Bending of thin periodic plates

TL;DR

This paper investigates the asymptotic behavior of thin, periodically structured elastic plates as their thickness approaches zero, revealing a novel discontinuous anisotropic effective energy due to microscale constraints.

Contribution

It introduces a new analysis of the nonlinear elasticity of thin periodic plates, highlighting a discontinuous anisotropic effective energy and an additional microscale isometric constraint.

Findings

Effective stored-energy density is discontinuously anisotropic.

Behavior depends on the relation between thickness and periodicity.

New microscale isometric constraint on deformation fields.

Abstract

We show that nonlinearly elastic plates of thickness $h\to 0$ with an $\varepsilon$-periodic structure such that $\varepsilon^{-2}h\to 0$ exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity: in general, their effective stored-energy density is "discontinuously anisotropic" in all directions. The proof relies on a new result concerning an additional isometric constraint that deformation fields must satisfy on the microscale.