Cumulative geometric frustration in physical assemblies

TL;DR

This paper introduces a framework based on compatibility conditions to analyze geometric frustration in physical assemblies, explaining diverse behaviors and energy scaling in such systems.

Contribution

It provides an intrinsic approach that links geometric frustration to compatibility conditions, predicting assembly behaviors and energy growth.

Findings

Compatibility conditions determine small assembly behavior.

Framework predicts super-extensive energy growth.

Applied to well-known frustrated assemblies.

Abstract

Geometric frustration arises whenever the constituents of a physical assembly locally favor an arrangement that cannot be realized globally. Recently, such frustrated assemblies were shown to exhibit filamentation, size limitation, large morphological variations and other exotic response properties. While these unique characteristics can be shown to be a direct outcome of the geometric frustration, some geometrically frustrated systems do not exhibit any of the above phenomena. In this work we exploit the intrinsic approach to provide a framework for directly addressing the frustration in physical assemblies. The framework highlights the role of the compatibility conditions associated with the intrinsic fields describing the physical assembly. We show that the structure of the compatibility conditions determines the behavior of small assemblies, and in particular predicts their super-extensive energy growth exponent. We illustrate the use of this framework to several well known frustrated assemblies.