关于非局部场论的两点注记

研究了非局部场论中尚未完全解决的两个基本问题：其一为局部化体力，力矩残余之间的相关性，由此得到了一个描述两者关系的定理；其二为线性非局部弹性理论的应力边界条件的提法；文中所得到的应力边界条件不仅解释了在裂纹混合边界值问题中线性非局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以给出裂纹尖端的分子内聚力模型。     

关于线性非局部弹性理论的应力边界条件

应力边界条件的提法是线性非局部弹性理论尚未解决的一个理论问题。文中针对这一问题进行了研究，所导出的应力边界条件包含了物体微观结构的长程相互作用，这个结果不仅解释了在裂纹混合边界值问题中非线性局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以自然地得到裂纹尖端的分子内聚力模型。     

用非局部弹性场论研究圆盘状裂纹问题

本文采用非局部弹性理论。用Love位移函数导出三维轴对称问题的非局部弹性应力的一般形式解,并求解了圆盘裂纹问题。得到了裂纹尖端区的应力是有界的,再次证实了非局部理论模型求解断裂力学问题的正确性。     

脆性断裂的非局部力学理论

本文提出一种脆性材料断裂的非局部力学理论,内容包括:Ⅰ、Ⅱ、Ⅲ型Griffith裂纹的非局部弹性应力场,裂纹尖端邻域非局部弹性应力场的渐近形式,脆性开裂的最大拉应力准则。文中给出了这种理论应用于三种基本型裂纹和Ⅰ-Ⅱ、Ⅰ-Ⅲ复合型裂纹临界开裂条件的计算结果,并把它们与一些试验资料和最小应变能密度因子理论进行了对比。     

非均布表面应力作用下薄板变形问题的求解

给出非均布表面应力作用下弹性薄板挠曲变形问题的控制方程及边界条件,通过与热应力问题进行物理比拟,对这一问题进行了求解,并采用这一方法对实验中观测到的局部弯曲现象进行了理论解释．     

小尺度对振动简支单层碳纳米管边界条件的影响

以非局部弹性理论为基础,考虑了碳纳米管的小尺度效应,采用欧拉-伯努利梁模型给出了单层碳纳米管的动力学控制方程.研究了小尺度效应对振动简支单层碳纳米管边界条件的影响,并通过具体算例与经典连续介质理论的简支边界条件进行比较.结果表明:简支条件下考虑小尺度效应的非局部弹性理论和经典连续介质理论的边界条件具有同一性.     

非局部杆纵向振动的特性研究

基于非局部理论，研究弹性杆在任意边界约束条件下的纵向振动特性．根据Chebyshev 谱级数建立非局部弹性杆的纵向位移形式．在杆的两端引入纵向约束弹簧，通过设置弹簧刚度系数，模拟经典边界及弹性边界．建立非局部杆的能量表达式，由瑞利-里兹法得到齐次线性方程组，求解对应的矩阵特征值与特征向量问题获得非局部杆的固有频率和振型．通过数值仿真计算，研究非局部特征系数与边界约束条件对非局部杆振动频率的影响．结果表明本文方法合理简便，具有良好的精度，且适用于任意弹性边界条件．     

基于非局部理论的分数阶黏弹性场地地震放大效应分析

基于非局部理论和分数阶导数理论,研究上覆黏弹性场地土的地震放大效应。利用Eringen非局部理论考虑土体颗粒尺度等非局部效应的影响,通过分数阶黏弹性本构模型刻画场地土的应力应变本构关系,建立基于非局部理论的分数阶黏弹性场地土的振动微分方程;考虑分数阶导数的性质和黏弹性场地土的边界条件,得到了简谐地震波作用下黏弹性场地土的位移和剪切应力的解析解,并在频率域内给出了位移放大系数和应力放大系数的表达式;最后通过数值算例分析了非局部效应、分数阶导数的阶数和土体黏性参数等对黏弹性场地地震放大效应的影响。数值分析结果表明,在低频时位移放大系数和应力放大系数随频率变化曲线存在波动,高频时逐渐趋于稳定;非局部效应对场地土位移放大系数的影响与频率有关,对应力放大系数的影响较大,在研究场地土振动效应时有必要考虑土体非局部效应的影响;分数阶导数的阶数越小,位移放大系数和应力放大系数随频率变化曲线波动越大;场地土的力学性质对场地土的振动效应的影响较大;上覆场地土的黏性对位移放大系数的影响与频率有关,高频时,土体黏性越大,位移放大系数越大;越接近基岩,土体的应力放大系数越大,且土体深度对应力放大系数的影响越大。     

具有非局部体力矩的非局部弹性理论

本文基于非局部连续统场论的公理系统,建立了具有非局部体力矩作用的非局部弹性理论,我们证明了,在非局部弹性固体中存在着非局部体力矩,非局部体力矩引起了应力的非对称和非局部体力矩是由材料中的共价键产生的。     

含裂纹纳米板振动问题的哈密顿体系方法

基于裂纹处范德华力效应,采用非局部弹性理论构造纳米板模型,并通过导入哈密顿体系建立含裂纹纳米板振动问题的对偶正则控制方程组。在全状态向量表示的哈密顿体系下,将含裂纹纳米板的固有频率和振型问题归结为广义辛本征值和本征解问题。利用哈密顿体系具有的辛共轭正交关系,得到问题解的级数解析表达式。结合边界条件,得到固有频率与辛本征值的代数方程关系式,进而直接给出固有频率的表达式。数值结果表明,非局部尺寸参数和裂纹长度对纳米板振动的各阶固有频率有直接的影响。对比表明,辛方法是准确且可靠的,可为工程应用提供依据。     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS (Ⅰ)──FUNDAMENTAL THEORY

In the linear nonlocal elasticity theory, the solution to the boundary-value problem of the crack with a constant stress boundary condition does not exist. This problem has been studied in this paper. The contents studied contain of examining objectivity of the energy balance, deducing the constitutive equations of nonlocal thermoelastic bodies, and determining nonlocal force and the linear nonlocal elasticity theory. Some new results are obtained. Among them, the stress boundary condition derived from the linear theory not only solves the problem mentioned at the beginning, but also contains the model of molecular cohesive stress on the sharp crack tip advanced by Barenblatt.     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS (Ⅰ)──FUNDAMENTAL THEORY

I.Intr0ductionNonlocallinearelasticitytheoryisp0ssible0fbuildingthebridgebetweenmicrostructuresofmaterialsandtheirmacrosc0picmechanicsbehaviorsduet0consideringthelong-rangeforcesamongmicroscopicparticles.SincenonIocalfieldtheorywasadvanced,aseriesresultsl…     

Two notes on nonlocal field theory

In this paper, two fundamental problems completely unsolved in nonlocal field theory are studied. The first is the dependence of nonlocal residuals. By studying this problem, a theorem concerning the relationship between the residuals of nonlocal body force and nonlocal moment of momentum is given and proven. The other problem is how to give the stress boundary conditions in the linear theory of nonlocal elasticity. The stress boundary conditions obtained in this paper can not only answer why the nonlocal stress solution satisfying the boundary conditionst _ji ^(s) n _j ¦O ₂ =p _i (p _i is a constant) on the surface of crack does not exist but also give a model of the molecular cohesive stress on the crack tip.     

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in anisotropic material plane by using the non-local theory

In this paper, the dynamic behavior of two collinear cracks in the anisotropic elasticity material plane subjected to the harmonic anti-plane shear waves is investigated by use of the nonlocal theory. To overcome the mathematical difficulties, a one-dimensional nonlocal kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The nonlocal elasticity solutions yield a finite hoop stress at the crack tips, thus allowing us to using the maximum stress as a fracture criterion. The magnitude of the finite stress field not only depends on the crack length but also on the frequency of the incident waves and the lattice parameter of the materials.     

On the study of a penny-shaped crack in nonlocal elasticity

This article is concerned with the penny-shaped crack in an infinite body subjected to a uniform pressure on the surface of the carck in nonlocal elasticity. Making use of Love function in classical elasticity, we reduce the stress solution of an axisymmetric problem of the penny-shaped crack. The result of this article shows the stress on the crack tip is finite and demonstrates again the correctness of the nonlocal model for solving problems in fracture mechanics.Project Supported by the Science Foundation of the Chinese Academy     

Bond-based peridynamic modelling of singular and nonsingular crack-tip fields

A static meshfree implementation of the bond-based peridynamics formulation for linearly elastic solids is applied to the study of the transition from local to nonlocal behavior of the stress and displacement fields in the vicinity of a crack front and other sources of stress concentration. The long-range nature of the interactions between material points that is intrinsic to and can be modulated within peridynamics enables the smooth transition from the square-root singular stress fields predicted by the classical (local) linear theory of elasticity, to the nonsingular fields associated with nonlocal theories. The accuracy of the peridynamics scheme and the transition from local to nonlocal behavior, which are dictated by the lattice spacing and micromodulus function, are assessed by performing an analysis of the boundary layer that surrounds the front of a two dimensional crack subjected to mode-I loading and of a cracked plate subjected to far-field tension.     

Dynamic modeling of preloaded size-dependent nano-crystalline nano-structures

The vibration behavior of size-dependent nano-crystalline nano-beams is investigated based on nonlocal, couple stress and surface elasticity theories. A nanocrystalline nano-beam is composed of three phases which are nano-grains, nano-voids,and interface. Nano-voids or porosities inside the material have a stiffness-softening impact on the nano-beam. A Eringen's nonlocal elasticity theory is applied in the analysis of nano-crystalline nano-beams for the first time. Residual surface stresses which are usually neglected in modeling nano-crystalline nano-beams are incorporated into nonlocal elasticity to better understand the physics of the problem. Also, a modified couple stress theory is used to capture rigid rotations of grains. Applying a differential transform method(DTM) satisfying various boundary conditions, the governing equations obtained from the Hamilton's principle are solved. Reliability of the proposed approach is verified by comparing the obtained results with those provided in the literature. The effects of the nonlocal parameter, surface effect, couple stress, grain size, porosities, and interface thickness on the vibration characteristics of nano-crystalline nano-beams are explored.