Fundamental Solutions For The One Dimensional And Two Dimensional Hexagonal Quasicrystals

Quasicrystal (QC) is a kind of new solids, which differs from the traditional crystalline and amorphous materials. Owing to its unique microstructure, QC has various excellent properties, such as a high strength, wear resistance and high temperature resistant. QC attracts more and more attention of scholars both at home and abroad. Theoretical analyses plays an important role in the research of the QC. Theoretical study of QC is crucial in analyzing and interpreting its the mechanical behavior and may provide a guidance for forthcoming experiments of QC. Fundamental solutions are an important part of theoretical studies of QC. The fundamental solution of QC can be used to construct analytical solutions of various more involved boundary value problems, such as crack, inclusions and contacts. Therefore, the research of fundamental solution of QC is essential for the further research. In the present work, three aspects concerning the fundamental solutions has been carried out.(1) The fundamental solutions in the framework of thermo-electro-elasticity in an infinite/half-infinite space of 1D hexagonal QC is derived. To this end, a set of three-dimensional static general solutions in terms of 5 quasi-harmonic functions are derived with the help of rigorous operator theory and the generalized Almansi’s theorem. For an infinite/half-infinite space subjected to an external thermal load, appropriate potential functions are set by a trail-and-error technique. The corresponding Green functions for the problems in question are explicitly obtained in the closed forms. These solutions also serve as benchmarks for various numerical simulations.(2) Fundamental phonon-phason field in a half-infinite space of two-dimensional hexagonal QC is derived, on the basis of general solutions in terms of quasi-harmonic functions, by virtue of the trial-and-error technique. The extended Boussinesq and Cerruti problems are studied. Appropriate potential functions are assumed and corresponding fundamental solutions are explicitly derived in terms of elementary functions. The boundary integral equations governing the contact and crack problems are constructed from the present fundament solutions. The obtained analytical solutions can serve as guidelines for future indentation tests via scanning probe microscopy (SPM) and atomic force microscopy (AFM) methods.(3) The fundamental solutions is presented for an infinite/half-infinite space of 2D hexagonal QC in the context of thermo-elasticity. With the help of the general solutions, corresponding fundamental solutions are explicitly derived in terms of elementary functions.