On extraordinary semisimple matrix $\textbf {N}(v)$ for anisotropic elastic materials

The $6 \times 6$ real matrix $N\left ( v \right )$ for anisotropic elastic materials under a two-dimensional steady-state motion with speed $v$ is extraordinary semisimple when $N\left ( v \right )$ has three identical complex eigenvalues $p$ and three independent associated eigenvectors. We show that such an $N\left ( v \right )$ exists when $v \ne 0$. The eigenvalues are purely imaginary. The material can sustain a steady-state motion such as a moving line dislocation. Explicit expressions of the Barnett-Lothe tensors for $v \ne 0$ are presented. However, $N\left ( v \right )$ cannot be extraordinary semisimple for surface waves. When $v = 0$, $N\left ( 0 \right )$ can be extraordinary semisimple if the strain energy of the material is allowed to be positive semidefinite. Explicit expressions of the Barnett-Lothe tensors and Green’s functions for the infinite space and half-space are presented. An unusual phenomenon for the material with positive semidefinite strain energy considered here is that it can support an edge dislocation with zero stresses everywhere. In the special case when $p = i$ is a triple eigenvalue, this material is an un-pressurable material in the sense that it can change its (two-dimensional) volume with zero pressure. It is a counterpart of an incompressible material (whose strain energy is also positive semidefinite) that can support pressure with zero volume change.