Spectral and evolution analysis of composite elastic plates with high contrast

TL;DR

This paper investigates the asymptotic behaviour of thin composite elastic plates with high contrast materials, deriving effective 2D models and analyzing their spectral and evolution properties across different regimes.

Contribution

It introduces a novel operator-theoretic approach to derive and compare asymptotic models for high-contrast composite plates under various scaling regimes.

Findings

Different asymptotic regimes lead to qualitatively distinct models.

The limit spectrum and evolution equations are explicitly characterized.

The approach unifies spectral and evolution analysis for complex composite structures.

Abstract

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.