Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals

Abstract

The first quasicrystal (QC) structure was observed in 1984. QCs possess long-range orientational and translational order while lacking the periodicity of crystals. An overview is given on some physical properties of QCs. It begins with group theory and symmetry. Then the thermodynamics of equilibrium properties and physical property tensors are discussed. Finally, the generalized elasticity theory of QCs and the elasticity theory of dislocations in QCs are presented.

General Solutions for Three-Dimensional Thermoelasticity of Two-Dimensional Hexagonal Quasicrystals and an Application

“…For 2D hexagonal QCs, the point groups 6 mm 622 6 m2 6/mmm belong to Laue class 10. The linear constitutive equations of the QCs take the following form [10,11] (3) where C 11 C 12 C 13 C 33 C 44 represent the elastic constants in phonon field, K 1 K 2 K 3 K 4 are the elastic constants in phason field, R 1 R 2 R 3 R 4 are the phonon-phason coupling elastic constants, 1 3 are the thermal constants, T is the variation of the temperature, and 2C 66 = C 11 − C 12…”

“…Ding et al [9] established the generalized linear elastic theory of QCs, which provides us with a fundamental theory based on the notion of a continuum model to describe the elastic behavior of QCs. For comprehensive and detailed presentation for the linear elasticity of QCs, the review by Hu et al [10] and a monograph by Fan [11] are recommended.…”

Elasto-Dynamics of Quasicrystals

“…So the total displacement field in a quasicrystal can be expressed by u = u ⊕ u ⊥ = u ⊕ w, in which u is in the parallel space, or the physical space; w is in the complement space, or the perpendicular space; which is an internal space and ⊕ denotes the direct sum. On the basis of the above physical framework and the extended methodology in mathematical physics from classical elasticity, the independent elastic constants for different symmetries of quasicrystals can be determined [16][17][18][19][20][21]. Then the mathematical elasticity theory of quasicrystals has been developed rapidly.…”

Electric-Elastic Field Induced by a Straight Dislocation in One-Dimensional Quasicrystals

“…The elasticity of QCs is described by the generalized elastic theory established by Ding et al [2] that has been proved to be a powerful and important tool to study the mechanical behavior of QCs. Expressions of physical properties of QCs, such as elasticity, thermal expansion, and piezoelectricity tensors have been obtained in [3] and [4].…”