Complex method of the plane elasticity in 2D quasicrystal with point group 10 mm tenfold rotational symmetry and holey problems

Abstract

The complex method of the plane elasticity in 2D quasicrystal with point group 10 mm tenfold rotational symmetry is established. First displacement potential function in the quasicrystal is represented by four analytic functions. Then by utilizing the properties of analytic function and through a great deal of derivation, the complex representations of stresses and displacements components of phonon fields and phason fields in the quasicrystal are given, which are the theoretical foundation for this method. From this theory, and by the help of conformal transformations in the theory of complex function, the problems of elliptic hole in the quasicrystal are solved. Its special cases are the solutions of well-known crack problem. Meanwhile, the results show that even if under the self-counterbalance force in the quasicrystal plane with elliptic hole, the stress components of phonon fields are also related to material constants of the quasicrystal when the phonon fields and phason fields are coupled, which is another distinctive difference from the properties of classical elastic theory. Besides, the present work is generalization and application of the complex method in the classical elastic theory established by Muskhelishvili to 2D quasicrystal. As in the classical elastic theory, if only conformal transformation from the quasicrystal plane to unit circle is found, any holey and crack problem in the quasicrystal plane could be solved.