Collinear periodic cracks and/or rigid line inclusions ofantiplane sliding mode in one-dimensional hexagonal quasicrystal

For a one-dimensional (1D) hexagonal quasicrystal (QC), there is the periodic (x1,x2)(x1,x2)-plane of atomic structures with the quasiperiodic direction x3x3-axis along which there exists a phason displacement. The macroscopically collinear periodic cracks and/or rigid line inclusions are placed on the periodic (x1,x2)(x1,x2)-plane for finding out the influence of phason displacement on the related physical quantities. These two models are reduced to the Riemann–Hilbert problem of periodic analytic functions to obtain the closed-form solutions for the antiplane sliding mode. It is found that the phonon and phason stress intensity factors of cracks as well as the phonon and phason stress field intensity factors of rigid line inclusions are not related to the coupling of phonon and phason fields. These mean that there is not the influence of phason displacement on both the phonon stress intensity factor (usual stress intensity factor) of cracks and the phonon stress field intensity factor of rigid line inclusions. However, the energy release rates of periodic cracks and/or rigid line inclusions are obtained and affected not only by the periodicity of cracks and/or rigid line inclusions but also by the phason displacement.