Operator-norm resolvent estimates for thin elastic periodically heterogeneous rods in moderate contrast

TL;DR

This paper derives operator-norm resolvent estimates for thin elastic rods with highly oscillating periodic material properties, analyzing asymptotics as the rod's thickness and oscillation period shrink together.

Contribution

It provides new resolvent asymptotics and operator-norm estimates for elastic rods with periodic heterogeneities, considering the interplay of thickness and oscillation period.

Findings

Derived resolvent asymptotics for thin elastic rods

Established operator-norm estimates in the high-oscillation regime

Analyzed displacements in invariant subspaces under various assumptions

Abstract

We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing the thin infinite elastic rod with material coefficients which periodically highly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the thickness of the rod and the period of material oscillations. These two parameters are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under some additional assumptions, as well as in the general case where these two sorts of displacements intertwine inseparably.