Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force

The asymmetric buckling of a shallow initially curved stress-free micro beam subjected to distributed nonlinear deflection-dependent electrostatic force is studied. In order to highlight the symmetry breaking phenomenon and the approach to its analysis, the subsidiary simplified problem of a curved beam attached to a linearly elastic foundation, and subjected to uniformly distributed “mechanical” load, which is independent of deflections, is addressed first. The analysis is based on a two degrees of freedom reduced order (RO) model resulting from the Galerkin decomposition with linear undamped eigenmodes of a straight beam used as the base functions. Simple approximate expressions are derived defining the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric limit point buckling curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the curve. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict non-symmetric buckling in electrostatically actuated bistable micro beams.     

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。     

含初缺陷裂纹损伤梁的冲击动力屈曲

由Hamilton原理导出考虑初始缺陷及横向剪切变形时裂纹梁的动力屈曲控制方程;应用断裂力学中常用的线弹簧模型将裂纹引入到屈曲控制方程中;基于B-R动力屈曲判断准则,采用数值方法求解了受轴向冲击载荷作用时裂纹梁的动力屈曲;对比讨论了不同冲击速度、初始几何缺陷大小以及分布形式等因素对梁冲击动力屈曲的影响。     

弹塑性梁在轴向应力波下的动态屈曲

本文考虑轴向应力波效应，利用分叉理论研究各种支承半无限长弹塑性梁的动态屈曲问题。在轴向阶梯载荷和脉冲载荷冲击下得到了梁的临界屈曲载荷及初始屈曲模态。其结果与实验现象相一致。同时也为研究结构动态屈曲问题提供了有效途径。     

Non-linear stability analysis of imperfect thin-walled composite beams

The static non-linear behavior of thin-walled composite beams is analyzed considering the effect of initial imperfections. A simple approach is used for determining the influence of imperfection on the buckling, prebuckling and postbuckling behavior of thin-walled composite beams. The fundamental and secondary equilibrium paths of perfect and imperfect systems corresponding to a major imperfection are analyzed for the case where the perfect system has a stable symmetric bifurcation point. A geometrically non-linear theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field. An initial displacement, either in vertical or horizontal plane, is considered in presence of initial geometric imperfection. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results are presented for a simply supported beam subjected to axial or lateral load. It is shown in the examples that a major imperfection reduces the load-carrying capacity of thin-walled beams. The influence of this effect is analyzed for different fiber orientation angle of a symmetric balanced lamination. In addition, the postbuckling response obtained with the present beam model is compared with the results obtained with a shell finite element model (Abaqus).     

横向载荷作用下轴向运动梁的屈曲失稳以及非线性振动特性

梁的轴向运动会诱发其产生横向振动并可能导致屈曲失稳,对结构的安全性和可靠性产生重大的影响。本文重点研究了横向载荷作用下轴向运动梁的屈曲失稳及横向非线性振动特性。基于Hamilton变分原理,建立了横向载荷作用下轴向运动梁的动力学方程,获得了梁的后屈曲构型。使用截断Galerkin法,将控制方程改写成Duffing方程的形式。用同伦分析方法确定载荷作用下轴向运动梁的非线性受迫振动的封闭形式的表达式。结果表明,后屈曲构型对轴向速度和初始轴向应力有明显的依赖性。通过同伦分析法得出非线性基频的显式表达式,获得了初始轴向力会影响非线性频率随初始振幅和轴向速度的线性关系。另外,轴向外激励的方向也会改变系统固有频率。     

粘贴压电层功能梯度材料Timoshenko梁的热过屈曲分析

研究了上下表面粘贴压电层的功能梯度材料Timoshenko梁在升温及电场作用下的过屈曲行为。在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了压电功能梯度Timoshenko层合梁在热-电-机械载荷作用下的几何非线性控制方程。其中,假设功能梯度的材料性质沿厚度方向按照幂函数连续变化,压电层为各向同性均匀材料。采用打靶法数值求解所得强非线性边值问题,获得了在均匀电场和横向非均匀升温场内两端固定Timoshenko梁的静态非线性屈曲和过屈曲数值解。并给出了梁的变形随热、电载荷及材料梯度参数变化的特性曲线。结果表明,通过施加电压在压电层产生拉应力可以有效地提高梁的热屈曲临界载荷,延缓热过屈曲发生。由于材料在横向的非均匀性,即使在均匀升温和均匀电场作用下,也会产生拉-弯耦合效应。但是对于两端固定的压电-功能梯度材料梁,在横向非均匀升温下过屈曲变形仍然是分叉形的。     

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material (FGM) structures is one of the major threats to their service safety in engineering applications. This paper aims to clarify a long-standing controversy on the thermal instability type of simply-supported FGM beams. First, based on the Euler-Bernoulli beam theory and von Kármán geometric nonlinearity, a nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang's two-variable method is formulated. Second, an approximate analytic solution to the nonlinear integro-differential boundary value problem under a thermal-induced inhomogeneous force boundary condition is obtained by using a semiinverse method when the coordinate axis is relocated to the bending axis (physical neutral plane), and then the analytical predictions are verified by the differential quadrature method (DQM). Finally, based on the free energy theorem, it is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes; furthermore, the nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect. These results are expected to provide new ideas and references for the design and regulation of FGM structures.     

Bifurcation phenomena in the rigid inclusion power law matrix composite sphere

Bifurcation of interface separation related to cavity nucleation is analyzed for a radially loaded composite sphere consisting of a rigid inclusion separated from a power law matrix by a uniform, non-linear cohesive zone. Equations for the spherically symmetric and non-symmetric problems are obtained from a hyperelastic finite strain theory by a limiting process that preserves non-linear matrix and interface response at infinitesimal strain. A complete solution to the symmetric problem is presented including bifurcation load, stresses, and evolution of elasto-plastic boundary and interface separation. An analysis of non-symmetric bifurcation, under symmetric conditions of geometry and loading, yields the bifurcation load and first non-symmetric mode shape associated with rigid inclusion displacement. An energy analysis is carried out for both symmetric and non-symmetric problems in order to assess stability of spherically symmetric states to spherically symmetric and non-symmetric “rigid body mode” perturbations.Results are provided for an interface force law that captures interface failure in normal mode and linear response in shear mode. For the symmetric problem, (i) there are threshold parameter values above which bifurcation will generally not occur, (ii) threshold values below which there do not exist equilibria in the post bifurcation regime, (iii) bifurcation occurs after attainment of the maximum interface strength. For the non-symmetric problem, (i) bifurcation always occurs, although it can be delayed by interfacial shear, (ii) for the smooth interface, non-symmetric bifurcation occurs after attainment of the maximum interface strength and always precedes symmetric bifurcation.     

Analysis of bending and buckling of pre-twisted beams: A bioinspired study

Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanical properties of elas- tic beams by using the Timoshenko beam model. Particular attention is paid to the transverse bending and axial buckling of a pre-twisted rectangular beam. The analytical solution is first derived for the deflection of a clamped-free beam under a uniformly or periodically distributed transverse force. The critical buckling condition of the beam subjected to its self- weight and an axial compressive force is further solved. The results show that the twisting morphology can significantly improve the resistance of beams to both transverse bending and axial buckling. This study helps understand some phenomena associated with twisting chirality in nature and provides inspirations for the design of novel devices and structures.     

Hamilton体系下功能梯度梁的热冲击动力屈曲分析

在Hamilton体系下,基于Euler梁理论研究了功能梯度材料梁受热冲击载荷作用时的动力屈曲问题;将非均匀功能梯度复合材料的物性参数假设为厚度坐标的幂函数形式,采用Laplace变换法和幂级数法解析求得热冲击下功能梯度梁内的动态温度场:首先将功能梯度梁的屈曲问题归结为辛空间中系统的零本征值问题,梁的屈曲载荷与屈曲模态分别对应于Hamilton体系下的辛本征值和本征解问题,由分叉条件求得屈曲模态和屈曲热轴力,根据屈曲热轴力求解临界屈曲升温载荷。给出了热冲击载荷作用下一类非均匀梯度材料梁屈曲特性的辛方法研究过程,讨论了材料的梯度特性、结构几何参数和热冲击载荷参数对临界温度的影响。     

Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).     

非保守功能梯度弹性组合曲梁的精确模型及其数值解

基于直法线假设,采用可伸长梁的几何非线性理论,建立了功能梯度材料弹性组合曲梁受切线均布随从力作用下的静态大变形数学模型。该模型不仅计及了轴线伸长,同时也精确地考虑了梁的初始曲率对变形的影响以及轴向变形与弯曲变形之间的耦合效应。用打靶法数值求解了由金属和陶瓷两相材料所构成的一种FGM组合曲梁在沿轴线均布切向随动载荷作用下的非线性平面弯曲问题,给出了不同梯度指标下FGM弹性曲梁随载荷参数大范围变化的平衡路径,并与金属和陶瓷两种单相材料曲梁的相应特性进行了比较。     

Buckling of a beam subjected to electromagnetic force and its stabilization by controlling the perturbation of the bifurcation

For a beam subjected to electromagnetic force, magnetoelastic buckling due to the increase of such force is theoretically investigated by taking account of the nonlinearity of the electromagnetic force and the elastic force of the beam. Using Liapunov-Schmidt method and center manifold theory, the equilibrium space, the bifurcation set and the bifurcation diagram are theoretically derived. Also, the effect of the higher modes other than the buckling mode on the mode shape of the postbuckling state is discussed. Furthermore, a control method to stabilize the magnetoelastic buckling is proposed, and the unstable equilibrium state of the beam in the postbuckling state, i.e., the straight position of the beam, is stabilized by controlling the perturbation of the bifurcation.     

THERMAL POST-BUCKLING OF FUNCTIONALLY GRADED MATERIAL TIMOSHENKO BEAMS

Analysis of thermal post-buckling of FGM (Functionally Graded Material) Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely non-uniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.     

On the plastic bifurcation and post-bifurcation of axially compressed beams

The paper presents two new results in the domain of the elastoplastic buckling and post-buckling of beams under axial compression. (i) First, the tangent modulus critical load, the buckling mode and the initial slope of the bifurcated branch are given for a Timoshenko beam (with the transverse shear effects). The result is derived from the 3D J₂ flow plastic bifurcation theory with the von Mises yield criterion and a linear isotropic hardening. (ii) Second, use is made of a specific method in order to provide the asymptotic expansion of the post-critical branch for a Euler-Bernoulli beam, exhibiting one new non-linear fractional term. All the analytical results are validated by finite element computations.     

复合材料脱层梁的屈曲分析

本文研究了具有任意位置透型脱层的复合材料梁的屈曲问题。基于弹性理论建立了复合材料脱层梁的基本方程式。对脱层梁进行了分区处理,利用B样条函数作为位移型函数的基函数,方便地描述了脱层长度、脱层位置。考虑边界条件、区间位移连续性条件和弯矩剪力的平衡条件以及纵向内力的附加条件,对基本方程式进行了求解。得出了脱层位置不同,脱层长度不同的屈曲荷载的变化规律,并与轴对称脱层时的屈曲荷载进行了比较,认为层合梁考虑脱层对屈曲的影响是非常必要的。     

Secondary buckling and tertiary states of a beam on a non-linear elastic foundation

An axially compressed beam resting on a non-linear foundation undergoes a loss of stability (buckling) via a supercritical pitchfork bifurcation. In the post-buckled regime, it has been shown that under certain circumstances the system may experience a secondary bifurcation. This second bifurcation destablizes the primary buckling mode and the system “jumps” to a higher mode; for this reason, this phenomenon is often referred to as mode jumping. This work investigates two new aspects related to the problem of mode jumping. First, a three mode analysis is conducted. This analysis shows the usual primary and secondary buckling events. But it also shows stable solutions involving the third mode. However, for the cases studied here, there is no natural loading path that leads to this solution branch, i.e. only a contrived loading history would result in this solution. Second, the effect of an initial geometric imperfection is considered. This breaks the symmetry of the system and significantly complicates the bifurcation diagram.     

Dynamic buckling of a 2-DOF imperfect system with symmetric imperfections

Non-linear dynamic buckling of a two-degree-of freedom (2-DOF) imperfect planar system with symmetric imperfections under a step load of infinite duration (autonomous system) is thoroughly discussed using energy and geometric considerations. This system under the same load applied statically exhibits (prior to limit point) an unstable symmetric bifurcation lying on the non-linear primary equilibrium path. With the aid of the total energy-balance equation of the system and the particular geometry (due to symmetric imperfections) of the plane curve corresponding to the zero level total potential energy “surface” exact dynamic buckling loads are obtained without solving the non-linear initial-value problem. The efficiency and the reliability of the technique proposed herein is demonstrated with the aid of various dynamic buckling analyses which are compared with numerical simulation using the Verner-Runge-Kutta scheme, the accuracy of which is checked via the energy-balance equation.     

Relationship between Euler buckling and unstable equilibria of buckled beams

Classic snap-through of curved beams, plates, and shells has long been an object of attention in structural engineering. Euler buckling under axial loading is perhaps an even more entrenched part of the canon of engineering education and practice. In this paper we introduce a relationship between the two phenomena, that to our knowledge has not been directly addressed before. The relationship shows that Euler buckling configurations are connected by the force–displacement curve under transverse loading. The results are used to develop a very simple metric to estimate the number of unstable static equilibria of a buckled structure based only on its geometry with no need for static or dynamic solvers. The study is focused on beams as this allows for an unambiguous discussion of the idea on the simplest possible structure.