Stress formalism for elastic waves in anisotropic solids

Abstract

Traditionally, the propagation of bulk elastic waves in anisotropic crystals or solids is governed by the displacement form of the elastodynamic wave equation. Phase velocities follow after transforming the wave equation into an eigenvalue problem, which gives the Christoffel equations, and solving for the roots of the characteristic polynomial. Alternatively, only a few researchers have considered a pure stress wave equation of motion to describe the propagation of waves in elastic solids. However, to our knowledge, the dual characteristic polynomial and its solutions based on a pure stress formalism has not been established previously. This presentation will outline the needed steps to reach and solve the dual characteristic polynomial. Phase velocity solutions follow from the formation of new principal invariants, which are functions of the elastic compliance constants and propagation directions. The dual formalism is proven consistent to the traditional displacement formalism through the use of sixth-rank Levi-Civitia identities. Lastly, possible extensions of this work will be highlighted.