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。     

岩体灌浆锚杆的非局部摩擦分析

本文对Oden等提出的非局部摩擦模型进行了修正,得到了修正后的非局部摩擦模型,并应用于岩土工程问题。文中利用Mindlin问题的位移解导出了岩体灌浆锚杆沿杆体表面所受的剪应力的弹性解,对岩体灌浆锚杆进行非局部摩擦分析,在简化的情况下,得到了在修正后的非局部摩擦模型下的岩体灌浆锚杆侧剪应力的积分形式,再用Maple程序求解,将其所得的结果与局部摩擦(库仑摩擦)模型下的侧剪应力进行比较,结果表明是合理有效的。     

全长粘结式锚杆的非局部摩擦分析

本文对Oden等提出的非局部摩擦模型进行了修正,得到了修正后的非局部摩擦模型.并应用于全长粘结式锚杆问题.文中利用Mindlin问题的位移解导出的全长粘结式锚杆沿杆体表面所受的剪应力的弹性解,对全长粘结式锚杆进行非局部摩擦分析,得到了在修正后的非局部摩擦模型下的全长粘结式锚杆的侧剪应力.文中修正的非局部摩擦模型下的全长粘结式锚杆的剪应力分布规律与试验得到的结果以及局部摩擦模型(库仑模型)下的计算结果进行了对比,结果表明修正的非局部摩擦模型下的计算结果与实验更符合.因此在描述此锚杆剪应力时,非局部摩擦模型比局部摩擦模型(库仑模型)更接近的实际.     

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。     

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。     

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．     

线性非局部弹性理论的修正

本文通过考虑局部化残余力的影响对线性非局部弹性理论进行了修正，由修正后的理论所导出的应力边界条件包含了物体微观结构的长程力的作用，这个结果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，而且可以自然地得到裂纹尖端的Ｂａｒｅｎｂｌａｔｔ分子内聚力模型。     

一维声子晶体带隙的非局部辛分析

周期性弹性复合结构(声子晶体)中传播的弹性波存在特殊的色散关系:弹性波只能在某段频率范围内无损耗的传播,该频率范围称为通带.一维声子晶体的色散问题可以看作分层介质中弹性波的传播问题,利用二维弹性理论予以分析.为了研究非局部效应对声子晶体带隙特性的影响,将Eringen的二维非局部弹性理论引入到Hamilton体系下,利用精细积分与扩展的Wittrick Williams算法可获取任意频率范围内的本征解.通过对不同算例的数值计算,分析和对比了非局部理论方法与传统局部理论方法的差别.并进一步指出了该套算法的适用性和优势所在.     

非局部弹性直杆振动特征及Eringen常数的一个上限

应用非局部连续介质理论推导了弹性直杆的振动方程,并采用分离变量法 进行求解,得到了振动方程的本征方程、模态函数及通解. 结果表明：非局部连续介质弹性 直杆的自振频率因非局部效应而降低,降低的幅度不仅与材料内禀长度相关,还与振动频率 的阶次相关;而且频率大小存在极限值,显示了与晶格点阵相同特性. 通过与Brillouin格 波结果比较,给出了Eringen非局部理论中材料常数的一个上限.     

梯度型非局部高阶梁理论与非局部弯曲新解法

针对文献中关于纳米结构刚度受非局部效应影响趋势的不一致预测,基于梯度型的非局部微分本构模型,利用迭代法及泰勒展开法求得了非局部高阶应力的无穷级数表达,非局部应力相当于经典弯曲应力与非局部挠度的逐阶梯度之和. 然后推导了梯度型非局部高阶梁弯曲的挠曲轴微分方程,并结合正则摄动思想,求解了非局部挠度的表达式. 最后给出数值算例,具体量化挠度受非局部尺度因子的影响大小. 研究表明:相比于其经典值,纳米结构的非局部弯曲挠度可呈现出或增大或减小或不变的趋势,考虑梯度型非局部高阶应力降低或提高或不影响纳米结构的刚度,具体结果依赖于外载和边界约束的类型. 算例显示外载形式和边界约束条件均各自独立地影响着纳米结构的非局部弯曲挠度,同时首次观察到非局部最大弯曲挠度的位置可能受非局部尺度因子的影响. 研究结论解决了非局部弹性力学应用于纳米结构的若干疑点,并为理论的发展和优化提供支持.      

Theory of nonlocal asymmetric elastic solids

In this paper, a nonlinear theory of nonlocal asymmetric, elastic solids is developed on the basis of basic theories of nonlocal continuum fieM theory and nonlinear continuum mechanics. It perfects and expands the nonlocal elastic fiteld theory developed by Eringen and others. The linear theory of nonlocal asymmetric elasticity developed in [1] expands to the finite deformation, We show that there is the nonlocal body moment in the nonlocal elastic solids. The noniocal body moment causes the stress asymmetric and itself is caused by the covalent bond formed by the reaction between atoms. The theory developed in this paper is applied to explain reasonably that curves of dispersion relation of one-dimensional plane longitudinal waves are not similar with those of transverse waves.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅰ) —MICROPOLAR CONTINUA

IntroductionThemicropolarcontinuumtheoryisatypicalandwideapplicatedtheoryinthegeneralizedcontinuumtheoriesandwasgenerallyrecognizedlongago .Manymonographsandalargenumberofscientificpapersconcerningthisfieldarepublished .Wehaverestudiedtheexistingpolarcontinuumtheoriesandfoundthatsomebasicbalancelawsandprinciplesofthemareincompleteandthereexistsometheoreticaldefectswhichshouldberemoved .Forcontrastandclaritywenowlistthetraditionalbasicbalancelawsandequationsformicropolarcontinuumtheoryasfollows…     

An asymmetric theory of nonlocal elasticity—Part 2. Continuum field

In this paper, an asymmetric theory of nonlocal elasticity with nonlocal body couple is developed on the basis of the axiom system in nonlocal continuum field theory. The Galileo invariance is used for determining the explicit form of the constitutive equations. It is shown that both continuum field theory and quasicontinuum theory give the same constitutive equations and field equations for the general theory of nonlocal elasticity. Finally, the relations among nonlocal theory, couple stress theory, and higher gradient theory are investigated.     

移动荷载作用下非局部地基梁动力响应分析的有限元法Dynamic response of beams with nonlocal foundations subjected to moving loads using finite element method

基于非局部地基理论,推导了移动荷载作用下非局部地基梁动力响应问题的有限元解,分别讨论了地基的非局部参数、刚度、阻尼系数以及移动荷载速度对非局部地基梁动力响应的影响,并比较了非局部结果与局部结果的差异。结果表明,地基的非局部参数、刚度和阻尼是地基梁的动力响应的主要影响参数,地基梁最大响应及其发生的时刻与移动荷载速度有关。研究成果可为轨道地基系统设计提供参考。     

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory： equilibrium, governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.     

TWISTING STATICS AND DYNAMICS FOR CIRCULAR ELASTIC NANOSOLIDS BY NONLOCAL ELASTICITY THEORY

The torsional static and dynamic behaviors of circular nanosolids such as nanoshafts, nanorods and nanotubes are established based on a new nonlocal elastic stress field theory. Based on a new expression for strain energy with a nonlocal nanoscale parameter, new higher-order governing equations and the corresponding boundary conditions are first derived here via the variational principle because the classical equilibrium conditions and/or equations of motion can- not be directly applied to nonlocal nanostructures even if the stress and moment quantities are replaced by the corresponding nonlocal quantities. The static twist and torsional vibration of circular, nonlocal nanosolids are solved and discussed in detail. A comparison of the conventional and new nonlocal models is also presented for a fully fixed nanosolid, where a lower-order governing equation and reduced stiffness are found in the conventional model while the new model reports opposite solutions. Analytical solutions and numerical examples based on the new nonlocal stress theory demonstrate that nonlocal stress enhances stiffness of nanosolids, i.e. the angular displacement decreases with the increasing nonlocal nanoscale while the natural frequency increases with the increasing nonlocal nanoscale.     

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.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅰ)—MICROPOLAR CONTINUA

Based on the restudies of existing polar continuum theories rather complete systems of basic balance laws and equations for micropolar continuum theory are presented. In these new systems not only the additional angular momentum, surface moment and body moment produced by the linear momentum, surface force and body force, respectively, but also the additional velocity produced by the angular velocity are considered. The new coupled balance laws of linear momentum, angular momentum and energy are reestablished. From them the new coupled local and nonlocal balance equations are naturally derived. Via contrast it can be clearly seen that the new results are believed to be rather general and complete. Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: Dupai Tian-min (1931≈)     

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.     

NEW POINTSOF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS( Ⅲ) ——RE_DISCUSS THE LINEAR THEORY OF NONLOCAL ELASTICITY

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