Wave impedance matrices for cylindrically anisotropic radially inhomogeneous elastic solids

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.

39 pages, 2 figures

References in corpus (1)

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