组合超弹性材料中的空穴生成与增长

本文研究了一种组合不可压超弹性材料圆柱体中空穴的生成与增长问题，得到了这种材料受表面均布拉伸死荷载和轴向拉压共同作用下空穴生成问题的解析解，得到了不同组合情况下圆柱体中空穴生成时的临界载荷及分叉曲线，发现组合材料可以发生右分叉，也可以发生左分叉；给出了空穴生成后的应力分布，并讨论了所存在的应力间断和应力集中问题；通过能量比较分析了解的稳定性，讨论了发生右分叉或左分叉的条件，并分析了材料中预存微孔的增长情况。     

Cavitated bifurcation for composed compressible hyper-elastic materials

The cavitated bifurcation problem in a solid sphere composed of two compressible hyper-elastic materials is examined. The bifurcation solution for the composed sphere under a uniform radial tensile boundary dead-load is obtained. The bifurcation curves and the stress contributions subsequent to the cavitation are given. The right and left bifurcation as well as the catastrophe and concentration of stresses are analyzed. The stability of solutions is discussed through an energy comparison. Project supported by the National Natural Science Foundation of China (No. 19802012).     

QUALITATIVE ANALYSIS OF SPHERICAL CAVITY NUCLEATION AND GROWTH FOR INCOMPRESSIBLE GENERALIZED VALANIS-LANDEL HYPERELASTIC MATERIALS

I. INTRODUCTION In practice, cavity formulation in materials is recognized as precursors to failure. Thus void nucleationand growth in solid materials have a great in?uence on failure mechanism. Gent and Lindley[1] have observed experimentally the phe…     

中厚矩形板的非线性动力屈曲分叉

建立考虑横向剪切与转动惯量影响的矩形板的动力控制方程，应用Ｇａｌｅｒｋｉｎ方法将其化为Ｍａｔｈｉｅｕ方程，然后根据Ｌｙａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ方法得到了系统在参数激励下的１／２亚谐分叉特性，并给出了四边简支与四边固支弹性薄板的非线性动力屈曲分叉条件。     

Qualitative study of cavitated bifurcation for a class of incompressible generalized neo-Hookean spheres

IntroductionIn 1 958,GentandLindleyobservedthephenomenonofsuddenvoidnucleationinsolidsexperimentallyintensioningahomogenousclose_grainedvulcanizedrubbercylinderforthefirsttime.ButthemathematicalmodelonvoidnucleationandgrowthhasnotbeendescribedasabifurcationproblembasedonthetheoryofnonlinearelasticmechanicsbyBall[1]until1 982 .Inrecentyears,manyinvestigationshavebeenmadeonthisaspect.Theproblemofcavitatedbifurcationforincompressibleisotropichyperelasticmaterialswithpower_lawtypehasbeeninvestig…     

Cavity formation at the center of a sphere composed of two compressible hyper-elastic materials

IntroductionHorgan[1] reviewedthecavitatedbifurcationproblemforhyper_elasticmaterials,includinginhomogeneousandanisotropicmaterialsaswellashomogeneousandisotropicmaterials .Forincompressiblematerials,HorganandPence[2 ,3 ] examinedtheeffectofmaterialinhomogeneityontheformationandgrowthofvoidandobtainedananalyticsolutionofthecavitatedbifurcationproblemforasolidspherecomposedoftwoneo_Hookeanmaterials.Thebifurcationmayoccurnotonlytotherightbutalsototheleftforthecomposedsphere .Thestabilitiesofth…     

非线性转子-机匣系统的分岔行为研究

建立了一类非线性转子-机匣系统的碰摩模型.应用数值分析的方法对其进行研究,得到了不同参数变化下系统响应随转速变化的分岔图,分析了系统参数变化对分岔过程的影响,并作出了在相应参数状态和特定转速下的Poincare截面图,揭示系统参数变化对非线性碰摩转子-机匣系统分岔特性的影响.     

Non-linear convection due to compositional and thermal buoyancy in Earth's outer core

The linear stability of convection due to compositional and thermal buoyancy in Earth's outer core has been investigated. We have obtained the values of Takens-Bogdanov bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters. We have derived a non-linear two-dimensional Landau-Ginzburg equation with real coefficients near the onset of stationary convection at a supercritical pitchfork bifurcation and two non-linear one-dimensional coupled Landau-Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation. We have studied Nusselt number contribution from a Landau-Ginzburg equation at the onset of stationary convection. We have discussed the stability regions of standing and travelling waves. We have also discussed the occurrence of secondary instabilities such as Eckhaus, zigzag and Benjamin-Feir instabilities. We have also derived the non-linear amplitude equation near the Takens-Bogdanov bifurcation point.     

一类非光滑映射的边界碰撞分岔

对于一类分段映射讨论了非线性幂次z导致的不同非光滑性, 推导了周期n 解的边界碰撞分岔及光滑flip和fold 分岔条件. 通过数值仿真验证了这些分岔条件的正确性, 发现存在稳定周期窗的加周期分岔序列是非光滑映射的一个普遍现象, 根本原因在于边界碰撞分岔和光滑flip 或fold 分岔相互作用. 当z取值不同分岔序列有很大的不同, 而参数γ 对于分岔序列的结构影响不大, 因此令参数γ=0 可简化映射的参数分析.     

A Finite Strain Analysis of Cavity Formation at a Rigid Inhomogeneity

The formation of a cavity by inclusion-matrix interfacial separation is examined by analyzing the response of a plane rigid inclusion embedded in an unbounded incompressible matrix subject to remote equibiaxial dead load traction. A vanishingly thin interfacial cohesive zone, characterized by normal and tangential interface force-separation constitutive relations, is assumed to govern separation behavior. Rotationally symmetric cavity shapes (circles) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation and the remote loading. Nonsymmetrical cavity formation, under rotationally symmetric conditions of geometry and loading, is treated within the theory of infinitesimal strain superimposed on a given finite strain state. Rotationally symmetric and nonsymmetric bifurcations are analyzed and detailed results, for the Mooney–Rivlin strain energy density and for an exponential interface force-separation law, are presented. For the nonsymmetric rigid body displacement mode, a simple formula for the critical load is presented. The effect on bifurcation behavior of interfacial shear stiffness and other interface parameters is treated as well. In particular we demonstrate that (i) for the smooth interface nonsymmetric bifurcation always precedes rotationally symmetric bifurcation, (ii) unlike rotationally symmetric bifurcation, there is no threshold value of interface parameter for which nonsymmetric bifurcation will not occur and (iii) interfacial shear may significantly delay the onset of nonsymmetric bifurcation. Also discussed is the range of validity of a nonlinear infinitesimal strain theory previously presented by the author (Levy [1]). This revised version was published online in July 2006 with corrections to the Cover Date.     

On the sensitivity of non-generic bifurcation of non-linear normal modes

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r=1,3,5,… . Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears, and generic bifurcation appears in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.     

The complicated bifurcation of an archetypal self-excited SD oscillator with dry friction

In this paper, we investigate the local and global bifurcation behaviors of an archetypal self-excited smooth and discontinuous oscillator driven by moving belt friction. The belt friction is described in the sense of Stribeck characteristic to formulate the mathematical model of the proposed system. For such a friction characteristic, the complicated bifurcation behaviors of the system are discussed. The bifurcation of the multiple sliding segments for this self-excited system is exhibited by analytically exploring the collision of tangent points. The Hopf bifurcation of this self-excited system with viscous damping is analyzed by making the examination of the eigenvalues at the steady state and discussing the stability of the limit cycles. The bifurcation diagrams and the corresponding phase portraits are depicted to demonstrate the complicated dynamical behaviors of double tangency bifurcation, the bifurcation of sliding homoclinic orbit to a saddle, subcritical Hopf bifurcation and grazing bifurcation for this system.     

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh–Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov–Takens bifurcation which includes a saddle-node bifurcation, an Andronov–Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.     

Non-normality and bifurcation in plane strain tension and compression

The bifurcations of a rectangular block subject to plane strain tension or compression are investigated. The block material is taken to be incompressible and is characterized by an incrementally linear constitutive law for which “normality” does not necessarily hold. The consequences of non-normality regarding bifurcation are given primary emphasis here. The characteristic regimes of the governing equations (elliptic, parabolic and hyperbolic) are detennined. In each of these regimes both symmetric and antisymmetric diffuse bifurcation modes are available. Additionally, in the hyperbolic and parabolic regimes, bifurcation into a localized shear band mode is also possible. Particular attention is given to the limiting cases of long wavelength and soon wavelength diffuse bifurcation modes. The range of parameter values is identified for which bifurcation into some localized mode may precede bifurcation into a long wavelength diffuse mode. Some difficulties associated with employing a linear incremental solid in a bifurcation analysis, when primary interest is in the bifurcation of an underlying elastic-plastic solid, are also discussed.     

Instabilities in bimetallic layers

The analysis of the thickness fluctuations in the extrusion process of bimetallic tubes has motivated the present work. For an orthotropic, incrementally-linear solid, the possibility of non-uniform solutions near the bimaterial interface is investigated by assuming an initial perturbation along the interface. The bifurcation equation for the problem is established and firstly solved numerically to obtain a critical strain in terms of a corresponding wavenumber. Then the bifurcation equation is solved for strains above the critical strain in order to analyse the growth of the perturbation, by means of an instability parameter introduced in the analysis. A set of values for several selected parameters of the process and three constitutive equations are considered in the computations. Their influence in the stability of the process is discussed in detail.     

Stability and bifurcation behaviors analysis in a nonlinear harmful algal dynamical model

A food chain made up of two typical algae and a zooplankton was considered. Based on ecological eutrophication, interaction of the algal and the prey of the zooplankton, a nutrient nonlinear dynamic system was constructed. Using the methods of the modern nonlinear dynamics, the bifurcation behaviors and stability of the model equations by changing the control parameter r were discussed. The value of r for bifurcation point was calculated, and the stability of the limit cycle was also discussed. The result shows that through quasi-periodicity bifurcation the system is lost in chaos.     

Equivariant multiparameter bifurcation via singularity theory

The theoretical machinery from singularity theory introduced by Golubitsky, Stewart, and Schaeffer, to study equivariant bifurcation problems, is completed and expanded while generalized to the multiple parameter context. In this setting the finite determinacy theorems or normal forms, the stability of equivariant bifurcation problems, and the structural stability of the universal unfolding are discussed.     

Singularity analysis of Duffing-van der Pol system with two bifurcation parameters under multi-frequency excitations

Bifurcation properties of a Duffing-van der Pol system with two parameters under multi-frequency excitations are studied. Three cases are discussed： （1） λ 1 is considered as bifurcation parameter, （2） λ 2 is considered as bifurcation parameter, and （3） λ 1 and λ 2 are both considered as bifurcation parameters. According to the definition of transition sets, the whole parametric space is divided into several different persistent regions by the transition sets for different cases. The bifurcation diagrams in different persistent regions are obtained, which provides a theoretical basis for optimal design of the system.     

多维磁浮柔性转子控制系统分岔与控制器设计

讨论了多维悬浮柔性转子控制系统局部及全局分岔问题,首先建立了该复杂系统动力学模型,应用中心流形和求规范形综合方法,得到此系统非半简双零特征值问题的规范形及其普适开折,并进一步讨论了此控制系统的分岔 行为（余维二分岔）及稳定性;给出了为实现稳定控制,控制器参数、转子系统结构参数的相互关系及稳定控制域,即给出分岔 参数条件、分岔曲线、转迁集,最后,给出此柔性转子控制系统的数值仿真结果。     

The bifurcation analysis on the circular functionally graded plate with combination resonances

The bifurcation and chaos of a clamped circular functionally graded plate is investigated. Considered the geometrically nonlinear relations and the temperature-dependent properties of the materials, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic excitation and thermal load are derived. The Duffing nonlinear forced vibration equation is deduced by using Galerkin method and a multiscale method is used to obtain the bifurcation equation. According to singularity theory, the universal unfolding problem of the bifurcation equation is studied and the bifurcation diagrams are plotted under some conditions for unfolding parameters. Numerical simulation of the dynamic bifurcations of the FGM plate is carried out. The influence of the period doubling bifurcation and chaotic motion with the change of an external excitation are discussed.