General solutions of equilibrium equations for 1D hexagonal quasicrystals

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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.

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Gao, Y., Xu, S. P., & Zhao, B. S. (2009). General solutions of equilibrium equations for 1D hexagonal quasicrystals. Mechanics Research Communications, 36(3), 302–308. https://doi.org/10.1016/j.mechrescom.2008.09.014

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