On anisotropic elastic materials that possess three identical Stroh eigenvalues as do isotropic materials

For anisotropic elastic materials for which the displacements ${u_i}$ depend on ${x_1}$ and ${x_2}$ only, a general solution for ${u_i}$ depends on one variable $z = {x_1} + p{x_2}$ where $p$ is an eigenvalue of the fundamental elasticity tensor of Stroh. There are six $p$’s which consist of three pairs of complex conjugates. For isotropic materials, $p = \pm i$ are the eigenvalues of multiplicity three. We point out trivial cases in which a completely anisotropic material has the eigenvalues $p = \pm i$ and has the solutions to two-dimensional elasticity problems that are identical to the solutions for isotropic materials. Excluding these trivial cases, we show that $p = \pm i$ can be the eigenvalues of multiplicity three for monoclinic materials with the symmetry plane at ${x_1} = 0$, at ${x_2} = 0$, or at any plane that contains the ${x_3}$-axis. If the symmetry plane is at ${x_3} = 0$, then $p = \pm i$ occur only when the material is transversely isotropic with the axis of symmetry at the ${x_3}$-axis. We also consider the general case in which the eigenvalues are arbitrary and are of multiplicity three. The eigenrelation associated with the triple eigenvalues is nonsemisimple for all cases studied here. There are only two independent eigenvectors associated with the triple eigenvalues.