A unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic material

For a two-dimensional deformation of an anisotropic elastic material, the Lekhnitski formalism applies to elastostatics only. It does not apply to steady state motion such as surface waves or a moving line dislocation. The Stroh formalism applies to elastostatics as well as steady state motion. However, it does not apply to an incompressible material because some of the elastic stiffnesses become unbounded in the incompressible limit. Modifying a new formalism studied recently, a more explicit and remarkably simple expression of the matrices in the new formalism is given, and a unified formalism that is applicable to elastostatics as well as steady state motion of a compressible or incompressible anisotropic elastic material is obtained. As an application, the secular equation is presented for surface waves propagating in a monoclinic material with the symmetry plane at x3 ¼ 0 that has a more explicit expression than what is available in the literature. For a general anisotropic elastic material, explicit expressions are obtained for the sextic equation for the eigenvalues p and the eigenvectors b and a for any steady state wave speed. The explicit expressions recover results related to one-component surface waves. When the material is incompressible, it is shown that the eigenvalues p remain complex for elastostatics. We also show that, when the material is incompressible, a monoclinic material with the symmetry plane at x3 ¼ 0 behaves like that of an orthotropic material.