A Stress Analysis of Some Fundamental Specimens of Soft-Matter Quasicrystals with Eightfold Symmetry Based on Generalized Dynamics

Fang Wang,Hai-Yun Hu,Tian-You Fan,Hui Cheng

doi:10.1155/2019/8789151

Abstract

This paper reports a stress analysis of some fundamental samples made of soft-matter quasicrystals with 8-fold symmetry based on the generalized dynamics. The most distinction from the hydrodynamics for solid quasicrystals is that the structure of soft matter belongs to a complex liquid, which is an intermediate phase between solid and liquid and behaves natures of both solid and liquid. In addition, the soft-matter quasicrystals possess high symmetry, and the symmetry breaking is of fundamental importance. So the Landau symmetry breaking theory and elementary excitation principle are therefore the paradigm of the study of soft-matter quasicrystals. Soft-matter quasicrystals belong to the complex fluid, in which the fluid phonon elementary excitation is introduced apart from the phonon and phason elementary excitations. With this model and the equation of state, the equations of motion for possible soft-matter quasicrystals of 8-fold symmetry are derived. The initial boundary value problems for the xy plane field are solved by applying the finite difference method, in which the z-direction represents the 8-fold symmetry axis. A complete hydrodynamics analysis is given to quantitatively explore the phonon, phason, and fluid fields as well as their interactions in the physical time-space domain. The analysis shows the governing equations are exact to the prediction of the dynamics of soft-matter quasicrystals. The computational results reveal the gigantic differences of physical properties between solid and soft-matter quasicrystals.

Highlights

Fan [1] suggested a theory of generalized hydrodynamics, or generalized dynamics in brief, for soft-matter quasicrystals observed in liquid crystals, colloids, and polymers during the period 2004–2011 [2,3,4,5,6] and in surfactants more recently [7] by analyzing and summarizing their symmetry, structure, and dynamic behaviour. e theory is inspired by the hydrodynamics of solid quasicrystals proposed by Lubensky et al [8], but there are principal differences between the hydrodynamics of soft-matter and solid quasicrystals.In the hydrodynamics of solid quasicrystals, the effects of solid viscosity and elasticity are considered, while in the hydrodynamics of soft-matter quasicrystals, the effects of fluid and elasticity are studied
In the hydrodynamics of soft-matter quasicrystals, one should introduce the fluid phonon elementary excitation apart from the phonon and phason excitations in which the latter are well known in the solid quasicrystal study, and the fluid phonon is introduced for the first time by Fan [1] for the soft-matter quasicrystal study which was originated from the Landau school [9], and they claimed that the fluid acoustic wave is a fluid phonon
A complete solution of equation set of hydrodynamics of possible soft-matter octagonal quasicrystals is constructed through the finite difference method

Summary

Introduction

Fan [1] suggested a theory of generalized hydrodynamics, or generalized dynamics in brief, for soft-matter quasicrystals observed in liquid crystals, colloids, and polymers during the period 2004–2011 [2,3,4,5,6] and in surfactants more recently [7] by analyzing and summarizing their symmetry, structure, and dynamic behaviour. e theory is inspired by the hydrodynamics of solid quasicrystals proposed by Lubensky et al [8], but there are principal differences between the hydrodynamics of soft-matter and solid quasicrystals.In the hydrodynamics of solid quasicrystals, the effects of solid viscosity and elasticity are considered, while in the hydrodynamics of soft-matter quasicrystals, the effects of fluid and elasticity are studied. We here discuss another interesting case that the soft-matter quasicrystals flow past an obstacle, for example, a circular cylinder, shown, and the governing equation (1) should be modified as the generalized Oseen equations.

Results

Conclusion