QUALITATIVE ANALYSIS OF DYNAMICAL BEHAVIOR FOR AN IMPERFECT INCOMPRESSIBLE NEO-HOOKEAN SPHERICAL SHELL

IntroductionIn applications, it is commonly considered that the phenomena of cavity formation,growth and run-through of adjacent cavities occur in materials as precursors to failure. Thesephenomena are mainly due to instability of materials. On the other …     

Dynamical formation of cavity in a composed hyper-elastic sphere

The dynamical formation of cavity in a hyper-elastic sphere composed of two materials with the incompressible strain energy function, subjected to a suddenly applied uniform radial tensile boundary dead-load, was studied following the theory of finite deformation dynamics. Besides a trivial solution corresponding to the homogeneous static state, a cavity forms at the center of the sphere when the tensile load is larger than its critical value. An exact differential relation between the cavity radius and the tensile land was obtained. It is proved that the evolution of cavity radius with time displays nonlinear periodic oscillations. The phase diagram for oscillation, the maximum amplitude, the approximate period and the critical load were all discussed.     

超弹性材料中空穴的动态生成

本文在有限变形动力学的框架下研究了一种不可压超弹性材料圆柱体在表面突加均布拉伸载荷作用下空穴的动态生成问题,除一个相应于均匀变形状态的平凡解外,当外加载荷超过其临界值时,柱体内部还有空穴的突然生成,得到了空穴半径和表面载荷之间的一个精确的微分关系,证明了空穴随时间的演化是非线性的周期性振动,给出了空穴振动的相图、最大振幅、临界载荷及近似的周期。     

Dynamical formation of cavity in transversely isotropic hyper-elastic spheres

The cavity formation in a radial transversely isotropic hyper-elastic sphere of an incompressible Ogden material, subjected to a suddenly applied uniform radial tensile boundary dead-load, is studied fllowing the theory of finite deformation dynamics. A cavity forms at the center of the sphere when the tensile load is greater than its critical value. It is proved that the evolution of the cavity radius with time follows that of nonlinear periodic oscillations. The project supported by the National Natural Science Foundation of China (10272069) and Shanghai Key Subject Program     

Cavity formation and singular periodic oscillations in isotropic incompressible hyperelastic materials

In this paper, a dynamical problem is considered for an incompressible hyperelastic solid sphere composed of the classical isotropic neo-Hookean material, where the sphere is subjected to a class of periodic step radial tensile loads on its surface. A second-order non-linear ordinary differential equation that describes cavity formation and motion is proposed. The qualitative properties of the solutions of the equation are examined. Correspondingly, under a prescribed constant dead-load, it is proved that a cavity forms in the sphere as the dead-load exceeds a certain critical value and the motion of the formed cavity presents a class of singular periodic oscillations. Under periodic step loads, the existence conditions for periodic oscillation of the formed cavity are determined by using the phase diagrams of the motion equation of cavity. In each section, numerical examples are also carried out.     

DYNAMIC CHARACTERISTICS IN INCOMPRESSIBLE HYPERELASTIC CYLINDRICAL MEMBRANES

In this paper, the dynamic characteristics are examined for a cylindrical membrane composed of a transversely isotropic incompressible hyperelastic material under an applied uniform radial constant pressure at its inner surface. A second-order nonlinear ordinary differential equation that approximately describes the radial oscillation of the inner surface of the membrane with respect to time is obtained. Some interesting conclusions are proposed for different materials, such as the neo-Hookean material, the Mooney-Rivlin material and the Rivlin-Saunders material. Firstly, the bifurcation conditions depending on the material parameters and the pressure loads are determined. Secondly, the conditions of periodic motion are presented in detail for membranes composed of different materials. Meanwhile, numerical simulations are also provided.     

电活性聚合物圆柱壳静态与动态电压下的响应及稳定性

摘要：在电活性聚合物圆柱壳内外表面施加电压,圆柱壳会变薄并且伸长,因此相同的电压会在圆柱壳内产生更大的电场。这个正反馈可能使圆柱壳厚度不断变薄,最终导致其失稳破坏。本文研究了电活性聚合物圆柱壳在静态和周期电压作用下的响应及稳定性问题。采用neo-Hookean材料模型得到描述圆柱壳表面运动的非线性常微分方程。给出了圆柱壳在不同厚度和边界条件下外加电压随圆柱壳变形的变化曲线,结果表明存在一个临界电压,当外加电压大于这一临界值时,圆柱壳将被破坏。同时,也讨论了厚度和边界条件对临界电压的影响。圆柱壳在正弦周期电压作用下,其运动随时间的变化是周期性的或拟周期性的非线性振动。给出了圆柱壳振动固有频率的计算结果,采用打靶法得到圆柱壳振动的周期解,并且用数值法研究了周期解的稳定性。采用数值仿真得到圆柱壳振动振幅随外加动态电压激励频率的变化曲线,结果表明圆柱壳会发生多频共振,共振时圆柱壳振幅发生跳跃,导致圆柱壳失稳破坏。最后给出共振点临近点的振动曲线和相图,并对其振动特性进行讨论。     

一种确定非线性裂纹转子解的形式的新方法

将小波变换与Ｐｏｉｎｃａｒｅ映射相结合，即用Ｐｏｉｎｃａｒｅ映射确定周期解，用谐波小波变换区分拟周期响应和混沌运动，提出了一种分析非线性裂纹转子系统解的形式随参数变化的新方法．结果表明这种方法是非常有效的，它比以前所用的计算Ｌｉａｐｕｎｏｖ指数的方法节约了计算时间，并且较易实施．     

横观各向同性超弹性球壳的有限振动

应用有限变形动力学理论研究了一种横观各向同性不可压超弹性材料球壳在表面突加均布拉伸载荷作用下的有限振动问题．给出了球壳振动的振幅和外加载荷之间的关系,得到了,球壳振动的相同和近似的周期,讨论了球壳振动的振幅、相图及振动的周期和材料各向异性程度的关系．     

Asymptotics of Homoclinic Bifurcation in a Three-Dimensional System

An analytical approach to predicting a critical parameter valueof homoclinic bifurcation in a three-dimensional system is reported. Themultiple scales method is first performed to construct a higher-orderapproximation of the periodic solution. A criterion based on a collisionbetween the periodic orbit and the fixed point involved in thebifurcation is applied. This criterion developed initially to predicthomoclinic bifurcations in planar autonomous systems, is adapted here toderive a critical value of the homoclinic bifurcation in a specificthree-dimensional system. To support our analytical predictions and todescribe the dynamical behaviour of the system, a complete numericalstudy is provided.     

NONLINEAR DYNAMICS OF AXIALLY ACCELERATING VISCOELASTIC BEAMS BASED ON DIFFERENTIAL QUADRATURE

This paper investigates nonlinear dynamical behaviors in transverse motion of an axially accelerating viscoelastic beam via the differential quadrature method. The governing equation, a nonlinear partial-differential equation, is derived from the viscoelastic constitution relation using the material derivative. The differential quadrature scheme is developed to solve numerically the governing equation. Based on the numerical solutions, the nonlinear dynamical behaviors are identified by use of the Poincare map and the phase portrait. The bifurcation diagrams are presented in the case that the mean axial speed and the amplitude of the speed fluctuation are respectively varied while other parameters are fixed. The Lyapunov exponent and the initial value sensitivity of the different points of the beam, calculated from the time series based on the numerical solutions, are used to indicate periodic motions or chaotic motions occurring in the transverse motion of the axially accelerating viscoelastic beam.     

Dynamic snapping of a suddenly loaded elastica with fixed end slopes

In this paper we study the dynamic snapping of an elastica with fixed end slopes. Both ends of the elastica are clamped, with one fixed in space and the other allowed to slide along a linear track. An edge thrust is applied suddenly on the sliding clamp, causing the elastica to undergo a snap-through jump. This setup can be used as a bistable device. The interest of this paper is to determine the critical value of the suddenly applied edge thrust causing snap-through, termed dynamic snapping load. With use of deformation potentials, the equations of motion of the elastica-slider assembly are rearranged, taking into account the effects of both the slider mass and damping. Finite difference method is adopted to discretize the resulted equations. In order to ensure the convergence of the numerical scheme, a linearization approach has to be adopted when the slider mass is non-zero. It is observed that for the setup studied in this paper, the dynamic snapping load is about 90% of its static counterpart. The viscous damping associated with the end slider in general has a favorable tendency of raising the dynamic snapping load. Somewhat unexpectedly, although the slider mass affects the transient response of the elastica-slider assembly significantly, it has no effect on the dynamic snapping load.     

简支饱和多孔弹性梁的非线性弯曲

基于饱和多孔介质理论和弹性梁的大挠度弯曲假设,在多孔弹性梁轴线不可伸长,孔隙流体仅沿轴向方向扩散的限制下,建立了微观不可压饱和多孔弹性梁大挠度拟静态响应的一维非线性数学模型.在此基础上,利用Galerkin截断法,分析了两端可渗透的简支多孔弹性梁在突加横向均布载荷作用下的非线性弯曲,给出了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶随时间的响应曲线.数值结果表明:当载荷较小时,大挠度非线性与小挠度线性理论的结果相差很小,而当载荷较大时,非线性大挠度理论的结果小于相应线性小挠度理论的结果,并且这种差异随着载荷的增大而增大.同时,在载荷突加于梁上时,多孔弹性梁骨架起初不变形,孔隙流体压力等效力偶由零突增为非零,其值与外载荷保持平衡.随着时间的增加,梁的挠度增加,等效力偶逐渐减小为零,最终多孔梁骨架承担全部的外载荷.     

Cavity Formation And Its Vibration for A Class of Generalized Incompressible Hyper-Elastic Materials

The problem of radial symmetric motion for a solid sphere composed of a class of generalized incompressible neo-Hookean materials, subjected to a suddenly applied surface tensile dead load, is examined.The analytic solutions for this problem and t…     

NONLINEAR DYNAMICAL BIFURCATION AND CHAOTIC MOTION OF SHALLOW CONICAL LATTICE SHELL

The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)     

On relative equilibria and integrable dynamics of point vortices in periodic domains

The motion of two point vortices defines an integrable Hamiltonian dynamical system in either singly or doubly periodic domains. The motion of three point vortices in these domains is also integrable when the net circulation is zero. The relative vortex motion in both domains can be reduced to advection of a passive particle by fixed vortices in an equivalent Hamiltonian system. A survey of the solutions for vortex motion in these systems is discussed. Some initial conditions lead to relative equilibria, or vortex configurations that move without change of shape or size. These configurations can be determined as stagnation points in the reduced problem or through explicit solution of the governing equations. These periodic point-vortex systems present a rich collection of interesting solutions despite the few degrees of freedom, and several questions on this subject remain open.     

对称线布载荷作用下圆底扁球壳的大挠度问题

本文研究对称线布载荷作用下圆底扁球壳的轴对称非线性弯曲和稳定性,讨论了当几何参数固定而载荷位置发生变化吋壳体的屈曲行为,以及当载荷作用位置固定而几何参数发生变化时壳体的屈曲行为,分析了屈曲模式对临界载荷的影响,并就ν=0.3的情形给出了数值结果。     

Stability of fluid-conveying periodic shells on an elastic foundation with external loads

This paper presents the beam-mode stability of a fluid-conveying periodic shell on an elastic foundation subjected to external loading. A transfer matrix (TM) method was developed to investigate the characteristics of steady-state waves in the system and the dynamic response of the periodic shell system. When subjected to external perturbations, including either a moving load or a stationary one, the shell may be subjected to instability for flow velocities exceeding a certain critical velocity. The system can also become unstable when a travelling load exceeds a certain critical value. The coupled effects of the speed of a moving load and the flow velocity of a fluid on the stability of the shell system were also investigated. A periodic structure was designed for such a shell system to enhance its dynamic stability. The periodic shell system produces innumerable velocity band gaps (VBGs), which could raise the critical velocity and extend the stable range for both the moving load and the flowing fluid. Finally, the formation mechanism of the VBGs was studied, as well as the effects of the thickness, length of the shell cells, Young׳s modulus and stiffness of the elastic foundation on modulating the VBGs.     

Critical speeds and sizes of liquid globules

Summary A liquid globule that has a translatory motion relative to air, is split up if the speed exceeds a critical value or, inversely, if at a given speed the size of the globule is greater than a critical value. The criterion for determining whether or not the antagonism between air forces and surface tension will lead to splitting-up, is the relevant value of Weber's number.It is shown that the critical value of Weber's number for falling drops is different from the one for drops suddenly exposed to a constant air flow, which latter case occurs, for instance, during disintegration processes of liquid jets. Moreover, the critical value depends on the effect of viscosity on the deformation of the globule. The cases of a slight and of a great viscosity effect have been considered. The theory has been applied to experimental results on falling drops. The consequences for disintegration processes have been discussed.