Certain problems in constrained cubic quasicrystals: General solutions and infinite-space Green's functions

The generalized elasticity is applied to the analysis of quasicrystals to account for the mechanical response of the phonon and phason fields. Using the operator method introduced by Lur'e, the general solutions are derived for three-dimensional constrained cubic quasicrystals, leading to the fundamental solution with the phonon-phason coupling effect. Based on the experimental observation that most three-dimensional quasicrystals are of weak anisotropy, we firstly introduce a weak anisotropy approximation on the phason field and the phono-phason coupling field to construct a new quasicrystal model with weak anisotropy. Using this new model, we derived the new general solutions like the generalized Boussinesq-Galerkin solution, the generalized Papkovich-Neuber solution and the generalized Kelvin solution. We identified the distinction between quasicrystals and classical elastic isotropic media when anisotropy is weak. These solutions present complicated mechanical behaviors that cannot be seen in traditional isotropic media, such as phonon-phason symmetry breaking. We find out that the phason field itself could lead to certain unique phenomena without involving complex anisotropy, and the loading ratio has a noticeable influence on the quasicrystals even with weak anisotropy. These findings could serve as the theoretical basis to guide the future experiments of quasicrystals.