How periodic surfaces bend without stretching

TL;DR

This work addresses how periodic shells bend without stretching by deriving a general constraint that links in-plane effective strains $E_{ij}$ to out-of-plane curvatures $\chi_{ij}$ for isometric deformations: $E_{11}\chi_{22}-2E_{12}\chi_{12}+E_{22}\chi_{11}=0$. The approach relies on Gauss-curvature invariance under isometries and differential geometry to unify effective membrane and flexure modes, extending beyond discrete origami unit cells. It generalizes Poisson-like identities observed in origami tessellations to curved and smoothly creased surfaces, with implications for the design of compliant shells and origami-inspired metamaterials. The paper also discusses nonlinear extensions, limitations to closed surfaces, and multi-scale considerations relevant to real periodic architectures.

Abstract

Many compliant shell mechanisms are periodically corrugated or creased. Being thin, their preferred deformation modes are inextensional, i.e., isometric. Here, we report on a recent characterization of the isometric deformations of periodic surfaces. In a way reminiscent of Gauss theorem, the result builds a constraint that relates the ways in which the periodic surface stretches, effectively but isometrically, to the ways in which it bends and twists. Several examples and use cases are presented.

Figures (1)

• Figure 1: A periodic surface of the form $(x,y,f(x)+g(y))$ where $f$ is a sine function and $g$ is a triangular pattern: (a-c) an effective membrane mode where the surface stretches without bending by bending without stretching; (d) the same surface with more unit cells to help appreciate the curvatures seen in (e) and (f). Theorem \ref{['thm']} states that (b) implies (e) and that the possibility of (f) implies the impossibility of an effective shear membrane mode. Surfaces are computed using a triangulation; source code available at https://github.com/nassarh/curvedCreaseOrigami.

Theorems & Definitions (1)

• proof