Fundamental elastic field in an infinite medium of two-dimensional hexagonal quasicrystal with a planar crack: 3D exact analysis

The present article concerns itself with three-dimensional analytical solutions for the phonon-phason field in an infinite space made of two-dimensional hexagonal quasi-crystal, which contains a planar crack subjected to a pair of equal but opposite phonon loadings on the upper and lower crack lips. Based on the general solutions in terms of harmonic functions, the method of potential theory is extended to the planar crack problems, in the context of elasticity of two-dimensional quasi-crystals. Five potentials are properly assumed and the boundary integro-differential equation governing the crack problem is established. For three common cracks (penny-shaped, external circular and half infinite planar), fundamental phonon-phason field variables in terms of elementary functions are obtained in a unified fashion. Important quantities in fracture mechanics, such as crack surface displacement, stress intensity factor and energy release rate, are explicitly presented. Furthermore, the fundamental solutions find some applications to the crack problems where distributed loadings are involved. Numerical calculations are preformed in order for a multiple purpose. The present analytical solutions can serve as benchmarks for numerous simplified analyses and various numerical codes in the future.