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

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.

Keywords

• quasicrystal
• crack
• stress
• conformal mapping