Exact Solution for Elastic Networks on Curved Surfaces

TL;DR

This paper presents an exact non-linear elasticity solution for elastic networks on curved surfaces, enabling precise analysis of effects like line tension and particle positioning, with implications for virus assembly.

Contribution

It introduces an exact non-linear elasticity framework for curved surface networks, surpassing previous approximate methods and allowing detailed effect analysis.

Findings

Exact solutions for several symmetric geometries

Agreement with linear elasticity beyond its usual range

Insights into virus assembly processes

Abstract

The problem of characterizing the structure of an elastic network constrained to lie on a frozen curved surface appears in many areas of science and has been addressed by many different approaches, most notably, extending linear elasticity or through effective defect interaction models. In this paper, we show that the problem can be solved by considering non-linear elasticity in an exact form without resorting to any approximation in terms of geometric quantities. In this way, we are able to consider different effects that have been unwieldy or not viable to include in the past, such as a finite line tension, explicit dependence on the Poisson ratio or the determination of the particle positions for the entire lattice. Several geometries with rotational symmetry are solved explicitly. Comparison with linear elasticity reveals an agreement that extends beyond its strict range of applicability. Implications for the problem of the characterization of virus assembly are also discussed.