Buckling transition and boundary layer in non-Euclidean plates

Details Buckling transition and boundary layer in non-Euclidean plates Buckling transition and boundary layer in non-Euclidean plates

2009-02-17

arxiv.org/abs/0902.2841v2

Physics

Condensed Matter

Soft Condensed Matter

Scientific paper

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.

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