Sharp operator-norm asymptotics for thin elastic plates with rapidly oscillating periodic properties

TL;DR

This paper derives sharp operator-norm asymptotics for thin elastic plates with rapidly oscillating periodic properties, combining homogenisation and dimension reduction techniques in the linear elasticity regime.

Contribution

It introduces a novel approach to obtain precise operator-norm estimates for elastic plates with periodic moduli in the asymptotic limit where period and thickness are small.

Findings

Established sharp operator-norm asymptotics for the elastic plate model.

Derived homogenisation dimension-reduction estimates under various energy scalings.

Applicable to wave propagation in thin elastic structures with periodic properties.

Abstract

We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order of smallness. Assuming that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response, such as the case in e.g. acoustic wave propagation, we derive a class of "hybrid", homogenisation dimension-reduction, norm-resolvent estimates for the plate, under different energy scalings with respect to the plate thickness.