cellular-solids

name	cellular-solids
description	Cellular solids — foams/honeycombs, relative density, Gibson-Ashby scaling laws, open/closed cell, compression plateau, energy absorption, metallic foams, lattice structures, AM-optimized topology.
metadata	{"priority":7,"promptSignals":{"phrases":["cellular solid","foam mechanics","honeycomb structure","foam energy absorption","Gibson Ashby","metallic foam","lattice structure"],"minScore":3}}

Cellular Solids — Complete Skill

Relative Density and Geometry

Relative density: ρ*/ρ_s = (mass of foam) / (mass of solid with same volume)

Typical values:

• Metallic foam: ρ*/ρ_s = 0.05–0.20
• Polymer foam: ρ*/ρ_s = 0.01–0.30
• Honeycomb (in-plane): ρ*/ρ_s = t/(l) for hexagonal cell (t = wall thickness; l = cell edge length)

Cell topology:

• Open-cell: cells interconnected; liquid/gas can permeate; e.g., reticulated foam
• Closed-cell: cells isolated; gas trapped; better compression; e.g., aluminum foam

Gibson-Ashby Scaling Laws (from "Cellular Solids" — Gibson and Ashby 1997)

Mechanical Properties

Open-cell foam: E*/E_s = C₁ × (ρ*/ρ_s)² [Young's modulus; C₁ ≈ 1] σ_pl / σ_ys = C₂ × (ρ/ρ_s)^1.5 [plastic collapse stress; C₂ ≈ 0.3] σ_cr / E_s = C₃ × (ρ/ρ_s)² [elastic buckling in compression; C₃ ≈ 0.05]

Closed-cell foam (additional gas pressure contribution): E*/E_s = φ²(ρ*/ρ_s)² + (1-φ)(ρ*/ρ_s) [φ = fraction solid in cell faces; ≈ 0 open; ≈ 1 closed]

Tensile fracture (brittle): σ_ts / σ_fs = C₄ × (ρ/ρ_s)^1.5 [tensile fracture; σ_fs = modulus of rupture of solid]

Key insight: doubling ρ*/ρ_s → quadruples E*; squares modulus-to-density

Thermal Conductivity

Combined conduction + radiation: k* = C_s × k_s × (ρ*/ρ_s) + C_r × T³ × l_cell [solid + radiation terms] Radiation dominates at high T; useful for thermal insulation with very low ρ*

Stress-Strain Behavior

Compression Behavior (Three Regimes)

1. Linear elastic: E* = C₁ E_s × (ρ*/ρ_s)² ; steep initial response
2. Plateau: long, flat region; plastic buckling (ductile) or brittle crushing (brittle)
   • Ductile metallic foam: σ_pl ≈ constant across large strain (10–60%)
   • Brittle ceramic foam: serrated; incremental crushing
3. Densification: all cells collapsed; approaches solid material response

Densification strain: ε_D ≈ 1 - 1.4 × (ρ*/ρ_s) [approximate; assumes uniform collapse]

Energy absorption: W = ∫₀^{ε_D} σ dε ≈ σ_pl × ε_D [J/m³]

Ideal energy absorber: maximum W / volume at given σ_max → higher density → higher plateau stress; tradeoff vs. transmitted force

Honeycomb Structures

In-Plane (2D Cellular)

Regular hexagonal honeycomb: E₁* = E_s × (t/l)³ [in-plane X₁-direction; t = wall thickness; l = cell edge] E₂* = E_s × (t/l)³ [same for regular hexagonal] G₁₂* = E_s × (t/l)³ × 1/(1+4h/l) × (h/l + sinα/cosα) ...

Effective density: ρ* = ρ_s × 2t/l × (h/l + sinα) / (cosα(h/l + sinα)) [for arbitrary cell angle α]

Out-of-Plane (Used as Core in Sandwich)

Compression modulus: E₃* = E_s × (t/l)

Shear modulus (for sandwich facing loads): G₃₁* = E_s × 1/√3 × (t/l) (for equilateral hexagonal) G₃₂* = G₃₁*

Sandwich panel stiffness: D = E_f × t_f³/6 + E_f × t_f × (d/2)² [bending; d = core thickness]

Honeycomb Materials

Nomex (aramid paper): aerospace sandwich panels; low density; fire resistant Aluminum (5052, 5056): higher strength; aircraft floor panels, ship superstructure Steel honeycomb: high crush strength; automotive crash structures Carbon fiber honeycomb: very lightweight; spacecraft structures

Metallic Foams

Aluminum Foam (Alulight, Alporas, Cymat)

Alporas (closed-cell): ρ* = 0.20–0.25 g/cm³; σ_pl = 1.5–3.0 MPa; ε_D = 0.65–0.70 Made by: foaming agent (TiH₂) added to molten Al → releases H₂ → foam solidifies

Alulight (powder metallurgy): higher purity; more controlled porosity; ρ* = 0.4–0.8 g/cm³

Applications:

• Crash energy absorbers: automotive bumpers, train buffers
• Acoustic panels: lightweight + damping
• Heat sinks: open-cell Al foam; surface area → enhanced convection
• Sandwich core: Al face sheets + Al foam core; blast protection

Nickel Foam (Recemat, Incofoam)

Open-cell; electrodeposition of Ni onto polyurethane foam template; template burned out ρ* = 0.015–0.10 g/cm³; porosity > 90% Applications: battery electrodes (NiMH, NiCd); filters; catalyst supports

Steel Foam (Fraunhofer sintered)

Higher strength; impact protection; automotive energy management

Lattice Structures (AM-Enabled)

AM lattice: additive manufacturing (LPBF, EBM) enables regular open-cell structures impossible to cast Cell types: BCC, FCC, octet-truss, Kelvin, rhombi-dodecahedron, gyroid (TPMS)

Octet-truss (stretching-dominated): E*/E_s = 0.35 × (ρ*/ρ_s) [linear scaling — more efficient than bending-dominated] σ_pl / σ_ys ≈ 0.30 × (ρ/ρ_s) [linear scaling]

BCC (bending-dominated): E*/E_s ≈ 0.10 × (ρ*/ρ_s)² [quadratic; less efficient for stiffness]

Design rule: stretching-dominated (octet, Kelvin) → higher stiffness/density Bending-dominated (BCC, FCC) → better energy absorption (large plateau strain)

Topology-Optimized Lattice

Gradient density lattice: spatially varying ρ*/ρ_s for load-path following SIMP method extended to cellular unit cells Software: nTopology; Altair Inspire; Materialise; COMSOL

TPMS (Triply Periodic Minimal Surface): Schwartz-P, gyroid, Diamond surfaces → mathematically smooth; uniform stress distribution Excellent for heat exchangers, bone scaffolds, energy absorption

Output

Provide: relative density ρ*/ρ_s, elastic modulus E* [MPa], plateau stress σ_pl [MPa], densification strain ε_D, energy absorption W [J/m³], mass efficiency index (σ/ρ*)^2/E* for comparison, cell topology (open/closed, hexagonal/lattice), manufacturing method (foaming agent/AM/powder metallurgy), material (Al/Ni/steel/polymer), honeycomb out-of-plane shear moduli G₃₁ G₃₂ [MPa], applicable standard (ASTM C273 for core shear; ISO 13314 for metallic foam compression).