A re-entrant honeycomb's inward-pointing struts hinge open under tension: instead of squeezing thinner (like a rubber band), the whole lattice widens in every direction — a negative Poisson's ratio.
Wx(θ) = 2·l·cos(θ), Hy(θ) = 2·(h + l·sin(θ))
θ(ε) = θ0 · (1 − 1.3·ε) [hinge straightens under strain ε]
ν = − εy / εx , εx = Wx(θ)/Wx(θ0) − 1, εy = Hy(θ)/Hy(θ0) − 1
- Auxetic / conventional toggle — flips the hinge angle's sign between a re-entrant (auxetic) and a normal convex honeycomb cell.
- Applied tensile strain — pulls the lattice, straightening the hinge struts and driving the transverse response.
- Cell hinge angle — the unstrained re-entrant angle magnitude; sharper angles give a more strongly negative Poisson's ratio.
- Reset strain — returns the lattice to its unstrained rest shape.
Real-world relevance: auxetic metamaterials are used in impact-absorbing helmet foam, expandable stents, and shape-morphing aircraft skins — all because they get thicker, not thinner, under load.