Stress potential function formulation and complex variable function method for solving the elasticity of quasicrystals of point group 10 and the exact solution for the notch problem

The plane elasticity equations of two-dimensional quasicrystals of point group 10 are reduced to a single partial differential equation with eighth order by introducing a stress potential function. Further, we develop the complex variable function method for classical elasticity theory to that of the quasicrystals. The complex representations of stress and displacement components of phonon and phason fields in the quasicrystals are given. With the help of conformal transformation, an exact solution for the elliptic notch of the quasicrystals is presented. The solution of the Griffith crack problem as a special case of the results is also observed. This work shows that the stress potential and complex variable function methods are powerful for solving the complicated boundary value problems of higher order partial differential equations originating from quasicrystal elasticity.