Stability of forced torsional vibrations of an equipped rod

A model of an equipped elastic rod is considered. In the average sense, this model shows the properties of the one-dimensional Cosserat continuum during longitudinal and torsional motions. Natural and forced torsional vibrations are studied in the case of flow loading. Several conditions for vibration stability and for the end of vibrations are formulated. The following distinctive features of motion are found: each vibration mode has two different shapes and two different frequencies and the onset of the divergence regime is observed when the external loads become more intensive.     

关于弹性梁的数学模型

叙述和比较一维弹性体的两种不同建模方法, 即弹性梁的传统建模方法和基于 Kirchhoff-Cosserat模型的建模方法. 应用精确Cosserat模型分析梁的三维运动. 考虑中 心线的拉伸压缩变形、截面的剪切变形、截面转动的惯性和端部载荷影响等因素, 建立精确 的弹性梁动力学方程. 讨论梁的静态和动态平衡稳定性. Kirchhoff杆、铁摩辛柯 梁和欧拉--伯努利梁等为Cosserat模型在各种简化条件下的特例.     

A Bridging Scale Method for Multi-scale Analysis of Granular Materials

本文提出了耦合细尺度上基于离散颗粒集合体模型的离散单元法（DEM）和粗尺度上基于Cosserat连续体模型的有限元法（FEM）的连接尺度方法（BSM）以研究颗粒材料的力学行为。采用Cosserat连续体模型和FEM模拟的粗尺度域覆盖全域,而采用离散颗粒集合体模型的DEM模拟的细尺度域仅限于需特别关注材料微结构演变和非连续变形行为的局部区域。对这两个区域间的界面提出了适当的界面条件及其实施方案。通过采用适当的连接尺度投影算子,空间离散的粗、细尺度耦合系统多尺度运动方程具有解耦和允许分别求解、因而也允许分别采用不同时间步长对粗、细尺度计算的特点,可极大地提高BSM的计算效率。文中二维地基数值算例结果说明了所陈述方法的可应用性,以及相对基于Cosserat连续体模型的FEM和基于离散颗粒集合体模型的DEM的优越性。     

颗粒材料多尺度分析的连接尺度方法

本文提出了耦合细尺度上基于离散颗粒集合体模型的离散单元法（DEM）和粗尺度上基于Cosserat连续体模型的有限元法（FEM）的连接尺度方法（BSM）以研究颗粒材料的力学行为。采用Cosserat连续体模型和FEM模拟的粗尺度域覆盖全域,而采用离散颗粒集合体模型的DEM模拟的细尺度域仅限于需特别关注材料微结构演变和非连续变形行为的局部区域。对这两个区域间的界面提出了适当的界面条件及其实施方案。通过采用适当的连接尺度投影算子,空间离散的粗、细尺度耦合系统多尺度运动方程具有解耦和允许分别求解、因而也允许分别采用不同时间步长对粗、细尺度计算的特点,可极大地提高BSM的计算效率。文中二维地基数值算例结果说明了所陈述方法的可应用性,以及相对基于Cosserat连续体模型的FEM和基于离散颗粒集合体模型的DEM的优越性。      

Nonlinear dynamics of oriented elastic solids

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.     

On periodic motions of an orbital dumbbell-shaped body with a cabin-elevator

The motion of a dumbbell-shaped body (a pair of massive points connected with each other by a weightless rod along which the elevator, i.e., a third point, is moving according to a given law) in an attractive Newtonian central field is considered. In particular, such a mechanical system can be considered as a simplified model of an orbital cable system equipped with an elevator. The practically most interesting case where the cabin performs periodic ??shuttle??motions is studied. Under the assumption that the elevator mass is small compared with the dumbbell mass, the Poincaré theory is used to determine the conditions for the existence of families of system periodic motions analytically depending on the arising small parameter and passing into some stable radial steady-state motion of the unperturbed problem as the small parameter tends to zero. It is also proved that, for sufficiently small parameter values, each of the radial relative equilibria generates exactly one family of such periodic motions. The stability of the obtained periodic solutions is studied in the linear approximation, and these solutions themselves are calculated up to terms of the firstorder in the small parameter. The contemporary studies of the motion of orbital dumbbell systems apparently originated in Okunev??s papers [1, 2]. These studies were continued in [3], where plane motions of an orbit tether (represented as a dumbbell-shaped satellite) in a circular orbit were considered in the satellite approximation. In [4], in the case of equal masses and in the unbounded statement, the energy-momentum method was used to perform the dynamic reduction of the problem and analyze the stability of relative equilibria. A similar technique was used in [5], where, in contrast to the above-mentioned problems, the massive points were connected by an elastic spring resisting to compression and forming a dumbbell with elastic properties. Under such assumptions, the stability of radial configurations was investigated in that paper. The bifurcations and stability of steady-state configurations of a deformable elastic dumbbell were also studied in [6]. Various obstacles arising in the construction of orbital cable systems, in particular, the strong deformability of known materials, were discussed in [7]. In [8], the problem of orbital motion of a pair of massive points connected by an inextensible weightless cable was considered in the exact statement. In other words, it was assumed that a unilateral constraint is imposed on themassive points. The conditions of stability of vertical positions of the relative equilibria of the cable system, which were obtained in [8], can be used for any ratio of the subsatellite and station masses. In turn, these results agree well with the results obtained earlier in the studies of stability of vertical configurations in the case of equal masses of the system end bodies [3, 4]. One of the basic papers in the dynamics of three-body orbital cable systems is the paper [9]. The steady-state motions and their bifurcations and stability were studied depending on the elevator cabin position in [10].     

A properly invariant theory of small deformations superposed on large deformations of an elastic rod

In the context of the direct or Cosserat theory of rods developed by Green, Naghdi and several of their co-workers, this paper is concerned with the development of a theory of small deformations which are superposed on large deformations. The resulting theory is properly invariant under all superposed rigid body motions. Furthermore, it is also valid for elastic rods which are subject to kinematical constraints, and it specializes to a linear theory of an elastic rod which is invariant under superposed rigid body motions. The construction of these theories is based on the method developed by Casey & Naghdi [1] who established similar theories for unconstrained nonpolar elastic bodies.     

A Double-Strand Elastic Rod Theory

Motivated by applications in the modeling of deformations of the DNA double helix, we construct a continuum mechanics model of two elastically interacting elastic strands. The two strands are described in terms of averaged, or macroscopic, variables plus an additional small, internal or microscopic, perturbation. We call this composite structure a birod. The balance laws for the macroscopic configuration variables of the birod can be cast in the form of a classic Cosserat rod model with coupling to the internal balance laws through the constitutive relations. The internal balance laws for the microstructure variables also take a mathematical form analogous to that for a Cosserat rod, but with coupling to the macroscopic system through terms corresponding to distributed force and couple loads.     

Secondary convective motions in a plane vertical layer

Secondary plane-parallel motion in a vertical layer between isothermal planes heated to different temperatures is unstable at low and moderate values of the Prandtl number with respect to monotonically increasing disturbances [1]. The results of numerical experiments carried out by the method of networks [2, 3] indicate that this instability leads to the development of stationary secondary motions; the secondary motions have also been investigated in [4] by averaging the original equations. In the present paper we consider plane and three-dimensional stationary spatially periodic secondary motions near the threshold at which the motions develop. We make use of the methods of branching theory which were used earlier for the investigation of isothermal flows [5–9]. We determine the regions of “soft∝ and “hard∝ instability of the plane-parallel motion and the region of stability of the secondary motions. We give the results obtained by calculation of the basic characteristics of the secondary motions.     

Substructure method in high-speed monorail dynamic problems

The study of actions of high-speed moving loads on bridges and elevated tracks remains a topical problem for transport. In the present study, we propose a new method for moving load analysis of elevated tracks (monorail structures or bridges), which permits studying the interaction between two strained objects consisting of rod systems and rigid bodies with viscoelastic links; one of these objects is the moving load (monorail rolling stock), and the other is the carrying structure (monorail elevated track or bridge). The methods for moving load analysis of structures were developed in numerous papers [1–15]. At the first stage, when solving the problem about a beam under the action of the simplest moving load such as a moving weight, two fundamental methods can be used; the same methods are realized for other structures and loads. The first method is based on the use of a generalized coordinate in the expansion of the deflection in the natural shapes of the beam, and the problem is reduced to solving a system of ordinary differential equations with variable coefficients [1–3]. In the second method, after the “beam-weight” system is decomposed, just as in the problem with the weight impact on the beam [4], solving the problem is reduced to solving an integral equation for the dynamic weight reaction [6, 7]. In [1–3], an increase in the number of retained forms leads to an increase in the order of the system of equations; in [6, 7], difficulties arise when solving the integral equations related to the conditional stability of the step procedures. The method proposed in [9, 14] for beams and rod systems combines the above approaches and eliminates their drawbacks, because it permits retaining any necessary number of shapes in the deflection expansion and has a resolving system of equations with an unconditionally stable integration scheme and with a minimum number of unknowns, just as in the method of integral equations [6, 7]. This method is further developed for combined schemes modeling a strained elastic compound moving structure and a monorail elevated track. The problems of development of methods for dynamic analysis of monorails are very topical, especially because of increasing speeds of the rolling stock motion. These structures are studied in [16–18].In the present paper, the above problem is solved by using the method for the moving load analysis and a step procedure of integration with respect to time, which were proposed in [9, 19], respectively. Further, these components are used to enlarge the possibilities of the substructure method in problems of dynamics. In the approach proposed for moving load analysis of structures, for a substructure (having the shape of a boundary element or a superelement) we choose an object moving at a constant speed (a monorail rolling stock); in this case, we use rod boundary elements of large length, which are gathered in a system modeling these objects. In particular, sets of such elements form a model of a monorail rolling stock, namely, carriage hulls, wheeled carts, elements of the wheel spring suspension, models of continuous beams of monorail ways and piers with foundations admitting emergency subsidence and unilateral links. These specialized rigid finite elements with linear and nonlinear links, included into the set of earlier proposed finite elements [14, 19], permit studying unsteady vibrations in the “monorail train-elevated track” (MTET) system taking into account various irregularities on the beam-rail, the pier emergency subsidence, and their elastic support by the basement. In this case, a high degree of the structure spatial digitization is obtained by using rods with distributed parameters in the analysis. The displacements are approximated by linear functions and trigonometric Fourier series, which, as was already noted, permits increasing the number of degrees of freedom of the system under study simultaneously preserving the order of the resolving system of equations.This approach permits studying the stress-strain state in the MTET system and determining accelerations at the desired points of the rolling stock. The proposed numerical procedure permits uniquely solving linear and nonlinear differential equations describing the operation of the model, which replaces the system by a monorail rolling stock consisting of several specialized mutually connected cars and a system of continuous beams on elastic inertial supports.This approach (based on the use of a moving substructure, which is also modeled by a system of boundary rod elements) permits maximally reducing the number of unknowns in the resolving system of equations at each step of its solution [11]. The authors of the preceding investigations of this problem, when studying the simultaneous vibrations of bridges and moving loads, considered only the case in which the rolling stock was represented by sufficiently complicated systems of rigid bodies connected by viscoelastic links [3–18] and the rolling stock motion was described by systems of ordinary differential equations. A specific characteristic of the proposed method is that it is convenient to derive the equations of motion of both the rolling stock and the bridge structure. The method [9, 14] permits obtaining the equations of interaction between the structures as two separate finite-element structures. Hence the researcher need not traditionally write out the system of equations of motion, for example, for the rolling stock (of cars) with finitely many degrees of freedom [3–18].We note several papers where simultaneous vibrations of an elastic moving load and an elastic carrying structure are considered in a rather narrow region and have a specific character. For example, the motion of an elastic rod along an elastic infinite rod on an elastic foundation is studied in [20], and the body of a car moving along a beam is considered as a rod with ten concentrated masses in [21].     

On a moment static problem in stresses

Recently one has often been speaking of problems with couple stresses. The theory in which such problems are considered is sometimes called micropolar, or the theory of Cosserat continuum [1]. In the case of elastic medium, such a theory is considered in [2].     

A finite element approach to the chaotic motion of geometrically exact rods undergoing in-plane deformations

The paper is concerned with a hybrid finite element formulation for the geometrically exact dynamics of rods with applications to chaotic motion. The rod theory is developed for in-plane motions using the direct approach where the rod is treated as a one-dimensional Cosserat line. Shear deformation is included in the formulation. Within the elements, a linear distribution of the kinematical fields is combined with a constant distribution of the normal and shear forces. For time integration, the mid-point rule is employed. Various numerical examples of chaotic motion of straight and initially curved rods are presented proving the powerfulness and applicability of the finite element formulation.     

Statics and kinematics of discrete Cosserat-type granular materials

A theoretical framework is presented for the statics and kinematics of discrete Cosserat-type granular materials. In analogy to the force and moment equilibrium equations for particles, compatibility equations for closed loops are formulated in the two-dimensional case for relative displacements and relative rotations at contacts. By taking moments of the equilibrium equations, micromechanical expressions are obtained for the static quantities average Cauchy stress tensor and average couple stress tensor. In analogy, by taking moments of the compatibility equations, micromechanical expressions are obtained for the (infinitesimal) kinematic quantities average rotation gradient tensor and average Cosserat strain tensor in the two-dimensional case. Alternatively, these expressions for the average Cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane. In analogy, the expressions for the average rotation gradient tensor and the average Cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two-dimensional case. It is shown that the average particle stress tensor is always symmetrical, contrary to the average stress tensor of an equivalent homogenized continuum. Finally, discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived.     

Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: Non-linear model and stability analysis

The stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end is analyzed. The beam is herein modeled as a non-linear Cosserat rod model. Non-linear, partial integro-differential equations of motion are derived expanded up to cubic terms in the transversal displacement and torsional angle of the beam. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Interaction between conservative and non-conservative loads with respect to the stability problem is discussed. The critical spectral properties are derived and the corresponding critical eigenspace is evaluated.     

Self-similar problems of propagation of an extended hydrofracture in a permeable medium

The propagation of an extended hydrofracture in a permeable elastic medium under the influence of an injected viscous fluid is considered within the framework of the model proposed in [1, 2]. It is assumed that the motion of the fluid in the fracture is turbulent. The flow of the fluid in the porous medium is described by the filtration equation. In the quasisteady approximation and for locally one-dimensional leakage [3] new self-similarity solutions of the problem of the hydraulic fracture of a permeable reservoir with an exponential self-similar variable are obtained for plane and axial symmetry. The solution of this two-dimensional evolution problem is reduced to the integration of a one-dimensional integral equation. The asymptotic behavior of the solution near the well and the tip of the fracture is analyzed. The difficulties of using the quasisteady approximation for solving problems of the hydraulic fracture of permeable reservoirs are discussed. Other similarity solutions of the problem of the propagation of plane hydrofractures in the locally one-dimensional leakage approximation were considered in [3, 4] and for leakage constant along the surface of the fracture in [5–7].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 91–101, March–April, 1992.     

An atomistic-continuum Cosserat rod model of carbon nanotubes

The focus of the present work is an atomistic-continuum model of single-walled carbon nanotubes (CNTs) based on an elastic rod theory which can exhibit geometric as well as material nonlinearity [Healey, T.J., 2002. Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7, 405-420]. In particular, the single-walled carbon nanotube (SWNT) is modeled as a one-dimensional elastic continuum with some finite thickness bounded by the lateral surface. Exploitation of certain symmetries in the underlying atomic structure leads to suitable representations of the continuum elastic strain energy density in terms of strain measures that capture extension, twist, bending, and shear deformations [Healey, T.J., 2002. Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7, 405-420]. Bridging between the atomic scale and the effective continuum is carried out by parameterization of the continuum elastic energy and determination of the parameters using unit cell atomistic simulations over a range of deformation magnitudes and types. Specifically, the proposed model takes into account (a) bending, (b) twist, (c) shear, (d) extension, (e) coupled extension and twist, and (f) coupled bending and shear deformations. The extracted parameters reveal benefits of accounting for important anisotropic and large-strain effects as improvements over employing traditional, linearly elastic, isotropic, small-strain, continuum models to simulate deformations of atomic systems such as SWNTs. It is envisioned that the proposed approach and the extracted model parameters can serve as a useful input to simulations of SWNT deformations using existing nonlinearly elastic continuum codes based, for example, on the finite element method (FEM).