立方准晶材料中的运动螺型位错

发展了立方准晶的位错弹性理论．通过引入位移势函数,使得立方准晶的反平面弹性动力学问题归结为求解两个波动方程,得到了运动螺型位错的位移场、应力场与能量的解析表达式及运动位错的速度极限．这些为研究此固体材料的塑性变形的物理机理提供了重要的信息．     

立方准晶压电材料的半空间问题北大核心CSCD

考虑了立方准晶压电材料的半空间问题.给出了反平面机械载荷和面内电载荷作用下立方准晶压电材料弹性问题的控制方程,结合半无限区域表面边界条件,利用算子理论和复变函数方法获得了立方准晶压电材料半空间问题一般解的表达式.基于一般解得到了集中线力作用下,半空间问题的声子场和相位子场的位移、应力以及电位移的解析表达式.     

一维正方准晶中半无限裂纹问题的解析解

运用广义复变函数方法,通过构造适当的广义保角映射,研究了含有沿准周期方向穿透的半无限裂纹的一维正方准晶的反平面弹性问题,给出了在部分裂纹面上受均匀面外剪切时应力场和裂纹尖端应力强度因子的解析解.将此方法进一步推广到半无限裂纹垂直于一维正方准晶的准周期方向穿透的情形中,得到了相应的平面弹性问题的解析解.当准晶体的对称性增加时,还可以得出一维四方准晶相应问题的解析解.     

十次对称二维准晶弹性半平面问题的复变函数方法

研究了集中力作用下二维十次对称准晶半平面弹性问题的复变函数方法.首先将Stroh公式推广到二维准晶中,这里保留了Stroh公式的本质特征,在此基础上,采用推广的Stroh公式给出了应力和位移的通解,结合边界条件,获得了应力和位移的解析表达式,为实际应用奠定了理论基础.表明复变函数方法是解决十次对称二维准晶复杂弹性边值问题的有力工具.     

对边简支十次对称二维准晶板弯曲问题的辛分析北大核心CSCD

该文讨论了对边简支十次对称二维准晶中厚板弹性问题的辛方法.将十次对称二维准晶弹性理论基本方程转化为Hamilton对偶方程,采用分离变量方法,获得了相应Hamilton算子矩阵的辛特征值及辛特征函数系.证明了Hamilton算子矩阵的辛特征函数系在Cauchy主值意义下的完备性,在此基础上,基于Hamilton系统的辛特征函数展开,给出了十次对称二维准晶板弯曲问题的解析表达式.     

带裂纹十次对称二维准晶平面弹性的无摩擦接触问题

借助经典平面弹性复变函数方法，研究了单个刚性凸基底压头作用下，带任意形状裂纹十次对称二维准晶半平面弹性的无摩擦接触问题.利用十次对称二维准晶位移、应力的复变函数表达式, 带任意形状裂纹的准晶半平面弹性无摩擦接触问题被转换为可解的解析函数复合边值问题，进而简化成一类可解的Riemann边值问题.通过求解Riemann边值问题，得到了应力函数的封闭解, 并给出了裂纹端点处应力强度因子和压头下方准晶体表面任意点处接触应力的显式表达式.从压头下方接触应力的表达式可以看出, 接触应力在压头边缘和裂纹端点处具有奇异性.当忽略相位子场影响时, 该文所得结论与弹性材料对应结果一致.数值算例分别给出了单个平底刚性压头无摩擦压入带单个垂直裂纹和水平裂纹的十次对称二维准晶下半平面的结果.该文所得结论为准晶材料的应用提供了理论参考.     

软物质准晶广义流体动力学方程组

建立了软物质准晶广义流体动力学方程组,其基础为广义Langevin方程,推导方法为Poisson括号,它参考了固体准晶的广义流体动力学方程组,但是两者存在原则的不同.固体准晶的广义流体动力学方程组考虑了固体粘性与声子弹性和相位子弹性的相互作用,没有状态方程问题;软物质准晶广义流体动力学方程组考虑的是软物质流体声子与声子弹性和相位子弹性的相互作用,按物理学术语多出了一种元激发,而且必须考虑状态方程问题,这是一个新课题,又增加了难点.实际应用的结果发现,软物质准晶广义流体动力学方程组大大激活了广义流体动力学的效能,为软物质准晶学科的发展提供了一个数学模型,为探讨有关物理问题的时间-空间演化提供了可操作的实际可行的求解体系和分析工具,求解的结果令人满意.     

直位错和线性力作用下点群10十次对称二维准晶的弹性场研究

本文研究了直位错和线性力作用下点群10十次对称二维准晶的弹性场.首先将Stroh公式推广到点群10十次对称二维准晶研究中,在此基础上,采用推广的Stroh公式给出了应力和位移的通解,结合边界条件,获得了应力的解析表达式,为实际应用奠定了理论基础.     

平面十次对称准晶中Ⅱ型Griffith裂纹的求解

应用应力函数法,求解了二维十次对称准晶中的Ⅱ型Griffith裂纹问题。特点是把二维准晶的弹性力学问题分解成一个平面应变问题与一个反平面问题的叠加,通过引入应力函数,把平面应变问题的十八个弹性力学基本方程简化成一个八阶偏微分方程,并且求出了其在Ⅱ型Griffith裂纹情况的混合边值问题的解,所有的应力分量和位移分量都用初等函数表示出来,并且由此得出了准晶中Ⅱ型Griffith裂纹问题的应力强度因子和能量释放率。     

准晶弹性理论边值问题的可解性

通过给出准晶弹性偏微分方程组边值问题的矩阵表示去定义弱解,利用Korn不等式和函数空间理论证明了这种弱解的存在性与唯一性,从而把经典弹性理论边值问题解的存在性定理推广到准晶弹性理论上,这种理论为发展极其复杂与困难的准晶弹性的偏微分方程的边值问题的数值解提供了一个基础．     

A model of the liquid-crystal phase transition and the quasicrystal model of liquid

We compare properties of the crystal structure and of the quasicrystal model of liquid. We show that if the Lennard-Jones potential is used to model the properties of argon, then the temperature of the phase transition between the densely packed face-centered and the body-centered cubic structure is very close to the liquid-crystal transition temperature. Based on this, we propose a model of the phase transition consisting in a change of just the short-range order, which then leads to a change of the long-range order. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 155–171, April, 2007.     

Buckling and vibration of the two-dimensional quasicrystal cylindrical shells under axial compression

In this paper, the buckling and the free vibration of the quasicrystal cylindrical shells under axial compression are investigated. Three quasi-periodicity cases of quasicrystal cylindrical shells are considered. The first-order shear displacement theory of the cylindrical shells is utilized to obtain the equations of motion and the boundary conditions. Numerical results for simply supported cylindrical shells at the two ends are calculated. The effects of the geometry, in-plane phonon and phason loads, and half-wave number of the quasicrystal cylindrical shells on both the buckling loads and the frequency are demonstrated.     

Three-dimensional exact solution of layered two-dimensional quasicrystal simply supported nanoplates with size-dependent effects

A size-dependent plate model is developed to investigate the elastic responses of the multilayered two-dimensional quasicrystal nanoplates based on the nonlocal strain gradient theory for the first time. A nonlocal stress field parameter and a length scale parameter are taken into account in the new model to capture both stiffness-softening and stiffness-hardening size effects. The exact solution for a single-layer two-dimensional quasicrystal simply supported nanoplate is derived by utilizing the pseudo-Stroh formalism in conjunction with the nonlocal strain gradient theory. Afterward, a dual variable and position method is used to deal with the multilayered case. Numerical examples are presented to study the dependence of size-dependent effect on nanoplate length and the influences of scale parameters on the quasicrystal nanoplate subjected to a z-direction mechanical load on its top surface. The proposed model should be useful to verify various nanoplate theories and other numerical methods.     

Three-dimensional exact thermo-elastic analysis of multilayered two-dimensional quasicrystal nanoplates

A three-dimensional thermo-elastic analytical solution for two-dimensional quasicrystal simply supported nanoplates subjected to a temperature change on their top surface is presented. The nonlocal theory and pseudo-Stroh formalism are used to obtain the exact solution for a homogeneous two-dimensional decagonal quasicrystal nanoplate with its thickness direction as a quasi-periodic direction. The propagator matrix method is introduced to deal with the corresponding multilayered nanoplates. Comprehensive numerical results show that nonlocal parameters, stress-temperature coefficients, stacking sequences have great influence on the stress, displacement components and heat fluxes of the nanoplates. In addition, the stacking sequences also influence the temperature and heat fluxes of the nanoplate. The exact thermo-elastic solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and multilayered plates. The mechanical behaviors of the nanoplates in numerical results can also serve as benchmarks to verify various thin-plate theories or other numerical methods.     

Two-dimensional solitons in irregular lattice systems

We compute and study localized nonlinear modes (solitons) in the semi-infinite gap of the focusing two-dimensional nonlinear Schrödinger (NLS) equation with various irregular lattice-type potentials. The potentials are characterized by large variations from periodicity, such as vacancy defects, edge dislocations, and a quasicrystal structure. We use a spectral fixed-point computational scheme to obtain the solitons. The eigenvalue dependence of the soliton power indicates parameter regions of self-focusing instability; we compare these results with direct numerical simulations of the NLS equation. We show that in the general case, solitons on local lattice maximums collapse. Furthermore, we show that the Nth-order quasicrystal solitons approach Bessel solitons in the large-N limit.     

Universal sampling,quasicrystals and bounded remainder sets

We examine the result due to Matei and Meyer that simple quasicrystals are universal sampling sets, in the critical case when the density of the sampling set is equal to the measure of the spectrum. We show that in this case, an arithmetical condition on the quasicrystal determines whether it is a universal set of “stable and non-redundant” sampling.     

Proving the deterministic period breaking of linear congruential generators using two tile quasicrystals

We describe the design of a family of aperiodic PRNGs (APRNGs). We show how a one-dimensional two tile cut and project quasicrystal (2TQC) used in conjunction with LCGs in an APRNG generates an infinite aperiodic pseudorandom sequence. In the suggested design, any 2TQC corresponding to unitary quadratic Pisot number combined with either one or two different LCGs can be used.

     

Analysis of anti-plane interface cracks in one-dimensional hexagonal quasicrystal coating

The displacement discontinuity method is extended to study the fracture behavior of interface cracks in one-dimensional hexagonal quasicrystal coating subjected to anti-plane loading. The Fredholm integral equation of the first kind is established in terms of displacement discontinuities. The fundamental solution for anti-plane displacement discontinuity is derived by the Fourier transform method. The singularity of stress near the crack front is analyzed, and Chebyshev polynomials of the second kind are numerically adopted to solve the integral equations. The displacement discontinuities across crack faces, the stress intensity factors, and the energy release rate are calculated from the coefficients of Chebyshev polynomials. In combination with numerical simulations, a comprehensive study of influencing factors on the fracture behavior is conducted.