Crack analysis in decagonal quasicrystals by the MLPG

Abstract

A meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary-value crack problems in decagonal quasicrystals. These quasicrystals belong to the class of two-dimensional (2-d) quasicrystals, where the atomic arrangement is quasiperiodic in a plane, and periodic in the perpendicular direction. The ten-fold symmetries occur in these quasicrystals. The 2-d crack problem is described by a coupling of phonon and phason displacements. Both stationary governing equations and dynamic equations represented by the Bak’s model with oscillations for phasons are analyzed here. Nodal points are spread on the analyzed domain, and each node is surrounded by a small circle for simplicity. The spatial variation of phonon and phason displacements is approximated by the moving least-squares scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method.