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.      

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.     

纳米圆轴在弹性介质中的扭转振动分析

基于非局部应变梯度理论,考虑周围弹性介质的影响,研究纳米圆轴的扭转自由振动。首先通过Hamilton原理推导纳米圆轴扭转振动的控制方程及边界条件,接着采用微分求积法得到控制方程及三类边界条件的离散形式,最后由数值计算结果分析扭转振动特性。讨论了两个小尺度参数和弹性介质刚度的变化对振动频率的影响,并通过小尺度参数比对振动频率的影响分析两个尺度参数的耦合作用。研究结果表明,扭转自由振动频率随非局部参数增加而减小,随应变梯度尺度参数、弹性介质刚度增加而增大;当非局部参数大于应变梯度尺度参数时,小尺度效应体现为非局部效应,相反则体现为应变梯度效应。     

基于非局部铁木辛柯梁模型的碳纳米管弯曲特性研究

本文基于哈密顿变分原理和非局部连续介质弹性理论,建立了新型非局部铁木辛柯梁模型(ANT),推导了碳纳米管的ANT弯曲平衡方程以及两端简支梁、悬臂梁和简直-固定梁的边界条件表达式,分析了剪切变形效应和非局部微观尺度效应对碳纳米管(CNT)弯曲特性的影响.数值计算结果显示,碳纳米管的弯曲刚度随着小尺度效应的增强而升高.其次,这种小尺度效应对自由端受集中力的悬臂梁碳纳米管有明显作用,其刚度变化规律和其它约束条件的碳纳米管一样,这一点是ANT模型区别于普通非局部纳米梁模型的主要特点.经分子动力学模拟验证,ANT模型是合理分析碳纳米管力学特性的有效方法.     

基于精化锯齿理论的功能梯度夹心微板静弯曲模型

基于精化锯齿理论和新修正偶应力理论,建立了能够准确预测功能梯度夹心微板挠度、位移和应力的静弯曲模型。为了描述微板不同方向上的尺度效应,将两个正交材料尺度参数引入本文模型。以受双向正弦载荷作用的简支板为例,探究了夹心微板弯曲行为中尺度效应对结构刚度的影响。算例结果表明,当微板几何参数与材料尺度参数接近时,基于本文模型所测微板的最大弯曲挠度、局部位移和应力均小于传统精化锯齿理论给出的结果,捕捉到了尺度效应;尺度效应随着微板几何尺寸的增大而逐渐减弱,当微板几何尺寸远大于材料尺度参数时,尺度效应消失。此外,板的跨厚比和功能梯度变化指数也会对尺度效应产生一定影响。     

基于非局部效应和表面效应的输流碳纳米管稳定性分析

应用非局部黏弹性夹层梁模型分析双参数弹性介质中输送脉动流碳纳米管的稳定性.新模型中同时考虑了由管道内、外壁上的薄表面层引起的表面弹性效应和表面残余应力,经典的欧拉梁模型因此通过引入非局部参数和表面参数得到了改进.用平均法对其控制方程进行求解,得到了管道稳定性区域.数值算例揭示了纳米材料的非局部效应、表面效应及两个弹性介质参数对管道固有频率、临界流速和动态稳定性的复杂影响,结论可为纳米流体机械的结构设计和振动分析提供理论基础.     

基于能量法的载流单层碳纳米管振动特性研究

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,基于能量法给出了载流单层碳纳米管的振动频率近似解;并通过具体算例,应用振动频率近似解公式求解了悬臂单层纳米管的频率值,进而研究了管内流体流速、碳纳米管小尺度参数对悬臂梁一阶振动频率及振型的影响,并将得到的结论与已有文献的结果进行比较,证明了振动频率近似解的正确性。     

基于近场动力学的梁结构振动特性分析方法研究

基于欧拉梁理论推导了两自由度梁的常规态型近场动力学（Peridynamics,PD）模型,并提出一种新的自由边界条件施加方法,对不同边界条件的PD梁进行了模态分析,与局部梁的有限元结果进行了对比,验证了模型的收敛性,分析了PD非局部参数对固有频率的影响。结果表明,当近场作用域内物质点密度较小时,PD梁模型非局部性较弱,与局部梁的有限元结果接近,随着物质点密度逐渐增大,非局部作用增强,PD梁的固有频率逐渐降低;当尺度参数趋于零时,PD梁的固有频率收敛到局部梁的有限元解,PD梁退化为局部梁。研究表明,本文提出的PD梁模型和自由边界施加方法适用于分析梁的振动特性,为采用PD方法分析梁结构的动力特性提供了参考。     

三种典型边界条件下受集中载荷梁大挠度弯曲精确解

利用Jacobi椭圆函数得到了自由端受集中载荷悬臂梁大挠度弯曲问题的显式精确解,不同于由传统椭圆积分公式得到的解,该显式精确解给出梁中任意点的转角,由此可方便的得到梁弯曲后各点的位移.研究表明:由该解出发,可得到任意位置受集中载荷悬臂梁问题的解;对称性分析表明该解可直接用于两端简支或两端固支梁中点受集中载荷的情况.最后分别给出了载荷取一系列值时以上三种边界条件下梁弯曲的挠度曲线.     

简支饱和多孔弹性梁的非线性弯曲

基于饱和多孔介质理论和弹性梁的大挠度弯曲假设,在多孔弹性梁轴线不可伸长,孔隙流体仅沿轴向方向扩散的限制下,建立了微观不可压饱和多孔弹性梁大挠度拟静态响应的一维非线性数学模型.在此基础上,利用Galerkin截断法,分析了两端可渗透的简支多孔弹性梁在突加横向均布载荷作用下的非线性弯曲,给出了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶随时间的响应曲线.数值结果表明:当载荷较小时,大挠度非线性与小挠度线性理论的结果相差很小,而当载荷较大时,非线性大挠度理论的结果小于相应线性小挠度理论的结果,并且这种差异随着载荷的增大而增大.同时,在载荷突加于梁上时,多孔弹性梁骨架起初不变形,孔隙流体压力等效力偶由零突增为非零,其值与外载荷保持平衡.随着时间的增加,梁的挠度增加,等效力偶逐渐减小为零,最终多孔梁骨架承担全部的外载荷.     

A well-posed Euler-Bernoulli beam model incorporating nonlocality and surface energy effect

This study shows that it is possible to develop a well-posed size-dependent model by considering the effect of both nonlocality and surface energy, and the model can provide another effective way of nanomechanics for nanostructures. For a practical but simple problem(an Euler-Bernoulli beam model under bending), the ill-posed issue of the pure nonlocal integral elasticity can be overcome. Therefore, a well-posed governing equation can be developed for the Euler-Bernoulli beams when considering both the pure nonlocal integral elasticity and surface elasticity. Moreover, closed-form solutions are found for the deflections of clamped-clamped(C-C), simply-supported(S-S) and cantilever(C-F) nano-/micro-beams. The effective elastic moduli are obtained in terms of the closed-form solutions since the transfer of physical quantities in the transition region is an important problem for span-scale modeling methods. The nonlocal integral and surface elasticities are adopted to examine the size-dependence of the effective moduli and deflection of Ag beams.     

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.     

A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli-Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli-Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.     

Thermo-electro-magneto-mechanical bending behavior of size-dependent sandwich piezomagnetic nanoplates

Thermo-electro-magneto-mechanical bending analysis of a sandwich nanoplate is presented in this paper based on Kirchhoff’s plate theory and nonlocal theory. The sandwich nanoplate includes an elastic nano-core and two piezomagnetic face-sheets actuated by applied electric and magnetic potentials. The governing equations for the electro-magneto-mechanical bending are derived in terms of the displacement components and electric and magnetic potentials. Then, the problem is solved analytically by using Navier’s method. A parametric study is presented to show the effects of the nonlocal parameter, temperature rise, applied electric and magnetic potentials on the bending behaviors of sandwich nanoplates for simply-supported boundary conditions. As a main result of study, it is concluded that the deflection decreases as applied electric potential increases and applied magnetic potential decreases. In addition, the increase of nonlocal parameter leads to increase of deflection and maximum electric potential through the thickness direction.     

Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory

By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material (FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.     

考虑尺度效应的夹层楔形多孔梁动力学分析

基于修正偶应力理论, 研究了具有大范围旋转中心刚体-功能梯度夹层Euler-Bernoulli楔形多孔柔性微梁系统的动力学特性.楔形梁是中间层为不完全功能梯度层, 两表层为均质材料的功能梯度夹层结构, 它可以减小传统夹层结构由于层与层之间材料属性的不同导致脱粘类型损伤的影响.采用假设模态法描述变形, 考虑具有捕捉动力刚化效应的非线性耦合项, 计及von Kármán几何非线性应变, 运用第二类Lagrange方程, 导出了适用于较大变形的高次刚柔耦合动力学方程.对在平面内做大范围运动的中心刚体-功能梯度夹层Euler-Bernoulli楔形多孔微梁的动力学特性进行了详细研究.研究表明: 功能梯度夹层楔形梁表层结构高度、旋转角速度、功能梯度幂指数、尺度参数、孔隙度以及各层结构的体积分数对系统的动力学特性都有很大的影响; 功能梯度夹层楔形梁综合了功能梯度直梁和楔形梁的特性, 其相对于功能梯度直梁的固有频率增大, 同时使得孔隙度对结构固有频率变化趋势的影响不再与功能梯度直梁相同; 由于柔性梁变形能中具有横向与轴向的耦合势能, 系统在稳态下的平衡位置发生了迁移现象; 系统随着尺度参数的变化发生了频率转向与振型转换.      

Static bending behaviors of nanoplate embedded in elastic matrix with small scale effects

In this paper, the bending behaviors of the nanoplate with small scale effects are investigated by the nonlocal continuum theory. The governing equations for the nonlocal Mindlin and Kirchhoff plate models are derived. The expressions of the bending displacement are presented analytically. The difference between the two models is discussed and bending properties of the nanoplate are illustrated. It can be observed that the small scale effects are obvious for bending properties of the nanoplate. The half wave numbers, width ratios and elastic matrix properties also have significant influence on bending behaviors.     

Influence of surface micro-beams with large deflection on the resonance frequency of a quartz crystal resonator in thickness-shear mode vibrations

We study the dynamic behavior of a quartz crystal resonator(QCR)in thickness-shear vibrations with the upper surface covered by an array of micro-beams(MBs)under large deflection.Through taking into account the continuous conditions of shear force and bending moment at the interface of MBs/resonator,dependences of frequency shift of the compound QCR system versus material parameter and geometrical parameter are illustrated in detail for nonlinear and linear vibrations.It is found that the frequency shift produces a little right(left)translation for increasing elastic modulus(length/radius ratio)of MBs.Moreover,the frequency right(left)translation distance caused by nonlinear deformation becomes more serious in the second-order mode than in the first-order one.     

Influence of size effect and elastic boundary condition on the pull-in instability of nano-scale cantilever beams immersed in liquid electrolytes

In this study, the static pull-in instability of nanocantilever beams immersed in a liquid electrolyte is theoretically investigated. In modeling the nanocantilever beam, the effects of van der Waals forces, elastic boundary condition and size dependency are considered. The modified couple stress theory, containing material length scale parameter, is used to interpret the size effect which appears in micro/nanoscale structures. The modified Adomian decomposition (MAD) method is used to gain an approximate analytical expression for the critical pull-in parameters which are essential for the design of micro/nanoactuators. The results show that the beam can deflect upward or downward, based on the values of the non-dimensional parameters. It is found that the size effect greatly influences the beam deflection and is more noticeable for small thicknesses. Neglecting size effect overestimates the deflection of the nanobeam. The findings reveal that the increase of ion concentration increases the pull-in voltage but decreases the pull-in deflection. Furthermore, an increase in ion concentration increases the influence of size-dependent effect on pull-in voltage.