Effect of Material Nonlinearity on Spatial Buckling of Nanorods and Nanotubes

We show the importance of incorporating material nonlinearity for accurate determination of spatial buckling of nanorods and nanotubes. Both the nanorods and nanotubes are modeled as a special Cosserat rod whose nonlinear material laws are obtained using the recently proposed helical Cauchy-Born rule. We first present Euler buckling of solid diamond nanorods whose normalized buckling load, obtained from fully atomistic calculations, exhibits an interesting trend. The buckling load starts from unity at large aspect ratio of the nanorod, then as the aspect ratio is decreased, the buckling load increases slowly and finally decreases rapidly. We attribute this trend to material nonlinearity of the nanorod’s core at large compressive strain. We also discuss how surface stress affects buckling in nanorods. We then present the effect of compression and twist on buckling of single-walled carbon nanotubes. Interestingly, for highly twisted nanotubes, fully atomistic calculations show the first buckled mode to be different from a typical Euler buckling mode. Both the observations about nanorods and nanotubes are accurately replicated in the finite element special Cosserat rod simulation when the material nonlinearity is also incorporated. However, the simulation results exhibit completely different trend when only linear material laws are incorporated.     

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.     

Helical Birods: An Elastic Model of Helically Wound Double-Stranded Rods

We consider a geometrically accurate model for a helically wound rope constructed from two intertwined elastic rods. The line of contact has an arbitrary smooth shape which is obtained under the action of an arbitrary set of applied forces and moments. We discuss the general form the theory should take along with an insight into the necessary geometric or constitutive laws which must be detailed in order for the system to be complete. This includes a number of contact laws for the interaction of the two rods, in order to fit various relevant physical scenarios. This discussion also extends to the boundary and how this composite system can be acted upon by a single moment and force pair. A second strand of inquiry concerns the linear response of an initially helical rope to an arbitrary set of forces and moments. In particular we show that if the rope has the dimensions assumed of a rod in the Kirchhoff rod theory then it can be accurately treated as an isotropic inextensible elastic rod. An important consideration in this demonstration is the possible effect of varying the geometric boundary constraints; it is shown the effect of this choice becomes negligible in this limit in which the rope has dimensions similar to those of a Kirchhoff rod. Finally we derive the bending and twisting coefficients of this effective rod.     

Kirchhoff’s Problem of Helical Equilibria of Uniform Rods

It is demonstrated that a uniform and hyperelastic, but otherwise arbitrary, nonlinear Cosserat rod subject to appropriate end loadings has equilibria whose center lines form two-parameter families of helices. The absolute energy minimizer that arises in the absence of any end loading is a helical equilibrium by the assumption of uniformity, but more generally the helical equilibria arise for non-vanishing end loads. For inextensible, unshearable rods the two parameters correspond to arbitrary values of the curvature and torsion of the helix. For non-isotropic rods, each member of the two-dimensional family of helical center lines has at least two possible equilibrium orientations of the director frame. The possible orientations are characterized by a pair of finite-dimensional, dual variational principles involving pointwise values of the strain-energy density and its conjugate function. For isotropic rods, the characterization of possible equilibrium configurations degenerates, and in place of a discrete number of two-parameter families of helical equilibria, typically a single four-parameter family arises. The four continuous parameters correspond to the two of the helical center lines, a one-parameter family of possible angular phases, and a one-parameter family of imposed excess twists. Mathematics Subject Classifications (2000) 74K10.     

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.     

The Influence of Moduli Slope of a Linearly Graded Matrix on the Bulk Moduli of Some Particle- and Fiber-Reinforced Composites

The stored energy functional of a homogeneous isotropic elastic body is invariant with respect to translation and rotation of a reference configuration. One can use Noether's Theorem to derive the conservation laws corresponding to these invariant transformations. These conservation laws provide an alternative way of formulating the system of equations governing equilibrium of a homogeneous isotropic body. The resulting system is mathematically identical to the system of equilibrium equations and constitutive relations, generally, of another material. This implies that each solution of the system of equilibrium equations gives rise to another solution, which describes the reciprocal deformation and solves the system of equilibrium equations of another material. In this paper we derive conservation laws and prove the theorem on conjugate solutions for two models of elastic homogeneous isotropic bodies – the model of a simple material and the model of a material with couple stress (Cosserat continuum). This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

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 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.     

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.     

Theory of thin thermoelastic rods made of porous materials

In this paper, we consider thin rods modeled by the direct approach, in which the rod-like body is regarded as a one-dimensional continuum (i.e., a deformable curve) with a triad of rigidly rotating orthonormal vectors attached to each material point. In this context, we present a model for porous thermoelastic curved rods, having natural twisting and arbitrary shape of cross-section. To describe the porosity, we employ the theory of elastic materials with voids. The basic laws of thermodynamics are applied directly to the one-dimensional continuum, and the nonlinear governing equations are established. We formulate the constitutive equations and determine the structure of constitutive tensors. We prove the uniqueness of solution to the boundary-initial-value problem associated with the deformation of porous thermoelastic rods in the framework of linear theory. Then, we show the decoupling of the bending-shear and extension-torsion problems for straight porous rods. Using a comparison with three-dimensional equations, we identify and give interpretations to the relevant fields introduced in the direct approach. Finally, we consider the case of orthotropic materials and determine the constitutive coefficients for deformable curves in terms of three-dimensional constitutive constants by means of comparison between simple solutions obtained in the two approaches for porous thermoelastic rods.     

A direct theory of affine rods

A direct theory of affine rods is developed from first principles. To concentrate on the central aspects of the model, we use an axiomatic format and tools from Lie group theory. To facilitate comparisons with other theories, we propose an identification procedure to derive the constitutive relations of the affine rod from those of a rod modeled as a three-dimensional body.     

Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity

The axial vibration of single walled carbon nanotube embedded in an elastic medium is studied using nonlocal elasticity theory. The nonlocal constitutive equations of Eringen are used in the formulations. The effect of various parameters like stiffness of elastic medium, boundary conditions and nonlocal parameters on the axial vibration of nanorods is discussed. It is obtained that, the axial vibration frequencies of the embedded nanorods are highly over estimated by the classical continuum rod model which ignores the effect of small length scale.     

A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point

The theory of a Cosserat point has been used to formulate a new 3-D finite element for the numerical analysis of dynamic problems in nonlinear elasticity. The kinematics of this element are consistent with the standard tri-linear approximation in an eight node brick-element. Specifically, the Cosserat point is characterized by eight director vectors which are determined by balance laws and constitutive equations. For hyperelastic response, the constitutive equations for the director couples are determined by derivatives of a strain energy function. Restrictions are imposed on the strain energy function which ensure that the element satisfies a nonlinear version of the patch test. It is shown that the Cosserat balance laws are in one-to-one correspondence with those obtained using a Bubnov–Galerkin formulation. Nevertheless, there is an essential difference between the two approaches in the procedure for obtaining the strain energy function. Specifically, the Cosserat approach determines the constitutive coefficients for inhomogeneous deformations by comparison with exact solutions or experimental data. In contrast, the Bubnov–Galerkin approach determines these constitutive coefficients by integrating the 3-D strain energy function using the kinematic approximation. It is shown that the resulting Cosserat equations eliminate unphysical locking, and hourglassing in large compression without the need for using assumed enhanced strains or special weighting functions.     

Statistical Foundations of Liquid-Crystal Theory. I: Discrete Systems of Rod-Like Molecules

We develop a mechanical theory for systems of rod-like particles. Central to our approach is the assumption that the external power expenditure for any subsystem of rods is independent of the underlying frame of reference. This assumption is used to derive the basic balance laws for forces and torques. By considering inertial forces on par with other forces, these laws hold relative to any frame of reference, inertial or noninertial. Finally, we introduce a simple set of constitutive relations to govern the interactions between rods and find restrictions necessary and sufficient for these laws to be consistent with thermodynamics. Our framework provides a foundation for a statistical mechanical derivation of the macroscopic balance laws governing liquid crystals.     

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.