# Printable lattice: negative Poisson's ratio at usable density

**Status: model only. No specimen printed or tested. Requires physical confirmation.**

## What was computed

Gibson–Ashby in-plane bending model for re-entrant honeycomb unit cells. Swept rib angle θ (−70° to −6°),
rib ratio h/l (0.5–3.0) and thickness ratio t/l (0.04–0.20): 765 valid geometries.

```
E*/Es = (t/l)^3 cosθ / ((h/l + sinθ) sin²θ)
ν12   = cos²θ / ((h/l + sinθ) sinθ)
ρ*/ρs = (t/l)(h/l + 2) / (2 cosθ (h/l + sinθ))
```

## Candidate A — most auxetic at ρ ≤ 0.15 and E*/Es ≥ 0.002 (55 qualify)

| θ | h/l | t/l | E*/Es | ν | ρ*/ρs |
|---:|---:|---:|---:|---:|---:|
| -16 | 1.0 | 0.06 | 0.0038 | -4.628 | 0.129 |
| -18 | 1.0 | 0.06 | 0.0031 | -4.236 | 0.137 |
| -20 | 1.0 | 0.06 | 0.0026 | -3.924 | 0.146 |
| -16 | 1.5 | 0.06 | 0.0022 | -2.738 | 0.089 |
| -16 | 1.5 | 0.08 | 0.0053 | -2.738 | 0.119 |
| -16 | 1.5 | 0.1 | 0.0103 | -2.738 | 0.149 |
| -18 | 1.5 | 0.08 | 0.0043 | -2.458 | 0.124 |
| -20 | 1.5 | 0.08 | 0.0036 | -2.230 | 0.129 |

## Candidate B — stiffest at ν ≤ −0.3 (347 qualify)

| θ | h/l | t/l | E*/Es | ν | ρ*/ρs | E/ρ |
|---:|---:|---:|---:|---:|---:|---:|
| -16 | 3.0 | 0.2 | 0.0372 | -1.230 | 0.191 | 0.1946 |
| -18 | 3.0 | 0.2 | 0.0296 | -1.088 | 0.195 | 0.1516 |
| -16 | 2.0 | 0.15 | 0.0248 | -1.944 | 0.181 | 0.1368 |
| -16 | 2.5 | 0.15 | 0.0192 | -1.507 | 0.158 | 0.1216 |
| -16 | 3.0 | 0.15 | 0.0157 | -1.230 | 0.143 | 0.1095 |

## Prediction to test

Best candidate A: θ = -16°, h/l = 1.0, t/l = 0.06 → ν = -4.63, E*/Es = 0.0038, ρ*/ρs = 0.129.

The falsifiable prediction is that a printed specimen of this geometry shows a negative in-plane Poisson's
ratio of similar magnitude under small strain. If the measured ν is ≥ 0, the model fails for this regime.

## Essential confirmation (not done)

1. Print the unit cell in a polymer at ≥3 t/l values and measure ν under quasi-static tension.

2. Compare measured E*/Es with the model; bending-dominated predictions typically overestimate stiffness

   at low density because of node compliance and manufacturing tolerance.

3. Test whether the auxetic response survives at the strains of interest — re-entrant cells often lose

   negative ν beyond a few percent strain as ribs rotate toward the non-re-entrant configuration.

## Limits

- 2D in-plane linear-elastic model only. Out-of-plane and 3D behaviour are not computed.
- Gibson–Ashby ignores node compliance, joint stiffening and print anisotropy, all of which matter at t/l < 0.1.
- No buckling, yield or fatigue analysis: the stiff candidates may be the first to buckle.
- Angles steeper than −14° were excluded: the Gibson–Ashby expression diverges as θ→0 and returns unphysical values (|ν| > 5) that no real specimen exhibits.
- Textbook formulas, not a new theory. The contribution is the swept design space and the named candidates.

## Verdict

A candidate geometry with a specific, falsifiable prediction. Not a discovered material — a hypothesis

with a test attached.