A STUDY OF THE CATASTROPHE AND THE CAVITATION FOR A SPHERICAL CAVITY IN HOOKE’S MATERIAL WITH 1/2 POISSON’S RATIO

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓ,ｔｈｅｒｅｓｅａｒｃｈｅｓｏｎｃａｖｉｔａｔｉｏｎａｎｄｃａｔａｓｔｒｏｐｈｅｏｆａｃａｖｉｔｙｈａｖｅｓｕｐｐｌｉｅｄａｎｅｗｍｅｔｈｏｄｆｏｒｉｎｖｅｓｔｉｇａｔｉｎｇｔｈｅｍｅｃｈａｎｉｃｓｏ...     

Bifurcation and stability of homogeneous deformations of an elastic body under dead load tractions with Z 2 symmetry

An analysis is given of bifurcation and stability of homogeneous deformations of a homogeneous, isotropic, incompressible elastic body subject to three perpendicular sets of dead-load surface tractions of which two have equal magnitude. A minimization problem is formulated within the framework of non-linear elasticity, which leads to a bifurcation problem with Z ₂ symmetry. Various bifurcation diagrams are deduced by using singularity theory, and stabilities of solution branches are examined.     

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…     

Dispersed Damages in Stability Problems for Doubly Curved Shells of Revolution

An approach is expounded to the study of the bifurcation stability of doubly curved shells of revolution. The microdamage of an isotropic material is considered as empty spherical pores randomly dispersed over the volume, their concentration increasing with load. A damaged inhomogeneous material is modeled by a continuous physically nonlinear medium whose nonlinear deformation depends on how the material fails and the microstrength is distributed. A bifurcation stability problem is formulated based on the concept of continuous loading within the framework of the Kirchhoff–Love hypotheses. As an example, a solution is given to the stability problem on shells of positive Gaussian curvature under external uniform pressure     

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。     

理想直杆中轴向应力波的传播和反射导致的分叉问题

对阶跃载荷作用下弹性理想完善直杆中应力波的传播及其反射引起民的分叉问题进行了理论分析和数值计算，讨论了此类动力屈曲问题中横向惯性效应的影响；给出了一种进行后分叉分析的近似方法，得到了一些结论，在各类结构由于应力波的传播及反射引起的分叉问题具有指导意义。     

Numerical Analysis of Axisymmetric Buckling of a Conical Shell under Radial Compression

The nonlinear boundary-value problem of the axisymmetric buckling of a simply supported conical shell (dome) under a radial compressive load applied to the supported edge is formulated for a system of six first-order ordinary differential equations for independent fields of finite displacements and rotations. Multivalued solutions are obtained using the shooting method with specified accuracy. For various values of the loading parameter, bifurcation of the solutions of the problem is studied and a parametric branching diagram is constructed. The buckling modes are obtained for three branches of the solution. Curves of the buckling modes corresponding to three isolated branches of the solution are given.     

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.     

Imperfect bifurcation of systems with slowly varying parameters and application to Duffing's equation

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On the non-linear Beck''s problem with external damping

The non-linear Beck's problem with external damping is analyzed for the first time. It is observed that the problem exhibits a Hopf bifurcation, and the character of this bifurcation is ascertained. The non-linear stability is examined, and previously unanswered questions are addressed. Comments on the discontinuous nature of the undamped limit are given.     

Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel

The evolution of steady-state viscous incompressible fluid flows in a plane divergent channel is investigated. For the classical formulation of the Jeffery-Hamel problem a complete solution is given as a function of the determining parameters. For a fixed angle of divergence the behavior of the main unimodal flow is determined as a function of the Reynolds number. Critical values at which the flow pattern bifurcates and the steady-state unimodal flow ceases to exist are found. The mechanism of bifurcation is established and its diagram is constructed. This mechanism and the diagram were not previously known in the scientific literature in connection with the investigation of the Jeffery-Hamel problem. The critical Reynolds number at which bifurcation occurs is given as a function of the channel divergence angle. The results may be of interest for hydromechanical, technological, and geophysical applications.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 25–36.Original Russian Text Copyright © 2005 by Akulenko and Kumakshev.     

剪切带状分叉

本文从扩散型解法出发，结合实验，揭示了剪切带状分叉的主要特征；进而，论述材料分叉与结构屈曲（分叉）的不同之处。文中结果表明：在诱发剪切带过程中，材料内空洞的激化速率是控制因素而不在于空洞体积比的自身数量。     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Global Bifurcation in Nonlinear Elasticity with an Application to Barrelling States of Cylindrical Columns

We present rigorous local and global bifurcation results for a concrete example from 3-dimensional nonlinear elastostatics - the problem of barrelling of compressed cylindrical columns. We use standard tools of bifurcation theory for the local analysis, already producing results that are rare in our field. For the global part we employ the generalized degree designed by Healey and Simpson to overcome the specific difficulties of 3-dimensional nonlinear elasticity. Ours are the first global bifurcation results for a problem from 3-dimensional nonlinear elastostatics not governed by ordinary differential equations. Moreover, our approach to the barrelling problem provides a paradigm for the solution of a large class of problems in nonlinear elastostatics concerning bifurcation from a homogeneously deformed state.     

CAVITATED BIFURCATION FOR INCOMPRESSIBLE HYPERELASTIC MATERIAL

ＩｎｔｒｏｄｕｃｔｉｏｎＣａｖｉｔａｔｉｏｎｐｈｅｎｏｍｅｎｏｎ ,ｔｈｅｓｕｄｄｅｎｆｏｒｍａｔｉｏｎａｎｄｇｒｏｗｔｈｏｆｖｏｉｄｓｉｎｓｏｌｉｄｍａｔｅｒｉａｌｓ,ｈａｓｌｏｎｇａｔｔｒａｃｔｅｄｍｕｃｈａｔｔｅｎｔｉｏｎｂｅｃａｕｓｅｏｆｔｈｅｆｕｎｄａｍｅｎｔａｌｒｏｌｅｉｔｐｌａｙｓｏｎｔｈｅｌｏｃａｌｆａｉｌｕｒｅａｎｄｆｒａｃｔｕｒｅｏｆｍａｔｅｒｉａｌｓ.Ｔｈｅａｐｐｅａｒａｎｃｅｏｆｖｏｉｄｉｎｔｈｅｉｎｔｅｒｉｏｒｏｆｒｕｂｂｅｒｏｒｒｕｂｂｅｒ＿ｌｉｋｅｍａｔｅｒｉａｌｓｗａ…     

黏弹性圆柱形壳动力学高余维分岔、普适开折问题

讨论两端受到谐波激励的黏弹性圆柱形壳的非线性动力学行为,利用奇异性理论,研究了分岔方程的普适开折问题,严格证明了它是一个高余维分岔问题。余维数为5（含有一个模参数）,给出了它的所有可能的普适开折形式。在分岔参数满足某些条件时得到该分岔问题的转迁集及分岔图,展示了一些新的动力学行为,改进和完善了奇异性分析方法。     

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…     

薄壁结构屈曲和初始后屈曲的局部分析及其精度

为了解决大型复杂薄壁结构稳定性有限元分析中方程阶数过高的问题,本文建议用结构局部代替整体进行计算,证明局部分析方法可以得到整体结构线性分支点的上限和下限,以及整体结构渐近后屈曲行为的近似解,给出了算例。     

Second-order bifurcation in elastic-plastic solidS

Second-order rate constitutive equations are formulated for a time-independent elastic-plastic material, obeying the normality flow rule with a smooth yield surface. Under specified regularity restrictions imposed on the involved fields, the regular second-order rate boundary value problem with quasistatic accelerations as unknowns is posed. It is shown that every solution of this generally non-linear rate problem is governed by a variational principle and that the corresponding functional reaches a strict absolute minimum, provided the solution satisfies a sufficient uniqueness condition. With the same incrementally linear comparison solid, Hill's exclusion condition rules out not only a first- but also a second-order bifurcation. The criticality of the exclusion condition is discussed and conditions are indicated under which a second-order bifurcation becomes possible, while the first-order rate problem is still uniquely solvable.