细长弹性杆在轴压下的二次分叉

基于[1]的弹性曲杆的平衡方程,本文研究了矩形横截面细长杆在轴压下的后屈曲行为。设横截面的边长比为 1:2δ,使用 Poincare-Keller 的打靶法并引进坐标的伸缩变换,研究了δ在 δ_0=1 附近的情形。当δ≠1 时,发现了杆平衡态的二次分叉。我们也给出了原始后屈曲解支及二次分支的渐近表示并分析了各个解支的稳定性。     

Hutchinson模型的塑性后分叉和缺陷敏感性

本文通过对Ｈｕｔｃｈｉｎｓｏｎ模型的初始塑性后分叉和缺陷敏感性的分析，介绍了一个分析塑性后屈曲的一般方法，这一方法非常类似Ｋｏｉｎｔｅｒ的弹性后屈曲理论。用这一方法分析缺陷敏感性和初始后分叉的过程是一致的，所得展开在渐近的意义上是精确的，和数值解符合得相当好。     

薄圆板的后屈曲响应及圆形脱层的能量释放率

研究了薄圆板及圆形脱层薄膜的轴对称后屈曲问题,并首次研究了轴对称后屈曲变形之后的二次非轴对称分叉屈曲问题.采用幂级数展开的方法,结合应用傅立叶级数得出非线性及非轴对称问题方程的解.并由此计算了圆形脱层薄膜与上述后屈曲变形的两个阶段对应的能量释放率.     

应用突变理论分析Euler压杆的稳定性

建立了Euler压杆问题的能量表达式。对此模型，应用突变理论进行稳定分析，得出了系统全部的分叉集与突变流形。突变流形为一族层层嵌套、互不交叉的抛物线。分析的结论要比有限差分法、有限单元法、大范围非线形分析等数值计算的结果简洁、明了，有助于压杆后屈曲问题的研究。     

受轴向冲击有限长弹性直杆中应力波引起的分叉问题

研究了有限长理想直杆受阶跃载荷作用的弹性动力屈曲问题。将直杆的屈问题归结为由轴向应力波传播而导致的杆的分叉问题，并考虑了应力波反射的影响。给出了分叉发生的临界条件并对具体实例进行了计算。最后对阶跃载荷及脉冲载荷对杆的动力屈曲的影响进行了讨论。     

环形板的非轴对称屈曲分析

本文利用打靶法研究了内外边界固支且在外边界受均匀径向压力作用下的极正交各向异性环形板的非轴对称屈曲和过屈曲,计算了临界载荷,讨论了分支解的存在性,得到了分支解的渐近表达式,分析了环形板的屈曲性态。     

环形板的非轴对称屈曲分析

本文利用打靶法研究了内外边界固支且在外边界受均匀径向压力作用下的极正交各向异性环形板的非轴对称屈曲和过屈曲,计算了临界载荷,讨论了分支解的存在性,得到了分支解的渐近表达式,分析了环形板的屈曲性态。     

二次分叉数值分析方法及在多孔介质热对流上的应用

本文给出了数值计算具有Ｚ２对称性和双参数的非线性动力学方程二次分叉问题的具体方法,并用此方法计算了不同长宽比矩形区域内多孔介质热对流的二次分叉点和相应的流场和温度场分布。     

直杆中应力波传播引起的分叉问题

讨论轴向矩形脉冲载荷作用下有限长理想弹性直杆中应力波的传播及其反射引起的分叉问题，讨论此类动力屈曲问题中横向惯性效应的影响；理论分析始终遵循着这样的思路：在分叉的那一瞬间，杆并没有横向惯性效应，目的是给出此类问题的一个合理提法，从理论分析和数值计算中得到了许多重要的结论，理论分析所遵循的思路和所采用的方法对各类结构由于应力波的传播及反射引起的分叉问题具有一定的意义．     

细长压杆二阶屈曲临界态的数值模拟

竹子轻盈坚韧的特性源自于竹子特殊的天然构造。根据对竹子构造观察所获得的启示,认识到细长压杆二阶屈曲的物理存在性,并通过数值模拟技术加以验证。针对细长压杆潜在的屈曲多向性,设计了一种含有双向引导槽的细长压杆,当杆件受到轴向压力时可以沿着二阶屈曲方向自动产生微曲,继而对细长压杆的二阶屈曲临界态进行探讨和数值模拟,从而进一步从物理角度揭示了细长压杆二阶屈曲临界态的潜在性。计算机数值模拟结果表明:两端铰支二阶屈曲态细长压杆的临界力为一阶屈曲态临界力的3.24倍;经过特殊设计的智能压杆可以自动向着压杆二阶曲线形态屈曲,从而大幅提了高细长压杆的临界压力。关注和挖掘细长压杆高阶屈曲的潜能对某些需使用昂贵材料制作压杆的特殊领域而言,可以大大降低工程成本,并具有着很好的启示意义。     

Rayleigh–Ritz analysis for localized buckling of a strut on a softening foundation by Hermite functions

This paper proposes a Rayleigh–Ritz procedure for localized buckling of a strut on a non-linear elastic foundation. Firstly, the deflected shape of a strut is expanded into a series of Hermite orthogonal functions, which are proved energy-integrable in an infinite region. Secondly, the errors of the numerical integrations of Hermite functions on the infinite region are investigated and the suitable integral limit is proposed. Through the numerical investigation, it is demonstrated that the first thirty Hermite functions are usually enough to approximate the localized buckling pattern. The proposed method overcomes the disadvantages of the traditional methods, in which the trial functions in either Rayleigh–Ritz or Galerkin analysis are based on the perturbation analyses of the corresponding non-linear differential equation.     

Experimental and theoretical study of the buckling of narrow thin plates on an elastic foundation under compression

The paper describes the processes of elastic deformation of thin films under mechanical loading. The film is modeled longitudinally by a compressed plate arranged on an elastic foundation. A computer model of the buckling of the narrow thin plate with a delamination portion located on an elastic foundation is constructed. This paper also touches upon the supercritical behavior of the plate–substrate system. The experiments on the axial compression of a metal strip adhered to a rubber plate are performed, and 2 to 7 buckling modes are obtained therein. The critical loads and buckling modes obtained in the numerical calculations are compared with the experimental data. It is shown that there is the possibility of progressive delamination of the metal plate from the foundation if the critical load is exceeded. It is found that the use of the proposed approach, which, in contrast to other approaches, accounts for the elastic deformation of the substrate, causes the dependence between the critical bending stress and the stiffness of the foundation.     

The plastica: The large elastic-plastic deflection of a strut

An extension of the theory of the Elastica is developed to determine the shape of a strut (or a cantilever) which undergoes large plastic deflection. The equations governing such a behaviour and known as the Plastica equations are set up and then solved by a perturbation method and by numerical integration. The post-buckling elastic-plastic deformation of a strut after initial elastic buckling is analysed, and some numerical results are given to show the variation of its load-carrying capacity with the development of the plastic region.     

Thermal buckling and post-buckling of pinned–fixed Euler–Bernoulli beams on an elastic foundation

In this article, both thermal buckling and post-buckling of pinned–fixed beams resting on an elastic foundation are investigated. Based on the accurate geometrically non-linear theory for Euler–Bernoulli beams, considering both linear and non-linear elastic foundation effects, governing equations for large static deformations of the beam subjected to uniform temperature rise are derived. Due to the large deformation of the beam, the constraint forces of elastic foundation in both longitudinal and transverse directions are taken into account. The boundary value problem for the non-linear ordinary differential equations is solved effectively by using the shooting method. Characteristic curves of critical buckling temperature versus elastic foundation stiffness parameter corresponding to the first, the second, and the third buckling mode shapes are plotted. From the numerical results it can be found that the buckling load-elastic foundation stiffness curves have no intersection when the value of linear foundation stiffness parameter is less than 3000, which is different from the behaviors of symmetrically supported (pinned–pinned and fixed–fixed) beams. As we expect that the non-linear foundation stiffness parameter has no sharp influence on the critical buckling temperature and it has a slight effect on the post-buckling temperature compared with the linear one.     

Finite element analysis of post-buckling dynamics in plates—Part I: An asymptotic approach

Various static and dynamic aspects of post-buckled thin plates, including the transition of buckled patterns, post-buckling dynamics, secondary bifurcation, and dynamic snapping (mode jumping phenomenon), are investigated systematically using asymptotical and non-stationary finite element methods. In part I, the secondary dynamic instability and the local post-secondary buckling behavior of thin rectangular plates under generalized (mechanical and thermal) loading is investigated using an asymptotic numerical method which combines Koiter’s nonlinear instability theory with the finite element technique. A dynamic multi-mode reduction method—similar to its static single-mode counterpart: Liapunov–Schmidt reduction—is developed in this perturbation approach. Post-secondary buckling equilibrium branches are obtained by solving the reduced low-dimensional parametric equations and their stability properties are determined directly by checking the eigenvalues of the resulting Jacobian matrix. Typical post-secondary buckling forms—transcritical, supercritical and subcritical bifurcations are observed according to different combinations of boundary conditions and load types. Geometric imperfection analysis shows that not only the secondary bifurcation load but also changes in the fundamental post-secondary buckling behavior are affected. The post-buckling dynamics and the global analysis of mode jumping of the plates are addressed in part II.     

梯度弹性基础上正交异性薄板的屈曲分析

基于双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板的屈曲问题. 首先,基于能量法与变分原理,给出了梯度弹性基础上正交异性薄板的屈曲控制方程,并得到了梯度弹性基础刚度系数K₁ 与K₂的计算式;进而,通过将位移函数采用三角函数展开的方法,给出了单向压缩载荷作用下、四边简支正交异性弹性基础板屈曲载荷的计算式;在算例中,通过将该文的解退化到单纯的正交异性板,并与经典弹性解比较,证明了理论的正确性;最后,求解了弹性模量在厚度方向上呈幂律分布的梯度基础上的薄板屈曲问题,分析了基础上下表层材料弹性模量比与体积分数指数对屈曲载荷的影响.     

THE TRANSVERSE NONLINEAR FREE VIBRATION AND THE BUCKLING OF A COMPRESSIVE BAR ON AN ELASTIC FOUNDATION

基于Hamilton 原理,运用假设时间模态法,得到了弹性基础上压杆的横向非线性自由振动与屈曲的位移型常微分控制方程. 考虑一端固定另一端可移简支边界条件,采用打靶法得到了结构第一至第三阶结构频率与一阶屈曲载荷的数值结果. 结果表明：随轴心压力增加,结构频率减小;随弹性基础刚度增加,结构频率与屈曲载荷均增加;弹性基础刚度对结构频率的影响随振型阶数增加在减小;在小振幅的情形下,不同振型对一阶屈曲载荷的影响很小.     

Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: an analytical approach

This paper presents an analytical investigation on the buckling analysis of symmetric sandwich plates with functionally graded material (FGM) face sheets resting on an elastic foundation based on the first-order shear deformation plate theory (FSDT) and subjected to mechanical, thermal and thermo-mechanical loads. The material properties of FGM face sheets are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. An analytical approach is used to reduce the governing equations of stability and then solved using an analytical solution which is named as power series Frobenius method for symmetric sandwich plates with six different boundary conditions. A detailed numerical study is carried out to examine the influence of the plate aspect ratio, side-to-thickness ratio, loading type, sandwich plate type, volume fraction index, elastic foundation coefficients and boundary conditions on the buckling response of FGM sandwich plates. This has not been done before and serves to fill the gap of knowledge in this area.     

Postbuckling analysis of columns resting on an elastic foundation

The postbuckling response of perfect and geometrically imperfect elastic columns resting on an elastic Winkler type foundation is thoroughly discussed. This is established by employing an approximate analytic technique leading to very reliable results in the vicinity of the critical state. It was found that the critical state of perfect columns is a stable symmetric bifurcation point and consequently there is no sensitivity to initial geometrical imperfections. Moreover, a simple but readily analyzed mechanical model is proposed to simulate the salient features of buckling mechanism of the column on elastic foundation with those of the model. The simplicity, reliability and efficiency of the proposed analysis as well as the successful modeling of the buckling mechanism of the column by that of a single mode mechanical model are illustrated with the aid of numerical examples.     

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.