13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- phonon and the phason fields in the vicinity of the crack tip are obtained. The exact solutions of the stress intensity factors for the phonon field and the phason field are obtained respectively, which are very useful in practice. 2. Basic equations When defects parallel to the quasi-periodic axis of 1D hexagonal QCs exist, the geometrical properties of the materials will be invariable along the quasi-periodic direction. In this case, the corresponding elasticity problem can be decomposed into two independent problems, i.e., a plane elasticity of conventional hexagonal crystal which can be solved by the route of the linear elastic theory [19] and an anti-plane phonon-phason field coupling elasticity problem [4]. Thus, we only need consider the latter one. The physical problem considered in this paper is shown in Fig. 1. 1x a a− ∞ 32σ ∞ 31 H 2x ∞ 31σ ∞ 32 H g g ih ih − o a x Figure 1. A lip-shape crack in 1D hexagonal QCs. It is assumed that the quasi-periodic direction of 1D hexagonal QCs is along the positive direction of 3x axis. In this case, all field variables are independent of 3x and we have the following deformation geometrical equations [4] 2, 3, 3 3 j j j u ε = = ε , 3, 3 j j w v = (1) the equilibrium equations 0, 3 , = j j σ 0, 3 , = j j H (2) and the generalized Hooke’s law , 3 3, 44 3, 3 j j j C u R v + = σ , 2 3, 3 3, 3 j j j H R u K v + = (3) where 1,2 j = ; the repeated indices denote summation; a comma in the subscripts stands for a
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