13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- phason strain and stress tensor, respectively. Symbols ij c , R and iK denote phonon elastic constants, phonon-phason coupling parameter and phason elastic constants, respectively. The phonon and phason strains are defined as ( ) , , 1 ( ) 2 ij i j j i u u ε = + x , (2) , ( ) ( ) ij i j w w= x x , (3) where ( ) iu x and ( ) iw x are the phonon and phason displacements, respectively. The phonon field describes the mechanical displacements of the crystal system, and the phason field represents the atom arrangement along the quasiperiodic direction. The phonon strains are the same as in classical elasticity and they are symmetric. However, the phason strains are new physical quantities used only in quasi-crystal elasticity and they are asymmetric. According to Bak`s model [7] the phason structure disorders are realized by fluctuations in quasicrystals. The balance of momentum is valid for phonon deformation and similarly for phason oscillations too. Then, the model is described by following governing equations: , ( , ) ( , ) ( , ) ij j i i X u σ τ τ ρ τ + = x x x & , (4) , ( , ) ( , ) ( , ) ij j i i H g w τ τ ρ τ + = x x x & , (5) where iu& , iw& , ρ, i X and ig denote the acceleration of the phonon and phason displacements, the mass density, and the body force vectors, respectively. Both governing equations have mathematically a similar structure. The MLPG method constructs a weak-form over the local fictitious subdomains such as sΩ , which is a small region taken for each node inside the global domain [16]. The local subdomains could be of any geometrical shape and size. In the present paper, the local subdomains are taken to be of a circular shape for simplicity. The local weak-form of the governing equations (4) and (5) can be written as * , ( , ) ( , ) ( , ) ( ) 0 s ij j i i ik u X u d σ τ ρ τ τ Ω ⎡ ⎤ − + Ω = ⎣ ⎦ ∫ x x x x & , (6) * , ( , ) ( , ) ( , ) ( ) 0 s ij j i i ik H w g u d τ ρ τ τ Ω ⎡ ⎤ − + Ω = ⎣ ⎦ ∫ x x x x & , (7) where * ( ) ik u x is a test function. Applying the Gauss divergence theorem to the first domain integrals in both equations one gets [ ] * * * , ( , ) ( ) ( ) ( , ) ( ) ( , ) ( , ) ( ) 0 s s s ij j ik ij ik j i i ik t n u d t u d u t X t u d σ σ ρ ∂Ω Ω Ω Γ− Ω+ − + Ω= ∫ ∫ ∫ x x x x x x x x & , (8) [ ] * * * , ( , ) ( ) ( ) ( , ) ( ) ( , ) ( , ) ( ) 0 s s s ij j ik ij ik j i i ik H t n u d H t u d w t g t u d ρ ∂Ω Ω Ω Γ− Ω+ − + Ω= ∫ ∫ ∫ x x x x x x x x & , (9)
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