Positive disclination in a thin elastic sheet with boundary

TL;DR

This paper investigates how a positive disclination affects the shape and stress of a finite elastic sheet, revealing deviations from ideal cone behavior due to elasticity and boundary conditions, with implications for buckling and curvature.

Contribution

It provides rigorous analytical and numerical analysis of disclination-induced deformations in finite elastic sheets, highlighting the effects of extensibility and boundary conditions.

Findings

Gaussian curvature is non-localized and negative away from the defect with elasticity.

Stress fields lack Dirac singularity when extensibility is considered.

Increasing Young's modulus leads to development of Dirac singularities in curvature and stress.

Abstract

An isolated positive wedge disclination deforms an initially flat elastic sheet into a perfect cone when the sheet is of infinite extent and is elastically inextensible. The latter requires the elastic stretching strains to be vanishingly small. In this paper, rigorous analytical and numerical results are obtained for the disclination induced deformed shape and stress field of a bounded F{\"o}ppl-von K{\'a}rm{\'a}n elastic sheet with finite extensibility, while emphasising the deviations from the perfect cone solution. In particular, the Gaussian curvature field is no longer localised as a Dirac singularity at the defect location whenever elastic extensibility is allowed and is necessarily negative in large regions away from the defect. The stress field, similarly, has no Dirac singularity in the presence of elastic extensibility. However, with increasing Young's modulus of the sheet, while keeping the bending modulus and the domain size fixed, both of these fields tend to develop a Dirac singularity. Noticeably, in this limiting behaviour, inextensibility eludes the bounded elastic sheet due to persisting regions of non-trivial Gaussian curvature away from the defect. Other results in the paper include studying the effect of specific boundary conditions (free, simply supported, or partially clamped) on the Gaussian curvature field away from the defect and on the buckling transition from the flat to a conical solution.