13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Transient Dynamic Crack Analysis in Decagonal Quasicrystal Jan Sladek1,*, Vladimir Sladek1, Slavomir Krahulec1, Chuanzeng Zhang2, Michael Wünsche2 1 Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia 2 Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany * Corresponding author: jan.sladek@savba.sk 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 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 two-dimensional (2-d) crack problem is represented by a coupling of phonon and phason displacements. Both stationary governing equations and dynamic equations represented by the Bak model with oscillations for phason 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 the phonon and phason displacements is approximated by the Moving Least-Squares (MLS) 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. Keywords: Meshless local Petrov-Galerkin method (MLPG), phonon, phason, intensity factors 1. Introduction Qusicrystals discovered in 1984 combine aperiodic long-range positional order with noncrystallographic rotational symmetry [1]. Decagonal quasicrystals (QC) belong to the class of two-dimensional quasicrystals, where the atomic arrangement is quasiperiodic in a plane, and periodic in the third direction. The problem can be decomposed into plane and anti-plane elasticity. Here, we consider only the plane elasticity, because the anti-plane elasticity is a classical one. Experimental observations [2] have shown that quasicrystals are brittle. Therefore, to understand the effect of cracks on the mechanical behaviour of a quasicrystal, the crack analysis of quasicrystals, including the determination of the stress intensity factors, the elastic field, the strain energy release rate and so on, is a prerequisite. Many crack investigations in the QC are focused on Griffith cracks in an infinite body, where analytical solutions are available for one and two-dimensional quasicrystals [3-6]. Elastodynamics of quasicrystals brings some additional problems. A unique opinion on governing equations for the phason field is missing. According to Bak [7] the phason describes particular structure disorders in qusicrystals, and it can be formulated in a six-dimensional space. Since there are six continuous symmetries, there exist six hydrodynamic vibration modes. Then, phonons and phasons play similar roles in the dynamics and both fields should be described by similar governing equations, namely the balance of momentum. Lubensky and his students [8] were thinking that the phason field should be described by a diffusion equation with very a large diffusion time. According to them, phasons are insensitive to spatial translations and phason modes represent the relative motion of the constituent density waves. Rochal and
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