Analytic solutions of problem about a circular hole with a straight crack in one-dimensional hexagonal quasicrystals with piezoelectric effects

• For the first time, piezoelectric effect is introduced in 1D hexagonal QCs.
• The electric impermeable and permeable conditions are adopted for the first time.
• The SIFs and EDIFs are obtained for the first time.
• In the limiting cases, the obtainable results match with the classical ones.
• Two numerical examples are given for the first time.

The anti-plane shear problem about a circular hole with a straight crack in one-dimensional (1D) hexagonal quasicrystals (QCs) with piezoelectric effects is investigated by means of the complex variable function method and using the technique of conformal mapping. The analytic solutions of the stress intensity factors (SIFs) and the electric displacement intensity factors (EDIFs) with electrically impermeable and permeable conditions were obtained, and these solutions have an important theoretical significance for the engineering application of QCs materials. When the circle radius tends to zero, the present results can be reduced to the cases of the Griffith crack. In the absence of the phason field, the obtainable results in this paper agrees well with the results for piezoelectric materials. Numerical analysis is then conducted to discuss the influences of geometric parameters and applied mechanical/electric loads on the field intensity factors and energy release rate.