New general solutions of plane elasticity of one-dimensional quasicrystals

A theory of general solutions of plane problems is developed for the coupled equations of plane elasticity of one-dimensional quasicrystals. By introducing a displacement function, very large numbers of complicated equations of the in-plane problem are simplified to a sixth-order partial differential governing equation, and then three general solutions are presented through an operator method. Since the displacement function is required to satisfy the sixth-order governing equation, it is still difficult to obtain rigorous analytic solutions from the higher-order equation directly and is not applicable in most cases. Therefore, a decomposition and superposition procedure is employed to replace the higher-order displacement function with three second-order displacement functions, and accordingly the general solutions are simplified in terms of these functions. In consideration of different cases of three characteristic toots, the general solutions involve three cases, but all are in simple forms that are conveniently applied. Furthermore, it is worth noting that the general solutions obtained here are complete in X-3-convex domains. To illustrate the application of the general solutions obtained, the closed form solutions are obtained for wedge problems subjected to point phonon force and phason force at the apex.