The Application of the Stroh Formalism to Prestressed Elastic Media

Abstract

The Stroh formalism is a six-dimensional representation of the equations governing plane motions of an elastic body, stemming from a juxtaposition of the displacement and a traction vector. Crucially, the formalism leads to a sextic eigenvalue problem which is the mainspring of far-reaching theoretical developments. It is known that the formalism extends to prestressed unconstrained elastic media subject to a restriction on the prestress. In this paper, the limitation is removed, and it is shown that the sextic eigenvalue problem can also be constructed for a prestressed elastic medium which is incompressible. The latter problem is exhibited as the limit of the former in a process in which the condition of incompressibility is reached through a one-parameter family of nearly incompressible elastic materials. As an application of the theory, the analysis of surface waves in a homogeneously prestressed semi-infinite body of incompressible elastic material is carried as far as the derivation of the secular equation, determining the speed of propagation. Complete results are obtained in the special case in which the material is orthotropic, with the symmetry axes aligned with the principal axes of prestress and the surface wave basis.