# The Griffith Crack and the Interaction between Screw Dislocation and Semi-Infinite Crack in Cubic Quasicrystal Piezoelectric Materials

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## Abstract

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## 1. Introduction

## 2. Basic Equations of Cubic Quasicrystal Piezoelectric Material

## 3. Basic Solution for the Fracture Problem of Cubic Quasicrystal Piezoelectric Materials

## 4. Griffith Crack Problem

## 5. Interaction between Screw Dislocation and Semi-Infinite Crack

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Hu, C.Z.; Wang, R.H.; Ding, D.H.; Yang, W.G. Piezoelectric effects in quasicrystals. Phys. Rev. B
**1997**, 56, 2463–2468. [Google Scholar] [CrossRef] - Altay, G.; Dökmeci, M.C. On the fundamental equations of piezoelasticity of quasicrystal media. Int. J. Solids Struct.
**2012**, 49, 3255–3262. [Google Scholar] [CrossRef] - Li, X.Y.; Li, P.D.; Wu, T.H.; Shi, M.X.; Zhu, Z.W. Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect. Phys. Lett. A
**2014**, 378, 826–834. [Google Scholar] [CrossRef] - Yu, J.; Guo, J.H.; Pan, E.N.; Xing, Y.M. General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics. Appl. Math. Mech. (Engl. Ed.)
**2015**, 36, 793–814. [Google Scholar] [CrossRef] - Zhou, Y.B.; Liu, G.T. Electroplastic analysis of anti-plane type ш crack in one-dimensional hexagonal quasicrystal piezoelectric materials. Chin. J. Solid Mech.
**2015**, 36, 63–68. [Google Scholar] - Guo, J.h.; Zhang, Z.Y.; Xing, Y.M. Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites. Philos. Mag.
**2016**, 96, 349–369. [Google Scholar] [CrossRef] - Cui, X.W.; Li, L.H. Anti-plane problem of finite large one-dimensional hexagonal quasicrystal piezoelectric wedge with screw dislocation. Chin. J. Appl. Mech.
**2019**, 36, 1058–1062. [Google Scholar] - Zhou, Y.B.; Liu, G.T.; Li, L.H. Effect of T-stress on the fracture in an infinite one-dimensional hexagonal piezoelectric quasicrystal with a Griffith crack. Eur. J. Mech.—A/Solids
**2021**, 86, 104184. [Google Scholar] [CrossRef] - Loboda, V.; Sheveleva, A.; Komarov, O.; Lapusta, Y. An interface crack with mixed electrical conditions at it faces in 1D quasicrystal with piezoelectric effect. Mech. Adv. Mater. Struct.
**2022**, 29, 3334–3344. [Google Scholar] [CrossRef] - Jiang, L.J.; Liu, G.T. The interaction between a screw dislocation and a wedge-shaped crack in one-dimensional hexagonal piezoelectric quasicrystals. Chin. Phys. B
**2017**, 26, 249–255. [Google Scholar] [CrossRef] - Zhang, F.; Li, X. The Analytic Solutions of a Circular Hole with Four Cracks of One-Dimensional Hexagonal Piezoelectric Quasicrystals. J. Jiangxi Norm. Univ. (Natural Sci. Ed.)
**2015**, 39, 50–54. [Google Scholar] - Bai, Q.M.; Ding, S.H. Anti-plane problem of hexagonal hole edge crack in one-dimensional hexagonal quasicrystal piezoelectric. Appl. Math. Mech.
**2019**, 40, 1071–1080. [Google Scholar] - Pi, J.D.; Zhao, Y.; Li, L.H. Interaction between a ScrewDislocation and Two Unequal Interface racks Emanating from an Elliptical Hole in One Dimensional Hexagonal Piezoelectric Quasicrystal Bi-Material. Crystals
**2022**, 12, 314. [Google Scholar] [CrossRef] - Zhou, W.M.; Fan, T.Y.; Yin, S.Y.; Wang, N.P. Anti-Plane Elasticity Problem and Mode III Crack Problem of Cubic Quasicrystal. J. Beijing Inst. Technol. (Engl. Ed.)
**2001**, 10, 250–254. [Google Scholar] - Zhang, L. Stress Intensity of Antiplane Conjugate Cracks in Cubic Quasicrystal. J. Southwest JT. Univ. (Engl. Ed.)
**2008**, 16, 285–289. [Google Scholar] - Gao, Y.; Ricoeur, A.; Zhang, L. Plane problems of cubic quasicrystal media with an elliptic hole or a crack. Phys. Lett. A
**2011**, 375, 2775–2781. [Google Scholar] [CrossRef] - Suo, Y.R.; Zhou, Y.B.; Liu, G.T. Effect of T stress on the presence of cross-shaped cracks in an optically shaped quasicrystal. Acta Mech. Solida Sin.
**2022**, 43, 95–110. [Google Scholar] - Zhang, J.M.; Mao, Z.H.; Feng, X.; Zhang, L.L.; Gao, Y. Free Vibration of Three-Dimensional Piezoelectric Cubic Quasicrystal Plates. In Proceedings of the 2020 15th Symposium on Piezoelectrcity, Acoustic Waves and Device Applications (SPAWDA), Zhengzhou, China, 16–19 April 2021. [Google Scholar]
- Wang, R.H.; Hu, C.Z.; Gui, J.N. Quasicrystal Physics; Science Press: Beijing, China, 2004. [Google Scholar]
- Gao, Y. Governing equations and general solutions of plane elasticity of cubic quasicrystals. Phys. Lett. A
**2009**, 8, 885–889. [Google Scholar] [CrossRef]

**Figure 2.**Influence of linear force ${Q}_{1}$ on the phonon field stress intensity factor ${K}_{\sigma}$.

**Figure 4.**Influence of coupling elastic coefficient ${R}_{3}$ on the phonon filed stress intensity factor.

**Figure 5.**Influence of coupling elastic coefficient ${R}_{3}$ on the phason field stress intensity factor.

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**MDPI and ACS Style**

Pi, J.; Li, L.
The Griffith Crack and the Interaction between Screw Dislocation and Semi-Infinite Crack in Cubic Quasicrystal Piezoelectric Materials. *Crystals* **2022**, *12*, 1250.
https://doi.org/10.3390/cryst12091250

**AMA Style**

Pi J, Li L.
The Griffith Crack and the Interaction between Screw Dislocation and Semi-Infinite Crack in Cubic Quasicrystal Piezoelectric Materials. *Crystals*. 2022; 12(9):1250.
https://doi.org/10.3390/cryst12091250

**Chicago/Turabian Style**

Pi, Jiandong, and Lianhe Li.
2022. "The Griffith Crack and the Interaction between Screw Dislocation and Semi-Infinite Crack in Cubic Quasicrystal Piezoelectric Materials" *Crystals* 12, no. 9: 1250.
https://doi.org/10.3390/cryst12091250