Symplectic approach for plane elasticity problems of two-dimensional octagonal quasicrystals

Abstract

The symplectic approach (SA) is used in this paper to analyze the plane elasticity problems of two-dimensional (2D) octagonal quasicrystals (QCs). The equilibrium equations for point group 8mm octagonal QCs are first transferred into Hamiltonian dual equations. Then the symplectic eigenvalue problem of the corresponding Hamiltonian operator matrix is derived by applying the method of separation of variables. Based on the eigenvalue analysis and expansion of symplectic eigenvectors, the exact analytic solutions for point group 8mm octagonal QCs with selected boundary conditions are established. Numerical results for the phonon and phason displacements are presented and validated by the finite integral transform method (FITM), which are useful for validation of other numerical methods. In addition, inspired by the SA, the equilibrium equations of plane elasticity of octagonal QCs with Laue class 15 are simplified, which is an open problem. The approach presented here is rational and systematic with clear step-by-step derivation procedure, thus it has potential for plane elasticity problems of other QCs. The results obtained in this paper can serve as benchmarks for future researches.