Buckling transition and boundary layer in non-Euclidean plates

TL;DR

This paper studies how non-Euclidean plates transition from flat to buckled shapes, revealing the critical thickness for buckling, the nature of the transition, and the formation of boundary layers influenced by residual stresses.

Contribution

It characterizes the buckling transition in non-Euclidean plates, including the critical thickness, transition type, and boundary layer formation, based on a new theoretical framework.

Findings

Buckling transition occurs at a critical thickness.

Transition can be continuous or discontinuous depending on the metric.

Boundary layer size scales with the square root of plate thickness.

Abstract

Non-Euclidean plates are thin elastic bodies having no stress-free configuration, hence exhibiting residual stresses in the absence of external constraints. These bodies are endowed with a three-dimensional reference metric, which may not necessarily be immersible in physical space. Here, based on a recently developed theory for such bodies, we characterize the transition from flat to buckled equilibrium configurations at a critical value of the plate thickness. Depending of the reference metric, the buckling transition may be either continuous or discontinuous. In the infinitely thin plate limit, under the assumption that a limiting configuration exists, we show that the limit is a configuration that minimizes the bending content, amongst all configurations with zero stretching content (isometric immersions of the mid-surface). For small but finite plate thickness we show the formation of a boundary layer, whose size scales with the square root of the plate thickness, and whose shape is determined by a balance between stretching and bending energies.