The appropriate edge conditions for two-dimensional quasicrystal semi-infinite strips with mixed edge-data

For two-dimensional quasicrystal semi-infinite strips with mixed edge-data, the Betti–Rayleigh reciprocal theorem and the general solution of plane elasticity of quasicrystals are applied in a novel way to obtain the appropriate edge conditions accurate to all order. By introducing two definitions for the decaying and regular states, the necessary conditions deduced form the reciprocal theorem, for the edge-data to induce only a decaying elastostatic are directly translated into the appropriate edge conditions for the existence of a rapidly decaying solution of the strips. Once a suitable regular state, which fulfills load-free boundary conditions, is constructed for the relevant edge-data, the translation is immediate. However, this is not the situation for general edge-data. For the case of transverse bending and in-plane extension of the strips, these decaying state conditions are obtained explicitly for the first time when the mixed edge-data are imposed on the strip edge. Besides, for a degenerated form, an analytical solution of the decaying state is formulated to verify validity of these edge conditions.