Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism

TL;DR

This paper develops a method to solve elastodynamic problems in radially inhomogeneous, spherically anisotropic elastic materials using an extension of the Stroh formalism, highlighting the conditions for variable separation and solutions.

Contribution

It extends the Stroh formalism to spherical coordinates and identifies conditions under which separation of variables is possible in radially inhomogeneous anisotropic materials.

Findings

Separation of variables is generally possible only for radially transverse isotropic materials.

The displacement amplitudes satisfy a first-order ODE system.

The formalism extends to certain lower symmetry displacement fields.

Abstract

A method is presented for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy, i.e.\ materials such that $c_{ijkl}= c_{ijkl}(r)$ in a spherical coordinate system ${r,\theta,\phi}$. The time harmonic displacement field $\mathbf{u}(r,\theta ,\phi)$ is expanded in a separation of variables form with dependence on $\theta,\phi$ described by vector spherical harmonics with $r$-dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as $\mathbf{u}(r,\theta)$, admit this type of separation of variables solutions for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.