Elastica of an euler rod with clamped ends

The stability of the postcritical states of equilibrium of a flexible rod with clamped ends loaded by an axial force is analyzed. It is shown that the existing Lagrange elliptic-integral solution has bifurcation points and branches of solution that have not been investigated thus far. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 184–186, May–June, 2000.     

Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios

Homoclinic snaking refers to the bifurcation structure of symmetric localised roll patterns that are often found to lie on two sinusoidal “snaking” bifurcation curves, which are connected by an infinite number of “rung” segments along which asymmetric localised rolls of various widths exist. The envelopes of all these structures have a unique maximum and we refer to them as symmetric or asymmetric 1-pulses. In this paper, the existence of stationary 1D patterns of symmetric 2-pulses that consist of two well-separated 1-pulses is established. Corroborating earlier numerical evidence, it is shown that symmetric 2-pulses exist along isolas in parameter space that are formed by parts of the snaking curves and the rungs mentioned above.     

On Bifurcations of Equilibria of Intrinsically Curved, Electrically Charged, Rod-like Structures that Model DNA Molecules in Solution

DNA molecules in the familiar Watson–Crick double helical B form can be treated as though they have rod-like structures obtained by stacking dominoes one on top of another with each rotated by approximately one-tenth of a full turn with respect to its immediate predecessor in the stack. These “dominoes” are called base pairs. A recently developed theory of sequence-dependent DNA elasticity (Coleman, Olson, & Swigon, J. Chem. Phys. 118:7127–7140, 2003) takes into account the observation that the step from one base pair to the next can be one of several distinct types, each having its own mechanical properties that depend on the nucleotide composition of the step. In the present paper, which is based on that theory, emphasis is placed on the fact that, as each base in a base pair is attached to the sugar-phosphate backbone chain of one of the two DNA strands that have come together to form the Watson–Crick structure, and each phosphate group in a backbone chain bears one electronic charge, two such charges are associated with each base pair, which implies that each base pair is subject to not only the elastic forces and moments exerted on it by its neighboring base pairs but also to long range electrostatic forces that, because they are only partially screened out by positively charged counter ions, can render the molecule’s equilibrium configurations sensitive to changes in the concentration c of salt in the medium. When these electrostatic forces are taken into account, the equations of mechanical equilibrium for a DNA molecule with N + 1 base pairs are a system of μN non-linear equations, where μ, the number of kinematical variables describing the relative displacement and orientation of adjacent pairs is in general 6; it reduces to 3 when base-pair steps are assumed to be inextensible and non-shearable. As a consequence of the long-range electrostatic interactions of base pairs, the μN × μN Jacobian matrix of the equations of equilibrium is full. An efficient numerically stable computational scheme is here presented for solving those equations and determining the mechanical stability of the calculated equilibrium configurations. That scheme is employed to compute and analyze bifurcation diagrams in which c is the bifurcation parameter and to show that, for an intrinsically curved molecule, small changes in c can have a strong effect on stable equilibrium configurations. Cases are presented in which several stable configurations occur at a single value of c.      

Symmetry Breaking and Turbulence in Perturbed Plane Couette Flow

Perturbed plane Couette flow containing a thin spanwise-oriented ribbon undergoes a subcritical bifurcation at Re≈230 to a steady three-dimensional state containing streamwise vortices. This bifurcation is followed by several others giving rise to a fascinating series of stable and unstable steady states of different symmetries and wavelengths. First, the backwards-bifurcating branch reverses direction and becomes stable near Re≈200. Then the spanwise reflection symmetry is broken, leading to two asymmetric branches which are themselves destabilized at Re≈420. Above this Reynolds number, time evolution leads first to a metastable state whose spanwise wavelength is halved and then to complicated time-dependent behavior. These features are in agreement with experiments. Received 15 December 2001 and accepted 29 March 2002 Published online: 2 October 2002 Communicated by H.J.S. Fernando     

Stability and folds

It is known that when one branch of a simple fold in a bifurcation diagram represents (linearly) stable solutions, the other branch represents unstable solutions. The theory developed here can predict instability of some branches close to folds, without knowledge of stability of the adjacent branch, provided that the underlying problem has a variational structure. First, one particular bifurcation diagram is identified as playing a special role, the relevant diagram being specified by the choice of functional plotted as ordinate. The results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram. In many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph. The theory is particularly useful in applications where a solution branch has a succession of folds.The theory is illustrated with applications to simple models of thermal selfignition and of a chemical reactor, both of which systems are of Émden-Fowler type. An analysis concerning an elastic rod is also presented.     

Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration.     

Writhing instabilities of twisted rods: from infinite to finite length

We use three different approaches to describe the static spatial configurations of a twisted rod as well as its stability during rigid loading experiments. The first approach considers the rod as infinite in length and predicts an instability causing a jump to self-contact at a certain point of the experiment. Semi-finite corrections, taken into account as a second approach, reveal some possible experiments in which the configuration of a very long rod will be stable through out. Finally, in a third approach, we consider a rod of real finite length and we show that another type of instability may occur, leading to possible hysteresis behavior. As we go from infinite to finite length, we compare the different information given by the three approaches on the possible equilibrium configurations of the rod and their stability. These finite size effects studied here in a 1D elasticity problem could help us guess what are the stability features of other more complicated (2D elastic shells for example) problems for which only the infinite length approach is understood.     

On the equilibrium configurations of an elastically constrained rotating disk: An analytical approach

According to the linear theory of vibration for spinning disks, the backward traveling waves of some of the modes may have zero natural frequency at what are called the critical speeds. At these speeds, the linear equations of motion cannot properly predict the amplitude response of the spinning disk, and nonlinear equations of motion must be used. In this paper, geometrical nonlinear equations of motion based on Von Karman plate theory are employed to study the dynamics of an elastically constrained disk near its critical speeds. A one-mode approximation is used to examine the effect of elastic constraint on the amplitude response. Presenting the equations in a space-fixed coordinate system, this study aims to find closed-form solutions for some of the equilibrium configurations of an elastically constrained spinning disk. Also, the stability of these configurations is studied using analytical techniques. It is shown that below the critical speed, one neutrally stable equilibrium solution exists, while above it, a bifurcation occurs. In this situation, two more branches of equilibrium configurations emerge, one of which is neutrally stable and the other unstable. Closed-form expressions for the bifurcation points are obtained. Due to the effect of an elastic constraint, a bifurcation occurs and the previously neutrally stable equilibrium configuration turns unstable. Also at this bifurcation point, two more branches of equilibrium solutions emerge.     

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele�CShaw Cell

We study the radial movement of an incompressible fluid located in a Hele–Shaw cell rotating at a constant angular velocity in the horizontal plane. Within an analytic framework, local existence and uniqueness of solutions is proved, and it is shown that the unique rotationally invariant equilibrium of the flow is unstable. There are, however, other time-independent solutions: using surface tension as a bifurcation parameter we establish the existence of global bifurcation branches consisting of stationary fingering patterns. The same results can be obtained by fixing the surface tension while varying the angular velocity. Finally, it is shown that the equilibria on a global bifurcation branch converge to a circle as the surface tension tends to infinity, provided they stay suitably bounded.     

A simplified method to investigate the stability of cantilever rod equilibrium forms

A straightforward application of the Jacobi test is used to establish the stability of all equilibrium configurations of a cantilever rod with a free end and sliding end. It is demonstrated that in both cases, only the forms produced by the first critical load can be stable, while all other equilibrium forms are unstable. The results are presented on bifurcation diagrams.     

Non-linear dynamic intertwining of rods with self-contact

Twisted marine cables on the sea floor can form highly contorted three-dimensional loops that resemble tangles. Such tangles or ‘hockles’ are topologically equivalent to the plectomenes that form in supercoiled DNA molecules. The dynamic evolution of these intertwined loops is studied herein using a computational rod model that explicitly accounts for dynamic self-contact. Numerical solutions are presented for an illustrative example of a long rod subjected to increasing twist at one end. The solutions reveal the dynamic evolution of the rod from an initially straight state, through a buckled state in the approximate form of a helix, through the dynamic collapse of this helix into a near-planar loop with one site of self-contact, and the subsequent intertwining of this loop with multiple sites of self-contact. This evolution is controlled by the dynamic conversion of torsional strain energy to bending strain energy or, alternatively, by the dynamic conversion of twist (Tw) to writhe (Wr).     

Flow Bifurcations in a Thin Gap Between Two Rotating Spheres

This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres. The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams. Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789. This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re ₁ , Re ₂ ) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states are accompanied by certain unstable states. Received 26 November 2001 and accepted 10 May 2002 Published online 30 October 2002 Communicated by M.Y. Hussaini     

Piezo rotary and axial vibrator (PRAV) characterization of a fresh coating during its drying

Using a piezo rotary and axial vibrator (PRAV) the viscoelastic properties of a fresh Al₂O₃ coating that is deposited on a well-chosen plate can be measured simultaneously during drying. There are three types of vibrations available to be excited and detected in their resonance modes, by measuring resonance frequencies f _k and half-widths h _k, before and after coating as a function of time: axial vibration, bending vibration to get the evolving Young’s modulus and rotary vibration to follow the viscosity increase during drying of the thin coating. This information is contained in the complex frequency shifts Ω_k = 2Δf _k /f _k + iΔh _k /f _k of the three vibration modes caused by the coating on the plate. A derivation of the relationships, their validation on Newtonian liquids and experimental applications carried out using the PRAV are given in the paper.     

Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist, and held at both endpoints at infinity. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically nonlinear. The problem is formulated as a nonlinear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. We obtain a family of equilibrium solutions depending on a single loading parameter (proportional to applied twisting moment divided by square root of pulling force), which are asymptotically valid in the limit of a loose knot, ε→0. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium.     

Cavitation for incompressible anisotropic nonlinearly elastic spheres

In this paper, the effect ofmaterial anisotropy on void nucleation and growth inincompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid sphere composed of an incompressible homogeneous nonlinearly elastic material which is transversely isotropic about the radial direction. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Closed form analytic solutions are obtained for a specific material model, which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials. In contrast to the situation for a neo-Hookean sphere, bifurcation here may occur locally either to the right (supercritical) or to the left (subcritical), depending on the degree of anisotropy. In the latter case, the cavity has finite radius on first appearance. Such a discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon of structural mechanics. Such dramatic cavitational instabilities were previously encountered by Antman and Negrón-Marrero [3] for anisotropiccompressible solids and by Horgan and Pence [17] forcomposite incompressible spheres.     

Moderately Strong Vorticity in a Bathtub-Type Flow

Theoretical and numerical analysis is performed for an inviscid axisymmetric vortical bathtub-type flow. The level of vorticity is kept high so that the image of the flow on the radial–axial plane (r–z plane)is not potential. The most significant findings are: (1) the region of validity of the strong vortex approximation is separated from the drain by a buffer region, (2) the power-law asymptote of the stream function, specified by Δψ∼r ^4/3Δz, appears near the axis when vorticity in the flow is sufficiently strong and (3) the local Rossby number in the region of the power-law is not very sensitive to the changes of the initial vorticity level in the flow and the global Rossby number. Received 3 April 2000 and accepted 29 September 2000     

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 Similarity Subgrid Model for Premixed Turbulent Combustion

The accuracy of large-eddy simulation (LES) of a turbulent premixed Bunsen flame is investigated in this paper. To distinguish between discretization and modeling errors, multiple LES, using different grid sizes h but the same filterwidth Δ, are compared with the direct numerical simulation (DNS). In addition, LES using various values of Δ but the same ratio Δ/h are compared. The chemistry in the LES and DNS is parametrized with the standard steady premixed flamelet for stochiometric methane-air combustion. The subgrid terms are closed with an eddy-viscosity or eddy-diffusivity approach, with an exception of the dominant subgrid term, which is the subgrid part of the chemical source term. The latter subgrid contribution is modeled by a similarity model based upon 2Δ, which is found to be superior to such a model based upon Δ. Using the 2Δ similarity model for the subgrid chemistry the LES produces good results, certainly in view of the fact that the LES is completely wrong if the subgrid chemistry model is omitted. The grid refinements of the LES show that the results for Δ = h do depend on the numerical scheme, much more than for h = Δ/2 and h = Δ/4. Nevertheless, modeling errors and discretization error may partially cancel each other; occasionally the Δ = h results were more accurate than the h ≤ Δ results. Finally, for this flame LES results obtained with the present similarity model are shown to be slightly better than those obtained with standard β-pdf closure for the subgrid chemistry.     

Euler-Lagrange Equations for Nonlinearly Elastic Rods with Self-Contact

 We derive the Euler-Lagrange equations for nonlinearly elastic rods with self-contact. The excluded-volume constraint is formulated in terms of an upper bound on the global curvature of the centre line. This condition is shown to guarantee the global injectivity of the deformation of the elastic rod. Topological constraints such as a prescribed knot and link class to model knotting and supercoiling phenomena as observed, e.g., in DNA-molecules, are included by using the notion of isotopy and Gaussian linking number. The bound on the global curvature as a nonsmooth side condition requires the use of Clarke's generalized gradients to obtain the explicit structure of the contact forces, which appear naturally as Lagrange multipliers in the Euler-Lagrange equations. Transversality conditions are discussed and higher regularity for the strains, moments, the centre line and the directors is shown. (Accepted December 20, 2002) Published online April 8, 2003 Communicated by S. S. Antman     

On the stability of an elastic column positioned on an elastic half space

Summary Stability of a heavy elastic column loaded by a concentrated force at the top is analysed. It is assumed that the column is fixed to a rigid circular plate that is positioned on a homogeneous, isotropic, linearly elastic half-space. The constitutive equations for the column are assumed in the form that allows axial compressibility and takes into account the influence of shear stresses. It is shown that eigenvalues of the linearized equations determine the bifurcation points of the full non-linear system of equilibrium equations. Also, the type of bifurcation at the lowest eigenvalue is examined and shown that it could be both super-and sub-critical. The post-critical shape of the column is determined by numerical integration of the equilibrium equations. Received 13 June 1998; accepted for publication 12 November 1998