Auxetic Lattice Simulator

What is Poisson’s ratio?

When you stretch a rubber band, it gets thinner in the middle. When you squeeze a ball of clay, it bulges outward. This coupling between longitudinal and lateral deformation is captured by Poisson’s ratio (ν), defined as the negative ratio of transverse strain to axial strain: ν = −(ε_x / ε_y). Most materials have a positive Poisson’s ratio, typically between 0.2 and 0.5. Rubber is nearly 0.5 (incompressible — it conserves volume, so stretching one direction forces contraction in the others). Cork is near zero, which is why it works as a bottle stopper: when you push it into the neck, it does not bulge outward. And then there are auxetic materials, where ν is negative.

Auxetic materials

The first engineered auxetic material was created by Rod Lakes at the University of Iowa in 1987. He took ordinary polyurethane foam, compressed it in a mold, and heated it. The heat treatment collapsed the cell walls inward, converting the regular honeycomb microstructure into a re-entrant one. The result: foam that gets fatter when stretched. The key insight was that the auxetic behavior comes entirely from geometry, not chemistry. The same polyurethane molecules, the same cell walls, the same material — but a different arrangement produces opposite mechanical behavior. This is the central idea: structure determines properties.

How re-entrant geometry works

In a regular honeycomb, the angled struts point outward like the letter V. When you pull vertically, the V’s flatten, and the horizontal ribs are pushed inward — the material contracts laterally. In a re-entrant honeycomb, the same struts point inward, like bow-ties or butterflies. When you pull vertically, the inverted V’s unfold outward, pushing the horizontal ribs apart. Vertical extension becomes horizontal expansion. The geometry converts pull in one direction into push in the perpendicular direction. Switch the preset between “Honeycomb” and “Re-entrant” above and stretch both — the difference is immediate and visceral.

Auxetics in nature

Nature discovered auxetic geometry long before Rod Lakes. Cat skin is auxetic: when a cat arches its back, the skin stretches and thickens rather than thinning, which helps protect against bites and scratches. Salamander skin shows similar behavior. Arterial endothelium — the inner lining of blood vessels — is auxetic, which helps it withstand the pulsatile stretching of blood flow without tearing. Perhaps most remarkably, the nuclei of embryonic stem cells are auxetic. The chromatin (packaged DNA) inside stem cell nuclei is loosely arranged in a geometry that produces a negative Poisson’s ratio. As cells differentiate and chromatin condenses, the nucleus loses its auxetic behavior. The genome’s packaging geometry produces a material property.

Mechanical metamaterials

Auxetic lattices belong to a broader class of mechanical metamaterials — materials whose mechanical properties are determined by structure rather than composition. The same principle underlies acoustic cloaking (lattices that bend sound waves around an object), energy-absorbing structures (lattices that collapse in controlled patterns), and deployable structures (flat-packed lattices that expand into 3D shapes). In 2024, researchers achieved a Poisson’s ratio of exactly −1, the theoretical isotropic limit, meaning the material expands in all directions equally when pulled in any direction. The design space of mechanical metamaterials is vast and largely unexplored.

Topological mechanics

At the frontier of lattice mechanics lies a surprising connection to quantum physics. Maxwell lattices — lattices where the number of degrees of freedom exactly equals the number of constraints — can exhibit topological edge modes analogous to electronic topological insulators. Just as electrons flow along the surface of a topological insulator without scattering, mechanical distortions propagate along the edges of a topological lattice without penetrating the bulk. This enables unidirectional crack steering: controlling where fractures propagate based purely on lattice topology. The mathematics of band theory, developed for electrons in crystals, turns out to describe vibrations in spring networks just as well. Geometry, once again, is the bridge.