Elastic analysis of a mode II crack in a decagonal quasi-crystal

The elastic analysis of a mode II Griffith crack penetrating through a decagonal quasi-crystal along the periodic axis is made within the context of the continuum theory. By using a general solution obtained previously, the problem in the case of uniform shear stress at infinity is solved, and the analytical expressions for the entire stress field disturbed by an internal crack are derived in an explicit form. The asymptotic fields of the displacement and stress around a crack tip in both phonon and phason fields indicate that the stresses near a crack tip exhibit the square-root singularity. The formula for evaluating the energy release rate is also given. If imposing that the phason field is absent, the well-known results of a mode II crack in a conventional material are recovered from the present results.