Study On The Moving Crack In 1D Hexagonal Quasicrystals

The discovery of quasicrystals brings a profound revolution not only in classical theory of crystals, but also in the mathematical methods for analyzing them. Because quasicrystals are promising materials with a complex structure and unusual properties, they attract various researchers in the field of theory and experiment.However, the mechanical properties of quasicrystals can be seriously affected by defects, such as dislocation, crack, inhomogeneity etc. Therefore, the study of quasicrystals’ defect problem has great significance. The anti-plane problems of moving cracks in one-dimensional hexagonal quasicrystal bi-material are investigated in this thesis.First, the anti-plane problem of an interfacial Griffith crack of fixed length moving with a constant speed in two bonded one-dimensional hexagonal quasicrystal materials is investigated. By applying fourier integral transformation, the present boundary problem is converted into dual integral equations, and then the equations are further reduced to singular integral equations by using segmentally integral transformation. The closed form solutions of this problem are obtained, and the degradation results are consistent with the conclusions in previous literature.Assuming the phonon stress ratios?? /0to be equal to the phason stress ratio0/sH H,the plastic zone size(PZS), the crack opening displacement(COD)and the strain energy release rate of the phonon field and the phason field are formulated after the boundary problem being exactly solved. The results show that the PZS and the COD are both related to the smaller yield stress of the two bonded materials and the crack length, and the COD is ruled by material moduli and the velocity of the moving crack.The unrealistic oscillatory stresses and interface interpenetration appeared in most literatures is eliminated by utilizing the D-B model. A criterion for crack extension is proposed to overcome the difficulty caused by two different CODs. Finally the stress,displacement fields and several generally useful integral formulae are presented, and the dimensionless distribution of the stresses on the interface ahead of the crack tip are plotted.Second, the anti-plane problem of periodical interfacial cracks moving with a constant speed in two bonded one-dimensional hexagonal quasicrystal materials is investigated. By applying similar methods, the plastic zone size(PZS), the crackopening displacement(COD)and the strain energy release rate of the phonon field and the phason field are formulated after the boundary problem being exactly solved. The results show that the PZS and the COD are both related to the smaller yield stress of the two bonded materials, the crack length and the space between two neighboring cracks, the COD is also related to material moduli and the velocity of the moving crack. Finally, two corresponding problems of static and of isotropic bi-material are treated as the specific cases.In this article, the closed form solutions of two dynamic problems are derived by using fourier integral transformation and segmentally integral transformation. And the unrealistic oscillatory stresses and interface interpenetration appeared in most literatures is eliminated by utilizing the D-B model. The results provide support for the relevant problems of interface fracture and engineering application.