Wave impedance matrices for cylindrically anisotropic radially inhomogeneous elastic solids

TL;DR

This paper derives and analyzes impedance matrices for radially inhomogeneous, cylindrically anisotropic elastic solids, providing explicit solutions and methods for their computation, which facilitate wave propagation and scattering analysis.

Contribution

It introduces the solid-cylinder impedance matrix ${f Z}(r)$ and its limit ${f Z}_0$, showing their properties and providing explicit solutions for certain symmetries, along with methods to compute them.

Findings

${f Z}_0$ depends only on elastic moduli and azimuthal order

${f Z}(r)$ is Hermitian, ${f Z}_0$ is negative semi-definite

Two methods for computing ${f Z}(r)$ are proposed

Abstract

Impedance matrices are obtained for radially inhomogeneous structures using the Stroh-like system of six first order differential equations for the time harmonic displacement-traction 6-vector. Particular attention is paid to the newly identified solid-cylinder impedance matrix ${\mathbf Z} (r)$ appropriate to cylinders with material at $r=0$, and its limiting value at that point, the solid-cylinder impedance matrix ${\mathbf Z}_0$. We show that ${\mathbf Z}_0$ is a fundamental material property depending only on the elastic moduli and the azimuthal order $n$, that ${\mathbf Z} (r)$ is Hermitian and ${\mathbf Z}_0$ is negative semi-definite. Explicit solutions for ${\mathbf Z}_0$ are presented for monoclinic and higher material symmetry, and the special cases of $n=0$ and 1 are treated in detail. Two methods are proposed for finding ${\mathbf Z} (r)$, one based on the Frobenius series solution and the other using a differential Riccati equation with ${\mathbf Z}_0$ as initial value. %in a consistent manner as the solution of an algebraic Riccati equation. The radiation impedance matrix is defined and shown to be non-Hermitian. These impedance matrices enable concise and efficient formulations of dispersion equations for wave guides, and solutions of scattering and related wave problems in cylinders.