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。     

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.     

非对称的非局部塑性理论

从非局部连续场论出发，假定形变与局部转动均属微小，当存在体力矩出现非对称应力场时，设对称应力的能量函数和非对称应力的能量函数可以独立计算。基于热力学原理和屈服面的概念，建立了一种新的非对称－非局部弹塑性力学理论。     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅱ)——RE-DISCUSS NONLINEAR CONSTITUTIVE EQUATIONS OF NONLOCAL THERMOELASTIC BODIES

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NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅱ)--RE-DISCUSS NONLINEAR CONSTITUTIVE EQUATIONS OF NON-LOCAL THERMOELASTIC BODIES

In this paper, nonlinear constitutive equations are deduced strictly according to the constitutive axioms of rational continuum mechanics. The existing judgments are modified and improved. The results show that the constitutive responses of nonlocal thermoelastic body are related to the curvature and higher order gradient of its material space, and there exists an antisymmetric stress whose average value in the domain occupied by thermoelastic body is equal to zero. The expressions of the antisymmetric stress and the nonlocal residuals are given. The conclusion that the directions of thermal conduction and temperature gradient are consistent is reached. In addition, the objectivity about the nonlocal residuals and the energy conservation law of nonlocal field is discussed briefly, and a formula for calculating the nonlocal residuals of energy changing with rigid motion of the spatial frame of reference is derived. Foundation item: the Natural Science Foundation of Province Jiangshu (BK97063)     

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.     

Band structures of transverse waves in nanoscale multilayered phononic crystals with nonlocal interface imperfections by using the radial basis function method

A radial basis function collocation method based on the nonlocal elastic continuum theory is developed to com-pute the band structures of nanoscale multilayered phononic crystals. The effects of nonlocal imperfect interfaces on band structures of transverse waves propagating obliquely or verti-cally in the system are studied. The correctness of the present method is verified by comparing the numerical results with those obtained by applying the transfer matrix method in the case of nonlocal perfect interface. Furthermore, the influ-ences of the nanoscale size, the impedance ratio and the incident angle on the cut-off frequency and band structures are investigated and discussed in detail. Numerical results show that the nonlocal interface imperfections have signif-icant effects on the band structures in the macroscopic and microscopic scale.     

High-frequency antiplane wave propagation in ultra-thin films with nanostructures

Ultrasonic wave propagation is one of powerful and popular methods for measuring mechanical properties of solids even at nano scales. The extraction of material constants from the measured wave data may not be accurate and reliable when waves of short wavelengths are used. The objective of this paper is to study the high-frequency antiplane wave propagation in ultra-thin films at nanoscale. A developed continuum microstructure theory will be used to capture the effect of nanostructures in ultra-thin films. This continuum theory is developed from assumed displacement fields for nanostructures. Local kinematic variables are introduced to express these local displacements and are subjected to internal continuity conditions. The accuracy of the theory is verified by comparing the results with those of the lattice model for the antiplane problem in an infinite elastic medium. Specifically, dispersion curves and corresponding displacement fields for antiplane wave propagation in the ultra-thin films are studied. The inadequacy of the conventional continuum theory is discussed.     

A continuum model with microstructures for wave propagation in ultra-thin films

Ultrasonic waves are powerful and popular methods for measuring mechanical properties of solids even at nanoscales. The extraction of material constants from the measured wave data requires the use of a model that can accurately describe the wave motion in the solid. The objective of this paper is to develop a continuum theory with microstructures that can capture the effect of the microstructure or nanostructure in ultra-thin films when waves of short wavelengths are used. This continuum theory is developed from assumed displacement fields for microstructures. Local kinematic variables are introduced to express these local displacements and are subjected to internal continuity conditions. The accuracy of the present theory is verified by comparing the results with those of the lattice model for the thin film. Specifically, dispersion curves for surface wave propagation and wave propagation in a thin film supported by an elastic homogeneous substrate are studied. The inadequacy of the conventional continuum theory is discussed.     

各向异性非线性固体力学的规范空间理论

本文在弹性规范空间概念基础上，利用非平衡态热力学理论，证明了各向异性固体力学非线性问题规范空间场以及不可逆过程本征解的存在。损伤对结构刚度的弱化效应和损伤诱发各向异性效应分别反映在本征弹性和相应的模态向量中。在简正坐标中考察各向异性体变形时，材料的行为以六个普通的粘弹性Ｍａｘｗｅｌｌ方程描述，总的响应由模态叠加得到。以此为基础给出的非线性本构方程具有坐标转换不变性，最后给出了二个具体的算例。     

NONLINEAR AND QUASI-LINEAR BEHAVIOR OF A CURVED CARBON NANOTUBE VIBRATING IN AN ELECTRIC FORCE FIELD; AN ANALYTICAL APPROACH

The nonlinear vibrational model of a slightly curved single-walled carbon nanotube （SWCNT） resting on a Winkler-type elastic foundation is developed using nonlocal Euler- Bernoulli elastic theory. The SWCNT is assumed to vibrate under an external harmonic electric force field and an analytical solution is proposed to obtain the nonlinear resonant frequencies. The results show good agreement with the numerical simulation and the obtained analytical frequency is com- pletely related to the curvature of the nanotube. Our model predicts that although the model is nonlinear in nature, the curved SWCNT could behave linearly in a certain amount of curvatures and this quasi-linear vibrational behavior of curved SWCNT is a function of aspect ratio, nonlocal parameter, and stiffness of the foundation.     

Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.     

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.     

力电耦合偏场理论：回顾、比较与展望

力电耦合固体的非线性连续介质理论最早出现于20世纪50年代,而成熟于70年代.80年代末、90年代初则因智能材料与结构的兴起而又得到了新的发展,引起了较为广泛的关注,但应用上以线性分析为主.21世纪初以来,力电耦合软材料因其潜在的应用前景激发了众多的研究兴趣.由于牵涉到大变形,必须在一般非线性连续介质力学的框架内进行问题的建模和开展定量分析,因此力电耦合固体的非线性理论重新得到了大家的重视,出现了很多新版本.本文旨在阐述力电耦合固体非线性连续介质理论一般框架的基础上,采用3个构型的表述方式,较为详细地给出拉格朗日描述和更新拉格朗日描述下的力电耦合偏场理论,甄别不同理论表述版本之间的异同,以廓清目前文献中的混乱现象,为今后的相关研究提供理论指导.最后,本文讨论和展望了力电耦合偏场理论在不同研究领域的若干研究重点及其未来发展趋势.     

Theory of electroelasticity accounting for biasing fields: Retrospect,comparison and perspective

力电耦合固体的非线性连续介质理论最早出现于20世纪50年代,而成熟于70年代.80年代末、90年代初则因智能材料与结构的兴起而又得到了新的发展,引起了较为广泛的关注,但应用上以线性分析为主.21世纪初以来,力电耦合软材料因其潜在的应用前景激发了众多的研究兴趣.由于牵涉到大变形,必须在一般非线性连续介质力学的框架内进行问题的建模和开展定量分析,因此力电耦合固体的非线性理论重新得到了大家的重视,出现了很多新版本.本文旨在阐述力电耦合固体非线性连续介质理论一般框架的基础上,采用3个构型的表述方式,较为详细地给出拉格朗日描述和更新拉格朗日描述下的力电耦合偏场理论,甄别不同理论表述版本之间的异同,以廓清目前文献中的混乱现象,为今后的相关研究提供理论指导.最后,本文讨论和展望了力电耦合偏场理论在不同研究领域的若干研究重点及其未来发展趋势.     

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.     

Correction of local elasticity for nonlocal residuals: application to Euler–Bernoulli beams

Complications exist when solving the field equation in the nonlocal field. This has been attributed to the complexity of deriving explicit forms of the nonlocal boundary conditions. Thus, the paradoxes in the existing solutions of the nonlocal field equation have been revealed in recent studies. In the present study, a new methodology is proposed to easily determine the elastic nonlocal fields from their local counterparts without solving the field equation. This methodology depends on the iterative-nonlocal residual approach in which the sum of the nonlocal fields is treaded as a residual field. Thus, in this study the corrections of the local linear and nonlinear elastic fields for the nonlocal residuals in materials are presented. These corrections are formed based on the general nonlocal theory. In the context of the general nonlocal theory, two distinct nonlocal parameters are introduced to form the constitutive equations of isotropic elastic continua. In this study, it is demonstrated that the general nonlocal theory outperforms Eringen’s nonlocal theory in accounting for the impacts of the material’s Poisson’s ratio on its mechanics. To demonstrate the effectiveness of the proposed approach, the corrections of the local static bending, vibration, and buckling characteristics of Euler–Bernoulli beams are derived. Via these corrections, bending, vibration, and buckling behaviors of simple-supported nonlocal Euler–Bernoulli beams are determined without solving the beam’s equation of motion.     

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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