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.     

Chiral effects in uniformly loaded rods

Examples of chiral materials include some auxetic materials, bones, some honeycomb structures, as well as composites with inclusions. The chiral effects cannot be described within classical elasticity. In the context of the linear theory of Cosserat elastic solids, we investigate the deformation of a chiral rod subjected to tractions on the lateral surface, to body loads, and to resultant forces and moments on the ends. The work is motivated by the recent interest in the using of the Cosserat elastic solid as model for auxetic composites, carbon nanotubes and bones. The three-dimensional problem is reduced to the study of some generalized plane strain problems. New chiral effects are presented. In the case of cylinders of arbitrary cross-section, the flexure produced by a transversal force, in contrast with the case of achiral materials, is accompanied by extension and bending by terminal couples. The body loads and the tractions on the lateral surface produce extension, flexure, torsion, bending by terminal couples and a plane strain. It is shown that a uniform pressure acting on the lateral surface of a chiral circular cylinder does not produce bending effects.     

Wave propagation in double helical rods

The propagation of waves in helical rods has been studied extensively. However, studying the wave propagation in double helical rods have received less attention although this can be useful in multiple fields of science and engineering. Obtaining an analytical model for a double helical rod is challenging since the curvature and tortuosity are not constant. Thus, resolving the wave behaviour analytically is nearly impossible. In this paper, wave propagation in a double helical rod will be studied using the wave and finite element method which is a technique that can be used to model homogeneous and periodic one and two dimensional structures based on the periodic structure theory. For modelling a double helical rod, the finite element model of a single turn is processed using Bloch waves. The dispersion curves and wavemodes are obtained and the similarities and differences of waves in helical and double helical rods are highlighted.     

On Shear and Torsion Factors in the Theory of Linearly Elastic Rods

Lower bounds for the factors entering the standard notions of shear and torsion stiffness for a linearly elastic rod are established in a new and simple way. The proofs are based on the following criterion to identify the stiffness parameters entering rod theory: the rod’s stored-energy density per unit length expressed in terms of force and moment resultants should equal the stored-energy density per unit length expressed in terms of stress components of a Saint-Venant cylinder subject to either flexure or torsion, according to the case. It is shown that the shear factor is always greater than one, whatever the cross section, a fact that is customarily stated without proof in textbooks of structure mechanics; and that the torsion factor is also greater than one, except when the cross section is a circle or a circular annulus, a fact that is usually proved making use of Saint-Venant’s solution in terms of displacement components.     

Variation in the length of perfectly elastic rods under torsion

On the basis of elastic constitutive relations that reflect geometrically nonlinear second-order effects, we refine the theory of torsion of rectilinear rods of an arbitrary transverse cross-section. In particular, we obtain a universal formula, independent of the material properties, that determines the longitudinal strain arising as the rod undergoes free torsion. According to this formula, the length of a rod made of an isotropic perfectly elastic material can, in contrast to the traditional concepts, either increase or decrease as the rod undergoes torsion. Moreover, the variation in the length depends only on the geometry of the transverse cross-section.     

Dynamic symmetry breaking and structure-preserving analysis on the longitudinal wave in an elastic rod with a variable cross-section

The longitudinal wave propagating in an elastic rod with a variable cross-section owns wide engineering background, in which the longitudinal wave dissipation determines some important performances of the slender structure. To reproduce the longitudinal wave dissipation effects on an elastic rod with a variable cross-section, a structure-preserving approach is developed based on the dynamic symmetry breaking theory. For the dynamic model controlling the longitudinal wave propagating in the elast...     

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.     

A relook at Reissner’s theory of plates in bending

Shear deformation and higher order theories of plates in bending are (generally) based on plate element equilibrium equations derived either through variational principles or other methods. They involve coupling of flexure with torsion (torsion-type) problem and if applied vertical load is along one face of the plate, coupling even with extension problem. These coupled problems with reference to vertical deflection of plate in flexure result in artificial deflection due to torsion and increased deflection of faces of the plate due to extension. Coupling in the former case is eliminated earlier using an iterative method for analysis of thick plates in bending. The method is extended here for the analysis of associated stretching problem in flexure.     

A novel semi-inverse solution method for elastoplastic torsion of heat treated rods

Torsion rods are a primary component of many power transmission and other mechanical systems. The behavior of these rods under elastoplastic torsion is of major concern for designers. Different methods have so far been proposed which deal with the elastoplastic torsion of rods, most of which assume constant yield stress. This assumption produces rough and inaccurate results when the rods are heat treated, since in the process of heat treatment the form of yield stress distribution across the rod cross section changes. We propose a new method for calculating elastoplastic torsion of rods of simply connected cross section which is based on heat treatment observations. In our method the full plastic stress function is obtained by using the semi-inverse method. Elastoplastic stress function is obtained by generalizing the idea of the membrane analogy and using a piecewise continuous stress function. Since the proposed form of yield stress distribution can not be handled by the current Finite Element packages, we produce a computer package with a 3D graphical interface capable of calculating and displaying the 3D elastoplastic stress function, shear stress contours, and torque-angle of rotation per unit length. We show that our method produces excellent agreement for several known cross sections in comparison to methods which assume constant yield stress.     

Torsional instability of stretched rubber cylinders

When cylindrical rubber rods are stretched and twisted to a sufficiently large degree, they suddenly form a sharply bent ring or “knot”, and more knots form as the rod is twisted further. This well-known phenomenon is ascribed here to an elastic instability. As a stretched rod is twisted, the tensile stress required to maintain the stretch drops dramatically in agreement with Rivlin's theory of large elastic deformations (Philos. Trans. R. Soc. London Ser. A 241 (1948) 379; Rheology, Theory and Applications, Chapter 10, Vol. 1, Academic Press, New York, 1956). The additional strain energy required to form a ring is shown to become zero at a critical amount of torsion. In experiments on cylindrical rubber rods of various diameters, stretched to various extents, good agreement was obtained between measured values of the amount of torsion at which a ring formed and values predicted by this simple stability analysis, based on Rivlin's theory.     

Experimental investigation of longitudinal wave propagation in an elastic rod with coulomb friction

This paper presents the experimental results obtained from the propagation of a compression stress wave generated by the longitudinal impact of two cylindrical elastic rods. One of these rods is subjected to a uniformly distributed coulomb-friction force. In order to determine the stress-wave shape and the decay rate, the rod is subjected to longitudinal impact for different values of coulomb friction. As the stress wave propagates along the rod, it is measured at strain-gage stations located on the stationary rod.In order to correlate the experimental results, the solution of the modified wave equation with coulomb friction is obtained for the longitudinal impact of two semi-infinite rods.     

A Variational Approach to Obstacle Problems for Shearable Nonlinearly Elastic Rods

We use variational methods to study obstacle problems for geometrically exact (Cosserat) theories for the planar deformation of nonlinearly elastic rods. These rods can suffer flexure, extension, and shear. There is a marked difference between the behavior of a shearable and an unshearable rod. The set of admissible deformations is not convex, because of the exact geometry used. We first investigate the fundamental question of describing contact forces, which we necessarily treat as vector‐valued Borel measures. Moreover, we introduce techniques for describing point obstacles. Then we prove existence for a very large class of problems. Finally, using nonsmooth analysis for handling the obstacle, we show that the Euler‐Lagrange equations are satisfied almost everywhere. These equations provide very detailed structural information about the contact forces. Accepted June 3, 1996     

A rod model for three dimensional deformations of single-walled carbon nanotubes

A continuum model for single-walled carbon nanotubes (SWCNT) is presented which is based on an extension to the special Cosserat theory of rods (Kumar and Mukherjee, 2011). The model allows deformation of a nanotube’s lateral surface in a one dimensional framework and hence is an efficient substitute to the commonly used two dimensional shell models for nanotubes. The model predicts a new coupling mode in chiral nanotubes – coupling between twist and cross-sectional shrinkage implying that the three deformation modes (extension, twist and cross-sectional shrinkage) are all coupled to each other. Atomistic simulations based on the density functional based tight binding method (DFTB) are performed on a (9, 6) SWCNT and the simulation data is used to estimate material parameters of this rod model. A peculiar behavior of the nanotube is observed when it is axially stretched – induced rotation of each cross-section is equal in magnitude but opposite to that of its two neighboring cross-sections. This is shown to be an effect of relative shift/inner-displacement between the two SWCNT sub-lattices.     

Nonlinear Weakly Curved Rod by Γ-Convergence

We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use an approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity. We do not impose any constitutional behavior of the material and work in a general framework. To derive the model, by means of ??-convergence, we need to set the order of strain energy (i.e., its relation to the thickness of the body h). We analyze the situation when the strain energy (divided by the order of volume) is of the order h ⁴. This is the same approach as the one used in F?ppl-von Kármán model for plates and the analogous model for rods. The obtained model is analogous to Marguerre-von Kármán for shallow shells and its linearization is the linear shallow arch model which can be found in the literature.     

Non-linear flexural-flexural-torsional-extensional dynamics of beams—I. Formulation

The non-linear differential equations of motion, and boundary conditions, for Euler-Bernoulli beams able to experience flexure along two principal directions (and, thus, flexure in any direction in space), torsion and extension are formulated. The beam's material is assumed to be Hookean but its properties may vary along its span. The nonlinearities present in the differential equations include contributions from the curvature expression and from inertia terms. A set of differential equations with polynomial nonlinearities to cubic order, suitable for a perturbation analysis of the motion, is also developed and the validity of the inextensional approximation is assessed. The equations developed here reduce to those for an inextensional beam. In Part II of this paper, a specific example of application is analyzed and the results obtained are compared with those available in the literature where several non-linear terms have been neglected a priori.     

Uniaxial and biaxial extensional viscosity measurements of dilute and semi-dilute solutions of rigid rod polymers

Effective uniaxial extensional and biaxial extensional viscosities of dilute and semi-dilute solutions of collagen, a rigid rod molecule, have been measured with an opposing jet apparatus. The concentration of collagen in the glycerin/water solvent ranged from 50 to 2300 ppm. The data agree quantitatively with a theory developed by Batchelor describing the extensional viscosity of perfectly aligned rigid rods. The viscosity measured for the dilute rigid rod solutions is independent of the rate of strain as predicted by Batchelor's theory. Data taken on the semi-dilute, strain-thinning solutions at strain rates sufficiently high to align the rods in the extension direction also agree with the predictions of Batchelor's theory. The measured viscosity of semi-dilute solutions at low strain rates agree qualitatively with a theory developed by Doi and Edwards describing the strain-thinning behavior of semi-dilute rigid rod solutions.     

Nonlinear analysis of elastic round rods

Summary The nonlinear theory of elastic round rods is split into a vector equation concerning the properties of finite inextensional flexural deflections and a scalar equation concerning torsional deformations. In particular, the related boundary conditions are examined for the case of a Cardan joint. The buckling configurations of a rod under tension and torsion, constrained at both ends by Cardan joints, are determined by the linearized form of the above theory and compared with previous findings about the rod with cylindrical end hinges.

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Sommario La teoria non lineare della verga elastica rotonda è compendiata in un'equazione vettoriale che esprime le proprietà della deformata flessionale inestensionale: le associate condizioni al contorno includono le proprietà torsionali della verga. In particolare si considerano le condizioni ai limiti per deformazioni finite nel caso di vincolo con giunto cardanico. Per la verga con giunti cardanici ai due estremi, soggetta a trazione e torsione si determinano le configurazioni critiche.

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Post-graduate Student, Department of Structural Engineering, Politecnico di Torino.

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Professor emeritus, Department of Structural Engineering, Politecnico di Torino.     

Second-order solution of Saint-Venant's problem for an elastic bar predeformed in flexure

We use the method of Signorini's expansion to analyze the Saint-Venant problem for an isotropic and homogeneous second-order elastic prismatic bar predeformed by an infinitesimal amount in flexure. The centroid of one end face of the bar is rigidly clamped. The complete solution of the problem is expressed in terms of ten functions. For a general cross-section, explicit expressions for most of these functions are given; the remaining functions are solutions of well-posed plane elliptic problems. However, for a bar of circular cross-section, all of these functions are evaluated and a closed form solution of the 2nd-order problem is given. The solution contains six constants which characterize the second-order flexure, bending, torsion and extension of the bar. It is found that when the total axial force vanishes, the second-order axial deformation is not zero; it represents a generalized Poynting effect. The second-order elasticities affect only the second-order axial force.     

Hamiltonian dynamics of an elastica and the stability of solitary waves

A method is presented for deriving unconstrained Hamiltonian systems of partial differential equations equivalent to given constrained Lagrangian systems. The method is applied to the theory of planar, finite-amplitude motions of inextensible and unshearable elastic rods. The constraints of inextensibility and unshearability become integrals of motion in the Hamiltonian formulation.It is known that in the theory of uniform, inextensible, unshearable rods of infinite length there arise solitary-wave solutions with the property that each profile can move at arbitrary speed. The Hamiltonian formulation is exploited to analyze the stability properties of these solitary waves. The wave profiles are first characterized as critical points of an appropriate time-invariant functional. It is then shown that for a certain range of wave speeds the solitary-wave profiles are actually nonisolatedminimizers of the functional, a fact with implications for nonlinear stability.