On 2D Green's functions for 1D hexagonal quasi-crystals

Abstract

This paper cokncerns itself with two-dimensional fundamental solutions for infinite and semi-infinite planes, subjected to point heat sources, of one-dimensional hexagonal quasi-crystals. From the basic equations for plane problem in the context of thermo-elasticity of QCs, the rigors operator theory and generalized Almansi's theorem are employed to derive the general solutions in terms of four quasi-harmonic functions. In the framework of the present general solutions, three concreted examples are investigated. Appropriate potentials are set by the trail-and-error technique, and corresponding fundamental thermo-elastic fields are obtained in a complete and exact fashion. Since the fundamental solutions are all in elementary functions, the present solutions not only bear theoretical merits, but also can serve as benchmarks to clarify various approximate methods.