基于应变梯度中厚板单元的石墨烯振动研究

基于应变梯度理论建立了单层石墨烯等效明德林(Mindlin) 板动力学方程,推导了四边简支明德林中厚板自由振动固有频率的解析解. 提出了一种考虑应变梯度的4 节点36 自由度明德林板单元,利用虚功原理建立了单层石墨烯的等效非局部板有限元模型. 通过对石墨烯振动问题的研究,验证了应变梯度有限元计算结果的收敛性. 运用该有限元法研究了尺寸、振动模态阶数以及非局部参数对石墨烯振动特性的影响. 研究表明,这种单元能够较好地适用于研究考虑复杂边界条件石墨烯的尺度效应问题. 基于应变梯度理论的明德林板所获得石墨烯的固有频率小于基于经典明德林板理论得到的结果. 尺寸较小、模态阶数较高的石墨烯振动尺度效应更加明显. 无论采用应变梯度理论还是经典弹性本构关系,考虑一阶剪切变形的明德林板模型预测的固有频率低于基尔霍夫(Kirchho) 板所预测的固有频率.      

基于广义应变梯度理论的纳米梁挠曲电效应研究

挠曲电效应是应变梯度与电极化的耦合，它存在于所有的电介质材料中。在纳米电介质结构的挠曲电效应研究中，应变梯度弹性对挠曲电响应的影响一直以来被低估甚至被忽略了。根据广义应变梯度理论，应变梯度弹性中独立的尺度参数只有三个，而文献中所采用的一个或两个尺度参数的应变梯度理论只是它的简化形式。基于该理论，论文建立了考虑广义应变梯度弹性的三维电介质结构的理论模型，并以一维纳米梁为例研究了其弯曲问题的挠曲电响应及其能量俘获特性。结果表明，纳米梁的挠曲电响应存在尺寸效应，并且弹性应变梯度会影响结构挠曲电的尺寸效应，特别是当结构的特征尺寸低于尺度参数时。论文的工作为更进一步理解纳米尺度下的挠曲电机理和能量俘获特性提供理论基础和设计依据。     

能量非局部模型和新的应变梯度理论

提出一种新的基于能量非局部模型的应变梯度理论,并应用此理论对多晶铜以及薄膜基底的微压痕硬度进行理论预测和数值分析. 首先,提出了能量非局部模型,并由此模型,得出新应变梯度理论的本构关系;其次,由变分原理,得出相应的有限元公式;再次,给出了微压痕硬度的有限元分析方法;最后,将该理论预测结果与经典理论预测结果以及实验结果进行了对比. 结果表明,计算结果与实验结果相符;而经典理论的预测结果远低于实验结果.      

INVESTIGATIONS ON THE MECHANICAL CHARACTERISTICS OF NANOBEAMS BASED ON THE NONLOCAL STRAIN GRADIENT THEORY1)

基于非局部应变梯度理论,建立了一种具有尺度效应的高阶剪切变形纳米梁的力学模型. 其中,考虑了应变场和一阶应变梯度场下的非局部效应. 采用哈密顿原理推导了纳米梁的控制方程和边界条件,并给出了简支边界条件下静弯曲、自由振动和线性屈曲问题的纳维级数解. 数值结果表明,非局部效应对梁的刚度产生软化作用,应变梯度效应对纳米梁的刚度产生硬化作用,梁的刚度整体呈现软化还是硬化效应依赖于非局部参数与材料特征尺度的比值. 梁的厚度与材料特征尺度越接近,非局部应变梯度理论与经典弹性理论所预测结果之间的差异越显著.     

非局部应变梯度理论下纳米梁的力学特性研究

基于非局部应变梯度理论,建立了一种具有尺度效应的高阶剪切变形纳米梁的力学模型.其中,考虑了应变场和一阶应变梯度场下的非局部效应.采用哈密顿原理推导了纳米梁的控制方程和边界条件,并给出了简支边界条件下静弯曲、自由振动和线性屈曲问题的纳维级数解.数值结果表明,非局部效应对梁的刚度产生软化作用,应变梯度效应对纳米梁的刚度产生硬化作用,梁的刚度整体呈现软化还是硬化效应依赖于非局部参数与材料特征尺度的比值.梁的厚度与材料特征尺度越接近,非局部应变梯度理论与经典弹性理论所预测结果之间的差异越显著.     

全应变梯度挠曲电纳米梁有限单元法研究

挠曲电效应是一种存在于所有电介质材料中的特殊的力电耦合效应，本质上是应变梯度与电极化之间的线性耦合。然而，应变梯度会引入位移的高阶偏量，常给挠曲电问题的理论求解带来困难。且已有研究表明应变梯度弹性项会影响纳米结构中的力电耦合响应，但是现有的挠曲电研究大多忽略了应变梯度弹性的影响。因此，本文提出了一种既考虑应变梯度弹性，又考虑挠曲电效应的有效数值方法。基于全应变梯度弹性理论，建立了包含3个独立材料尺度参数的纳米欧拉梁的理论模型和有限元模型，提出了满足C₂弱连续的两节点六自由度单元。基于本文的有限单元法，以简支欧拉梁为例，通过分析讨论挠度、电势和能量效率，得到了挠曲电效应和应变梯度弹性项对梁的力电响应的影响。结果表明，挠曲电效应存在尺寸依赖性，且应变梯度弹性项在纳米电介质结构的挠曲电研究中的影响不可忽略。     

关于两种二阶应变梯度理论

对常见的两种应变梯度理论进行了深入的理论分析,比较了它们在公式推导、能量表达、物理解释和应用领域方面的差异. 选择钢筋拉拔弹性阶段和超薄悬臂梁受压弯曲两个典型的算例,采用可以通过C^{0\hbox{-}1}分片检验的18自由度应变梯度平面三角形单元和轴对称三角形单元,通过数值计算比较了两种理论在描述细观力学性能方面的差异.      

颗粒增强复合材料的界面开裂与尺度效应

采用基于Huang等提出的塑性应变梯度传统理论发展的有限元方法,模拟了颗粒增强金属基复合材料的界面开裂与颗粒尺度效应.分别针对考虑颗粒与基体间界面开裂和不开裂两种情况进行分析,并将考虑界面开裂的模拟结果与实验结果进行比较,证明了模型的有效性,同时也获得应变梯度理论中所包含的材料特征尺度参量的取值.     

基于应变率阶跃测试的晶体铜压入应变率敏感性研究

纳米压入测试可以原位获取材料的诸多力学性能,包括弹性模量,硬度,屈服应力,应变率敏感指数等。本文利用应变率阶跃测试技术对多晶铜试样的应变率敏感性进行测试分析,硬度-位移曲线表明压头下方所存在的变形梯度对各阶跃应变率下的硬度值存在明显影响;采用基于晶体细观机制的塑性应变梯度理论对压入变形梯度效应予以修正,比较了修正与未修正数据所得的应变率敏感指数,在有效剔除压入变形梯度影响的基础上,应变率阶跃测试可实现单次压入下材料应变率敏感性的测试表征。     

计及表面能的应变梯度理论本构参数识别

本文提出利用静态位移信息对一种计及表面能的应变梯度理论本构参数进行识别的求解策略.基于Vardoulakis和Sulem的计及表面能的简单线性应变梯度理论,文献[13]给出了伯努力-欧拉梁弯曲问题的正演解析模型,本文将其反演归结为两个带有不等式约束的非线性规划问题.在此基础上,采用黄金分割一维搜索方法进行求解,给出了数值验证,讨论了信息误差对反演结果的影响.结果显示,这种方法可以用来对应变梯度理论本构参数进行识别,即使在体积和表面能常数非常小的情况下,仍然能够得到满意的结果.     

A finite deformation theory of strain gradient plasticity

Plastic deformation exhibits strong size dependence at the micron scale, as observed in micro-torsion, bending, and indentation experiments. Classical plasticity theories, which possess no internal material lengths, cannot explain this size dependence. Based on dislocation mechanics, strain gradient plasticity theories have been developed for micron-scale applications. These theories, however, have been limited to infinitesimal deformation, even though the micro-scale experiments involve rather large strains and rotations. In this paper, we propose a finite deformation theory of strain gradient plasticity. The kinematics relations (including strain gradients), equilibrium equations, and constitutive laws are expressed in the reference configuration. The finite deformation strain gradient theory is used to model micro-indentation with results agreeing very well with the experimental data. We show that the finite deformation effect is not very significant for modeling micro-indentation experiments.     

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.     

Analysis of plate in second strain gradient elasticity

The bending analysis of a thin rectangular plate is carried out in the framework of the second gradient elasticity. In contrast to the classical plate theory, the gradient elasticity can capture the size effects by introducing internal length. In second gradient elasticity model, two internal lengths are present, and the potential energy function is assumed to be quadratic function in terms of strain, first- and second-order gradient strain. Second gradient theory captures the size effects of a structure with high strain gradients more effectively rather than first strain gradient elasticity. Adopting the Kirchhoff’s theory of plate, the plane stress dimension reduction is applied to the stress field, and the governing equation and possible boundary conditions are derived in a variational approach. The governing partial differential equation can be simplified to the first gradient or classical elasticity by setting first or both internal lengths equal to zero, respectively. The clamped and simply supported boundary conditions are derived from the variational equations. As an example, static, stability and free vibration analyses of a simply supported rectangular plate are presented analytically.     

Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nano-beams under periodic hard excitations

Abstract

Functionally graded porous materials (FGPMs) have a wide range of applications as hollow members in biomedical and aeronautical engineering. In the FGPMs, the porosity is varied over the material volume because of the density change of pores. In the present work, an analytical treatment on the size-dependent nonlinear secondary resonance of FGPM micro/nano-beams subjected to periodic hard excitations is proposed in the simultaneous presence of the nonlocality and strain gradient size dependencies. Based upon the closed-cell Gaussian-random field scheme, the mechanical properties of the FGPM micro/nano-beams are extracted corresponding to the uniform and three different functionally graded patterns of the porosity dispersion. The nonlocal strain gradient theory of elasticity is applied to the classical beam theory to formulate a newly combined size-dependent beam model. Thereafter, an analytical solving methodology based on the multiple time-scales together with the Galerkin technique is adopted to achieve the nonlocal strain gradient frequency–response and amplitude–response curves associated with the subharmonic and superharmonic external excitations. For the subharmonic excitation, it is observed that the nonlocality causes to shift the junction point of the stable and unstable branches to the higher value of the detuning parameter. However, the strain gradient size dependency plays an opposite role. For the superharmonic one, it is illustrated that the nonlocal size effect makes an increment in the height of jump phenomenon and shifts the peak to higher value of the detuning parameter. However, the strain gradient small scale effect leads to decrease the height of the jump phenomenon and shifts the peak to lower value of the detuning parameter.     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

Dislocations in second strain gradient elasticity

A second strain gradient elasticity theory is proposed based on first and second gradients of the strain tensor. Such a theory is an extension of first strain gradient elasticity with double stresses. In particular, the strain energy depends on the strain tensor and on the first and second gradient terms of it. Using a simplified but straightforward version of this gradient theory, we can connect it with a static version of Eringen’s nonlocal elasticity. For the first time, it is used to study a screw dislocation and an edge dislocation in second strain gradient elasticity. By means of this second gradient theory it is possible to eliminate both strain and stress singularities. Another important result is that we obtain nonsingular expressions for the force stresses, double stresses and triple stresses produced by a straight screw dislocation and a straight edge dislocation. The components of the force stresses and of the triple stresses have maximum values near the dislocation line and are zero there. On the other hand, the double stresses have maximum values at the dislocation line. The main feature is that it is possible to eliminate all unphysical singularities of physical fields, e.g., dislocation density tensor and elastic bend-twist tensor which are still singular in the first strain gradient elasticity.     

Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem

A detailed variational formulation is provided for a simplified strain gradient elasticity theory by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the complete boundary conditions of the theory for the first time. To supplement the stress-based formulation, the coordinate-invariant displacement form of the simplified strain gradient elasticity theory is also derived anew. In view of the lack of a consistent and complete formulation, derivation details are included for the tutorial purpose. It is shown that both the stress and displacement forms of the simplified strain gradient elasticity theory obtained reduce to their counterparts in classical elasticity when the strain gradient effect (a measure of the underlying material microstructure) is not considered. As a direct application of the newly obtained displacement form of the theory, the problem of a pressurized thick-walled cylinder is analytically solved. The solution contains a material length scale parameter and can account for microstructural effects, which is qualitatively different from Lamé’s solution in classical elasticity. In the absence of the strain gradient effect, this strain gradient elasticity solution reduces to Lamé’s solution. The numerical results reveal that microstructural effects can be large and Lamé’s solution may not be accurate for materials exhibiting significant microstructure dependence.     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

Gradient elasticity and nonstandard boundary conditions

Gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality. The pertinent thermodynamic restrictions upon the gradient constitutive equations are derived, which are shown to include, besides the field (differential) stress–strain laws, a set of nonstandard boundary conditions. Consistently with the latter thermodynamic requirements, a surface layer with membrane stresses is envisioned in the strained body, which together with the above nonstandard boundary conditions make the body constitutively insulated (i.e. no long distance energy flows out of the boundary surface due to nonlocality). The total strain energy is shown to include a bulk and surface strain energy. A minimum total potential energy principle is provided for the related structural boundary-value problem. The Toupin–Mindlin polar-type strain gradient material model is also addressed and compared with the above one, their substantial differences are pointed out, particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations. A gradient one-dimensional bar sample in tension is considered for a few applications of the proposed theory.