Granular Mechanics Lab

Balls

count to drop

diameter min (m) 0.030

diameter max (m) 0.050

disk size increment (m) 0.0050

density (kg/m³) 1500

drop height (m) 0.30

bounciness 0.30

roughness 0.60

color mode

immovable

Balls spawn as a rain from random (x, y) starting drop height above the container rim. Mass comes from density: m = ρ·(4/3)π(d/2)³ (spheres in the 2D plane), so bigger balls 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).

Container

type

width / bore (m) 2.0

height / length (m) 2.0

position x (m) 0.0

rotation (°) 0

wall roughness 0.60

wall bounciness 0.30

slidable walls

lid / open side

sidebars (left/right)

sidebar stiffness 10^3.5

max sidebar deflection (m)–

inner width at mid-height (m)–

Red walls are slidable pistons (1 degree of freedom along their outward normal). Bendable sidebars are elastic chains pinned at both ends (stiffness in N/m per node, log slider) — use them to study applied force vs deformation of the ensemble. The container rests on a fixed ground plane.

Piston force — experiment 2

force inward (N) 0

piston mass (kg) 2.0

chamber size (m)–

piston reaction (N)–

Void distribution — experiment 1

packing fraction–

void fraction–

Void fraction per height slab (exact disk–slab areas). Auto-measured at steady state.

v(r): increment (× ball diameter) 0.50

v(r): max probe (× ball diameter) 12

v(r): samples

v(r): probe centers

v(r): center particle size

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

Tapping — ensemble statistics

tap intensity Γ (g) 4.0

taps in protocol

mean specific volume ῡ–

variance σ²–

DOS shape k / scale θ–

υ_min–

compactivity X–

partition function Z(X)–

entropy S−S_RCP–

bed height / Δh (m)–

Volume distribution P(υ) per Chang, Chang & Chao, PRE 114, 015410 (2026): specific volumes υ_i from Voronoi cells (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: P(υ) histogram + factorized-Gamma prediction; below it, ῡ vs tap number.

Physics

static friction model

settle assist extra damping when nearly still — helps hard-to-reach steady states

gravity (m/s²) 9.81

sim speed 1.0×

Time step is set automatically from contact stiffness and the lightest ball. Scroll to zoom, drag to pan, double-click to re-fit. For runs beyond ~5,000 balls use the Python engine (same physics, C core, up to 125,000 balls).