539.3

interface crack, stress, quasicrystal, antiplane loading, limited electric permeability, problem of linear relationship.

The present study considers a mode III interface crack in a one-dimensional (1D) piezoelectric quasicrystal subjected to antiplane phonon and phason loading, as well as an in-plane electric field. Due to the complex function approach, all required electromechanical parameters are presented through vector-functions analytic in the entire complex plane, except in the crack region. The cases of electrically impermeable (insulated) and electrically limited permeable conditions on the crack faces are considered. In the first case, a vector Hilbert problem in the complex plane is formulated and solved exactly. In the second case, the quadratic equation with respect to the electric flux through the crack region is also obtained. Its solution enables the determination of phonon and phason stresses, displacement jumps (sliding), and also electric characteristics along the material interface. Analytical formulas are also derived for the corresponding stress intensity factors associated with each field. Numerical computations for three selected variants of the loading conditions were conducted, and the resulting field distributions are visualised to show crack continuation beyond the crack and also inside the crack region.

1. Shechtman D., Blech I., Gratias D., Cahn J. W. (1984) Metallic phase with long‑range orientational order and no translational symmetry. Phys. Rev. Lett., 53, pp. 1951–1953.
2. Ding D. H., Yang W., Hu C. Z., Wang R. (1993) Generalized elasticity theory of quasicrystals. Phys. Rev. B, 48, pp. 7003–7010.
3. Fan T. Y. (2011). Theory of Elasticity of Quasicrystals and Its Applications. Springer.
4. Shi W. C., Li H. H., Gao Q. H. (2007) Interfacial cracks of antiplane sliding mode between usual elastic materials and quasicrystals. Key Eng. Mater, 340–341, pp. 453–458.
5. Zhou Y.‑B., Li X.‑F. (2018) Exact solution of two collinear cracks normal to the boundaries of a 1D layered hexagonal piezoelectric quasicrystal. Philos. Mag, 98, pp. 1780–1798.
6. Zhou Y. B.; Li X. F. (2018) Two collinear mode‑III cracks in one‑dimensional hexagonal piezoelectric quasicrystal strip. Engineering Fracture Mechanics, 189, pp.  133–147. Doi: 10.1016/j.engfracmech.2017. 10.030.
7. Tupholme G. E. (2018) A non‑uniformly loaded anti‑plane crack embedded in a half‑space of a one‑dimensional piezoelectric quasicrystal. Meccanica, 53, pp. 973–983. Doi: 10.1007/S11012‑017‑0759‑1.
8. Kletskov O. M., Silich‑Balhabaieva V. B., Sheveleva A. E., Loboda V. V. (2025) On deformation peculiarities of two thin strips with a microcrack at the interface. Scientific Journal of TNTU, 118 (2), pp. 153–167.
9. Hu K. Q., Jin H., Yang Z., Chen X. (2019) Interface crack between dissimilar one‑dimensional hexagonal quasicrystals with piezoelectric effect. Acta Mech, 230, pp. 2455–2474.
10. Govorukha V., Kamlah M. (2024) Interface crack with mixed electric boundary conditions in quasicrystals. Arch. Appl. Mech, 94, pp. 589–607.
11.  Loboda V., Sheveleva A., Komarov O., Chapelle F., Lapusta Y. (2022) Arbitrary number of electrically permeable cracks on the interface between two one‑dimensional piezoelectric quasicrystals with piezoelectric effect. Eng. Fract. Mech, 276 pp. 108878.
12. Hao T. H., Shen Z. Y. (1994) A new electric boundary condition of electric fracture mechanics and its applications. Eng. Fract. Mech, 47, pp. 793–802.
13. Yang J., Li X. (2014) The anti‑plane shear problem of two symmetric cracks originating from an elliptical hole in 1D hexagonal piezoelectric QCs. Adv. Mater. Res, 936, pp. 127–135. Doi: 10.4028/www.scientific. net/AMR.936.127.
14. Muskhelishvili N.  I. (1975). Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen.
15. Zhou Y.‑B., Li X.‑F. (2018) Exact solution of two collinear cracks normal to the boundaries of a 1D layered hexagonal piezoelectric quasicrystal. Philos. Mag, 98, pp. 1780–1798. Doi: 10.1080/14786435.2018.1459057.
16.  Li X. Y., Li P. D., Wu T. H. (2014) Three‑dimensional fundamental solutions for one‑dimensional hexagonal quasicrystal with piezoelectric effect. Phys. Lett. A, 378, pp. 826–834. Doi: 10.1016/j.physleta.2014.01.016.

1. Shechtman D., Blech I., Gratias D., Cahn J. W. (1984) Metallic phase with long‑range orientational order and no translational symmetry. Phys. Rev. Lett., 53, pp. 1951–1953.
2. Ding D. H., Yang W., Hu C. Z., Wang R. (1993) Generalized elasticity theory of quasicrystals. Phys. Rev. B, 48, pp. 7003–7010.
3. Fan T. Y. (2011). Theory of Elasticity of Quasicrystals and Its Applications. Springer.
4. Shi W. C., Li H. H., Gao Q. H. (2007) Interfacial cracks of antiplane sliding mode between usual elastic materials and quasicrystals. Key Eng. Mater, 340–341, pp. 453–458.
5. Zhou Y.‑B., Li X.‑F. (2018) Exact solution of two collinear cracks normal to the boundaries of a 1D layered hexagonal piezoelectric quasicrystal. Philos. Mag, 98, pp. 1780–1798.
6. Zhou Y. B.; Li X. F. (2018) Two collinear mode‑III cracks in one‑dimensional hexagonal piezoelectric quasicrystal strip. Engineering Fracture Mechanics, 189, pp.  133–147. Doi: 10.1016/j.engfracmech.2017. 10.030.
7. Tupholme G. E. (2018) A non‑uniformly loaded anti‑plane crack embedded in a half‑space of a one‑dimensional piezoelectric quasicrystal. Meccanica, 53, pp. 973–983. Doi: 10.1007/S11012‑017‑0759‑1.
8. Kletskov O. M., Silich‑Balhabaieva V. B., Sheveleva A. E., Loboda V. V. (2025) On deformation peculiarities of two thin strips with a microcrack at the interface. Scientific Journal of TNTU, 118 (2), pp. 153–167.
9. Hu K. Q., Jin H., Yang Z., Chen X. (2019) Interface crack between dissimilar one‑dimensional hexagonal quasicrystals with piezoelectric effect. Acta Mech, 230, pp. 2455–2474.
10. Govorukha V., Kamlah M. (2024) Interface crack with mixed electric boundary conditions in quasicrystals. Arch. Appl. Mech, 94, pp. 589–607.
11.  Loboda V., Sheveleva A., Komarov O., Chapelle F., Lapusta Y. (2022) Arbitrary number of electrically permeable cracks on the interface between two one‑dimensional piezoelectric quasicrystals with piezoelectric effect. Eng. Fract. Mech, 276 pp. 108878.
12. Hao T. H., Shen Z. Y. (1994) A new electric boundary condition of electric fracture mechanics and its applications. Eng. Fract. Mech, 47, pp. 793–802.
13. Yang J., Li X. (2014) The anti‑plane shear problem of two symmetric cracks originating from an elliptical hole in 1D hexagonal piezoelectric QCs. Adv. Mater. Res, 936, pp. 127–135. Doi: 10.4028/www.scientific. net/AMR.936.127.
14. Muskhelishvili N.  I. (1975). Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen.
15. Zhou Y.‑B., Li X.‑F. (2018) Exact solution of two collinear cracks normal to the boundaries of a 1D layered hexagonal piezoelectric quasicrystal. Philos. Mag, 98, pp. 1780–1798. Doi: 10.1080/14786435.2018.1459057.
16.  Li X. Y., Li P. D., Wu T. H. (2014) Three‑dimensional fundamental solutions for one‑dimensional hexagonal quasicrystal with piezoelectric effect. Phys. Lett. A, 378, pp. 826–834. Doi: 10.1016/j.physleta.2014.01.016.