Fig. 1. z plane.

Fig. 1. Regular hexagonal nano-hole with six nano-cracks in 1D hexagonal piezoelectric QCs.

Fig. 2. Conformal mapping (ζ = ξ + iη).

Fig. 2. z₁ plane.

Fig. 3. z₁ plane, 0<θ<π3.

Fig. 3. With surface effect: when only the phonon field stress is applied, variations of Kτ*/τzy∞ with a; when only the phason field stress is applied, variations of KH*/Hzy∞ with a; when only the electric field stress is applied, variations of KD*/Dy∞ with a. Without surface effect: classical elasticity theory.

Fig. 4. Variations of Kτ*/Hzy∞ with a.

Fig. 4. z₂ plane.

Fig. 5. Variations of Kτ*/Dy∞ with a.

Fig. 5. z₃ plane.

Fig. 6. Variations of KH*/τzy∞ with a.

Fig. 6. z₄ plane.

Fig. 7. Variations of KH*/Dy∞ with a.

Fig. 7. z₅ plane.

Fig. 8. z₆ plane.

Fig. 8. Variations of KD*/τzy∞ with a.

Fig. 9. Variations of KD*/Hzy∞ with a.

Fig. 9. ζ plane.

Fig. 10. Variations of K with L for some given a.

Fig. 10. ζ plane.

Fig. 11. Variations of J/J_cr with τzy∞ for some given Hzy∞.

Fig. 12. Variations of J/J_cr with Hzy∞ for some given Dy∞.

Fig. 13. Variations of J/J_cr with Dy∞ for some given τzy∞.