Three-dimensional Green's functions for two-dimensional quasi-crystal bimaterials

Abstract

Owing to their specific structure, which can neither be classified as crystalline nor amorphous, quasi-crystals (QCs) exhibit properties that are interesting to both material science and mathematical physics or continuum mechanics. Within the framework of a mathematical theory of elasticity, one major focus is on features evolving from the coupling of phonon and phason fields, which is not observed in classical crystalline or amorphous materials. This paper deals with the problems of combinations of point phonon forces and point phason forces, which are applied to the interior of infinite solids and bimaterial solids of two-dimensional hexagonal QCs. By using the general solution of QCs, a series of displacement functions is adopted to obtain the analytical results when the two half-spaces are supposed to be ideally bonded or to be in smooth contact. In the final expressions, we provide three-dimensional Green's functions for infinite bimaterial QC solids in the closed form, which are very convenient to be used in the study of dislocations, cracks and inhomogeneities of the new solid phase. Furthermore, the paper is concluded by a discussion of some special cases, in which Green's functions for infinite transversely isotropic solids and Green's functions for a half-space with free or fixed boundary are given.

Fundamental solutions of penny-shaped and half-infinite plane cracks embedded in an infinite space of one-dimensional hexagonal quasi-crystal under thermal loading

“…As pointed out by Duffy [48], Ding et al [49] and Gao & Ricoeur [31] for Green's functions of specific problems, the present fundamental solutions are of high significance: they themselves can serve as benchmarks for various numerical codes and simplified solutions; in addition, when distributive thermal loadings are exerted to the crack surfaces, the corresponding field variable can be obtained by integrating the present solutions.…”

“…According to Mariano [29] and Colli & Mariano [30], the absence of a conservative self-action in the standard theory of quasi-crystal may lead to non-physical results. However, the topic is not discussed further here and the common assumptions are accepted, as in the recent studies for static problems [24,31] and dynamic problems [24,32].…”

Bending Analysis of Laminated Two-Dimensional Piezoelectric Quasicrystal Plates with Functionally Graded Material Properties

“…The 2D QC refers to a three-dimensional structure with atomic arrangement being quasi-periodic in a plane and periodic along one direction normal to the plane [27]. A 2D PQC with x 1 -x 2 plane being the quasi-periodic plane, x 3 being the periodic and poling directions referring to the Cartesian coordinates (x 1 , x 2 , x 3 ) is considered in this paper.…”

Three-dimensional analysis of a spheroidal inclusion in a two-dimensional quasicrystal body

“…Chen's method was generalized by Kogan et al [35], Ding and Chen [36] and Chen et al [37] to analyze the inclusion problem of an infinite piezoelectric material. The purpose of this paper is to further develop previous work [6,31,32], and derive an exact solution for an infinite 2D QC medium with a spheroidal inclusion. To achieve this, the general solution of 2D hexagonal QCs [38] is used to uncouple the system of equations of equilibrium.…”

“…Since the discovery of quasicrystals (QC) in an Al-Mn alloy [1,2], the elastic theory of QCs continues to attract investigators' attention [3][4][5][6][7][8][9][10]. A two-dimensional (2D) QC refers not to a real plane but to a three-dimensional (3D) solid with 2D quasiperiodic and one-dimensional periodic structure [11].…”