Flexural instabilities of elastic rods

In the framework of Antman's theory of elastic rods, it is shown that the behavior of a rod subject to end couples is ruled by a non-linear ordinary differential equation whose solutions describe the instability phenomena observed in severe bending tests of tubes, such as ovalization and necking of the cross section.

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Sommario Nell'ambito della teoria di Antman per le travi elastiche, si mostra che il comportamento di una trave soggetta a coppie concentrate agli estremi è regolato da un'equazione differenziale ordinaria non lineare, le cui soluzioni descrivono fenomeni di instabilità osservati nelle prove di grande flessione di tubi, quali l'ovalizzazione e la strizione della sezione trasversale.

     

Purely elastic instabilities in three-dimensional cross-slot geometries

Creeping and low Reynolds number flows of an upper-convected Maxwell (UCM) fluid are investigated numerically in a three-dimensional orthogonal cross-slot geometry. We analyze two different flow configurations corresponding to uniaxial extension and biaxial extension, and assess the effects of extensional flow type, Deborah and Reynolds numbers on flow dynamics near the interior stagnation point. Using these two flow arrangements the amount of stretch and compression near the stagnation point can be varied, providing further insights on the viscoelastic flow instability mechanisms in extensionally dominated flows with an interior stagnation point. The uniaxial extensional flow arrangement leads to the onset of a steady flow asymmetry, followed by a second purely elastic flow instability that generates an unsteady flow at higher flow rates. On the other hand, for the biaxial extension flow configuration a symmetric flow is observed up to the critical Deborah number when the time-dependent purely elastic instability sets in, without going through the steady symmetric to steady asymmetric transition.     

Bifurcation of rotating thick-walled elastic tubes

The deformation of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthai deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (Haughton and Ogden, 1980) for membrane tubes.Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function.Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.     

Buckling instabilities in dilute polymer solution elastic strands

A microfluidic oscillatory cross-slot flow is used to visualize birefringent strands in dilute polystyrene solutions with high temporal resolution, focusing specifically on reversals in the flow direction. Due to polymer conformation hysteresis, the elastic strands are slow to relax and demonstrate a compressive modulus, resulting in a “buckling instability.” An elastic loop of birefringence forms in an exit channel of the cross, seeding the development of a new birefringent strand in the perpendicular direction. These observations have major relevance to cyclic flows of polymer solutions (e.g., porous media flows where stagnation points are present) and for finitely extensible nonlinear elastic dumbbell modeling of such flows.     

Bending of beams on three-parameter elastic foundation

The bending of a Timoshenko beam resting on a Kerr-type three-parameter elastic foundation is introduced, its governing differential equations are formulated and analytically solved, and the solutions are discussed and applied to particular problems. Parametric analyses of elastically supported beams of infinite and finite length are carried out and comparisons are made between one, two or three-parameter foundation models and more accurate 2D finite element models. In order to estimate the necessary soil parameters, an analytical procedure based on the modified Vlasov model is proposed. The presented solutions and applications show the superiority of the Kerr-type foundation model compared to one or two-parameter models.     

Slumping instabilities of elastic membranes holding liquids and gases

This paper treats the plane-strain free-boundary problem describing the equilibrium of a closed cylindrical non-linearly elastic membrane that contains a heavy liquid and a weightless compressible gas and that sits upon a rigid horizontal plane. The material points of the membrane in contact with the rigid plane, the liquid, and the gas are unknowns of the problem. We carefully formulate the geometrically exact equations, and then show that solutions must have attractive regularity, symmetry, and similarity properties, which are not intuitively obvious. There is an interesting interplay between the material behavior and the live pressure loadings generated by gases and heavy liquids. The properties of the equilibrium states depend crucially on the material behaviour of the membrane. In particular, for certain materials the number of equilibrium states changes abruptly as certain parameters describing the liquid and gas change. Thus the system can suffer instabilities of snapping type (with their attendant hysteresis effects), which we term slumping instabilities.     

Bending and stability analysis of gradient elastic beams

The problems of bending and stability of Bernoulli–Euler beams are solved analytically on the basis of a simple linear theory of gradient elasticity with surface energy. The governing equations of equilibrium are obtained by both a combination of the basic equations and a variational statement. The additional boundary conditions are obtained by both variational and weighted residual approaches. Two boundary value problems (one for bending and one for stability) are solved and the gradient elasticity effect on the beam bending response and its critical (buckling) load is assessed for both cases. It is found that beam deflections decrease and buckling load increases for increasing values of the gradient coefficient, while the surface energy effect is small and insignificant for bending and buckling, respectively.     

Purely elastic interfacial instabilities in superposed flow of polymeric fluids

Purely elastic interfacial stability of superposed plane Poiseuille flow of polymeric liquids has been investigated utilizing both asymptotic and numerical techniques. It is shown that these instabilities are caused by an unfavorable jump in the first normal stress difference across the fluid interface. To determine the significance of these instabilities in finite experimental geometries, a comparison between the maximum growth rates of purely elastic instabilities with instabilities driven primarily by a viscosity or a combined viscosity and elasticity difference is made. Based on this comparison, it is shown that purely elastic interfacial instabilities can play a major role in superposed flow of polymeric liquids in finite experimental geometries.     

Bending and flexure of cylindrically monoclinic elastic cylinders

We consider a circular cylinder of linearly elastic material with cylindrically monoclinic material symmetry. This represents a model for a helically wound composite cable or wire rope. The elastic moduli are allowed to be arbitrary functions of the radius r. The cylinder undergoes deformation in which the axis of the cylinder is bent into a plane quartic curve. For the resulting stress field, we obtain exact integrals of the equilibrium equations, and derive simplified expressions for the shear stress resultants and bending moments.     

Bending of elastic plates with a physically nonlinear inclusion

An infinite elastic isotropic plate with an elliptical, physically nonlinear inclusion loaded at infinity by uniformly distributed moments is considered. Surface loads are absent. The problem of the stress-strain state of the plate is solved in a closed form. It is shown that, for reasonably general stress-strain relations for the inclusion, the bending-moment field (and the corresponding curvatures) in the inclusion is homogeneous. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 152–157, November–December, 2006.     

Bending and buckling of thin strain gradient elastic beams

Bending of strain gradient elastic thin beams is studied adopting Bernoulli-Euler principle. Simple linear strain gradient elastic theory with surface energy is employed. The governing beam equations with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin beams. Those terms are missing from the existing strain gradient beam theories. Those terms increase highly the stiffness of the thin beam. The buckling problem of the thin beams is also discussed.     

Solitary waves in initially stressed thin elastic tubes

In the present work, we study the propagation of non-linear waves in an initially stressed thin elastic tube filled with an inviscid fluid. Considering the physiological conditions of the arteries, in the analysis, the tube is assumed to be subjected to a uniform inner pressure P₀ and an axial stretch ratio λ_z. It is assumed that due to blood flow, a finite dynamical displacement field is superimposed on this static field and, then, the non-linear governing equations of the elastic tube are obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the longwave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. It is observed that the present model equations give two solitary wave solutions. The results are also discussed for some elastic materials existing in the literature.     

The nonlinear dynamics of elastic tubes conveying a fluid

The Kirchhoff equations for elastic tubes are modified to include the effect of fluid flow. Using the techniques of linear and nonlinear analysis specially developed for the Kirchhoff equations, the effect of the fluid flow on the basic twist-to-writhe instability is investigated. The results suggest an intriguing modification of the bifurcation threshold due to the flow. Beyond threshold the buckled tube acquires a slight curvature which modifies the flow rate and results in a correction to nonlinearity of the amplitude equation governing the deformation dynamics.     

The behavior of elastic tubes in a cohesion-free continuum

A model is made of a plain cross section of a tube (tunnel lining) in a cohesion-free continuum (rolling material, sand) which is represented by steel rollers of different diameters. The compactibility of sand in this model is represented by rubber inserts around the tunnel lining. The stress of the tube is measured by photoelasticity. These experiments are the start of a large program of investigation to calculate the stress deformation and buckling of elastic tubes under different loading conditions in rolling material.     

Finite elastic deformations of transversely isotropic circular cylindrical tubes

We consider the finite radially symmetric deformation of a circular cylindrical tube of a homogeneous transversely isotropic elastic material subject to axial stretch, radial deformation and torsion, supported by axial load, internal pressure and end moment. Two different directions of transverse isotropy are considered: the radial direction and an arbitrary direction in planes normal locally to the radial direction, the only directions for which the considered deformation is admissible in general. In the absence of body forces, formulas are obtained for the internal pressure, and the resultant axial load and torsional moment on the ends of the tube in respect of a general strain-energy function. For a specific material model of transversely isotropic elasticity, and material and geometrical parameters, numerical results are used to illustrate the dependence of the pressure, (reduced) axial load and moment on the radial stretch and a measure of the torsional deformation for a fixed value of the axial stretch.     

弹性结构的分岔点失稳和极值点失稳

以浅桁架为例, 介绍了弹性结构两类不同的失稳形态: 分岔点失稳和极值点失稳. 浅桁架失稳形态与斜杆的柔度λ和倾角α0 有关. 当α0λ 3/4π 时为极值点失稳, 发生突跳; 否则为分岔点型失稳.