General solutions of equilibrium equations for 1D hexagonal quasicrystals

Abstract

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. © 2008 Elsevier Ltd. All rights reserved.