Analysis of an interface crack with multiple electric boundary conditions on its faces in a one-dimensional hexagonal quasicrystal bimaterial

An interface crack between dissimilar one-dimensional hexagonal quasicrystals with piezoelectric effect under anti-plane shear and in-plane electric loadings is considered. Mixed boundary conditions at the crack faces are studied. Using special representations of field variables via sectionally analytic vector-functions, a homogeneous combined Dirichlet–Riemann boundary value problem and a Hilbert problem are formulated. Exact analytical solutions of both these problems are obtained, and analytical expressions for the phonon and phason stresses and the electric field as well as for the derivative jumps of the phonon and phason displacements and also the electrical displacement jump along the bimaterial interface are derived. The field intensity factors are determined as well. The dependencies of the mentioned values on the magnitude and direction of the external electric loading and different ratios of electrically conductive and electrically permeable crack face zone lengths are demonstrated in graph and table forms.