Boundary value problems of holomorphic vector functions in 1D QCs

By means of the generalized Stroh formalism, two-dimensional (2D) problems of one-dimensional (1D) quasicrystals (QCs) elasticity are turned into the boundary value problems of holomorphic vector functions in a given region. If the conformal mapping from an ellipse to a circle is known, a general method for solving the boundary value problems of holomorphic vector functions can be presented. To illustrate its utility, by using the necessary and sufficient condition of boundary value problems of holomorphic vector functions, we consider two basic 2D problems in 1D QCs, that is, an elliptic hole and a rigid line inclusion subjected to uniform loading at infinity. For the crack problem, the intensity factors of phonon and phason fields are determined, and the physical sense of the results relative to phason and the difference between mechanical behaviors of the crack problem in crystals and QCs are figured out. Moreover, the same procedure can be used to deal with the elastic problems for 2D and three-dimensional (3D) QCs.