new surface of fracture energy G_c = K²/E, η_k = 1.5 % becomes fragment kinetic energy, the rest is heat and sound. For cubic pieces of side s: 3 G_c V / s = η_s (E − U) The mass passing follows Rosin–Rammler, F(d) = 1 − exp[−(d/d₆₃)^n], n = 1.2, d₆₃ = 1.2 s.
Simplifications. Rigid anvil; one-dimensional wave stress; constant κ, η and n; effective (flaw-limited, dynamic) strengths chosen to give plausible tabletop heights, not handbook values; no fatigue, no crack-path modelling. Fragments are drawn as irregular chunks (at most 96) with the mass-conserving size spectrum; for very large N the pieces shown are representatives and the rest is dust. The containment ring (r = 1.35 m) reflects pieces with restitution 0.3. Rebounding hammer, fragment-on-floor impacts and the pile are rigid-body spheres with restitution and Coulomb friction.