A Theory of General Solutions of Plane Problems in Two-Dimensional Octagonal Quasicrystals

Abstract

A theory of general solutions of plane problems is developed for the coupled equations in plane elasticity of two-dimensional octagonal quasicrystals. In virtue of the operator method, the general solutions of the antiplane and inplane problems are given constructively with two displacement functions. The introduced displacement functions have to satisfy higher order partial differential equations, and therefore it is difficult to obtain rigorous analytic solutions directly and is not applicable in most cases. In this case, a decomposition and superposition procedure is employed to replace the higher order displacement functions with some lower order displacement functions, and accordingly the general solutions are further simplified in terms of these functions. In consideration of different cases of characteristic roots, the general solution of the antiplane problem involves two cases, and the general solution of the inplane problem takes three cases, but all are in simple forms that are convenient to be applied. Furthermore, it is noted that the general solutions obtained here are complete in x3-convex domains.