Granular Mechanics Lab

Balls rain from random (x, y) starting drop height above the container rim. Mass comes from density: m = ρ·(4/3)π(d/2)³, so bigger spheres are proportionally heavier. With a size increment > 0, diameters take the discrete values d_min, d_min+Δd, … d_max (0 = continuous). Friction: μ_slide=√(r₁r₂), μ_roll=0.05√(r₁r₂). Elasticity: each ball carries a contact stiffness k; a contacting pair uses the softer member's k (walls are rigid). Rigid balls are the special case at the top of the slider (k = 2·10⁴); softer balls visibly flatten (overlap) at contacts and pack denser. The timestep adapts automatically.

Red walls are slidable pistons (1 degree of freedom along their outward normal; they exist only while piston mass > 0). Cylinders are true 3D cylinders: a curved shell plus two end caps. Bendable walls turn the box side walls (amber) or the cylinder shell into elastic membranes: node grids pinned at their edges, deflecting along the wall normal under load (stiffness in N/m per node, log slider). Rigid is the special case of a very stiff membrane — at the top of the slider the deflection is negligible. Pistons and the bottom/top stay rigid. The container rests on a fixed ground plane. Cutaway hides the front half of the packing so you can see inside.

Pick which walls are slidable in the Container section, then give the piston a mass > 0 — the piston (red) only exists and appears while its mass is positive.

Void fraction per height slab (exact sphere–slab volumes; slab cross-section follows the container shape). Auto-measured at steady state.

v(r): pick a random particle center, measure the void fraction inside probe spheres of diameter r = Δ, 2Δ, … where the increment Δ is set in units of the mean ball diameter d̄ (exact sphere–sphere intersection volumes; probes must lie fully inside the bed region), averaged over all samples. Chart plots v against r/d̄. Set center particle size to restrict picked centers to one size class ("for a given size").

Volume distribution P(υ) per Chang, Chang & Chao, PRE 114, 015410 (2026), now with true 3D cells: specific volumes υ_i = (Voronoi cell volume)/(ball volume), computed by exact half-space clipping (wall & surface layers excluded); free volume v = υ − υ_min, Gamma DOS Ω(v), Z(X) = (X/(X+θ))^k, X = θv̄/(kθ−v̄). k and υ_min come from the structural relation ῡ = υ_min + √k·σ fitted across recorded states (tap at several Γ values for a good fit); θ from the loosest state. Chart & CSV: P(v) over the FREE volume v = υ − υ_min (Eq. 1), not the total specific volume, with the factorized-Gamma prediction. Keep the bed ≥8 ball diameters deep so enough interior cells survive the exclusion margins.

Time step is set automatically from contact stiffness and the lightest ball. Sim speed asks for a multiple of real time; steps/frame caps how much physics each display frame may run. Raise both to fast-forward long protocols — the display fps will drop while the simulation races ahead (the HUD shows the achieved ×real-time). Drag to orbit, wheel to zoom, shift-drag (or right-drag) to pan, double-click to re-fit. Best kept to ~1–2,000 balls in the browser; for larger 3D runs use the Python engine (same physics, OpenMP C core).