准晶数学弹性力学和缺陷力学

对准晶数学弹性理论的基本概念和基本框架作了介绍,在此基础上分别针对目前已经发现的几类一维准晶、二维准晶和三维准晶讨论了其数学弹性的理论体系．为了求解准晶弹性的边值问题或初值一边值问题,还必须发展相应的方法论．物理工作者在研究准晶位错弹性问题中发展了Ｇｒｅｅｎ函数方法．针对一维与二维准晶弹性中几类问题提出了分解与叠加程序,这一程序的使用,使极其复杂的准晶弹性问题得到简化,进而引进位移函数或应力函数,把数目。庞大的准晶弹性基本方程化成一个或少数几个高阶偏微分方程,进一步使求解步骤大为简化．对三维立方准晶弹性也采用了类似步骤使求解过程大为简化．在以上化简的基础上,发展了准晶弹性的边值问题或初值一边值问题的复交函数方法和 Ｆｏｕｒｉｅｒ分析方法,求得了一系列准晶位错问题和裂纹问题的分析解（古典解）．在研究准晶弹性的边值问题古典解的同时,也讨论了同这些边值问题相对应的变分问题和广义解（弱解）以及这种弱解的数值方法──有限元法．在物理学家工作基础上开展的这些工作可以看作对经典数学弹性理论和方法、经典Ｖｏｌｔｅｒｒａ位错理论、普通结构材料断裂力学和经典有限元的某些发展．此外,还把一维六方准晶弹性动力学的结果与统计物理的某些     

Solvability on boundary-value problems of elasticity of three-dimensional quasicrystals

Weak solution (or generalized solution) for the boundary-value problems of partial differential equations of elasticity of 3D (three-dimensional) quasicrystals is given, in which the matrix expression is used. In terms of Korn inequality and theory of function space,we prove the uniqueness of the weak solution.This gives an extension of existence theorem of solution for classical elasticity to that of quasicrystals,and develops the weak solution theory of elasticity of 2D quasicrystals given by the second author of the paper and his students.     

Displacement function and simplifying of plane elasticity problems of two-dimensional quasicrystals with noncrystal rotational symmetry

The paper systematically investigates the plane elasticity problems of two-dimensional quasicrystals with noncrystal rotational symmetry. First, applying their independent elastic constants, the equilibrium differential equations of the problems in terms of displacements are derived and it is found that the plane elasticity of pentagonal quasicrystals is the same as that of decagonal. Then by introducing displacement functions, huge numbers of complicated partial differential equations of the problems are simplified to a single or a pair of partial differential equations of higher order, which is called governing equations, such that the problems can be easily solved. Finally, some solving methods of these governing equations obtained are introduced briefly.     

准晶数学弹性力学和缺陷力学

对准晶数学弹性理论的基本概念和基本框架作了介绍,在此基础上分别针对目前已经发现的几类一维准晶、二维准晶和三维准晶讨论了其数学弹性的理论体系．为了求解准晶弹性的边值问题或初值一边值问题,还必须发展相应的方法论．物理工作者在研究准晶位错弹性问题中发展了Ｇｒｅｅｎ函数方法．针对一维与二维准晶弹性中几类问题提出了分解与叠加程序,这一程序的使用,使极其复杂的准晶弹性问题得到简化,进而引进位移函数或应力函数,把数目。庞大的准晶弹性基本方程化成一个或少数几个高阶偏微分方程,进一步使求解步骤大为简化．对三维立方准晶弹性也采用了类似步骤使求解过程大为简化．在以上化简的基础上,发展了准晶弹性的边值问题或初值一边值问题的复交函数方法和 Ｆｏｕｒｉｅｒ分析方法,求得了一系列准晶位错问题和裂纹问题的分析解（古典解）．在研究准晶弹性的边值问题古典解的同时,也讨论了同这些边值问题相对应的变分问题和广义解（弱解）以及这种弱解的数值方法──有限元法．在物理学家工作基础上开展的这些工作可以看作对经典数学弹性理论和方法、经典Ｖｏｌｔｅｒｒａ位错理论、普通结构材料断裂力学和经典有限元的某些发展．此外,还把一维六方准晶弹性动力学的结果与统计物理的某些      

On the fundamental equations of piezoelasticity of quasicrystal media

The three-dimensional fundamental equations of elasticity of quasicrystals with extension to quasi-static electric effect are expresses in both differential and variational invariant forms for a regular region of quasicrystal material. The principle of conservation of energy is stated for the regular region and the constitutive relations are obtained for the piezoelasticity of material. A theorem is proved for the uniqueness in solutions of the fundamental equations by means of the energy argument. The sufficient boundary and initial conditions are enumerated for the uniqueness. Hamilton’s principle is stated for the regular region and a three-field variational principle is obtained under some constraint conditions. The constraint conditions, which are generally undesirable in computation, are removed by applying an involutory transformation. Then, a unified variational principle is obtained for the regular region, with one or more fixed internal surface of discontinuity. The variational principle operating on all the field variables generates all the fundamental equations of piezoelasticity of quasicrystals under the symmetry conditions of the phonon stress tensor and the initial conditions. The resulting equations, which are expressible in any system of coordinates and may be used through simultaneous approximation upon all the field variables in a direct method of solutions, pave the way to the study of important dislocation, fracture and interface problems of both elasticity and piezoelasticity of quasicrystal materials.     

On the screw dislocation in a functionally graded material

This paper presents the stress field of a screw dislocation in a medium graded in y-direction. The medium is exponentially graded. For such a graded material theories of elasticity as well as gradient elasticity are applied. By means of the stress function technique we found exact analytical solutions of the corresponding master equations. Using the stress field, the Peach–Koehler force is given. The axial symmetry of a screw dislocation is lost due to the gradation in the y-direction.     

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.     

Nonlocal elasticity and dislocation image force due to cracks

Summary   Fundamental field equations of nonlocal elasticity are presented. With these equations, the image force on a screw dislocation due to a crack is analyzed using the conformal mapping technique. Two cases are considered: one is for a finite-length crack, the other is for an infinite one. All classical singularities of the dislocation image force are eliminated when the dislocation tends to the crack tip. The maximum of the force is obtained at the crack tip. Received 10 June 1999; accepted for publication 8 February 2000     

A Mode- II Griffith Crack in Decagonal Quasicrystals

By using the method of stress functions, the problem of mode-II Griffith crack in decagonal quasicrystals was solved. First, the crack problem of two-dimensional quasicrystals was decomposed into a plane strain state problem superposed on anti-plane state problem and secondly, by introducing stress functions, the18 basic elasticity equations on coupling phonon-phason field of decagonal quasicrystals were reduced to a single higher-order partial differential equations. The solution of this equation under mixed boundary conditions of mode-II Griffith crack was obtained in terms of Fourier transform and dual integral equations methods. All components of stresses and displacements can be expressed by elemental functions and the stress intensity factor and the strain energy release rate were determined. Biography: GUO Yu-cui (1962-), Associate professor, Doctor     

反平面弹性中边缘内分叉裂纹的奇异积分方程解法

利用复变函数和奇异积分方程方法,求解反平面弹性中半平面边缘内分叉裂纹问题。提出了满足半平面边界自由的由分布位错密度表示的半平面中单裂纹的基本解,此基本解由主要部分和辅助部分组成。将半平面边缘内分叉裂纹问题看作是许多单裂纹问题的叠加,建立了以分布位错密度为未知函数的Cauchy型奇异积分方程组。然后,利用半开型积分法则求解奇异积分方程,得到了裂纹端处的应力强度因子。文中给出两个数值算例的计算结果。     

T应力对立方准晶中含十字交叉型裂纹的影响

论文研究了立方准晶中含十字交叉型裂纹尖端附近的T应力.采用了积分变换法,将相关边值问题转换为求解奇异积分方程,得到声子场和相位子场的T应力,且可以表示为裂纹张开位移和施加在无穷远处的载荷之和.裂纹张开位移在确定声子场和相位子场的T应力中起着主导作用.同时强调了T应力在立方准晶脆性断裂中的作用.此外,还研究了裂纹臂长度比...     

Defects in gradient micropolar elasticity—I: screw dislocation

A gradient micropolar elasticity is proposed based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium. This theory is an extension of the theory of micropolar elasticity with couple stresses together with gradient elasticity in a way that in addition to hyper stresses, hyper couple stresses also appear. In particular, the strain energy, besides its dependence upon the distortion and bend-twist terms of a micropolar medium (Cosserat continuum), depends also on distortion and bend-twist gradients. Using a simplified but rigorous version of this gradient theory, we can connect it to Eringen's nonlocal micropolar elasticity. In addition, it is used to study a screw dislocation in gradient micropolar elasticity. One important result is that we obtained nonsingular expressions for the force and couple stresses. The components of the force stress have maximum values near the dislocation line and those of the couple stress have maximum values at the dislocation line.     

Defects in gradient micropolar elasticity—II: edge dislocation and wedge disclination

A theory of gradient micropolar elasticity based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium has been proposed in Part I of this paper. Gradient micropolar elasticity is an extension of micropolar elasticity such that in addition to double stresses double couple stresses also appear. The strain energy depends on the micropolar distortion and bend-twist terms as well as on distortion and bend-twist gradients. We use a version of this gradient theory which can be connected to Eringen's nonlocal micropolar elasticity. The theory is used to study a straight-edge dislocation and a straight-wedge disclination. As one important result, we obtained nonsingular expressions for the force and couple stresses. For the edge dislocation the components of the force stress have extremum values near the dislocation line and those of the couple stress have extremum values at the dislocation line and for the wedge disclination the components of the force stress have extremum values at the disclination line and those of the couple stress have extremum values near the disclination line.     

二维Nonlocal线弹性理论的变分原理与对偶方程

基于Eringen提出的Nonlocal线弹性理论的微分形式本构关系,导出了相应的能量密度表达式,进而得到二维Nonlocal线弹性理论的变分原理.利用变分原理导出了对偶平衡方程和相应的边界条件.进而给出了非局部动力问题的Lagrange函数,并引入对偶变量和Hamilton函数,得到了对偶体系下的变分方程.在Hamilton体系下,通过变分得到了二维Nonlocal线弹性理论的对偶平衡方程和相应的边界条件.     

Stroh Formalism and Rayleigh Waves

The Stroh formalism is a powerful and elegant mathematical method developed for the analysis of the equations of anisotropic elasticity. The purpose of this exposition is to introduce the essence of this formalism and demonstrate its effectiveness in both static and dynamic elasticity. The equations of elasticity are complicated, because they constitute a system and, particularly for the anisotropic cases, inherit many parameters from the elasticity tensor. The Stroh formalism reveals simple structures hidden in the equations of anisotropic elasticity and provides a systematic approach to these equations. This exposition is divided into three chapters. Chapter 1 gives a succinct introduction to the Stroh formalism so that the reader could grasp the essentials as quickly as possible. In Chapter 2 several important topics in static elasticity, which include fundamental solutions, piezoelectricity, and inverse boundary value problems, are studied on the basis of the Stroh formalism. Chapter 3 is devoted to Rayleigh waves, for long a topic of utmost importance in nondestructive evaluation, seismology, and materials science. There we discuss existence, uniqueness, phase velocity, polarization, and perturbation of Rayleigh waves through the Stroh formalism.

---

The Table of Contents and Index are also provided as Electronic Supplementary Material for online readers at doi:      

Lattice with long-range interaction of power-law type for fractional non-local elasticity

Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the transform operation, we map the lattice equations into continuum equation with Riesz derivatives of non-integer orders. The continuum equations that are obtained from the lattice model describe fractional generalization of non-local elasticity models. Particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case.     

THE DISLOC ATION EQU ATIONS OF A SIMPLE CUBIC CRYSTAL IN THE ISOTROPIC APPROXIMATION-A SOLVABLE MODEL

The dislocation equations of a simple cubic lattice have been obtained by using Green's function method based on the discrete lattice theory with the coefficients of the secondorder differential terms ...     

准晶数学弹性理论和某些有关研究的进展(上)

本文对固体准晶力学性能和准晶数学弹性, 塑性, 断裂以及有关研究的进展作了评论, 尤其对材料常数和塑性变形行为的测量, 一维、二维、三维准晶弹性理论, 动力学、非线性、缺陷理论、准晶弹性新型偏微分方程的推导和精确分析解, 复分析方法, 变分原理和有限元方法, 有限差分方法这些宏观问题和它们的数学方法进行了分析, 同时对准晶晶格动力学问题的数学理论也作了初步讨论. 近来在软物质中发现了12 次和18次对称准晶, 意义重大, 这里也做了初步介绍. 文中重点讨论此领域最近这些年来中国科学工作者的工作.      

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

The present study aims at determining the elastic stress and displacement fields around the tips of a finite-length crack in a microstructured solid under remotely applied plane-strain loading (mode I and II cases). The material microstructure is modeled through the Toupin-Mindlin generalized continuum theory of dipolar gradient elasticity. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain tensor (as in classical elasticity) and the gradient of the strain tensor (additional term). A simple but yet rigorous version of the theory is employed here by considering an isotropic linear expression of the elastic strain-energy density that involves only three material constants (the two Lamé constants and the so-called gradient coefficient). First, a near-tip asymptotic solution is obtained by the Knein-Williams technique. Then, we attack the complete boundary value problem in an effort to obtain a full-field solution. Hypersingular integral equations with a cubic singularity are formulated with the aid of the Fourier transform. These equations are solved by analytical considerations on Hadamard finite-part integrals and a numerical treatment. The results show significant departure from the predictions of standard fracture mechanics. In view of these results, it seems that the classical theory of elasticity is inadequate to analyze crack problems in microstructured materials. Indeed, the present results indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the standard result and the strain field is bounded. Finally, the J-integral (energy release rate) in gradient elasticity was evaluated. A decrease of its value is noticed in comparison with the classical theory. This shows that the gradient theory predicts a strengthening effect since a reduction of crack driving force takes place as the material microstructure becomes more pronounced.     

Soliton-like solutions based on geometrically nonlinear Cosserat micropolar elasticity

The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotations, which is also known as micropolar elasticity. We present the geometrically nonlinear theory taking into account all possible interaction terms between the elastic and microelastic structures. This is achieved by considering the irreducible pieces of the deformation gradient and of the dislocation curvature tensor. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small. By choosing a particular ansatz we are able to reduce the system of equations to a sine–Gordon type equation which is known to have soliton solutions.