Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals

doi:10.1088/1674-1056/ab9ddf

Abstract

By constructing a new conformal mapping function, we study the surface effects on six edge nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional (1D) hexagonal piezoelectric quasicrystals under anti-plane shear. Based on the Gurtin–Murdoch surface/interface model and complex potential theory, the exact solutions of phonon field, phason field and electric field are obtained. The analytical solutions of the stress intensity factor of the phonon field, the stress intensity factor of the phason field, the electric displacement intensity factor and the energy release rate are given. The interaction effects of the nano-cracks and nano-hole on the stress intensity factor of the phonon field, the stress intensity factor of the phason field and the electric displacement intensity factor are discussed in numerical examples. It can be seen that the surface effect leads to the coupling of phonon field, phason field and electric field. With the decrease of cavity size, the influence of surface effect is more obvious.

Keywords: one-dimensional quasicrystals piezoelectricity surface effect; energy release rate

Received: 01 January 1900
Revised: 01 January 1900
Accepted manuscript online: 01 January 1900

PACS:	46.05.+b	(General theory of continuum mechanics of solids)
	46.25.-y	(Static elasticity)
	46.50.+a	(Fracture mechanics, fatigue and cracks)

Corresponding Authors: ^†Corresponding author. E-mail: guantingliu@imnu.edu.cn

About author:

†Corresponding author. E-mail: guantingliu@imnu.edu.cn

* Project supported by the National Key R&D Program of China (Grant No. 2017YFC1405605), the Innovation Youth Fund of the Ocean Telemetry Technology Innovation Center of the Ministry of Natural Resources, China (Grant No. 21k20190088), the Natural Science Foundation of Inner Mongolia, China (Grant No. 2018MS01005), and the Graduate Students’ Scientific Research Innovation Program of Inner Mongolia Normal University (Grant No. CXJJS19098).

Cite this article:

Dongsheng Yang(杨东升) and Guanting Liu(刘官厅)† Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals 2020 Chin. Phys. B 29 104601

Fig. 1.

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

Fig. 2.

Conformal mapping (ζ = ξ + iη).

Fig. 3.

With surface effect: when only the phonon field stress is applied, variations of $ {K}_{\tau }^{* }/{\tau }_{zy}^{\infty } $ with a; when only the phason field stress is applied, variations of $ {K}_{H}^{* }/{H}_{zy}^{\infty } $ with a; when only the electric field stress is applied, variations of $ {K}_{D}^{* }/{D}_{y}^{\infty } $ with a. Without surface effect: classical elasticity theory.

Fig. 4.

Variations of $ {K}_{\tau }^{* }/{H}_{zy}^{\infty } $ with a.

Fig. 5.

Variations of $ {K}_{\tau }^{* }/{D}_{y}^{\infty } $ with a.

Fig. 6.

Variations of $ {K}_{H}^{* }/{\tau }_{zy}^{\infty } $ with a.

Fig. 7.

Variations of $ {K}_{H}^{* }/{D}_{y}^{\infty } $ with a.

Fig. 8.

Variations of $ {K}_{D}^{* }/{\tau }_{zy}^{\infty } $ with a.

Fig. 9.

Variations of $ {K}_{D}^{* }/{H}_{zy}^{\infty } $ with a.

Fig. 10.

Variations of K with L for some given a.

Fig. 11.

Variations of J/J_cr with $ {\tau }_{zy}^{\infty } $ for some given $ {H}_{zy}^{\infty } $ .

Fig. 12.

Variations of J/J_cr with $ {H}_{zy}^{\infty } $ for some given $ {D}_{y}^{\infty } $ .

Fig. 13.

Variations of J/J_cr with $ {D}_{y}^{\infty } $ for some given $ {\tau }_{zy}^{\infty } $ .

Fig. A1.

z plane.

Fig. A2.

z₁ plane.

Fig. A3.

z₁ plane, $ 0\lt \theta \lt \displaystyle \frac{\pi }{3} $ .

Fig. A4.

z₂ plane.

Fig. A5.

z₃ plane.

Fig. A6.

z₄ plane.

Fig. A7.

z₅ plane.

Fig. A8.

z₆ plane.

Fig. A9.

ζ plane.

Fig. A10.

ζ plane.

[1]	Shechtman D, Blech I, Gratias D et al. 1984 Phys. Rev. Lett. 53 1951 DOI: 10.1103/PhysRevLett.53.1951
[2]	Liu G T, Fan T Y, Guo R P 2003 Mech. Res. Commun. 30 335 DOI: 10.1016/S0093-6413(03)00034-X
[3]	Liu G T, Fan T Y 2003 Sci. Chin. E 46 326 DOI: 10.1360/03ye9036
[4]	Liu G T, Fan T Y, Guo R P 2004 Int. J. Solids Struct. 41 3949 DOI: 10.1016/j.ijsolstr.2004.02.028
[5]	Guo J H, Liu G T 2008 Appl. Math. Mech-Engl. 29 485 DOI: 10.1007/s10483-008-0406-x
[6]	Guo J H, Liu G T 2007 Acta Math. Appl. Sin. 30 1066 in Chinese
[7]	Guo J H, Liu G T 2008 Chin. Phys. B 17 2610 DOI: 10.1088/1674-1056/17/7/044
[8]	Chen Z, Liu G, Guan L 2011 Chin. J. Solid Mech. 29 412 in Chinese
[9]	Zhong H Y, Liu G T 2015 Chin. J. Solid Mech. 36 179 in Chinese
[10]	Hou X L, Wang J X, Jia L G 2018 Chin. J. Appl. Mech. 35 484 in Chinese
[11]	Liu G T, Yang L Y 2017 Chin. Phys. B 26 094601 DOI: 10.1088/1674-1056/26/9/094601
[12]	Li L H, Liu G T 2012 Acta Phys. Sin. 61 086103 in Chinese DOI: 10.7498/aps.61.086103
[13]	Li L H, Liu G T 2009 Mod. Phys. Lett. B 23 3397 DOI: 10.1142/S0217984909021430
[14]	Wang X 2006 Mech. Res. Commun. 33 576 DOI: 10.1016/j.mechrescom.2005.02.022
[15]	Li X, Huo H S, Shi P P 2014 Chin. J. Solid Mech. 35 135 DOI: 10.19636/j.cnki.cjsm42-1250/o3.2014.02.004
[16]	Fu X L, Wang G F, Feng X Q 2010 Eng. Fract. Mech. 77 1048 DOI: 10.1016/j.engfracmech.2010.02.022
[17]	Kim C I, Schiavone P, Ru C Q 2011 Arch. Mech. 63 267
[18]	Gurtin M E, Murdoch A I 1975 Arch. Ration. Mech. Anal. 57 291 DOI: 10.1007/BF00261375
[19]	Gurtin M E, Murdoch A I 1978 Int. J. Solids Struct. 14 431 DOI: 10.1016/0020-7683(78)90008-2
[20]	Xiao J H, Xu Y L, Zhang F C 2018 Theor. Appl. Fract. Mech. 96 476 DOI: 10.1016/j.tafmec.2018.06.012
[21]	Guo J H, Li X F 2018 Acta Mech. 229 4251 DOI: 10.1007/s00707-018-2232-1
[22]	Liu Y Z, Guo J H, Zhang X Y 2019 ZAMM-Z Angew. Math. Mech. 99 e201900043 DOI: 10.1002/zamm.201900043
[23]	Guo J H, He L T, Liu Y Z, Li L H 2020 Theor. Appl. Fract. Mech. 107 102553 DOI: 10.1016/j.tafmec.2020.102553
[24]	Song J Z, Li H, He Y Z, Ou G B 2004 J. Harbin Eng. Univ. 25 558
[25]	Jia L G, Sun H D, Wang C G 2012 Eng. Mech. 29 147 in Chinese
[26]	Jiang L J, Liu G T 2017 Chin. Phys. B 26 044601 DOI: 10.1088/1674-1056/26/4/044601

[1]	First-principles calculation of influences of La-doping on electronic structures of KNN lead-free ceramics Ting Wang(王挺), Yan-Chen Fan(樊晏辰), Jie Xing(邢洁), Ze Xu(徐泽), Geng Li(李庚), Ke Wang(王轲), Jia-Gang Wu(吴家刚), Jian-Guo Zhu(朱建国). Chin. Phys. B, 2020, 29(6): 067702.
[2]	First-principles study of polarization and piezoelectricity behavior in tetragonal PbTiO₃-based superlattices Zhenye Zhu(朱振业). Chin. Phys. B, 2018, 27(2): 027701.
[3]	Multiferroic and enhanced microwave absorption induced by complex oxide interfaces Cuimei Cao(曹翠梅), Chunhui Dong(董春晖), Jinli Yao(幺金丽), Changjun Jiang(蒋长军). Chin. Phys. B, 2018, 27(1): 017503.
[4]	Piezoelectricity in K_1-xNa_xNbO₃: First-principles calculation Li Qiang (李强), Zhang Rui (张锐), Lv Tian-Quan (吕天全), Zheng Li-Mei (郑立梅). Chin. Phys. B, 2015, 24(5): 053101.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals

Cite this article:

Online attention