Anti-plane problem of a nano-sharp crack in one-dimensional hexagonal piezoelectric quasicrystals with the electrically semi-permeable condition

Based on the Gurtin–Murdoch surface/interface theory and the complex potential theory, the problem of an electrically semi-permeable nano-sharp crack in one-dimensional (1D) hexagonal piezoelectric quasicrystals is analyzed. By constructing conformal mapping of the sharp crack, the analytic solutions of the electroelastic field are determined. Meanwhile, the field intensity factors and the energy release rate (ERR) under the electric boundary conditions of semi-permeable are obtained. Numerical examples are given to analyze the effects of the crack size, dielectric constant, electric loading, and mechanic loadings on the dimensionless field intensity factors and the dimensionless ERR. The results show that the impact of the surface effect becomes weaker along with the growth of the crack length, but increases with the increase in the crack angle. The change of the dielectric coefficient has little influence on the dimensionless stress intensity factors (SIFs), but has a strong influence on the dimensionless electric displacement intensity factor (EDIF). The obtained results are helpful to promote the development of nano-quasicrystal composite mechanics and provide an important theoretical support for the design and application of nano-QC materials.