Certain problems in constrained cubic quasicrystals: Half-space Green's functions

The half-space Green's functions of phonon and phason fields in the constrained cubic quasicrystals are derived for various boundary conditions. The analytical solutions of the generalized Lorentz problem, Mindlin problem, and two mixed boundary problems, are directly derived by using the Phan-Thien method, the general solutions, and infinite-space Green's functions. Unique phenomena are observed in stress fields, which show their significant effect on mechanical response of quasicrystals by the phonon and phason force. Furthermore, we observe, without considering the phason force, an interesting distinction between quasicrystals and traditional isotropic media in the Lorentz problem. We further notice that the unique stress distribution is induced by the phonon force rather than by the phason force. Because the phonon force is much more convenient to be applied than the phason force in experiments, this result provides us an efficient channel to study quasicrystals involving the phonon force only, and then to verify the current linear elasticity of quasicrystals.