Hamiltonian System for Two-Dimensional Decagonal Quasicrystal Plates and Its Analytical Solutions

This paper deals with the three-dimensional (3D) elasticity problem of two-dimensional (2D) decagonal quasicrystal (QC) plates by an analytical symplectic approach. Based on the elasticity theory of QCs, a system of Hamiltonian governing equations for a 2D decagonal QC plate is constructed with the fundamental mechanical quantities as the symplectic variables, such that the symplectic eigenvalue problem is formed. Combined with the properties of trigonometric functions of two variables, the adjoint symplectic orthogonality relations among the eigenvectors of the Hamiltonian operator are derived, and the expansion theorem of the double Fourier trigonometric series is proven. The symplectic analytical solutions of stresses and displacements in the phonon and phason fields of the 2D decagonal QC plate with four lateral simply supported edges are given without any assumptions. Furthermore, the numerical results are plotted to verify the correctness of the analytical solutions by comparison with those acquired by the pseudo-Stroh formalism. The proposed approach has broad applicability and high precision and can be extended to the 3D elasticity analysis of more QCs.