Frustration and Functionality : Geometry and Topology in Mechanical Metamaterials

Date of Award

5-11-2025

Date Published

June 2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics

Advisor(s)

Christian D Santangelo

Keywords

Differential Geometry;Elasticity;Geometric Frustration;Mechanical Metamaterials

Subject Categories

Physical Sciences and Mathematics | Physics

Abstract

This dissertation is concerned with a study of the effect of geometry in complex few body mechanical systems. The problems that we set out to solve lie on the interface of soft condensed matter physics, geometry and topology. In the first problem, I developed an effective elasticity theory of thin mechanical metamaterial sheets. This work explored the elastic response of thin metamaterial sheets draped on a uniformly and weakly curved substrate. The model spells out an elegant screening law that shows that a metamaterial alleviates elastic stresses by actuating the soft modes accessible to it. In the second problem, we extend this framework and show how geometrically frustrated metamaterials leverage their inherent floppy modes to reshape stress accumulation and self-assembly thermodynamics. These metamembranes exhibit a significantly extended size range for accommodating Gaussian curvature and self-limiting assembly compared to conventional elastic systems. This opens up new design possibilities in sub-micron level organizations like DNA origami or colloidal assemblies. In the third problem, we zoom into the micro-scale mechanics of metamaterial chains and sheets with varying unitcell geometries to understand the macro-scale consequences. This approach guides us towards a coarse-grained elasticity theory of disordered metamaterials. This model serves as a unifying framework for both topological kinematics of un-frustrated mechanical systems as well as topological features of mechanically incompatible frustrated systems. This model helps us understand deformations in disordered systems and capture their effective non-linear mechanical behavior. All these problems exemplify how geometry affects the topological and mechanical properties of these floppy materials and alter their macro-scale elastic response.

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