A Critical Study of Efrati et al.'s Elastic Theory of Unconstrained non-Euclidean Plates

TL;DR

This paper critically examines Efrati et al.'s elastic theory for non-Euclidean plates, identifying mathematical inconsistencies and proposing corrections to improve its physical and mathematical validity.

Contribution

It highlights the inconsistencies in Efrati et al.'s theory and suggests rectifications involving diffeomorphisms and external loadings for accurate modeling.

Findings

Efrati et al.'s theory is inconsistent with plate theory mathematics.

The theory aligns more with shell theory but uses an incorrect strain tensor.

Numerical results imply unrealistic stretching with minimal force.

Abstract

In our analysis, we show that Efrati et al.'s publication is inconsistent with the mathematics of plate theory. However it is more consistent with the mathematics of shell theory, but with an incorrect strain tensor. Thus, the authors' numerical results imply that a thin object can be stretched substantially with very little force, which is physically unrealistic and mathematically disprovable. All the theoretical work of the authors, i.e. nonlinear plate equations in curvilinear coordinates, can easily be rectified with the inclusion of both a sufficiently differentiable diffeomorphism and a set of external loadings, such as an external strain field.