Finite element formulation of slender structures with shear deformation based on the Cosserat theory

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.     

Regularity for Shearable Nonlinearly Elastic Rods in Obstacle Problems

Based on the Cosserat theory describing planar deformations of shearable nonlinearly elastic rods we study the regularity of equilibrium states for problems where the deformations are restricted by rigid obstacles. We start with the discussion of general conditions modeling frictionless contact. In particular we motivate a contact condition that, roughly speaking, requires the contact forces to be directed normally, in a generalized sense, both to the obstacle and to the deformed shape of the rod. We show that there is a jump in the strains in the case of a concentrated contact force, i.e., the deformed shape of the rod has a corner. Then we assume some smoothness for the boundary of the obstacle and derive corresponding regularity for the contact forces. Finally we compare the results with the case of unshearable rods and obtain interesting qualitative differences. (Accepted January 21, 1998)     

A constrained theory of a Cosserat point for the numerical solution of dynamic problems of non-linear elastic rods with rigid cross-sections

A constrained theory of a Cosserat point has been developed for the numerical solution of non-linear elastic rods. The cross-sections of the rod element are constrained to remain rigid but tangential shear deformations and axial extension are admitted. As opposed to the more general theory with deformable cross-sections, the kinetic coupling equations in the numerical formulation of the constrained theory are expressed in terms of the simple physical quantities of force and mechanical moment applied to the common ends of neighboring elements. Also, in contrast with standard finite element methods, the Cosserat element uses a direct approach to the development of constitutive equations. Specifically the kinetic quantities are determined by algebraic expressions which are obtained by derivatives of a strain energy function. Most importantly, no integration is needed over the element region. A number of example problems have been considered which indicate that the constrained Cosserat element can be used to model large deformation dynamic response of non-linear elastic rods.     

Modeling of properties and motions of an inhomogeneous one-dimensional continuum of a complicated Cosserat-type microstructure

We consider an approach to modeling the properties of the one-dimensional Cosserat continuum [1] by using the mechanical modeling method proposed by Il’yushin in [2] and applied in [3]. In this method, elements (blocks, cells) of special form are used to develop a discrete model of the structure so that the average properties of the model reproduced the properties of the continuum under study. The rigged rod model, which is an elastic structure in the form of a thin rod with massive inclusions (pulleys) fixed by elastic hinges on its elastic line and connected by elastic belt transmissions, is taken to be the original discrete model of the Cosserat continuum. The complete system of equations describing the mechanical properties and the dynamical equilibrium of the rigged rod in arbitrary plane motions is derived. These equations are averaged in the case of a sufficiently smooth variation in the parameters of motion along the rod (the long-wave approximation). It was found that the average equations exactly coincide with the equations for the one-dimensional Cosserat medium [1] and, in some specific cases, with the classical equations of motion of an elastic rod [4–6]. We study the plane motions of the one-dimensional continuum model thus constructed. The equations characterizing the continuum properties and motions are linearized by using several assumptions that the kinematic parameters are small. We solve the problem of natural vibrations with homogeneous boundary conditions and establish that each value of the parameter distinguishing the natural vibration modes is associated with exactly two distinct vibration mode shapes (in the same mode), each of which has its own frequency value.     

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.     

Experiments on snap buckling, hysteresis and loop formation in twisted rods

We give the results of large deflection experiments involving the bending and twisting of 1 mm diameter nickel-titanium alloy rods, up to 2 m in length. These results are compared to calculations based on the Cosserat theory of rods. We present details of this theory, formulated as a boundary value problem. The mathematical boundary conditions model the experimental setup. The rods are clamped in aligned chucks and the experiments are carried out under rigid loading conditions. An experiment proceeds by either twisting the ends of the rod by a certain amount and then adjusting the slack, or fixing the slack and varying the amount of twist. In this way, commonly encountered phenomena are investigated, such as snap buckling, the formation of loops, and buckling into and out of planar configurations. The effect of gravity is discussed.     

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 Helical Cauchy-Born Rule for Special Cosserat Rod Modeling of Nano and Continuum Rods

We present a novel scheme to derive nonlinearly elastic constitutive laws for special Cosserat rod modeling of nano and continuum rods. We first construct a 6-parameter (corresponding to the six strains in the theory of special Cosserat rods) family of helical rod configurations subjected to uniform strain along their arc-length. The uniformity in strain then enables us to deduce the constitutive laws by just solving the warping of the helical rod’s cross-section (smallest repeating cell for nanorods) but under certain constraints. The constraints are shown to be critical in the absence of which, the 6-parameter family reduces to a well known 2-parameter family of uniform helical equilibria. An explicit formula for the 6-parameter helical map is derived which maps atoms in the repeating cell of a nanorod to their images for the purpose of repeating cell energy minimization. A scheme for the passage from nano to continuum scale is also presented to derive the constitutive laws of a continuum rod via atomistic calculations of nanorods. The bending, twisting, stretching and shearing stiffnesses of diamond nanorods and carbon nanotubes are computed to demonstrate our theory. We show that our scheme is more general and accurate than existing schemes allowing us to deduce shearing stiffness and several coupling stiffnesses of a nanorod for the first time.     

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.     

Post-buckling of a cantilever rod with variable cross-sections under combined load

IntroductionSinceEuler[1] ,Lagrange[2 ] ,Love[3 ] etal.investigatedtheslenderrod ,asoneofthebasicstructuralstabilityproblems ,manyattentionshavebeenpaidtothepost_bucklingofelasticrodsforalongtime .Today ,flexiblerodshavebeenwidelyusedasspring ,linkages ,robot’sarms,largeantennasandsoon .Hence ,thestudiesofpost_bucklingofelasticrodshavewideengineeringandapplyingbackgroundsinrecentdays .Basedontheassumptionthattheaxiallineoftherodisinextensible ,Timoshenkoetal.[4] examinedthepost_bucklingofco…     

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.     

Mechanics of an elastic solid reinforced with bidirectional fiber in finite plane elastostatics: complete analysis

A continuum-based model is presented for the mechanics of bidirectional composites subjected to finite plane deformations. This is framed in the development of a constitutive relation within which the constraint of material incompressibility is augmented. The elastic resistance of the fibers is accounted for directly via the computation of variational derivatives along the lengths of bidirectional fibers. The equilibrium equation and necessary boundary conditions are derived by virtue of the principles of virtual work statement. A rigorous derivation of the corresponding linear theory is developed and used to obtain a complete analytical solution for small deformations superposed on large. The proposed model can serve as an alternative 2D Cosserat theory of nonlinear elasticity.     

Paradoxical Bending Behavior of Shearable Nonlinearly Elastic Rods

In nonlinear elasticity the exact geometry of deformation is combined with general constitutive relations. This allows a very sophisticated interaction of deformations in different material directions. Based on the Cosserat theory for planar deformations of nonlinearly elastic rods we demonstrate some paradoxical bending effects caused by a nontrivial interaction of extension, flexure, and shear. The analytical results are illustrated by numerical examples.     

A Theory of Chiral Cosserat Elastic Plates

The mechanical behaviour of chiral materials is of interest for the investigation of carbon nanotubes, honeycomb structures, auxetic materials and bones. This paper is concerned with a theory of chiral Cosserat elastic plates. In this theory, in contrast with the case of achiral plates, the stretching and flexure cannot be treated independently of each other. First, we derive the basic equations which characterize the deformation of chiral plates. Then we establish a uniqueness result in the dynamical theory. In the equilibrium theory we establish conditions under which the Neumann problem admits solutions. Finally, the deformation of an infinite plate with a circular hole is studied. It is shown that, in contrast with the theory of Cosserat achiral plates a uniform pressure acting on the boundary of the hole produces a microrotation of the material particles.     

Kirchhoff弹性杆分析动力学的准坐标表达

作为DNA的力学模型,依据Kirchhoff动力学比拟思想建立的弹性细杆的分析力学方法已从静力学深入到动力学。由于静力学平衡微分方程与刚体动力学相当,因此,弹性细杆动力学的分析力学方程必是以弧坐标和时间为双自变量的偏微分方程。以横截面的形心速度以及弯扭度和角速度沿主轴的分量为准速度,定义了准坐标,导出了准坐标的微分和变分运算的交换关系。从Hamilton原理出发,利用准坐标的微分和变分运算的交换关系,导出了Kirchhoff弹性杆动力学准坐标下的Boltzmann-Hamel方程,并由此导出Lanrange方程。指出了Boltzmann-Hamel方程显式即为弹性杆动力学的Kirchhoff方程。定义关于弧坐标和时间的正则变量和Hamilton函数,导出Boltzmann-Hamel方程的正则形式。本文结果是以弹性杆静力学和刚性杆动力学为其特例。作为例子,建立了垂挂的在重力作用下作平面运动的弹性细杆的动力学微分方程以说明本文方法的应用。     

On the stability problem of Nicolai with variable cross-section and compressibility effect

We consider the problem of Nicolai on dynamic stability of an elastic cantilever rod loaded by an axial compressive force and tangential twisting torque in continuous formulation. The rod is assumed to be non-uniform, i.e. having variable cross-section with non-equal principal moments of inertia. New linear equations and boundary conditions are derived from nonlinear governing equations. These equations form the basis for analytical and numerical studies. The important new details of this formulation include the pre-twisting effect due to the torque and compressibility of the rod. General formulae for the influence of small geometrical imperfections to the stability region are derived and numerical examples are presented.     

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.     

A further work on directed rods

In this paper, we discuss the field equations of a rod with three deformable directors. We then deal with the rod subjected to internal constraints. Finally, we compare the theory of the constrained directed rod with that of an unconstrained rod with two deformable directors and with that of Cosserat rods.