Elastic properties of two-dimensional quasicrystals

Quasicrystals (QC) with two-dimensional quasiperiodic and one-dimensional periodic structure are considered. Their symmetry can be described by embedding the three-dimensional physical space V(E) in a five-dimensional superspace V, which is the direct sum of V(E) and a two-dimensional internal space V(I). A displacement v in V can be written as v = u + w, where u in V(E) and w in V(I). If the QC has a point group P in V(E) that is crystallographic, it is assumed that w and a vector u' in V(E) lying in the plane in which the crystal is quasiperiodic transform under equivalent representations of P, inequivalent ones if the point group is 5-, 8-, 10- or 12-gonal. From the Neumann principle follow restrictions on the form of the phonon, phason and phonon-phason coupling contributions to the elastic stiffness matrix that can be determined by combining the restrictions obtained for a set of elements generating the point group of interest. For the phonon part, the restrictions obtained for the generating elements do not depend on the system to which the point group belongs. This remains true for the phason and coupling parts in the case of crystallographic point groups but, in general, breaks down for the non-crystallographic ones. The form of the symmetric 12 x 12 matrix giving the phonon, phason and phonon-phason coupling contributions to the elastic stiffness is presented in graphic notation.