Elastic theory of unconstrained non-Euclidean plates

TL;DR

This paper develops a covariant linear elasticity framework for non-Euclidean plates, capturing large displacements and intrinsic geometries, and demonstrates a buckling transition in hemispherical plates as thickness varies.

Contribution

It introduces a generalized thin plate theory for non-Euclidean bodies, extending linear elasticity to large displacements and arbitrary intrinsic geometries.

Findings

Identification of residual stresses in non-Euclidean plates

Derivation of a generalized thin plate theory for large displacements

Observation of a buckling transition in hemispherical plates

Abstract

Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.