Euler型梁-柱结构的非线性稳定性和后屈曲分析

在有限变形的假设下,建立了位于非线性弹性基础上非线性弹性Euler型梁-柱结构的广义Hamilton变分原理,并由此导出了任意变截面Euler型梁-柱结构的3维非线性数学模型,其中考虑了转动惯性、几何非线性、材料非线性等因素的影响．作为模型的应用,分析了弹性基础上一端完全固支另一端部分固支,并受轴力作用的均质等截面线性弹性Euler型梁的非线性稳定性和后屈曲;结合打靶法和Newton法,给出了一种计算平凡解(前屈曲状态)、分叉点(临界载荷)和分叉解(后屈曲状态)的数值方法,对前两个分支点和相应分支解,成功地实现了数值计算,并考虑了基础反力和惯性矩对分支点的影响．     

加热弹性杆的热过屈曲分析

基于轴线可伸长细杆的过屈曲变形几何理论,建立了两端轴向不可移的均匀加热直杆热弹性过屈曲行为的精确数学模型.这是一个包含杆轴线弧长在内的多未知函数的强非线性一阶常微分方程两点边值问题.采用打靶法和解析延拓法直接数值求解上述非线性边值问题,分别获得了两端横向简支和夹紧杆的热过屈曲状态解,给出了具有不同细长比杆的热过屈曲平衡路径.     

钢管混凝土柱抗爆性能数值模拟与实验验证

利用LS-DYNA非线性有限元程序,基于多物质流固耦合方法,建立了爆炸荷载作用下钢管混凝土柱的动态响应数值模型.对比分析了模拟结果与足尺构件的爆炸破坏实验结果,验证了数值模型和计算方法的有效性,并运用参数化分析方法,研究了截面形式、比例距离、混凝土强度及钢材等级、截面形状特性等关键参数对钢管混凝土柱抗爆性能的影响.研究结果表明:钢管混凝土柱具有优越的抗爆性能,所建立的数值模型能够有效地分析钢管混凝土柱在爆炸荷载作用下的动态影响及破坏形态;圆形截面钢管混凝土柱的抗爆性能优于方形截面;提高材料等级、减小圆形截面钢管混凝土柱的径厚比、增大矩形截面钢管混凝土柱的长宽比,均有利于提升钢管混凝土柱的抗爆特性.     

非线性弹性梁中的混沌带现象

研究了非线性弹性梁的混沌运动,梁受到轴向载荷的作用。非线性弹性梁的本构方程可用三次多项式表示。计及材料非线性和几何非线性,建立了系统的非线性控制方程。利用非线性Galerkin法,得到微分动力系统。采用Melnikov方法对系统进行分析后发现,当载荷P₀和f满足一定条件时,系统将发生混沌运动,且混沌运动的区域呈现带状。还详尽分析了从次谐分岔到混沌的路径,确定了混沌发生的临界条件。     

不可压饱和多孔弹性梁的大挠度非线性数学模型

在孔隙流体仅存在沿梁轴线方向扩散的假定下,建立了微观不可压饱和多孔弹性梁大挠度问题的非线性数学模型．利用Galerkin截断法,研究了固定端不可渗透、自由端可渗透的饱和多孔弹性悬臂梁在自由端突加集中载荷作用下的非线性弯曲,得到了梁骨架的挠度、弯矩以及孔隙流体压力等效力偶等的时间响应和沿轴线的分布．比较了大挠度非线性和小挠度线性理论的结果,揭示了两者间的差异．研究发现大挠度理论的结果小于相应的小挠度理论结果,并且,大挠度理论的结果趋于其稳态值的时间小于相应的小挠度理论结果趋于其稳态值的时间．     

静脉壁的力学特性及负压失稳问题

应用一类超弹性应变能函数,通过非线性弹性理论,研究了静脉壁在跨壁压及轴向拉伸联合作用下的变形和应力分布等力学特性,并分析了静脉壁的负压失稳问题．首先利用超弹性材料薄壁圆筒模型,得到了静脉壁在跨壁压及轴向拉伸联合作用下的变形方程,给出了正常静脉压下静脉壁的变形曲线和应力分布曲线,讨论了静脉壁的变形和应力分布规律．然后给出了负跨壁压下静脉壁的变形曲线,并由能量比较讨论了静脉壁的负压失稳问题．     

多夹层板壳的非线性理论及应用（Ⅲ）：扁壳的弯曲和稳定

本文根据文[1]获得的多夹层扁壳非线性基本方程，求解了各种载荷及边界条件下矩形底面多夹层扁壳的非线性弯曲问题，多夹层板、扁柱亮在轴向压力作用下的稳定问题，以及一般形状的扁壳在边界作用力下的变形．     

螺旋弹簧临界屈曲失稳的平衡方程

目前,针对螺旋弹簧失稳现象的研究主要基于当量柱模型,将弹簧等效为柱模型,忽略其绕轴线的转动．该文建立了三维螺旋弹簧模型,通过曲线Frenet坐标系和主轴坐标系,建立了描述弹簧螺旋中心线的空间变形和截面扭转变形的平衡方程．采用小变形假设,通过对弹簧挠度变量采用Taylor级数展开,并忽略其高阶小量,平衡方程可以被简化为扭转角和弧长的函数,使获得方程的数值解成为可能．同时,讨论了作用于弹簧圆心位置的轴向力引起的弹簧约束端反作用力,为平衡方程的求解确定了边界载荷条件．该文的研究工作为进一步研究受压螺旋弹簧的后屈曲性奠定了基础．     

轴向运动三参数黏弹性梁弱受迫振动的渐近分析

研究了轴向运动三参数黏弹性梁的弱受迫振动．建立了轴向运动三参数黏弹性梁受迫振动的控制方程．使用多尺度法渐近分析了运动梁的稳态响应,导出了解稳定性边界方程、稳态振幅的表达式以及稳态响应非零解的存在条件．依据Routh-Hurwitz定律决定了非线性稳态响应非零解的稳定性．     

轴向运动导电导磁梁的磁弹性振动方程

针对磁场环境中轴向运动导电导磁梁磁弹性耦合振动的理论建模问题进行研究.基于Timoshenko(铁木辛柯)梁理论并考虑几何非线性因素,给出轴向运动弹性梁在横向双向振动下的形变势能、动能计算式以及电磁力和机械力的虚功表达式.应用Hamilton(哈密顿)变分原理,推得磁场中轴向运动Timoshenko梁的非线性磁弹性耦合振动方程,并给出了简化形式的Euler-Bernoulli(欧拉 伯努利)梁磁弹性振动方程.根据电磁理论和相应的电磁本构关系,得到载流导电弹性梁所受电磁力的表达式,基于磁偶极子-电流环路模型给出铁磁弹性梁所受磁体力和磁体力偶的表述形式.通过算例,分析了轴向运动导电弹性梁的奇点分布及其稳定性问题.     

联合载荷作用下压杆稳定性分析

利用数学分析的基本知识建立了联合载荷作用下压杆的一般控制方程.基于两点边值问题的打靶法计算了悬臂梁在某些特定联合载荷作用下的后屈曲载荷.通过与T im oshenko用椭圆积分法所得结果进行比较,证明了方法的可行性.     

功能梯度材料杆的热后屈曲分析

对两端不可移简支陶瓷-金属功能梯度材料(FGM)杆建立了在热载荷作用下的非线性控制微分方程,采用打靶法分析了由二氧化锆和Ti-6Al-4V两种材料组成的FGM杆的热后屈曲行为．首先给出了在均匀温度场中不同梯度指标的FGM杆的热后屈曲平衡路径,并与二氧化锆和Ti-6Al-4V两种均质材料杆的相应特性进行了比较,同时讨论了不同端部转角下梯度指标对FGM杆稳定性的影响;然后分别研究了在温差一定、下表面温度变化时和在下表面温度一定、温差变化时FGM杆的热后屈曲特性,也与两种均质材料杆的后屈曲特性进行了比较．     

Mathematical modeling of the stress state of a transverse plastic layer in a round rod

We construct mathematical models of plastic deformations of a continuous round rod containing a transverse (less strong) inhomogeneous layer under an axial load. We thoroughly study the local strengthening of such layers by involving the base material of the rod in the plastic deformation process. We obtain explicit formulas for the critical stress states in the layer and the critical axial load on the rod.     

Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation

In this paper, post-buckling and nonlinear vibration analysis of geometrically imperfect beams made of functionally graded materials (FGMs) resting on nonlinear elastic foundation subjected to axial force are studied. The material properties of FGMs 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 assumptions of a small strain and moderate deformation are used. Based on Euler–Bernoulli beam theory and von-Karman geometric nonlinearity, the integral partial differential equation of motion is derived. Then this partial differential equation (PDE) problem, which has quadratic and cubic nonlinearities, is simplified into an ordinary differential equation (ODE) problem by using the Galerkin method. Finally, the governing equation is solved analytically using the variational iteration method (VIM). Some new results for the nonlinear natural frequencies and buckling load of the imperfect functionally graded (FG) beams such as the effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogeneity are presented for future references. Results show that the imperfection has a significant effect on the post-buckling and vibration response of FG beams.     

功能梯度材料Timoshenko梁的热过屈曲分析

研究了功能梯度材料Timoshenko梁在横向非均匀升温下的热过屈曲．在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了功能梯度Timoshenko梁在热-机械载荷作用下的几何非线性控制方程,将问题归结为含有7个基本未知函数的非线性常微分方程边值问题A·D2其中,假设功能梯度梁的材料性质为沿厚度方向按照幂函数连续变化的形式．然后采用打靶法数值求解所得强非线性边值问题,获得了横向非均匀升温场内两端固定Timoshenko梁的静态非线性热屈曲和热过屈曲数值解．绘出了梁的变形随温度载荷及材料梯度参数变化的特性曲线,分析和讨论了温度载荷及材料的梯度性质参数对梁变形的影响．结果表明,由于材料在横向的非均匀性,均匀升温时的梁中存在拉-弯耦合变形．     

Buckling and stability analysis of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces

A study on the buckling and dynamic stability of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces is performed in this research. The static and dynamic governing equations of the nanobeam are established with Galerkin method and under Euler–Bernoulli hypothesis. The buckling, post-buckling and nonlinear dynamic stability character of the nanobeam is presented. The quasi-elastic method, Leibnitz’s rule, Runge–Kutta method and the incremental harmonic balanced method are employed for obtaining the buckling voltage, post-buckling characteristics and the boundaries of the principal instability region of the dynamic system. Effects of the electrostatic load, van der Waals force, creep quantity, inner damping, geometric nonlinearity and other factors on the post-buckling and the principal region of instability are investigated.     

Mathematical modelling of the development of mechanical instability in fracturing systems

The dynamics of processes accompanying a loss of stability in a mechanical system are investigated. The mechanical system is in the form of an elastic rod, stretched by an axial load, with one of its lateral surfaces “glued” to a rigid wall. The “glue” is a low-strength elastic material which is subject to brittle fracture at a certain value of the load acting on it. In a fractured segment, the rod surface slides over the wall surface under the action of a dry friction force which is less than the breaking stress. The high sensitivity of the process of the development of instability to small perturbations which initiate the development of instability is established. The system considered is the simplest model of the zone of contact between lithospheric plates which generate earthquakes.     

Magnetically-Induced Buckling of a Whirling Conducting Rod with Applications to Electrodynamic Space Tethers

We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure, motivated by stability problems of electrodynamic space tethers. Both static (buckling) and dynamic (whirling) instability are considered and we also compute post-buckling configurations. The equations used are the geometrically exact Kirchhoff equations. Magnetic buckling of a welded rod is found to be described by a surprisingly degenerate bifurcation, which is unfolded when both transverse anisotropy of the rod and angular velocity are considered. By solving the linearised equations about the (quasi-) stationary solutions, we find various secondary instabilities. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano- and molecular wires.     

Static and dynamic beam forms of the loss of stability of a long orthotropic cylindrical shell under external pressure

A cylindrical shell with end sections which are closed and supported by hinges, in accordance with the concepts of the rod theory, is considered to be under the action of an omnidirectional external pressure which remains normal to the lateral surface during the deformation process. It is shown that, for such shells, the previously constructed consistent equations of the momentless theory, reduced using the Timoshenko shear model to the one-dimensional equations of the rod theory, describe three forms of loss of stability: (1) static loss of stability, which occures through a bending mode from the action of the total end axial compression force since, under the clamping conditions considered, its non-conservative part cannot perform work on deflections of the axial line; (2) also a static loss of stability but one which occurs through a purely shear mode with the conversion of a cylinder with normal sections into a cylinder with parallel sloping sections and a corresponding critical load which is independent of the length of the shell; (3) dynamic loss of stability which occurs through a bending-shear form and can only be revealed by a dynamic method using an improved shear model.     

加肋双曲冷却塔的非线性稳定分析*

本文采用有限元数值分析方法对土木工程中的大型钢筋混凝土双曲冷却塔壳的非线性稳定问题作了具体的数值计算.文中为了讨论双曲塔壳在屈曲后的特性,采用了将普通的载荷增量迭代法与改进的R.C.弧长法相结合的混合算法.为了与实际工程相符合,文中除了将钢筋混凝土当作非均匀体以外,还考虑了塔自重、底部的离散支承、及环向加肋等因素的影响.