The present study deals with the problem of interfacial cracks of antiplane sliding mode between a usual elastic material and a one-dimensional hexagonal quasicrystal. Based on the physical facts that balance of the phason stress field is foreign to the real force in or out of quasicrystal in physical space, and the quasicrystal is bonded to a usual elastic material without both phason displacement and stress fields, the problem is described by analytic functions and attributed to find solutions of the Riemann-Hilbert problem. It is found that the stress intensity factor is not related to phason strain field, and the phason stress field does not exist. The discontinuity of phonon displacement field across crack is related to the phason displacement field because of the coupling of phonon and phason strain fields. Although there is not the phason displacement on the bonded portion of interface, it exists on the crack’s surface. The energy release rate obtained from interfacial crack’s propagating is different from that of an interfacial crack between two different pure elastic materials.