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Chin. Phys. B, 2020, Vol. 29(10): 104601    DOI: 10.1088/1674-1056/ab9ddf
ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS Prev   Next  

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

Dongsheng Yang(杨东升) and Guanting Liu(刘官厅)†
1 College of Mathematics Science, Inner Mongolia Normal University, Hohhot 010022, China
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:  19 March 2020      Revised:  18 June 2020      Published:  05 October 2020
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/Jcr with $ {\tau }_{zy}^{\infty } $ for some given $ {H}_{zy}^{\infty } $ .

Fig. 12.  

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

Fig. 13.  

Variations of J/Jcr 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.

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