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.      

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.     

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。     

纳米板振动特性的两类尺度效应研究

利用非局部应变梯度理论研究了纳米板横向自由振动特性。通过迭代法获得非局部应力的渐近表达式,利用哈密顿变分原理推导了纳米板的振动控制方程。针对四边简支边界条件,运用双重三角级数法给出了板固有频率的表达式,然后研究了非局部参数、材料特征参数、几何尺寸对纳米板自振频率的影响。数值结果表明:非局部效应会弱化纳米板的等效刚度,因而使板的固有频率降低,应变梯度效应则与之相反,两类效应仅在纳米尺度下对自振频率有显著影响;板几何尺寸的改变也会对其振动频率产生重要影响。     

TWO KINDS OF SCALE EFFECTS OF VIBRATION CHARACTERISTICS OF NANO-PLATE1)

利用非局部应变梯度理论研究了纳米板横向自由振动特性。通过迭代法获得非局部应力的渐近表达式,利用哈密顿变分原理推导了纳米板的振动控制方程。针对四边简支边界条件,运用双重三角级数法给出了板固有频率的表达式,然后研究了非局部参数、材料特征参数、几何尺寸对纳米板自振频率的影响。数值结果表明：非局部效应会弱化纳米板的等效刚度,因而使板的固有频率降低,应变梯度效应则与之相反,两类效应仅在纳米尺度下对自振频率有显著影响;板几何尺寸的改变也会对其振动频率产生重要影响。     

石墨烯增强功能梯度材料板条的热弹性响应

研究了石墨烯增强功能梯度复合材料板条的热弹性问题。基于推广后的Mian和Spencer功能梯度板理论,在板厚度方向上考虑热传导引起的稳态温度场,沿板厚方向考虑了三种石墨烯纳米片的分布形式,最终给出了不同边界条件下石墨烯增强功能梯度板条在温度场作用下的解析解。通过算例分析验证了本文方法的有效性,并讨论了边界条件和石墨烯的分布形式等因素对板条热弹性响应的影响。计算结果表明:石墨烯增强功能梯度材料板条在温度场作用下的热响应与在机械力作用下的弹性响应有较大不同。     

基于非局部弹性理论的纳米板横向振动

本文利用非局部弹性理论研究了单层石墨烯的纳米板的横向自由振动响应．通过迭代法推导了非局部应力表达,进一步通过哈密顿原理推导了纳米板的控制方程,应用纳维解法得到四边简支纳米板振动固有频率的数值解,并将本文研究结果与已有文献结果进行对比,进一步讨论了小尺寸效应,以及纳米板的三维尺寸和半波数对振动频率的影响．结果表明：非局部效应的存在使得纳米板的等效刚度和固有频率降低;半波数的增加则使得纳米板的固有频率提高．相关分析结果对基于二维纳米材料的新设备的设计和优化具有重要意义．     

微纳米尺度效应作用下充流微通道系统波动特性研究

结合非局部弹性应力/应变梯度耦合本构关系和流体非局部应力关系式,基于Euler梁理论,建立了充流微通道流固耦合波传导模型;根据耦合固体非局部应力/应变梯度弹性效应以及流体非局部效应,分别模拟了微通道和管腔内流体的尺度效应,推导得出了充流微通道在微纳米尺度的波动控制方程和边界条件。通过对控制方程的求解,分析了不同类型尺度效应对微通道的波动和振动特性的影响。结果显示,各类尺度效应对系统的动力学特性影响不同。微通道非局部弹性效应对波动产生阻尼,特别是对波长较短的波传导;而应变梯度弹性效应对波传导有促进作用,且该效应对波动的影响与波长无关;非局部效应和应变梯度效应对微通道刚度产生不同影响,非局部效应降低刚度,应变梯度效应增加刚度。     

Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method

The elastic buckling behavior of quadrilateral single-layered graphene sheets (SLGS) under bi-axial compression is studied employing nonlocal continuum mechanics. Small-scale effects are taken into consideration. The principle of virtual work is employed to derive the governing equations. The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis. The buckling load of skew, rhombic, trapezoidal, and rectangular nanoplates considering various geometrical parameters are obtained. It is shown that nonlocal effects are very important in arbitrary quadrilateral graphene sheets and their inclusion results in smaller buckling loads. Also the effects of geometrical parameters such as aspect ratio, angle, and mode number on the buckling load decrease when scale coefficient increases, for all arbitrary quadrilateral SLGS.     

Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model

A higher-order nonlocal strain-gradient model is presented for the damped vibration analysis of single-layer graphene sheets (SLGSs) in hygrothermal environment. Based on Kirchhoff plate theory in conjunction with a higher-order (bi-Helmholtz) nonlocal strain gradient theory, the equations of motion are obtained using Hamilton's principle. The higher-order nonlocal strain gradient theory has lower- and higher-order nonlocal parameters and a material characteristic parameter. The presented model can reasonably interpret the softening effects of the SLGS, and indicates a reasonably good match with the experimental flexural frequencies. Finally, the roles of viscous and structural damping coefficients, small-scale parameters, hygrothermal environment and elastic foundation on the vibrational responses of SLGSs are studied in detail.     

Free vibration analysis of quadrilateral nanoplates based on nonlocal continuum models using the Galerkin method: the effects of small scale

Free vibration analysis of quadrilateral single-layered graphene sheets (SLGS) is carried out employing nonlocal continuum mechanics. The equations of motion of the nonlocal theory are derived using the principle of virtual work. The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis. The non-dimensional natural frequencies of skew, rhombic, trapezoidal and rectangular nanoplates considering various geometrical parameters and mode numbers are obtained and for each case the effects of the small length scale are investigated.     

Vibration of quadrilateral embedded multilayered graphene sheets based on nonlocal continuum models using the Galerkin method

Free vibration analysis of quadrilateral multilayered graphene sheets(MLGS) embedded in polymer matrix is carried out employing nonlocal continuum mechanics.The principle of virtual work is employed to derive the equations of motion.The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis.The dependence of small scale effect on thickness,elastic modulus,polymer matrix stiffness and interaction coefficient between two adjacent sheets is illustrated.The non-dimensional natural frequencies of skew,rhombic,trapezoidal and rectangular MLGS are obtained with various geometrical parameters and mode numbers taken into account,and for each case the effects of the small length scale are investigated.     

Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory

A nonlocal strain gradient theory(NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients,which makes it benefit from both hardening and softening effects in small-scale structures.In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported(SS) boundary, the clamped-clamped(CC) boundary, the clamped-free(CF)boundary, and the clamped-simply supported(CS) boundary. The effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. The results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam.The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.     

Isogeometric nonlocal strain gradient quasi-three-dimensional plate model for thermal postbuckling of porous functionally graded microplates with central cutout with different shapes

This study presents the size-dependent nonlinear thermal postbuckling characteristics of a porous functionally graded material(PFGM) microplate with a central cutout with various shapes using isogeometric numerical technique incorporating nonuniform rational B-splines. To construct the proposed non-classical plate model, the nonlocal strain gradient continuum elasticity is adopted on the basis of a hybrid quasithree-dimensional(3D) plate theory under through-thickness deformation conditions by only four variables. By taking a refined power-law function into account in conjunction with the Touloukian scheme, the temperature-porosity-dependent material properties are extracted. With the aid of the assembled isogeometric-based finite element formulations,nonlocal strain gradient thermal postbuckling curves are acquired for various boundary conditions as well as geometrical and material parameters. It is portrayed that for both size dependency types, by going deeper in the thermal postbuckling domain, gaps among equilibrium curves associated with various small scale parameter values get lower, which indicates that the pronounce of size effects reduces by going deeper in the thermal postbuckling regime. Moreover, we observe that the central cutout effect on the temperature rise associated with the thermal postbuckling behavior in the presence of the effect of strain gradient size and absence of nonlocality is stronger compared with the case including nonlocality in absence of the strain gradient small scale effect.     

Free vibration of thermo-elastic microplate based on spatiotemporal fractional-order derivatives with nonlocal characteristic length and time

A fractional-order thermo-elastic model taking into account the small-scale effects of the thermo-elastic coupled behavior is developed to study the free vibration of a higher-order shear microplate. The nonlocal strain gradient theory is modified with the introduction of the fractional-order derivatives and the nonlocal characteristic length. The Fourier heat conduction is replaced by the non-Fourier heat conduction with the introduction of the fractional order and the memory characteristic tim...     

Nonlocal Galerkin Strip Transfer Function Method for Vibration of Double-Layered Graphene Mass Sensor

Double-layered graphene sheets(DLGSs) can be applied to the development of a new generation of nanomechanical sensors due to their unique physical properties. A rectangular DLGS with a nanoparticle randomly located in the upper sheet is modeled as two nonlocal Kirchhoff plates connected by van der Waals forces. The Galerkin strip transfer function method which is a semi-analytical method is developed to compute the natural frequencies of the massplate vibrating system. It can give exact closed-form solutions along the longitudinal direction of the strip. The results obtained from the semi-analytical method are compared with the previous ones, and the differences between the single-layered graphene sheet(SLGS) and the DLGS mass sensors are also investigated. The results demonstrate the similarity of the in-phase mode between the SLGS and DLGS mass sensors. The sensitivity of the DLGS mass sensor can be increased by decreasing the nonlocal parameter, moving the attached nanoparticle closer to the DLGS center and making the DLGS smaller. These conclusions are helpful for the design and application of graphene-sheet-based resonators as nano-mass sensors.     

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.