Cosserat生长弹性杆动力学的Gauss最小拘束原理

以自然界中具有生长、变形和运动特征的细长体为背景,用经典力学中的Gauss最小拘束原理研究生长弹性杆的动力学建模问题.在为生长弹性杆动力学建模提供新方法的同时,扩大了Gauss原理的应用范围.以Cosserat弹性杆为对象,分析弹性杆生长和变形的几何规则,表明生长应变和弹性应变是非线性耦合的;本构方程给出了截面的内力与弹性变形的线性关系;利用逆并矢,将经典力学中的Gauss原理和Gauss最小拘束原理用于生长弹性杆动力学,得到等价的两种表现形式,反映了时间和弧坐标在表述上的对称性,由此导出了封闭的动力学微分方程.给出了两种形式的最小拘束函数,表明生长弹性杆的实际运动使拘束函数取驻值,且为最小值.最后讨论了生长弹性杆的约束与条件极值等问题.     

Recovery of a Heat Equation by Four Measurements at One End

We consider the reconstruction of a heat equation modeling the temperature distribution of a rod from readings at one end only. We assume that the rod, whose length is unknown, is totally buried except for a neighborhood of one of its ends, which is accessible for temperature measurements. We show that, under very mild restrictions, the heat coefficient and the length of the rod can be reconstructed from at most four measurements at one end.     

Étude des paramètres hétérogènes d'une surface active simulant une couronne d'éléments combustibles

In the method of the active surfaces a whole of nuclear fuel rods is described by a surface possessing suitable properties characterized in terms of ‘heterogeneous parameters’. This study gives a calculus allowing the theoretical determination of the heterogeneous parameters concerning a cylindrical active surface. The field in which an active surface is equivalent to fuel rods is discussed with particular references to 5 different fuel arrangements. A method for reaching the heterogeneous parameters from measurements in the sub-critical assembly CACTUS is described. This method takes into account the particular energy spectrum and locations of the neutron supply sources. The values of the heterogeneous parameters theoretically obtained are in the error limits of the experimental results.     

A reflection on the fractional order nuclear reactor dynamics

Bifurcation and stability analysis in the coupled integer-fractional order dynamic equations of a nuclear reactor is carried out in this work. To this end, the dynamics of a Pressurized Water Reactor (PWR) is taken into account as a mainstay design in the water reactor technology. The effect of fractional derivative order on the stability threshold and the onset of bifurcation phenomena is inspected therein with the temperature feedback coefficient taken as the bifurcation parameter. Overall, the transport of neutrons inside the nuclear reactor core, especially in the high neutron absorbing spaces such as the fuel or control rod, resembles that of a sub-diffusion phenomenon. As such, the pertaining equations which comprise neutron diffusion terms are more carefully treated within a fractional order framework. In this work, a formal approach is examined to help readily compute system poles and the associated stable half plane. Results confirm a sensible tendency towards instability as the value of the fractional order is decreased and a more sub-diffusive regime is established.     

On a nonlinear theory of thin rods

In this paper, a nonlinear theory for a straight rod is presented from the general theory of the three-dimensional deformable-body in the Cartesian coordinate frame. A set of nonlinear strains is presented, and the stretch on central curve exactly satisfies the deformation geometrical relations. The relations between the Euler angles and deformation are given from the curvatures and torsion curvatures of the central curves, which can easily explain the existing theories of rods and beams. Full nonlinear equations of motion for a nonlinear rod are developed via the vector form. Such a treatise is different from the traditional treatises of nonlinear rods, based on the Cosserat’s theory (e.g., Cosserat and Cosserat [1] in 1896) or the Kirchhoff assumptions (e.g., Kirchhoff [18] in 1859; Love [3] in 1944). This paper extends the ideas of Galerkin [4] in 1915. The nonlinear theory of thin rods can reduce to the existing theories for thin rods and beams, such ideas presented in this paper can be applied for development of the nonlinear theory for plates and shells as well.     

Designing of layered rod structures from viscoelastic materials at a given service life

The problem on the rational design of a layered rod structure subjected to force and temperature actions under the conditions of creep is considered. The linear hereditary theory is used for the viscoelastic materials of layers. The rods are packages of several homogeneous layers with a rectangular cross section. The variable geometrical parameters of the rods are determined from the energy equal-strength condition at a given in stant of time. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 581–594, September–October, 2007.     

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．     

Numerical Locking Problems for Hyperbolic Equations

We investigate a time-dependent model concerning the “locking phenomenon,” namely a model for three-dimensional clamped curved rods. We show that the convergence of the numerical schemes is independent of the discretization parameters and the thickness of the rod. We present corresponding numerical experiments.     

Analysis of contact of elastic rods subject to large displacements

We present a mathematical model describing the motion of two elastic rods in contact. The model allows for large displacements and is essentially based on Cosserat's modelling of rods. Using a penalty technique, we prove existence of a static solution in the case of a unique rod. The contact modelling involves unilateral constraints on the central lines of the rods. Existence is also proved for this contact problem. 2003 Elsevier Science Ltd. All rights reserved.     

Analysis of eccentric unbonded bimetal rod in ECAP based on different arrangements of soft and hard metals

In this paper, an analytical model of ECAP process with an eccentric unbonded bimetal rod is proposed by slab method. Based on the discussion, the effects of the eccentricity ratio and the diameter of inner rod as well as the geometrical parameters of the mold on the bonding process are obtained. Also this is the first time that different material arrangements are both taken into the consideration in ECAP. Two kinds of eccentric bimetal rods are researched, when outer pipe is hard and inner core rod is soft, the eccentricity ratio will be relieved. When the outer is soft and inner is hard, the eccentricity ratio will be enlarged. These changes can be more than 10% in some situations. In order to verify these conclusions, some experiments with aluminum and copper are designed and conducted. In these experiments, 3 different molds and 10 different eccentricity ratios as well as 5 different inner diameters are discussed with two kinds of eccentric bimetal rods in ECAP. The measured results are in agreement with those calculated by analytical model. Most of the errors between the measurements and calculations are less than 6%.     

Dynamical problems for geometrically exact theories of nonlinearly viscoelastic rods

Summary This paper surveys recent results and open problems for the equations of motion for geometrically exact theories of nonlinearly viscoelastic and elastic rods. These rods can deform in space by undergoing not only flexure and torsion, but also extension and shear. The paper begins with a derivation of the governing equations, which for viscoelastic rods form a quasilinear system of hyperbolic-parabolic partial differential equations of high order. It then derives the energy equation and discusses difficulties that can arise in getting useful energy estimates. The paper next treats constitutive assumptions precluding total compression. The paper then discusses the curious asymptotic problems that arise when the inertia of the rod is small relative to that of a rigid body attached to its end. The paper concludes with discussions of traveling waves and shock structure, Hopf bifurcation problems, and problems of control. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

Global Injectivity and Topological Constraints for Spatial Nonlinearly Elastic Rods

Summary. In this paper we study the local and global injectivity of spatial deformations of shearable nonlinearly elastic rods. We adopt an analytical condition introduced by Ciarlet & Nečas in nonlinear elasticity to ensure global injectivity in that case. In particular we verify the existence of an energy-minimizing equilibrium state without self-penetration which may also be restricted by a rigid obstacle. Furthermore we consider the special situation where the ends of the rod are glued together. In that case we can still impose topological restrictions such as, e.g., that the shape of the rod belongs to a given knot type. Again we show the existence of a globally injective energy minimizer which now in addition respects the topological constraints. Note that the investigation of super-coiled DNA molecules is an important application of the presented results.     

Bending, extension, and torsion of naturally twisted rods

Saint-Venant /1/ established that the spatial problem of linear elasticity theory of the deformation of straight rods with a load-free side surface allows of practically complete investigation: the extension problem is solved exactly (if the boundary layer is ignored), and the bending and torsion problems reduce to Neumann problems for the Laplace equation in the region of the rod cross-section (see /2, 3/). It is shown below that an analogous situation holds for a naturally twisted rod: the spatial problem is successfully reduced to a Neumann-type problem for a certain system of second-order elliptic equations in the cross-section. It is essential that this can be done for an arbitrary value of the rod twist. For zero twist the problem in the section reduces to the Saint-Venant problem. In the case of centrally-symmetric sections, the problem decomposes into two independent problems, on bending and on extensiontorsion. Variational principles and certain bilateral estimates of the extension and torsion stiffness are constructed for the latter, and the case of oblong sections is investigated.

The extension-torsion problem for naturally twisted rods was examined earlier in /4/. The difference from this research is discussed in Sect.4.     

Analyse asymptotique des modes de hautes fréquences dans les poutres minces

In this work, we show that a class of high frequency modes of the three-dimensional linearized elasticity system in a thin rod and their associated eigenfunctions converge, as the thickness of the rods goes to zero, and the limit model is a coupled onedimensional problem giving the classical equations for torsion and stretching vibrations.     

Biegeschwingungen verwundener,einseitig eingespannter und am andern Ende gelenkig gelagerter Stäbe

Summary The lateral and torsional vibrations of twisted rods can be treated separately if we consider as usual only first order terms. The eigen-frequencies of the lateral vibrations can be calculated exactly if we restrict ourselves to isotropic homogeneous rods with constant mass and twist per unit length and constant principal flexural rigidities. In this paper the eigen-frequencies for a rod built in at one end and supported at the other are given for 3 different cross-sections.     

由纵向振动引起的质点与圆锥形杆碰撞问题的解析解

给出细长圆锥形的截面杆受到质点纵向碰撞时的精确解析解．提出了一种新方法用于分析质点-圆锥形杆碰撞,使用了叠加法给出杆的响应．其结果可验证数值解和其他解析解．所提出方法的优点之一是响应解的解析形式简洁．结论是质量比和一些描述杆几何形状的变量,如倾斜度、杆长和半径在撞击分析中具有重要作用．     

Experimental reconsideration of spatio‐temporal dynamics observed in fluid‐elastic oscillator arrays from complex system viewpoint: From vibrating pipes in heat exchangers to waving plants in agricultural fields

Transition from local complexity to global spatio‐temporal dynamics in a two‐dimensional array of fluid‐elastic oscillators is examined experimentally with an apparatus comprising 90‐1000 cantilevered rods in a wind tunnel as the Reynolds number (based on rod diameter) is increased from 200 to 900. A cluster‐pattern entropy measure is introduced as a quantitative measure of local complexity. As the intensity of interaction among neighboring elements (in this case, frequency of collisions among rods) increases, a set of the elements (in this case, a rod‐array) achieves globally better‐organized behavior. On the basis of accelerometer data, the rod impact rate versus flow velocity shows a power‐law scaling relation. Video images reveal that, initially, each rod moves individually; then clusters consisting of several rods emerge. Finally, global wave‐like motion occurs at higher flow velocities. Each wave‐like motion has its specific frequency and spatial wavelength, which vary according to wind velocity. © 2007 Wiley Periodicals, Inc. Complexity 12: 36–47, 2007     

载电流夹紧杆的非线性稳定性分析*

本文研究了载电流夹紧杆在磁场作用下的非线性稳定性,其磁场由两根无限长相互平行的刚性直导线产生.杆的自然状态在刚性导线所在的平面内,并且与两刚性导线等距.首先,在空间变形的假定下,给出了问题的数学描述,讨论了线性化问题和临界电流.其次,证明了杆的过屈曲状态总是平面的.最后,数值计算了分支解的全局响应,得到了杆过屈曲状态的挠度、内力和弯矩的分布.结果表明,载电流杆既可发生超临界屈曲,又可发生次临屈界曲,其性态依赖于杆与导线间的距离;同时,在超临界的过屈曲状态上还存在极限点型的失稳,这与通常的压杆失稳有着本质的区别.