Axisymmetric compressive buckling of multi-walled carbon nanotubes under different boundary conditions

The paper studies the axisymmetric compressive buckling behavior of multi-walled carbon nanotubes (MWNTs) under different boundary conditions based on continuum mechanics model. A buckling condition is derived for determining the critical buckling load and associated buckling mode of MWNTs, and numerical results are worked out for MWNTs with different aspect ratios under fixed and simply supported boundary conditions. It is shown that the critical buckling load of MWNTs is insensitive to boundary conditions, except for nanotubes with smaller radii and very small aspect ratio. The associated buckling modes for different layers of MWNTs are in-phase, and the buckling displacement ratios for different layers are independent of the boundary conditions and the length of MWNTs. Moreover, for simply supported boundary conditions, the critical buckling load is compared with the corresponding one for axial compressive buckling, which indicates that the critical buckling load for axial compressive buckling can be well approximated by the corresponding one for axisymmetric compressive buckling. In particular, for axial compressive buckling of double-walled carbon nanotubes, an analytical expression is given for approximating the critical buckling load. The present investigation may be of some help in further understanding the mechanical properties of MWNTs.     

轻质金属点阵夹层板热屈曲临界温度分析

本文针对均匀温度场下四边简支和四边固支金属点阵夹层板的临界热屈曲温度进行了求解和参数影响分析。将点阵夹芯等效为均匀连续体,并且将夹层板的剪切刚度近似为点阵夹芯的抗剪切刚度,忽略夹芯的抗弯刚度且认为夹层板主要由面板来提供抗弯刚度。对于无法获得解析解的四边固支条件,通过对未知变量进行双傅里叶展开的方法求解了Ressiner夹层板模型的临界屈曲温度,理论分析结果与有限元计算结果吻合良好。进一步分析了不同边界条件、点阵胞元构型、点阵材料相对密度、面板厚度等对临界屈曲温度的影响规律。     

基于新修正偶应力理论的Mindlin层合板热稳定性分析

以新修正偶应力理论为基础,首次提出了机械载荷与热载荷共同作用下的微尺度Mindlin层合板热稳定性模型,该模型只引入一个材料尺度参数,通过虚功原理推导出了控制方程和边界条件,以四边简支方板为例,进行了热稳定性分析,应用纳维叶解法得到解析解。结果表明,所建模型可以捕捉到尺度效应。材料尺度参数值越大,屈曲临界温度越高;当跨厚比增大时,屈曲临界温度下降;随着板几何参数的增大,模型将退化为宏观模型;温度变化量越大,考虑热载荷作用下的屈曲临界载荷越大,尺度效应体现越显著。     

Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method

The size-dependent effect on the biaxial and shear nonlinear buckling analysis of an isotropic and orthotropic micro-plate based on the surface stress,the modified couple stress theory(MCST),and the nonlocal elasticity theories using the differential quadrature method(DQM)is presented.Main advantages of the MCST over the classical theory(CT)are the inclusion of the asymmetric couple stress tensor and the consideration of only one material length scale parameter.Based on the nonlinear von K′arm′an assumption,the governing equations of equilibrium for the micro-classical plate considering midplane displacements are derived based on the minimum principle of potential energy.Using the DQM,the biaxial and shear critical buckling loads of the micro-plate for various boundary conditions are obtained.Accuracy of the obtained results is validated by comparing the solutions with those reported in the literature.A parametric study is conducted to show the effects of the aspect ratio,the side-to-thickness ratio,Eringen’s nonlocal parameter,the material length scale parameter,Young’s modulus of the surface layer,the surface residual stress,the polymer matrix coefficients,and various boundary conditions on the dimensionless uniaxial,biaxial,and shear critical buckling loads.The results indicate that the critical buckling loads are strongly sensitive to Eringen’s nonlocal parameter,the material length scale parameter,and the surface residual stress effects,while the effect of Young’s modulus of the surface layer on the critical buckling load is negligible.Also,considering the size dependent effect causes the increase in the stiffness of the orthotropic micro-plate.The results show that the critical biaxial buckling load increases with an increase in G12/E2and vice versa for E1/E2.It is shown that the nonlinear biaxial buckling ratio decreases as the aspect ratio increases and vice versa for the buckling amplitude.Because of the most lightweight micro-composite materials with high strength/weight and stiffness/weight ratios,it is anticipated that the results of the present work are useful in experimental characterization of the mechanical properties of micro-composite plates in the aircraft industry and other engineering applications.     

Thermal buckling of nanorod based on non-local elasticity theory

The buckling of nanostructures including as a nanobeam, nanorod, and nanotube in a temperature field is investigated based on the non-local elasticity field theory with non-linear strain gradients first proposed by Eringen. New higher-order governing differential equations both in transverse and axial direction for buckling of such nanostructures are derived based on the exact variational principle approach with corresponding higher-order non-local boundary conditions. Based on these new governing equations and boundary conditions, new analytical solutions for some practical examples on buckling of nanostructures are presented and analyzed in detail. Subsequently, the effects of non-local nanoscale and temperature change on critical buckling load are analyzed and discussed. It is observed that those factors have great influence on the critical buckling load of the nanostructures. In particular, the non-local stress very much affects the stiffness of nanostructures and the critical buckling load is significantly increased in the presence of non-local stress. The paper concludes that at low and room temperature the critical buckling load of nanostructures increases with increasing temperature change, while at high temperature the critical buckling load decreases with increasing temperature change. A critical temperature change which causes buckling without external load is also derived and discussed.     

Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets

Wrinkles are commonly observed in stretched thin sheets and membranes. This paper presents a numerical study on stretch-induced wrinkling of hyperelastic thin sheets based on nonlinear finite element analyses. The model problem is set up for uniaxial stretching of a rectangular sheet with two clamped ends and two free edges. A two-dimensional stress analysis is performed first under the plane-stress condition to determine stretch-induced stress distribution patterns in the elastic sheets, assuming no wrinkles. As a prerequisite for wrinkling, development of compressive stresses in the transverse direction is found to depend on both the length-to-width aspect ratio of the sheet and the applied tensile strain in the longitudinal direction. A phase diagram is constructed with four different distribution patterns of the stretch-induced compressive stresses, spanning a wide range of aspect ratio and tensile strain. Next, an eigenvalue analysis is performed to find the potential buckling modes of the elastic sheet under the prescribed boundary conditions. Finally, a nonlinear post-buckling analysis is performed to show evolution of stretch-induced wrinkles. In addition to the aspect ratio and tensile strain, it is found that the critical condition for wrinkling and the post-buckling behavior both depend sensitively on the sheet thickness. In general, wrinkles form only when both the magnitude and the distribution area of the compressive stresses are sufficiently large. The wrinkle wavelength decreases with increasing strain, in good agreement with the prediction by a scaling analysis. However, as the tensile strain increases, the wrinkle amplitude first increases and then decreases, eventually flattened beyond a moderately large critical strain, in contrast to the scaling analysis.     

Elasto-plastic buckling analysis for perforated steel plates subject to uniform compression

This study investigates the elasto-plastic buckling behaviour of simply supported square and rectangular thin steel plates having elliptic cut-outs by means of finite element method. Plates with simply supported in the out-of-plane direction are applied uniform compression in long-edge direction. A50 steel was used in the analysis and the focus was on the effect of plate aspect ratio, elliptical hole size, elliptical hole angle, elliptical hole location and slenderness ratio on buckling behaviour. It was found in the study that as the plate slenderness ratio increases, the critical buckling stress decreases for all the perforated plates.     

褶皱与晶界耦合作用对石墨烯断裂行为的影响

由CVD方法制备的石墨烯含有大量的晶界,通常还带有许多褶皱,本文通过分子动力学方法研究了具有褶皱和晶界的石墨烯 平面拉伸断裂行为,结果显示,在垂直晶界方向,褶皱能够显著提高小角度双晶石墨烯的断裂应力,断裂应力增幅最 大约为50%,褶皱对断裂应力的影响随晶界角的增大减弱,导致双晶石墨烯断裂应力对晶界角不敏感,只略低于单晶石墨 烯,和实验结果完全吻合;在沿晶界方向,褶皱对双晶石墨烯断裂应力影响不明显. 另外,褶皱可以显著提高双晶石墨烯的断 裂应变,增幅最大约为100%. 增强机制归纳主要如下:通过面外变形,褶皱可以部分释放晶界5-7环中C---C键的预拉伸变形, 提高双晶石墨烯的断裂应力;褶皱可以降低相邻5-7环之间相互作用,导致断裂应力对晶界角不敏感;在拉伸作用下,褶皱被部分 拉平,这可以显著降低C---C键面内拉伸变形,导致断裂应变显著增大. 本研究为准确理解多晶石墨烯断裂行为提供重要帮助.      

A novel model for analysis of multilayer graphene sheets taking into account the interlayer shear effect

In this study, a multiplate shear model is developed for dynamic analysis of multilayer graphene sheets with arbitrary shapes considering the interlayer shear effect. By utilizing the model, then some free-vibration analysis is presented. According to the experimental results, the weak interlayer van der Waals interaction cannot maintain the integrity of carbon atoms in adjacent layers. Therefore, it is required that the interlayer shear effect is accounted to study multilayer graphene mechanical behavior. The governing differential equation of motion is derived for the multilayer graphene sheets utilizing a variational approach based on the Kirchhoff plate model. The essential and natural boundary conditions are also obtained at both the smooth periphery parts of the multilayer graphene sheets and the possible sharp corners. By considering cantilever and simply supported multilayer rectangular graphene sheets as two case studies, the results for the free-vibration analysis are presented based on the developed model, and these results are compared with those of molecular dynamics simulations as some sort of verification. These results show that when the layers number increases, the natural frequency also increases up to a specific number, and afterward the influence of layers number on the natural frequency significantly decreases. Moreover, the natural frequency decreases with increase in the sheet aspect ratio up to a specific value, then the changes in the aspect ratio have no considerable effect in the natural frequency.     

The reinforcement and defect interaction of two-dimensional lattice materials with imperfections

The defect interaction and reinforcement of imperfect two-dimensional lattice materials are studied by theoretical investigations and finite element (FE) simulations. An analytical model is proposed to predict the interaction of two defects in lattice materials based on a single defect model. An interaction coefficient is introduced to characterize the degree of interaction. The effects of defect type and defect distance on interaction coefficients are studied. The critical interaction distance of defects, beyond which the interaction of two defects can be neglected, is derived. FE calculations are performed to validate the theoretical model. The simulated results indicate that increasing the number of defects can reduce the stress concentration rather than weakening the strength of the residual parts in certain circumstances. Subsequently, several reinforcement methods are proposed to reduce the stress concentration in the triangular and Kagome lattice for the single-bar-missing defect and single-joint-missing defect. An analytical model is developed for the reinforced lattices, and the predicted stress concentration factors are in good agreement with those of FE simulations. By theoretical studies and FE simulations, optimal reinforcement methods are derived for the triangular and Kagome lattice under planar loading conditions.     

Effects of local thickness defects on the buckling of micro-beam

A buckling model of Timoshenko micro-beam with local thickness defects is established based on a modified gradient elasticity. By introducing the local thickness defects function of the micro-beam, the variable coe-cient differential equations of the buckling problem are obtained with the variational principle. Combining the eigensolution series of the complete micro-beam with the Galerkin method, we obtain the critical load and buckling modes of the micro-beam with defects. The results show that the depth and location of the defect are the main factors affecting the critical load, and the combined effect of boundary conditions and defects can significantly change the buckling mode of the micro-beam. The effect of defect location on buckling is related to the axial gradient of the rotation angle, and defects should be avoided at the maximum axial gradient of the rotation angle. The model and method are also applicable to the static deformation and vibration of the micro-beam.     

Buckling analysis of a single-layer graphene sheet embedded in an elastic medium based on nonlocal Mindlin plate theory

The effect of length scale on buckling behavior of a single-layer graphene sheet embedded in a Pasternak elastic medium is investigated using a nonlocal Mindlin plate theory. An explicit solution is extracted for the buckling loads of graphene sheet and the influence of the nonlocal parameter and aspect ratio on dimensionless buckling loads is presented. It is found that the nonlocal assumptions exhibit larger buckling loads and stiffness of elastic medium in comparison to classical plate theory.     

DYNAMIC BUCKLING BEHAVIOR OF MULTI-WALLED CARBON NANOTUBES SUBJECTED TO STEP AXIAL LOADING

This paper studies the dynamic buckling behavior of multi-walled carbon nanotubes (MWNTs) subjected to step axial loading.A buckling condition is derived,and numerical results are presented for MWNTs u...     

Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory

In this paper, a size-dependent first-order shear deformable shell model is developed based upon the modified strain gradient theory (MSGT) for the axial buckling analysis of functionally graded (FG) circular cylindrical microshells. It is assumed that the material properties of FG materials, which obey a simple power-law distribution, vary through the thickness direction. The principle of virtual work is utilized to formulate the governing equations and corresponding boundary conditions. Numerical results are presented for the axial buckling of FG circular cylindrical microshells subject to simply-supported end conditions and the effects of material length scale parameter, material property gradient index, length-to-radius ratio and circumferential mode number on the size-dependent critical buckling load are extensively studied. For comparison purpose, the critical buckling loads predicted by modified couple stress theory (MCST) and classical theory (CT) are also presented. Results show that the size effect plays an important role for lower values of dimensionless length scale parameter. Moreover, it is observed that the critical buckling loads obtained based on MSGT are greater than those obtained based on MCST and CT.     

Loading direction-dependent shear behavior at different temperatures of single-layer chiral graphene sheets

The loading direction-dependent shear behavior of single-layer chiral graphene sheets at different temperatures is studied by molecular dynamics(MD) simulations.Our results show that the shear properties(such as shear stress–strain curves, buckling strains, and failure strains) of chiral graphene sheets strongly depend on the loading direction due to the structural asymmetry. The maximum values of both the critical buckling shear strain and the failure strain under positive shear deformation can be around 1.4 times higher than those under negative shear deformation. For a given chiral graphene sheet, both its failure strain and failure stress decrease with increasing temperature. In particular, the amplitude to wavelength ratio of wrinkles for different chiral graphene sheets under shear deformation using present MD simulations agrees well with that from the existing theory. These findings provide physical insights into the origins of the loading direction-dependent shear behavior of chiral graphene sheets and their potential applications in nanodevices.     

Analytical solutions for buckling of size-dependent Timoshenko beams

The inconsistences of the higher-order shear resultant expressed in terms of displacement(s) and the complete boundary value problems of structures modeled by the nonlocal strain gradient theory have not been well addressed. This paper develops a size-dependent Timoshenko beam model that considers both the nonlocal effect and strain gradient effect. The variationally consistent boundary conditions corresponding to the equations of motion of Timoshenko beams are reformulated with the aid of the weighted residual method. The complete boundary value problems of nonlocal strain gradient Timoshenko beams undergoing buckling are solved in closed forms. All the possible higher-order boundary conditions induced by the strain gradient are selectively suggested based on the fact that the buckling loads increase with the increasing aspect ratios of beams from the conventional mechanics point of view. Then, motivated by the expression for beams with simply-supported(SS) boundary conditions, some semiempirical formulae are obtained by curve fitting procedures.     

薄宽带材局部纵向屈曲的辛弹性力学求解方法

局部纵向屈曲是普遍存在于薄宽带材生产过程的板形缺陷,是屈曲研究的难点,精确的解析求解方法对局部纵向屈曲形成机理的研究和板形质量的提高具有重要意义。本文将任意位置的局部纵向屈曲分为带材边部和内部两类,采用辛弹性力学方法直接推导得到了局部纵向屈曲区域承受不同边界约束条件时的临界屈曲应力和屈曲挠度函数,并将求解结果与有限元和相关文献结果进行了对比。结果表明：辛弹性力学方法与有限元方法相比具有相同计算精度和更高的计算效率,计算精度高于传统能量法;带材边界的约束条件对临界屈曲应力、屈曲区域几何形状和屈曲挠度函数均存在显著影响,验证了传统能量法求解的不足,有利于提高局部屈曲计算精度。     

Magnetoelastic buckling of a rectangular block in plane strain

Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value.This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors.The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block's aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0.25) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio.     

基于等几何有限元法的功能梯度微板热力耦合屈曲预测

基于修正偶应力理论和Kirchhoff板理论,建立了功能梯度微板热力耦合屈曲等几何有限元模型。该模型仅包含一个材料尺度参数,能够描述尺度效应现象,且满足修正偶应力理论的高阶连续性要求。基于虚功原理推导了功能梯度微板热力耦合屈曲等几何有限元方程。通过对板的典型算例分析,讨论了材料尺度参数、边长比及梯度指数对板稳定性的影响。结果表明,本文模型预测的屈曲载荷总是大于宏观理论的结果,即捕捉到了尺度效应现象;随着临界屈曲力的增加,临界屈曲热载荷线性减少;此外,边长比和梯度指数也对微板的稳定性产生一定影响。     

一维六方准晶层合简支梁自由振动与屈曲的精确解

伪Stroh型公式能够将多场耦合材料的控制方程转化为线性特征系统来求解,从而获得多层结构简支边界条件的精确解.本文利用伪Stroh型公式,研究一维六方准晶层合简支梁的自由振动和屈曲问题,通过传递矩阵法,获得准晶层合梁自由振动固有频率与临界屈曲载荷的精确解.通过与已有梁的剪切变形理论结果比较,验证了本文伪Stroh型公式的正确性和有效性.通过数值算例,分析由两种不同准晶材料组成的三明治层合梁的叠层方式、高跨比、层厚比及层数对梁的固有频率、临界屈曲载荷及其模态的影响规律.结果表明,叠层顺序和梁的高跨比、层厚比对准晶层合梁的自由振动固有频率和临界屈曲载荷有很大影响,可通过调整梁的几何尺寸和叠层顺序得到准晶层合梁的最佳固有频率和临界屈曲载荷.本文给出的精确解可为工程上研究准晶梁的各种数值解法和实验方法提供理论参考.