13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Lorman [9] suggested the minimal model of the phonon-phason dynamics in quasicrystals to reconcile contradictions between Bak´s and Lubensky´s arguments. In literature, analyses of dynamic crack problems are very seldom [10-12]. The purpose of this paper is to develop a reliable computational method for a general crack problem in quasicrystals with a finite size. Up to date we have practically only analytical solutions for simple boundary value problems of elasticity for quasicrystals. However, there are some limitations to apply analytical approaches for complicated boundary value problems. The finite difference method has been applied to elasto-hydrodynamics problems by Fan [12]. The basic equations for the finite element formulation can be found in the Fan`s book too. Meshless methods for solving PDE in physics and engineering sciences are a powerful new alternative to the traditional mesh-based techniques. Focusing only on nodes or points instead of elements used in the conventional FEM, meshless approaches have certain advantages. The meshless local Petrov-Galerkin (MLPG) method is a fundamental base for the derivation of many meshless formulations, since trial and test functions can be chosen from different functional spaces. The MLPG method with a Heaviside step function as the test functions has been successfully applied to multi-field coupled and crack problems [13,14]. In the present paper, the MLPG is applied to crack analysis in decagonal quasicrystals under static and transient dynamic loads. The MLPG formulation is developed for the Bak`s model. The coupled governing partial differential equations are satisfied in a weak-form on small fictitious subdomains. Nodal points are introduced and spread on the analyzed domain and each node is surrounded by a small circle for simplicity, but without loss of generality. The spatial variations of the phonon and phason displacements are approximated by the Moving Least-Squares (MLS) scheme. After performing the spatial MLS approximation, a system of ordinary differential equations for certain nodal unknowns is obtained. Then, the system of the ordinary differential equations of the second order resulting from the equations of motion is solved by the Houbolt finite-difference scheme [15] as a time-stepping method. 2. Local integral equations Two displacement fields named phonon and phason displacements are used for the deformation theory of quasicrystals [12]. The generalized Hooke`s law for plane elasticity of decagonal QC is given as 11 11 11 12 22 11 22 ( ) c c R w w σ ε ε = + + + , 22 12 11 22 22 11 22 ( ) c c R w w σ ε ε = + − + , 12 21 66 12 21 12 2 ( ) c R w w σ σ ε = = + − , 11 1 11 2 22 11 22 ( ) H Kw K w R ε ε = + + − , 22 1 22 2 11 11 22 ( ) H Kw K w R ε ε = + + − , 12 1 12 2 21 12 2 H Kw K w R ε = − − , 21 1 21 2 12 12 2 H Kw K w R ε = − + , (1) where ijε and ijσ correspond to the phonon strain and stress tensor, and ij w and ij H denote the
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