十次对称二维准晶中的圆弧形裂纹

采用复变函数法研究了在远场承受均布声子场机械载荷作用下十次对称二维准晶中穿透周期轴的一个圆弧形裂纹；报导了表征声子场和相位子场的4个复应力函数的解析表达式，并由复应力函数获得了两个裂尖处声子场和相位子场应力强度因子、裂面张开位移以及能量释放率的表达式．     

十次对称二维准晶材料的位错动力学问题

论文采用Bak准晶弹性动力学模型研究了十次对称二维准晶材料的位错动力学问题。将Stroh公式推广到了准晶材料动力学问题的研究中,给出了位移和应力一般解的表达式。讨论了位错移动速度、声子场和相位子场耦合弹性常数对声子场、相位子场位移和应力的影响。     

一维正方准晶椭圆孔口平面弹性问题的解析解

利用复变方法,引入广义保角映射,研究了一维正方准晶中具有椭圆孔口的平面弹性问题,给出了各应力分量的复变表示,并在特殊情况下转化为Griffith裂纹,得到该裂纹尖端处的应力强度因子的解析解.当准晶体的对称性增加时,正方准晶椭圆孔口平面弹性问题退化为一维四方准晶中具有椭圆孔口的平面弹性问题,同样在特殊情况下转化为Griffith裂纹,得到裂纹尖端处的应力强度因子的解析解.     

三维立方准晶椭球夹杂的Eshelby张量

准晶体颗粒复合智能材料具有优异的物化性能和应用前景。不同于传统的各向同性材料,三维立方准晶体材料包含声子场,相位子场,及声子-相位子耦合场。为更好地研究准晶体颗粒夹杂问题,揭示准晶体材料夹杂问题的物理现象,本文利用本征应变公式和柯西留数定理,考虑椭球体夹杂,获得了三维立方准晶材料夹杂问题的Eshelby张量,并给出了统一的表达式。进而,当三维立方准晶夹杂形状为球形、棒状、扁平状和带状时,获得了封闭形式的三维立方准晶Eshelby张量表达式。同时,给出了椭球体长径比变化时Eshelby张量的变化规律,这对研究准晶体颗粒夹杂问题具有重要的理论意义。     

压电陶瓷中圆币形裂纹在横向剪力下的机－电耦合行为

以弹性位移分量和电势函数为基本未知量时，横观各向同性压电介质三维问题的场方程可化为四个联立的二阶线性偏微分方程组，本文导出了用四个调和函数表示位移分量及电势函数的表达式，即得到了该场方程的势函数通解，作为通解的应用举例，文中求解了圆币形裂纹受横向剪切作用的问题，得到了裂尖附近应力场及电位移场的解析表达式，结果表明，在横向剪切载荷下圆币形裂纹的尖端场及应力、电位移强度因子均具有明显的机－电耦合性质，而应力和电位移分量在裂尖仍具有－１／２的奇异性。     

一维六方准晶中螺型位错偶极子与直线纳米裂纹的干涉效应

关于准晶中螺型位错偶极子与直线纳米裂纹干涉效应的问题，提出了一种可获得精确解的新方法。首先，将Gurtin-Murdoch表/界面理论推广运用到准晶材料中的纳米裂纹问题，建立了裂纹表面上含声子场、相位子场的应力边界模型；然后，利用复变函数中的共形映射方法，将直线裂纹问题化为单位圆孔问题；再借助于柯西积分方法获得了问题的封闭解；最后，数值结果表明：当裂纹的尺寸减小到纳米量级时，表面效应将对声子场和相位子场中的应力场、像力及像力偶矩产生显著的影响，从而改变其力学性能，即裂纹尺寸越小，表面效应越强，声子场和相位子场中的应力场、像力及像力偶矩的数值变化越大；螺型位错偶极子力偶臂的方向对像力的极值和自身平衡亦有较大影响。     

压电介质中受拉伸与弯曲联合作用的圆币形裂纹问题

以弹性位移分量和电势函数为基本未知量时，横观各向同性压电介质非轴对称三维问题的控制微分方程是四个二阶线性偏微分方程相联立的方程组。本文导出了用四个调和函数表示位移及电势的该方程组的势函数通解。作为通解的应用举例，文中求解了压电陶瓷材料中受拉伸与弯曲联合作用的圆币形裂纹问题，得到了裂纹尖端附近应力场及电位移场的解析表达式。结果表明裂尖场以及应力强度因子和电位移强度因子均表现出复杂的机－电耦合行为。     

弹性椭圆夹杂纵向剪切问题

获得纵向剪切下弹性椭圆夹杂问题的精确解。将复变函数的分区全纯函数理论，Cauchy型积分和Riemann边值问题相结合，求得各复势函数之间的解析关系，从而得到问题的封闭形式解，并给出了界面应力的解析表达式。本文解答与已有文献结果一致。本文发展的分析方法，为求解复杂多连通域的平面弹性问题提供了一条有效途径。     

含微裂纹弹性体的应力应变关系

本义建立了考虑裂纹闭合和裂纹表面摩擦影响的含微裂纹弹性体的应力应变关系，给出了柔度张量增量的显式表达式。对于二维平面应力和平面应变状态，给出了等效工程弹性系数。数值计算结果表明，裂纹闭合和裂纹面摩擦对裂纹体的应力应变关系和等效工程弹性系数有重要影响。     

含微裂纹弹性体的应力应变关系

本义建立了考虑裂纹闭合和裂纹表面摩擦影响的含微裂纹弹性体的应力应变关系，给出了柔度张量增量的显式表达式。对于二维平面应力和平面应变状态，给出了等效工程弹性系数。数值计算结果表明，裂纹闭合和裂纹面摩擦对裂纹体的应力应变关系和等效工程弹性系数有重要影响。     

A DECAGONAL QUASICRYSTAL WITH AN ARC-SHAPED CRACK

A decagonal quasicrystal,which is weakened by an arc-shaped crack penetratingthrough the solid in the period direction,and which is subjected to remote uniform phonon stresses,is investigated by applying the complex variable method which is just developed by the authors.It is found that the phonon and phason stresses near the crack tips exhibit inverse square rootsingularities.The four complex stress functions characterizing the phonon and phason fields arederived.Explicit expressions for the phonon and phason stress intensity factors,crack openingdisplacements and energy release rate are also presented.     

An exact solution for a multilayered two-dimensional decagonal quasicrystal plate

By extending the pseudo-Stroh formalism to two-dimensional decagonal quasicrystals, an exact closed-form solution for a simply supported and multilayered two-dimensional decagonal quasicrystal plate is derived in this paper. Based on the different relations between the periodic direction and the coordinate system of the plate, three internal structure cases for the two-dimensional quasicrystal layer are considered. The propagator matrix method is also introduced in order to treat efficiently and accurately the multilayered cases. The obtained exact closed-form solution has a concise and elegant expression. Two homogeneous quasicrystal plates and a sandwich plate made of a two-dimensional quasicrystal and a crystal with two stacking sequences are investigated using the derived solution. Numerical results show that the differences of the periodic direction have strong influences on the stress and displacement components in the phonon and phason fields; different coupling constants between the phonon and phason fields will also cause differences in physical quantities; the stacking sequences of the multilayer plates can substantially influence all physical quantities. The exact closed-form solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and laminated plates. The numerical results can also be employed to verify the accuracy of the solution by numerical methods, such as the finite element and difference methods, when analyzing laminated composites made of quasicrystals.     

Crack solutions and weight functions for plane problems in three-dimensional quasicrystals

The plane problems of an elliptic hole and a crack in three-dimensional quasicrystals subject to far-field loadings are studied. The generalized Stroh formalism is adopted here, and the explicit solutions for the coupled fields are obtained in the closed form. When the elliptic hole reduces to a crack, the analytical expressions for both the entire fields and the asymptotic fields near the crack tip are determined. The crack theory of quasicrystals, including the determination of the field intensity factors, crack opening displacements, crack tip energy release rates and so on, is a prerequisite. Applying Betti’s theorem of reciprocity, the weight functions for a quasicrystal body with a crack are derived. The weight functions provide a means of calculating the intensity factors for the crack when both phonon and phason point forces are imposed at arbitrary locations.     

集中载荷作用下各向异性平面内的椭圆孔或裂纹问题

应用复变函数Cauchy积分的方法,对含有椭圆孔或裂纹的各向异性平面,系统地导出了当其在面内受任意集中载荷作用时的复应力函数解或裂纹应力强度因子解析解,即基本解;并通过基本解的迭加,得到了在椭圆孔周或裂纹表面作用一般外载时的解,其特例证实了上述解的正确性。     

Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip

The anti-plane fracture problem for a finite crack in a one-dimensional hexagonal quasicrystal strip is analyzed. By using Fourier transforms, the mixed boundary value problems are reduced to the dual integral equations. The solution of the dual integral equations is then expressed by the complete elliptic integrals of the first and the third kinds. The expressions for stress, strains, displacements and field intensity factors of the phonon and phason fields near the crack tip are obtained exactly. The path-independent integral derived by a conservation law equals the energy release rate, which can be used as a fracture criterion for a mode III fracture problem.     

Axially symmetric thermal stress of an external circular crack under general thermal conditions

Summary The paper presents a solution for the linear thermoelastic problem of determining axisymmetric stress and displacement fields in an isotropic elastic solid of infinite extent weakened by an external circular crack under general mechanical loadings and general thermal conditions. The mechanical loadings and thermal conditions applied on the crack faces are axisymmetric, being non-symmetric about the crack plane. In similar lines of [7], equations of equilibrium of an elastic solid conducting heat have been solved using Hankel transforms and Abel operators of the first kind. Expressions for stress, displacement, temperature and heat flux functions are obtained in terms of Abel transforms of the first kind of the jumps of stress, displacement, temperature and heat flux at the crack plane. Two types of thermal conditions, that is, general surface temperatures and general heat flux on faces of the crack are considered. In both the cases, closed form solutions have been obtained for the unknown functions solving Abel type of integral equations. Explicit expressions for stresses, displacements, temperature fields, stress intensity factors have been obtained. Two special cases of thermal conditions in which: (i) crack faces are subjected to constant non-symmetric temperatures over a circular ring area, (ii) crack faces are subjected to constant non-symmetric heat flux over a circular ring area, have been considered. In some special cases, results have been compared with those from the literature.     

A steady line heat source in a decagonal quasicrystalline half-space

The problem of a steady line heat source within a decagonal quasicrystalline half-space, in which the boundary is traction-free, is investigated in detail by applying the complex variable technique. The line heat source is infinitely long in the period direction, then the plane strain condition prevails. The four complex stress functions are derived. It is found that the determination of the complex stress functions is independent of the thermal boundary conditions. The stress fields induced by the heat source are explicitly given based on the complex stress functions. Furthermore, we consider, both qualitatively and quantitatively, the distribution of σ₁₁ on the boundary y = 0 of the half-space to demonstrate the influence of the phonon–phason coupling constant R on the stress field.     

Elastic field in an infinite medium of one-dimensional hexagonal quasicrystal with a planar crack

This present work is concerned with planar cracks embedded in an infinite space of one-dimensional hexagonal quasicrystals. The potential theory method together with the general solutions is used to develop the framework of solving the crack problems in question. The mode I problems of three common planar cracks (a penny-shaped crack, an external circular crack and a half-infinite crack) are solved in a systematic manner. The phonon and phason elastic fundamental fields along with some important parameters in crack analysis are explicitly presented in terms of elementary functions. Several examples are given to show the applications of the present fundamental solutions. The validity of the present solutions is discussed both analytically and numerically. The derived analytical solutions of crack will not only play an important role in understanding the phonon–phason coupling behavior in quasicrystals, but also serve as benchmarks for future numerical studies and simplified analyses.     

On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies

Based on rigorous operator theory, a general solution of the three-dimensional elasticity equations of equilibrium for 1D hexagonal quasicrystals is obtained. The solution is expressed in terms of four quasiharmonic functions, which is very simple and useful. The point phonon (phason) force solution of an infinite 1D hexagonal quasicrystal body is derived, all in terms of elementary functions. They can play an important role in numerical simulations such as boundary element method.