Geometrically frustrated, mechanical metamaterial membranes: Large-scale stress accumulation and size-selective assembly

TL;DR

This paper investigates how geometric frustration influences the shape and stress distribution in mechanical metamaterial membranes, revealing size-independent hyperbolic curvatures and enhanced self-limiting assembly properties compared to elastic membranes.

Contribution

It introduces a continuum elastic theory and numerical model for frustrated metamembranes, demonstrating size-independent curvature accumulation and altered energy scaling.

Findings

Hyperbolic curvatures accumulate up to mesoscopic scales independently of microscopic size.

Elastic strain energy growth with size is weaker than in elastic membranes.

Self-limiting assembly range is significantly increased in metamembranes.

Abstract

We study the effect of geometric frustration on dilational mechanical metamaterial membranes. While shape frustrated elastic plates can only accommodate non-zero Gaussian curvature up to size scales that ultimately vanish with their elastic thickness, we show that frustrated {\it metamembranes} accumulate hyperbolic curvatures up to mesoscopic length scales that are ultimately independent of the size of their microscopic constituents. A continuum elastic theory and discrete numerical model describe the size-dependent shape and internal stresses of axisymmetric, trumpet-like frustrated metamembranes, revealing a non-trivial crossover to a much weaker power-law growth in elastic strain energy with size than in frustrated elastic membranes. We study a consequence of this for the self-limiting assembly thermodynamics of frustrated trumpets, showing a several-fold increase the size range of self-limitation of metamembranes relative to elastic membranes.