Analytic Solutions to Two-Dimensional Decagonal Quasicrystals with Defects Using Complex Potential Theory

Haobai Cao,Yiqing Shi,Wu Li

doi:10.3390/cryst9040209

Abstract

An analytical treatment for two-dimensional point group 10 mm decagonal quasicrystals with defects was suggested based on the complex potential method. On the basis of the assumption of linear elasticity, two new conformal maps were applied to two examples: the first was an arc with an elliptic notch inner surface in a decagonal quasicrystal, where the complex potentials could be exactly obtained; and the second was concerned with a decagonal point group 10 mm quasicrystalline strip weakened by a Griffith crack, which was subjected to a pair of uniform static pressures. Using the basic idea underlying crack theory, the extent of the stress intensity factors was analytically estimated. If the height was allowed to approach infinity, these results can be turned into the known results of an “ordinary” crystal with only phonon elastic parameters when the phason and phonon-phason elastic constants are eliminated.

Highlights

Quasicrystal is a new structure as well as a novel material that has presented an important application prospect in engineering [1]
The study of the crack and fracture problems of the material is significant. It is well-known that the deformation of quasicrystals is governed by two different displacement fields: one is the phonon field, which is similar to the conventional displacement field u(ux, u y, uz ) under the long-wave length approximation; and the other is the phason field w(wx, w y, wz ), which is an unusual physical quantity compared to the traditional condensed matter physics and materials science [2,3,4,5,6,7,8,9]
In the over 200 individual quasicrystals observed to date, there are about 70 individual quasicrystals belonging to two-dimensional decagonal quasicrystals

Summary

Introduction

Quasicrystal is a new structure as well as a novel material that has presented an important application prospect in engineering [1]. The study of the crack and fracture problems of the material is significant. To investigate the notch/crack and fracture problems of the material, Fan introduced a mathematical theory of the elasticity of quasicrystals, where one of the mathematical theories can be found in his recently published monograph [10]. In the over 200 individual quasicrystals observed to date, there are about 70 individual quasicrystals belonging to two-dimensional decagonal quasicrystals These kinds of quasicrystals are very important from a fundamental point and from their applications. Notch/crack problems for conventional structural materials were studied by Muskhelishvili [28]. We further developed the complex analysis for so-called Saint-Venant problems of quasicrystalline materials, which may extend the methodology to more worthwhile engineering applications

Governing Equations of Elasticity of Decagonal Quasicrystals

An Arc of Elliptic Notch Inner Surface in a Decagonal Quasicrystal

Solutions to a Decagonal Quasicrystalline Strip Containing a Centric Crack

Conclusions and Discussion