Generalization of the Stroh Formalism

It was shown in Chapter 12 that the Stroh formalism can be extended to elastodynamics when the problem is a steady state motion. In this chapter we will see several areas where the Stroh formalism can be generalized. First, the formalism is extended to a group of general boundary conditions that include as special cases the boundary conditions for a traction-free, a rigid, and a slippery surface. Explicit solutions are obtained for the Green's function and the image force in a half-space with a slippery boundary surface subjected to a line dislocation in the half-space. The Stroh formalism is then extended to thermo-anisotropic elasticity where we point out that applications to the interfacial crack may lead to a new and higher order stress singularity. Generalization of the formalism to piezoelectric materials results in an octet formalism for which there are four pairs of complex eigenvalues. Extensions of the Stroh formalism to three-dimensional deformations of anisotropic body require a special attention and are investigated in the next chapter. The Stroh formalism is in terms of the displacement u and the stress function ϕ, both 3-vectors. It is therefore most suitable for boundary conditions that are in terms of the displacement or the surface traction. For a slippery surface the normal component of the displacement and the two tangential components of the surface traction vanish. The boundary conditions are not in terms of u or ϕ, but a mixture of one component from u and two components from ϕ. This destroys the elegant sextic formalism of Stroh. We present here a generalized formalism that is applicable to a class of general boundary conditions. They include as special cases the boundary conditions for a traction-free, a rigid, and a slippery surface. The material presented below is taken from Ting and Wang (1992).