Antiplane Fracture Problem of Three Nanocracks Emanating from an Electrically Permeable Hexagonal Nanohole in One-Dimensional Hexagonal Piezoelectric Quasicrystals

Dongsheng Yang,Guanting Liu,Ivan D Rukhlenko

doi:10.1155/2020/1372474

Abstract

Based on the Gurtin-Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping, the electrically permeable boundary condition with surface effect is established, and the antiplane fracture problem of three nanocracks emanating from a hexagonal nanohole in one-dimensional hexagonal piezoelectric quasicrystals with surface effect is studied. The exact solutions of the stress intensity factor of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate are obtained under the two electrically permeable and the electrically impermeable boundary conditions. The numerical examples show the influence of surface effect on the stress intensity factors of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate under the two boundary conditions. It can be seen that the surface effect leads to the coupling of the phonon field, phason field, and electric field, and with the decrease of cavity size, the influence of surface effect is more obvious.

Highlights

Quasicrystals (QCs) [1] were a new kind of structural material which was discovered in 1984
Erefore, the research on QCs fracture mechanics is of great significance. e presence of the phason field makes quasicrystals different from traditional crystals
In reference [5], though the researchers studied elasticity from different descriptions, the unit-cell description based on the Penrose tiling is adopted too, but the density wave description based on the Laudau [6] phenomenological theory on symmetry-breaking of condensed matter has played the central role and been widely acknowledged. is means there are two elementary excitations of low energy, phonon u and phason w for quasicrystals, in which vector u is in the parallel space and vector w is in the perpendicular space, respectively

Summary

Introduction

Quasicrystals (QCs) [1] were a new kind of structural material which was discovered in 1984. A great deal of research has been done on the antiplane fracture of different defects in 1D hexagonal QCs. In references [17,18,19,20,21], the exact solutions of asymmetric cracks emanating from circular and elliptical hole in 1D hexagonal QCs have been obtained. For the study of the antiplane problem of nanoscale defect in one-dimensional hexagonal piezoelectric quasicrystals, the analytical solutions under the electrically permeable boundary condition is considered in this paper; this new study was not available in previous studies. By constructing a new conformal mapping and using the analytical function theory, the fracture problem of three cracks emanating from a hexagonal hole in one-dimensional hexagonal piezoelectric quasicrystals is studied for the first time at nanoscale. It has a good reference value for analyzing the change of stress intensity factor and energy release rate with material defect size

Basic Equation

Field Intensity Factor and Energy Release Rate

Numerical Analysis

Conclusion