Length dependence of critical measures in single-walled carbon nanotubes

This paper presents an investigation on the buckling behaviour of single-walled carbon nanotubes under various loading conditions (compression, bending and torsion) and unveils several aspects concerning the dependence of critical measures (axial strain, bending curvature and twisting angle) on the nanotube length. The buckling results are obtained by means of an atomistic-scale generalized beam theory (GBT) that incorporates local deformation of the nanotube cross-section by means of independent and orthogonal deformation modes. Moreover, some estimates are also obtained by means of non-linear shell finite element analyses using Abaqus code. After classifying the buckling modes of thin-walled tubes (global, local and distortional), the paper addresses the importance of the two-wave distortional mode (flattening or ovalization mode) in their structural behaviour. Then, the well known expression to determine the critical strain of compressed nanotubes, which is based on Donnell theory for shallow shells, is shown to be inadequate for moderately long tubes due to warping displacements appearing in the distortional buckling modes. After that, an in-depth study on the buckling behaviour of nanotubes under compression, bending and torsion is presented. The variation of the critical kinematic measures (axial strain, bending curvature and twisting angle) with the tube length is thoroughly investigated. Concerning this dependence, some uncertainties that exist in the specific literature are meticulously explained, a few useful expressions to determine critical measures of nanotubes are proposed and the results are compared with available data collected from several published works (most of them, obtained from molecular dynamics simulations).     

Enhanced generalised beam theory buckling formulation to handle transverse load application effects

This work deals with the development, finite element implementation and application of a generalised beam theory (GBT) formulation intended to analyse the localised, local, distortional and global buckling behaviour of thin-walled steel beams and frames subjected to transverse loads applied at various member cross-section points (away from its shear centre). In order to take into account the effects stemming from the transverse load position, the GBT buckling formulation must incorporate geometrical stiffness terms stemming from either (i) the internal work of the pre-buckling transversal normal stresses (“exact” formulation) or (ii) the external work of the applied transverse loads (approximate/simplified formulation). After presenting the main concepts and procedures involved in the development of the above “exact” and simplified formulations, the paper addresses the corresponding numerical implementations. Then, in order to illustrate their application and capabilities, as well as the limitations of the simplified formulation, various numerical result sets are presented and discussed. The accuracy of the GBT-based results is assessed through the comparison with “exact” values, yielded by rigorous shell finite element analyses carried out in the code Ansys.     

GBT pre-buckling and buckling analyses of thin-walled members under axial and transverse loads

This paper presents an analytical approach for pre-buckling and buckling analyses of thin-walled members implemented within the framework of the Generalised Beam Theory (GBT). With the proposed GBT cross-sectional analysis, the set of deformation modes used in the analysis is represented by the dynamic modes obtained for an unrestrained frame representing the cross-section. In this manner, it is possible to account for the deformability of the cross-section in both pre-buckling and buckling analyses. Different loading conditions, including both axial and transverse arrangements, are considered in the applications to highlight under which circumstances the use of the GBT deformation modes is required for an adequate representation of the pre-buckling and buckling response. The numerical results have been validated against those determined using a shell element model developed in the finite element software ABAQUS.     

On the relationship of the shear deformable Generalized Beam Theory with classical and non-classical theories

The possibility to establish clear relationships between the results of the Generalized Beam Theory (GBT) and those of the classical beam theories is a crucial issue for a correct theoretical positioning of the GBT within the other existing beam theories as well as for the application of the GBT in the current engineering practice. With this in mind, the recovery of classical and non-classical beam theories within the framework of the GBT is presented in this paper. To this purpose, a new formulation of the GBT with shear deformation is conceived. Particularly, the formulation recently proposed by the authors is here modified by introducing new definitions of the kinematic parameters and of the generalized deformations, and extended to the dynamic case. Firstly, it is shown that a suitable choice of the flexural deformation modes allows recovering the Vlasov beam theory, both with and without shear deformation. Also, the analytical solution of the non-uniform torsion problem with shear deformation is given. Then, the recovery of the Capurso beam theory using the nonlinear warping deformation modes is illustrated.     

Plastic buckling analysis of thick plates using p-Ritz method

This paper is concerned with the application of the p-Ritz method for the plastic buckling analysis of thick plates. In order to allow for the effect of transverse shear deformation in thick plates, the Mindlin plate theory is adopted. The plastic buckling behaviour of the plate is captured by using the incremental and deformation theories of plasticity. The material property of the plate is assumed to obey the Ramberg–Osgood stress–strain relation. The p-Ritz method will be applied to obtain the governing eigenvalue equation for the plastic buckling analysis of uniformly stressed plates with edges defined by polynomial functions. In the p-Ritz method, the displacement functions of the plate are approximately represented by the product of mathematically complete two-dimensional polynomial functions and boundary equations raised to appropriate powers that ensure the satisfaction of the geometric boundary conditions. The validity, convergence and accuracy of the method were demonstrated for various plate shapes such as rectangular, triangular and elliptical shapes. A parametric study was also undertaken to study the plastic buckling behaviour and the effect of transverse shear deformation.     

Global structural behaviour of thin and moderately thick monoclinic spherical shells using a mixed shear deformation model

Summary The static and dynamic responses of anisotropic spherical shells under a uniformly distributed transverse load are investigated. Analytical solutions using the mixed variational formulation are presented for spherical shells subjected to various boundary conditions. Numerical results of a refined mixed first-order shear deformation theory for natural frequencies, critical buckling, center deflections and stresses are compared with those obtained using the classical shell theory. A variety of simply-supported and clamped boundary conditions are considered and comparisons with the existing literature are made. The sample numerical results presented herein for global structural behaviour of monoclinic spherical shells should serve as references for future comparisons.     

含脱层损伤复合材料层合梁的冲击动力屈曲

针对含初始缺陷和脱层损伤的复合材料层合梁的轴向冲击动力屈曲问题进行了分析。基于Hamilton原理导出了考虑初始缺陷、轴向和横向惯性、横向剪切变形以及转动惯性影响时含脱层损伤复合材料梁的非线性动力屈曲控制方程;基于B-R准则,采用有限差分方法求解了受轴向冲击载荷作用下含脱层损伤复合材料梁的动力屈曲问题;讨论了冲击速度、初始几何缺陷、铺层角度以及脱层长度等因素对复合材料层合梁动力屈曲的影响。     

Mechanics of instability-induced pattern transformations in elastomeric porous cylinders

In this paper, we combine experiments and numerical simulations to investigate the large deformation mechanics of periodically patterned cylindrical structures under uniaxial compression. Focusing on cylinders with a square array of circular pores, we show that their buckling behavior is not only controlled by the porosity (as for the case of the corresponding infinitely large planar structures), but also by the length and thickness of the shell and the number of pores along the full circumference. While infinitely long cylindrical shells only support long wavelength (global) modes, by reducing the length and tuning the thickness, short wavelength (local) modes can be observed. Furthermore, frustrated short wavelength modes are triggered when a local instability is critical, but the buckling pattern is not compatible with the number of pores along the circumference.     

Energy method solution for the postbuckling response of an axially loaded bilaterally constrained beam

Bistable and multistable structures have shown great usefulness in many applications such as MEMS actuation and energy harvesting. Bistability of structures can be achieved through buckling. Confining a buckled beam between two lateral constraints allows it to buckle into higher modes as the axial load increases. This paper presents a theoretical study of the postbuckling response of a bilaterally constrained elastica subjected to gradually increased axial load. Equilibrium states are determined using an energy method. Under small deformation assumptions, the total potential energy is minimized under the defined constraints. The presented model allows for an accurate representation of the flatting behavior and the increase in the length of contact areas with the lateral constraints before the sudden snapping between equilibrium states. Mode transitions are manifested by jumps in the response curves. Previously developed models based on geometry and symmetries overestimate the required forces for higher equilibrium modes and do not match experimental observations. Results are validated with experimental force–displacement measurements under both force- and displacement-control. The kinetic energy released during buckling mode transitions is determined by a dynamic analysis.     

Sandwich column buckling – A hyperelastic formulation

The macro-buckling equations for a sandwich column are developed. A layer-wise Timoshenko beam displacement approximation is assumed. The constitutive relationships and equilibrium equations for the core and face sheets are derived using a consistent hyperelastic neo-Hookean formulation. The derivations in this paper are consistent with that of Haringx’s and Reissner’s proposal for beam actions. The buckling formulation includes the axial deformation prior to buckling and the transverse shear deformation of the core and face sheets. The buckling equations derived agree with the equation of [Allen, H.G., 1969. Analysis and Design of Structural Sandwich Panels, Pergamon, Oxford] for thick faces but are also applicable to any ratio of face sheet to core thickness and material properties. The formulation is compared to experimental results for sandwich columns and shows good comparison except for very short columns. The formulation is also compared to the buckling experimental results for short rubber rods and also compared well. The formulation does not predict a shear buckling mode.     

Buckling of delaminated bi-layer beam-columns

An improved analytical model is presented to analyze the delamination buckling of a bi-layer beam-column with a through-the-width delamination. Both the transverse shear deformation and local delamination tip deformations are taken into consideration, and two delaminated sub-layers as well as two substrates in the intact (un-delaminated) regions are modeled as individual Timoshenko beams. A deformable interface is introduced to establish the continuity condition between the two substrates in the intact regions. Consequently, a flexible joint is formed at the delamination tip, and it is different from the conventional rigid joint given in most of studies in the literature, in which the local delamination tip deformations are completely ignored. In contrast to the local delamination buckling in our previous study (Qiao et al., 2010), the present model accounts for the global deformations of the intact region in the delaminated composite beam-column, thus capable of capturing the buckling mode shape transitions from the global, to global–local coexistent, and to local buckling for asymmetric delamination as the interface delamination increases. Good agreement of the present analytical solutions with the full 2-D elastic finite element analysis demonstrates the local deformation effects around the delamination tip and verifies the accuracy of the present model. Parametric studies are conducted to investigate the effects of loading eccentricity, delaminated sub-layer thickness ratio, and interface compliance on the critical buckling load for the delaminated composite beam-column. Transitions of buckling modes from the global to local delamination buckling are also disclosed as the thickness of one sub-layer reduces from the thick sub-layer to a thin film. The developed delamination buckling solution facilitates the design analysis and optimization of laminated composite structures, and it can be used with confidence in buckling analysis of delaminated composite structures.     

冲击载荷作用下矩形薄板的弹性动力屈曲

为了研究冲击载荷作用下考虑应力波效应弹性矩形薄板的动力屈曲,根据动力屈曲发生瞬间的能量转换和守恒准则,导出板的屈曲控制方程和波阵面上的补充约束条件,真实的屈曲位移应同时满足控制方程和波阵面上的附加约束条件。满足上述条件,建立了该问题的完整数值解法,对屈曲过程中冲击载荷、屈曲模态和临界屈曲长度之间的关系进行研究,定量计算了横向惯性效应对提高薄板动力屈曲临界应力的贡献。研究表明:板的厚宽比一定时,临界屈曲长度随冲击载荷的增大而减小;由于屈曲时的横向惯性效应,应力波作用下薄板一阶临界力参数是相应边界板的静力失稳临界力参数的1.5倍;随着边界约束逐渐减弱,板临界力参数逐渐减小,动力特征参数逐渐增大。     

腹板V形加劲冷弯薄壁卷边槽钢柱试验研究

为了解腹板V形加劲冷弯薄壁卷边槽钢构件的受压性能,对8根短柱和中长柱进行了轴心受压试验研究。在试验之前首先用有限条和有限元方法对柱试件进行了详细的设计,以确保发生畸变屈曲失效。柱受压试验结果发现：所有试件均发生畸变屈曲失效破坏,同时在峰值荷载时中长柱试件有绕弱轴的弯曲产生,畸变屈曲和局部屈曲一样也有一定的屈曲后强度。在总结分析现有国内外规范计算理论的基础上,提出了基于直接强度法设计腹板V形加劲冷弯薄壁卷边槽钢轴压构件畸变屈曲承载力的计算方法。采用该建议方法计算所得结果与试验值和参数分析值吻合较好,其设计过程简便且安全可靠。     

On static and dynamic buckling modes of a rod-strip loaded by follower forces

In [1], it was shown that, under the action of compressing transverse forces of constant (in the deformation process) direction on the rod-strip, there are two statically possible buckling modes (for the adjacent neutral equilibrium), one of which is purely shear and the second is purely flexural and is realized without transverse strains.In the present paper, we consider problems about static and dynamic buckling modes of a rod-strip under the separate action of longitudinal and transverse compressing and also shear forces, which belong to the class of follower forces of two types. The first type corresponds to the conservation of directions of the above forces along the basis vectors of the strained state; the second, to the conservation of one of the components of the surface forces acting along the normal to the deformed boundary surface. We show that if the transverse compressing forces are follower forces, i.e., if in the deformation process they remain normal to the surfaces to which they are applied, then the flexural buckling mode realized in the rod can be found only by the dynamic method [2] based on the use of the refined shear Timoshenko-type model for rods.     

Interactive buckling in long thin-walled rectangular hollow section struts

An analytical model describing the nonlinear interaction between global and local buckling modes in long thin-walled rectangular hollow section struts under pure compression founded on variational principles is presented. A system of nonlinear differential and integral equations subject to boundary conditions is formulated and solved using numerical continuation techniques. For the first time, the equilibrium behaviour of such struts with different cross-section joint rigidities is highlighted with characteristically unstable interactive buckling paths and a progressive change in the local buckling wavelength. With increasing joint rigidity within the cross-section, the severity of the unstable post-buckling behaviour is shown to be mollified. The results from the analytical model are validated using a nonlinear finite element model developed within the commercial package Abaqus and show excellent comparisons. A simplified method to calculate the local buckling load of the more compressed web undergoing global buckling and the corresponding global mode amplitude at the secondary bifurcation is also developed. Parametric studies on the effect of varying the length and cross-section aspect ratio are also presented that demonstrate the effectiveness of the currently developed models.     

对桁架结构稳定分析经典理论的讨论

通过算例讨论了桁架结构稳定分析的经典理论,指出用该理论算出的临界荷载远远大于屈曲临界荷载,而且压杆的应力远远超过压缩强度极限。文中分析了问题的来源,提出了桁架结构临界荷载的屈曲理论计算方法,通过比较说明了屈曲理论的正确性。     

Column buckling with shear deformations—A hyperelastic formulation

Column constitutive relationships and buckling equations are derived using a consistent hyperelastic neo-Hookean formulation. It is shown that the Mandel stress tensor provides the most concise representation for stress components. The analogous definitions for uniaxial beam plane stress and plane strain for large deformations are established by examining the virtual work equations. Anticlastic transverse curvature of the beam cross-section is incorporated when plane stress or thick beam dimensions are assumed. Column buckling equations which allow for shear and axial deformations are derived using the positive definiteness of the second order work. The buckling equations agree with the equation derived by Haringx and are extended to incorporate anticlastic transverse curvature which is important for low slenderness, high buckling modes and with increasing width to thickness ratio. The work in this paper does not support the existence of a shear buckling mode for straight prismatic columns made of an isotropic material.     

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.     

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.     

Helical coil buckling mechanism for a stiff nanowire on an elastomeric substrate

When a stiff nanowire is deposited on a compliant soft substrate, it may buckle into a helical coil form when the system is compressed. Using theoretical and finite element method (FEM) analyses, the detailed three-dimensional coil buckling mechanism for a silicon nanowire (SiNW) on a polydimethylsiloxane (PDMS) substrate is studied. A continuum mechanics approach based on the minimization of the strain energy in the SiNW and elastomeric substrate is developed. Due to the helical buckling, the bending strain in SiNW is significantly reduced and the maximum local strain is almost uniformly distributed along SiNW. Based on the theoretical model, the energy landscape for different buckling modes of SiNW on PDMS substrate is given, which shows that both the in-plane and out-of-plane buckling modes have the local minimum potential energy, whereas the helical buckling model has the global minimum potential energy. Furthermore, the helical buckling spacing and amplitudes are deduced, taking into account the influences of the elastic properties and dimensions of SiNWs. These features are verified by systematic FEM simulations and parallel experiments. As the effective compressive strain in elastomeric substrate increases, the buckling profile evolves from a vertical ellipse to a lateral ellipse, and then approaches to a circle when the effective compressive strain is larger than 30%. The study may shed useful insights on the design and optimization of high-performance stretchable electronics and 3D complex nano-structures.