General solutions of equilibrium equations for 1D hexagonal quasicrystals

Three general solutions are presented for the coupled equilibrium equations in one-dimensional (1D) hexagonal quasicrystals (QCs). These solutions are expressed in terms of two displacement functions, which satisfy a quasi-harmonic equation and a sixth-order partial differential equation, respectively, as well as the theory of differential operator matrix developed by Wang (2006) [Wang, X., 2006. The general solution of one-dimensional hexagonal quasicrystal. Mech. Res. Commun. 33, 576–580]. However, it is difficult to obtain rigorous analytic solutions and not applicable in most cases, since a displacement function satisfies a higher-order equation. By utilizing a theorem, a decomposition and superposition procedure is taken to replace the sixth-order function with three quasi-harmonic functions, and the general solutions are simplified in terms of these quasi-harmonic functions. In consideration of different cases of three characteristic roots, each general solution possesses three cases, but all are in simple forms that are convenient to be used. As a special case, the general solutions for 1D hexagonal QCs can be degenerated into Elliott–Lodge (E–L) solutions of transversely isotropic elasticity when phonon–phason fields coupling effect is absent. Furthermore, it should be pointed out that the general solutions obtained here are complete in z-convex domains.