Bifurcating electric wires

Summary The stationary states of a string through which an electric current is sent and which is placed in an axial magnetic field, are investigated. Using methods of constrained variational principles, it is shown that, in case the string is inextensible, only those stationary states which have least total potential energy are stable.     

Critical states of drill strings in the channels of inclined boreholes

The problem of determining the critical states and the postbuckling deformation of drill strings in the cavities of curvilinear boreholes is posed. The process of elastic bending of the drill string is associated with the motion of its axial line along the corresponding channel surface. On the basis of the theory of flexible curvilinear rods, a specially chosen moving system of axes is used to construct nonlinear ordinary differential equations describing the contact interaction between the drill string tube and the borehole wall. Themoving reference system allows us to separate the desired variables and decrease the order of the resolving equations. As an example, we solve the problem of stability of drill strings lying at the bottom of a cylindrical cavity in a rectilinear inclined borehole. The critical values of the axial forces are determined and the buckling modes are constructed. It is also shown that they have the form of edge effects typical of singularly perturbed equations. The developedmethods can be used in design of a curvilinear borehole and its possible driving conditions to determine the admissible values of the axial force and the torque at the point of the drill string suspension so as to prevent its bifurcation buckling.     

On the free shapes of elastic rods

The problem of free shape consists in finding the form that an elastic body must have in a natural state in order that it shall assume a given form in an equilibrium configuration under the action of assigned loads. The problem, that is of interest in itself, arises in some practical applications and can constitute a preliminary step in the study of some mechanical properties of classes of equilibrium configurations that are not natural states. This paper examines the problem of free shape for inextensible elastic rods which in equilibrium are subject only to the action of forces and couples applied to the ends, and whose deformations can be described by the theory of finite displacements of thin rods due to Kirchhoff. After the general equations governing the problem have been deduced, they are employed to give a classification of the free shapes of rods that in equilibrium are circular rings.     

Classification of the Spatial Equilibria of the Clamped Elastica: Symmetries and Zoology of Solutions

We investigate the configurations of twisted elastic rods under applied end loads and clamped boundary conditions. We classify all the possible equilibrium states of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods. We show that all solutions of the clamped boundary value problem exhibit a π-flip symmetry. The Kirchhoff equations which describe the equilibria of these rods are integrated in a formal way which enable us to describe the boundary conditions in terms of 2 closed form equations involving 4 free parameters. We show that the flip symmetry property is equivalent to a reversibility property of the solutions of the Kirchhoff differential equations. We sort these solutions according to their period in the phase plane. We show how planar untwisted configurations as well as circularly closed configurations play an important role in the classification. This revised version was published online in June 2006 with corrections to the Cover Date.     

The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies

This paper treats the radially symmetric equilibrium states of aeolotropic nonlinearly elastic solid cylinders and balls under constant normal forces on their boundaries. It is shown that the aeolotropy gives rise to solutions describing both intact and cavitating states, which exhibit an array of remarkable new phenomena, not suggested by the solutions for isotropic bodies. E.g., it is shown that there are materials having a critical pressure such that for applied pressures on the boundary below the critical value, the normal pressures at the center of the body are zero and for applied pressures above the critical value, the normal pressures at the center are infinite. There are also materials for which there is no equilibrium state with center intact when the boundary is subjected to uniform tension. It is also shown that the equilibrium states treated here are the only radially symmetric equilibrium states. Thus the strange phenomena discovered here must be present in such stable equilibrium states.     

On Maximally Dissipative Shock Waves in Nonlinear Elasticity

Shock waves in nonlinearly elastic solids are, in general, dissipative. We study the following question: among all plane shock waves that can propagate with a given speed in a given one-dimensional nonlinearly elastic bar, which one—if any—maximizes the rate of dissipation? We find that the answer to this question depends strongly on the qualitative nature of the stress-strain relation characteristic of the given material. When maximally dissipative shocks do occur, they propagate according to a definite kinetic relation, which we characterize and illustrate with examples.     

Stability of an elastic cytoskeletal tensegrity model

An elastic cytoskeletal tensegrity structure composed by six inextensible elastic struts and 24 elastic cables is considered. The model is studied, adopting delay convention for stability. Critical conditions for simple and compound instabilities are defined. Post-critical behavior is also described. Equilibrium states with buckling of the struts are also considered. It is revealed that critical Euler buckling load of the struts is a necessary but not a sufficient condition for the existence of bifurcated equilibrium states, caused by buckling of the struts.     

Coexistent phases in nonlinear thermoelasticity: Radially symmetric equilibrium states of aeolotropic bodies

We determine the detailed qualitative behavior of radially symmetric equilibrium states having coexistent phases for general classes of aeotropic nonlinearly thermoelastic materials. We treat both structured and non-structured interfaces. The aeolotropy is responsible for many novel effects. We show that linearly elastic materials cannot sustain coexistent radially symmetric phases unless the interfaces are structured. Our analysis is largely elementary, being based on a combination of geometric constructions with phase-plane methods. A few results, however, depend on our development of appropriate versions of the theory of asymptotically autonomous ordinary differential equations.This paper is dedicated to Mort Gurtin on the occasion of his sixtieth birthday     

Energetics and conserved quantity of an axially moving string undergoing three-dimensional nonlinear vibration

Nonlinear three-dimensional vibration of axially moving strings is investigated in the view of energetics. The governing equation is derived from the Eulerian equation of motion of a continuum for axially accelerating strings. The time-rate of the total mechanical energy associated with the vibration is calculated for the string with its ends moving in a prescribed way. For a string moving in a constant axial speed and constrained by two fixed ends, a conserved quantity is proved to remain unchanged during three-dimensional vibration, while the string energy is not conserved. An approximate conserved quantity is derived from the conserved quantity in the neighborhood of the straight equilibrium configuration. The approximate conserved quantity is applied to verify the Lyapunov stability of the straight equilibrium configuration. Numerical simulations are performed for a rubber string and a steel string. The results demonstrate the variation of the total mechanical energy and the invariance of the conserved quantity.     

Buckling of slender string in cylindrical tube under axial load: Experiments and theoretical analysis

Buckling of drill and tubing strings in drill or casing holes will affect the life of the string and cause difficulties in drilling and in oil production. A number of theoretical models have been developed to study the buckling load and the post-buckling configuration of the strings under some idealized conditions. However, verification of these theories in practice or in laboratory has not been extensively reported.In this paper work on laboratory buckling tests of strings is presented and the results are compared with theoretical formulas. The results indicate that the process of string buckling can be divided into two stages, namely plane buckling and spatial buckling. In the spatial buckling the string-buckling configuration can roughly be represented by a helix. The relation between the buckling force and the helical pitch, obtained from the laboratory tests, is similar to that predicted by helical buckling theories. The friction between the string and the wall of the confining tube causes a nonuniform buckling shape and prohibits further buckling at some critical loads. The buckling theories do not cover these effects and may thus induce significant errors in predicting the string-post-buckling configuration. An approximate formula for estimating the friction force is proposed, which provides more accurate results. Further studies of the friction effect are suggested.     

Relative equilibrium stability of a mechanical system with deformable elements in a circular orbit

The relative equilibrium stability is analyzed for a mechanical system in an orbit. The system consists of two rigid bodies connected by a thin inextensible elastic rod. The problem of stability of steady motions is reduced to the minimization problem for the system’s potential energy consisting of the potential energy of elastic, gravitational, and centrifugal forces.     

Propagation of small waves in inextensible strings

The model of an inextensible uniform string subject to constant gravitation is used to study the propagation of transversal waves in one-dimensional continua. Perturbation analysis of the equations of motion yields as a result the local representation of small waves in terms of a normalized Riemann function. By means of the latter, shape and speed of propagating waves may be discussed. A refined analysis confirms that on first order, small waves travel along characteristics of the unperturbed equilibrium configuration. An explicit power law for the waves’ amplitudes is given, and the findings are supported by the numerical results.     

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)     

Finite Deformations and Motions of Radially Inextensible Hollow Spheres

Radial inflation–compaction and radial oscillation solutions are presented for hollow spheres of isotropic elastic material that are radially inextensible. The solutions for radial inflation–compaction and radial oscillation are obtained also for everted radially inextensible hollow spheres of isotropic elastic material. The static and dynamic results for everted and uneverted radially inextensible hollow spheres are then compared. Harmonic and compressible Varga materials are used to demonstrate the solutions.      

Bifurcation theory of an elastic conducting wire subject to magnetic forces

In this paper we study the equilibrium states of a nonlinearly elastic wire in a magnetic field. The wire is perfectly flexible, is suspended between fixed supports and carries an electric current. We consider two problems. The first in which the magnetic field is constant can be solved exactly. The set of solutions illustrates the phenomenon of symmetry breaking which is a chapter in the theory of imperfect bifurcation. The second problem is one in which the magnetic field is produced by current flowing in a pair of infinitely long parallel wires. When the line of supports of the elastic wire is parallel to these and equidistant from them we may apply the global bifurcation results of Crandall and Rabinowitz to study the set of solutions. We also consider perturbations of this case. This is another example of imperfect bifurcation.     

Modeling of planar nonshallow prestressed beams towards asymptotic solutions

Employing the geometrically exact approach, the governing equations of nonlinear planar motions around nonshallow prestressed equilibrium states of slender beams are derived. Internal kinematic constraints and approximations are introduced considering unshearable extensible and inextensible beams. The obtained approximate models, incorporating quadratic and cubic nonlinearities, are amenable to a perturbation treatment in view of asymptotic solutions. The different perturbation schemes for the two mechanical beam models are discussed.     

Elastic bodies reinforced with inextensible surfaces

The constraint of in-plane rigidity is examined within the general framework of the theory of internally constrained materials. It is shown that, for in-plane rigid materials, local strain and active stress are both defined by vectorial quantities. Representation formulae for the elastic response mapping are established in the cases of transverse isotropy and maximal symmetry, compatible with the constraint manifold. The equilibrium problem for an elastic body reinforced with parallel inextensible planes is also considered. In particular, universal solutions for bodies with maximal material symmetry are determined within the class of deformations which leave rigid every reinforcing plane.     

Nets with Hexagonal Cell Structure

We establish a continuum theory of inextensible nets with hexagonal cell structure. Some qualitative properties of the equilibrium problem and possible singularities are discussed.      

Dynamics of strings made of phase-transforming materials

This paper presents a theory to describe the dynamical behavior of a string made of a phase-transforming material like a shape-memory alloy. The study of phase boundaries, the driving force acting on them and the kinetic relation governing their propagation is of central concern. The paper proposes a qualitative experimental test of the notion of a kinetic relation, as well as a simple experimental method for measuring it quantitatively. It presents a numerical method for studying general initial and boundary value problems in strings, and concludes by exploring the use of phase transforming strings to generate motion at very small scales.     

DIVERGENCE INSTABILITY OF VARIABLE-ARC-LENGTH ELASTICA PIPES TRANSPORTING FLUID

A flexible elastic pipe transporting fluid is held by an elastic rotational spring at one end, while at the other end, a portion of the pipe may slide on a frictional support. Regardless of the gravity loads, when the internal flow velocity is higher than the critical velocity, large displacements of static equilibrium and divergence instability can be induced. This problem is highly nonlinear. Based on the inextensible elastica theory, it is solved herein via the use of elliptic integrals and the shooting method. Unlike buckling with stable branching of a simply supported elastica pipe with constant length, the variable arc-length elastica pipe buckles with unstable branching. The friction at the support has an influence in shifting the critical locus over the branching point. Alteration of the flow history causes jumping between equilibrium paths due to abrupt changes of direction of the support friction. The elastic rotational restraint brings about unsymmetrical bending configurations; consequently, snap-throughs and snap-backs can occur on odd and even buckling modes, respectively. From the theoretical point of view, the equilibrium configurations could be formed like soliton loops due to snapping instability.