General solutions of three-dimensional problems for two-dimensional quasicrystals

Three-dimensional problems are systematically investigated for the coupled equations in two-dimensional hexagonal quasicrystals, and two new general solutions, which are called generalized Lekhnitskii–Hu–Nowacki (LHN) solutions and generalized Elliott–Lodge (E–L) solutions, are presented, respectively. By introducing two higher-order displacement functions, an operator analysis technique is applied in a novel way to obtain generalized LHN solutions. For further simplification, a decomposition and superposition procedure is taken to replace the higher-order displacement functions with five quasi-harmonic displacement functions, and then generalized E–L solutions are simplified in terms of these functions. In consideration of different cases of characteristic roots, generalized E–L solutions take different forms, but all are in simple forms that are conveniently applied. To illustrate the application of the general solutions obtained, the closed form solution is obtained for an infinite quasicrystal medium subjected to a point force at an arbitrary point.