Bending instabilities of elastic tubes

The present paper examines instabilities of long thin elastic tubes. Both initially straight and initially bent tubes are analyzed under in-plane bending. Tube response, a combination of ovalization instability and bifurcation instability (buckling), is investigated using a nonlinear finite element (FE) technique, which employs polynomial functions in the longitudinal tube direction and trigonometric functions to describe cross-sectional deformation. It is demonstrated that the interaction between the two instability modes depends on the value and the sign of the initial tube curvature. The ovalization of initially bent tubes is examined in detail and, in particular, the case of opening moments. Furthermore, the paper emphasizes on bifurcation instability. It is shown that buckling may occur prior to or beyond the ovalization limit point, depending on the value of the initial curvature. Using the nonlinear FE formulation, the location of bifurcation on the primary path is detected, post-buckling equilibrium paths are traced, and the corresponding wavelengths of the buckled configurations are calculated. Moreover, results over a wide range of initial curvature values are presented, extending the findings of previous works. Finally, several analytical approaches, introduced in previous research works, are also employed to estimate the moments causing ovalization and bifurcation instability. These approaches are based on nonlinear flexible shell theory or simplified ring analysis. The efficiency and accuracy of those analytical methods with respect to the nonlinear FE formulation are examined.     

Collapse of cylindrical, elastic tubes under combined bending, pressure and axial loads

A study of the non-linear pre-buckling state and the bifurcation and initial post-buckling behaviour of infinitely long, cylindrical, elastic tubes subjected to bending, pressure and axial loads is presented. The collapse behaviour is analysed by determining both the limit load and the possibility and significance of axial wrinkling of the compressed region of the shell prior to the limit load.     

Yield anisotropy effects on buckling of circular tubes under bending

Relatively thin-walled tubes bent into the plastic range buckle by axial wrinkling. The wrinkles initially grow stably but eventually localize and cause catastrophic failure in the form of sharp local kinking. The onset of axial wrinkling was previously established by bifurcation analyses that use instantaneous deformation theory moduli. The curvatures at bifurcation were predicted accurately, but the wrinkle wavelengths were consistently longer than measured values. The subject is revisited with the aim of resolving this discrepancy. A set of new bending experiments is conducted on aluminum alloy tubes. The results are shown to be in line with previous ones. However, the tubes used were found to exhibit plastic anisotropy, which was measured and characterized through Hill’s quadratic anisotropic yield function. The anisotropy was incorporated in the flow theory used for prebuckling and postbuckling calculations as well as in the deformation theory used for bifurcation checks. With the anisotropy accounted for, calculated tube responses are found to be in excellent agreement with the measured ones while the predicted bifurcation curvatures and wrinkle wavelengths fall in line with the measurements also. The postbuckling response is established using a finite element model of a tube assigned an initial axisymmetric imperfection with the calculated wavelength. The response develops a limit moment that is followed by a sharp kink that grows while the overall moment drops. The curvature at the limit moment agrees well with the experimental onset of failure. From parametric studies of the various instabilities it is concluded that, for optimum predictions, anisotropy must be incorporated in both bifurcation buckling as well as in postbuckling calculations.     

Stability of long transversely-isotropic elastic cylindrical shells under bending

The present paper investigates buckling of cylindrical shells of transversely-isotropic elastic material subjected to bending, considering the nonlinear prebuckling ovalized configuration. A large-strain hypoelastic model is developed to simulate the anisotropic material behavior. The model is incorporated in a finite-element formulation that uses a special-purpose “tube element”. For comparison purposes, a hyperelastic model is also employed. Using an eigenvalue analysis, bifurcation on the prebuckling ovalization path to a uniform wrinkling state is detected. Subsequently, the postbuckling equilibrium path is traced through a continuation arc-length algorithm. The effects of anisotropy on the bifurcation moment, the corresponding curvature and the critical wavelength are examined, for a wide range of radius-to-thickness ratio values. The calculated values of bifurcation moment and curvature are also compared with analytical predictions, based on a heuristic argument. Finally, numerical results for the imperfection sensitivity of bent cylinders are obtained, which show good comparison with previously reported asymptotic expressions.     

Curvature effects on axially compressed buckling of a small-diameter double-walled carbon nanotube

The curvature effects of interlayer van der Waals (vdW) forces on axially compressed buckling of a double-walled carbon nanotube (DWNT) of diameter down to 0.7 nm are studied. Unlike most existing models which assume that the interlayer vdW pressure at a point between the inner and outer tubes depends merely on the change of the interlayer spacing at that point, the present model considers the dependence of the interlayer vdW pressure on the change of the curvatures of the inner and outer tubes at that point. A simple expression is derived for the curvature-dependence of the interlayer vdW pressure in which the curvature coefficient is determined. Based on this model, an explicit formula is obtained for the axial buckling strain. It is shown that neglecting the curvature effect alone leads to an under-estimate of the critical buckling strain with a relative error up to −7%, while taking the average radius of two tubes as the representative radius and the curvature effect leads to an over-estimate of the critical buckling strain with a relative error up to 20% when the inner radius downs to 0.35 nm. Therefore, the curvature effects play a significant role in axially compressed buckling problems only for DWNTs of very small radii. In addition, our results show that the effect of the vdW interaction pressure prior to buckling of DWNTs under pure axial stress is small enough and can be negligible whether the vdW interaction curvature effects are neglected or not.     

高浓度氩气稀释气体爆轰波临界管径和临界间距关系

建立圆管及环形管道系统研究临近极限下爆轰波在管道内传播失效机理。选用C₂H₂+2.5O₂+70%Ar气体,采用光纤探针测量爆轰波在管道内传播速度,用烟迹法记录管道内爆轰波胞格结构。结果表明:初始压力远大于爆轰极限压力时,爆轰波在管道内以稳定速度传播;随着初始压力的减小,爆轰波速度逐渐降低;当初始压力一定时,爆轰波速度随着管道尺寸的减小而逐渐减小;当初始压力达到临界压力时,爆轰波在进入到管道内后其速度会逐渐衰减直至爆轰波完全失效。对于不同几何尺寸的圆管与环管,通过引入无量纲参数d/λ及w /λ(d为圆管管径,w为环管间距,λ为爆轰胞格尺寸)得出,爆轰波在管道内传播的临界圆管直径为环形间距的2倍,与理论模型结果相吻合,验证了稳态气体基于爆轰波波面曲率的失效机理。     

不同角度分叉管道内氢气-空气爆轰传播特性

在3种角度分叉管道内开展化学计量比氢气-空气爆轰实验,采用自制的火焰传感器和烟迹法分别获得了爆轰波传播速度和胞格结构,探究了不同角度管道分叉对爆轰传播的影响。结果表明:氢气-空气爆轰在经过分叉三通时受分叉口稀疏波影响导致爆轰波衰减解耦,但随着入射激波与下游管道壁面碰撞,逐渐由规则反射向马赫反射转变,最终完成重起爆过程。其中,直通支管内爆轰衰减主要受支管入口面积的影响,随着分叉角度增大,入口面积减小,爆轰衰减程度和重起爆距离也随之减小;而分叉支管内,爆轰衰减受支管入口面积与入口渐扩程度共同影响,但随着分叉角度的增大,入口面积变为主要影响因素。不同角度分叉管内的实验结果均表明,初始压力升高能显著提高爆轰稳定性,从而削弱分叉几何结构的影响。     

Mechanics and full-field deformation study of the Ring Hoop Tension Test

We investigate the mechanics of the Ring Hoop Tension Test (RHTT), as a means to assess the properties of anisotropic tubes in the hoop direction. This test involves placing a ring extracted from the tube over two close-fitting D-shaped mandrels that are then parted using a universal testing machine. Since the curvature of the ring does not change during loading, the ring undergoes only stretching. We determine the effects of contact pressure, radial stress, and friction between the tube and mandrels using FEA simulations. The effects of the pre-existing thickness eccentricity and of the specimen-making procedure on the recorded RHTT response are also assessed with a combination of experiments and analysis. We tested tubes from Al-6061-T4 with a diameter/thickness ratio of 20. We found that as the friction increases beyond 0.1–0.15, the state of uniaxial tension is deteriorated, indicating that care must be taken to minimize the tube-mandrel friction. We determined that although these tubes have a relatively large thickness eccentricity (±4% of the nominal thickness), this had no effect on the recorded results. We showed that the tubes should not be turned to remove that eccentricity, as the machining process induces damage that is noticeable in the results. We found that the contact pressure and the contact-induced radial stress cause limited deviations from uniaxial tension, comparable to the case of a tube under axial load and internal pressure which is often used for assessing the material properties in the hoop direction. Central in our analyses is the knowledge of the hoop strain field, which was assessed using 3D Digital Image Correlation. We propose a data reduction procedure for RHTT that accounts for all the above effects. Finally, with all effects accounted for, we establish the anisotropy of the extruded Al-6061-T4 tubes studied.     

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.     

钢管混凝土圆弧桁架拱平面内徐变稳定分析

核心混凝土的徐变会增加钢管混凝土拱肋的屈曲前变形,降低结构的稳定承载力,因此只有计入屈曲前变形的影响,才能准确得到钢管混凝土拱的徐变稳定承载力。基于圆弧形浅拱的非线性屈曲理论,采用虚功原理,建立了考虑徐变和剪切变形双重效应的管混凝土圆弧桁架拱的平面内非线性平衡方程,求得两铰和无铰桁架拱发生反对称分岔屈曲和对称跳跃屈曲的徐变稳定临界荷载。探讨了钢管混凝土桁架拱核心混凝土徐变随修正长细比、圆心角和加载龄期对该类结构弹性稳定承载力的影响,为钢管混凝土桁架拱长期设计提供理论依据。     

Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes

Bifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers.Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.     

In-plane stability of arches

Classical buckling theory is mostly used to investigate the in-plane stability of arches, which assumes that the pre-buckling behaviour is linear and that the effects of pre-buckling deformations on buckling can be ignored. However, the behaviour of shallow arches becomes non-linear and the deformations are substantial prior to buckling, so that their effects on the buckling of shallow arches need to be considered. Classical buckling theory which does not consider these effects cannot correctly predict the in-plane buckling load of shallow arches. This paper investigates the in-plane buckling of circular arches with an arbitrary cross-section and subjected to a radial load uniformly distributed around the arch axis. An energy method is used to establish both non-linear equilibrium equations and buckling equilibrium equations for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained. Approximations to the symmetric buckling of shallow arches and formulae for the in-plane anti-symmetric bifurcation buckling load of non-shallow arches are proposed, and criteria that define shallow and non-shallow arches are also stated. Comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate, and that classical buckling theory can correctly predict the in-plane anti-symmetric bifurcation buckling load of non-shallow arches, but overestimates the in-plane anti-symmetric bifurcation buckling load of shallow arches significantly.     

Relative Permeability Analysis of Tube Bundle Models,Including Capillary Pressure

The analytical equations for calculating two-phase flow, including local capillary pressures, are developed for the bundle of parallel capillary tubes model. The flow equations that are derived were used to calculate dynamic immiscible displacements of oil by water under the constraint of a constant overall pressure drop across the tube bundle. Expressions for averaged fluid pressure gradients and total flow rates are developed, and relative permeabilities are calculated directly from the two-phase form of Darcy's law. The effects of pressure drop and viscosity ratio on the relative permeabilities are discussed. Capillary pressure as a function of water saturation was delineated for several cases and compared to a steady-state mercury-injection drainage type of capillary pressure profile. The bundle of serial tubes model (a model containing tubes whose diameters change randomly at periodic intervals along the direction of flow), including local Young-Laplace capillary pressures, was analyzed with respect to obtaining relative permeabilities and macroscopic capillary pressures. Relative permeabilities for the bundle of parallel tubes model were seen to be significantly affected by altering the overall pressure drop and the viscosity ratio; relative permeabilities for the bundle of serial tubes were seen to be relatively insensitive to viscosity ratio and pressure, and were consistently X-like in profile. This work also considers the standard Leverett (1941) type of capillary pressure versus saturation profile, where drainage of a wetting phase is completed in a step-wise steady fashion; it was delineated for both tube bundle models. Although the expected increase in capillary pressure at low wetting-phase saturation was produced, comparison of the primary-drainage capillary pressure curves with the pseudo-capillary pressure profiles, that are computed directly using the averaged pressures during the displacements, revealed inconsistencies between the two definitions of capillary pressure.     

Postbuckling of double-walled carbon nanotubes with temperature dependent properties and initial defects under combined axial and radial mechanical loads

Buckling and postbuckling analysis is presented for a double-walled carbon nanotube subjected to combined axial and radial loads in thermal environments. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual orthotropic shell with presence of van der Waals interaction forces and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity and include thermal effects. Temperature-dependent material properties, which come from molecular dynamics simulations, and initial point defect, which is simulated as a dimple on the tube wall, are both taken into account. A singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, double-walled carbon nanotubes subjected to combined axial and radial mechanical loads under different sets of thermal environments. The results reveal that temperature change only has a small effect on the postbuckling behavior of the double-walled carbon nanotube. The axially-loaded double-walled carbon nanotube subjected to radial pressure has an unstable postbuckling path, and the structure is imperfection–sensitive. In contrast, the pressure-loaded double-walled carbon nanotube subjected to axial compression has a very weak “snap-through” postbuckling path, and the structure is virtually imperfection–insensitive.     

Instability and failure of internally pressurized ductile metal cylinders

Forlong, ductile, thick-walled tubes under internal pressure instabilities and final failure modes are studied experimentally and theoretically. The test specimens are closed-end cylinders made of an aluminum alloy and of pure copper and the experiments have been carried out for a number of different initial external radius to internal radius ratios. The experiments show necking on one side of the tubes at a stage somewhat beyond the maximum internal pressure. All tubes, except for one aluminum alloy tube, failed by shear fracture under decreasing pressure. The aluminum alloy tubes exhibited localized shear deformations in the neck region prior to fracture and also occasionally surface wave instabilities. The numerical investigation is based on an elastic-plastic material model for a solid that develops a vertex on the yield surface, using representations of the uniaxial stress-strain curves found experimentally. In contrast to the simplest flow theory of plasticity this material model predicts shear band instabilities at a realistic level of strain. A rather sharp vertex is used in the material model for the aluminum alloy, while a more blunt vertex is used to characterize copper. The theoretically predicted bifurcation into a necking mode, the cross-sectional shape of the neck, and finally the initiation and growth of shear bands from the highly strained internal surface in the neck region are in good agreement with the experimental observations.     

Effect of the nonlinear pre-buckling state on the bifurcation point of conical shells

The effect of pre-buckling nonlinearity on the bifurcation point of a conical shell is examined on the basis of three shell theories: Donnell’s, Sanders’ and Timoshenko’s. The eigenvalue problem is solved iteratively about the nonlinear equilibrium state up to the bifurcation point. A new algorithm is presented for the real buckling behavior, dispensing with the need to cover the entire nonlinear pattern. This algorithm is very important for structures characterized by a softening process, in which the pre-buckling nonlinearity depresses the buckling level relative to the classical one.The procedure involves nonlinear partial differential equations, which are separated into two sets (using the perturbation technique) for the pre-buckling and buckling states, respectively and solved with the variable expanded in Fourier series in the circumferential direction, and by finite differences in the axial direction. A general computer code was developed and used in studying the effect of the pre-buckling nonlinearity on the buckling level, of the shell under axial compression, in the context of the three shell theories.     

考虑范德华力曲率效应的双壁碳纳米管外压屈曲

针对双壁碳纳米管外压屈曲问题,研究了层间范德华力的曲率效应对临界外压的影响。应用弹性双层圆柱壳模型,考虑层间范德华力不仅与层间距有关而且与挠度曲率的变化有关,导出了外压屈曲临界压力解析公式。计算得出在不同半径、不同长细比下,外压屈曲临界压力的数值结果,并与经典壳的结果和忽略范德华力曲率效应的结果做了比较。结果显示,对于小半径的双壁碳纳米管曲率效应对外压屈曲有效明显的影响。     

边界条件对甲烷预混气爆轰特性的影响

通过实验研究及数字化处理研究了边界条件对CH₄预混气体爆轰特性的影响。在内径为63.5、50.8 mm圆柱形管道及长方体管道进行爆轰实验,得到胞格结构和爆轰速度曲线。烟膜数字化处理量化了预混气体的爆轰不稳定性,并计算出胞格尺寸。3种管道内测得的平均爆轰速度与CJ速度接近,边界条件的影响不明显。分析爆轰速度曲线发现,极限压力受到边界条件的影响,?50.8和?63.5 mm管道内预混气的极限压力分别为5和4.05 kPa,即随着管径增大,爆轰极限压力降低。数字化处理所得不同管道内烟膜轨迹的不规则程度无明显差别,因此可以认为不稳定性是预混气固有的性质。在相同爆轰初始压力下,管径增大,胞格数量变多,表明爆轰传播时爆轰螺旋头数增多以维持传播。     

Localization and propagation of curvature under pure bending in steel tubes with Lüders bands

The paper examines the plastic bending of steel tubes exhibiting Lüders bands through a combination of experiments and analyses. In pure bending experiments on tubes with diameter-to-thickness ratio of 18.8 tested under end-rotation control, following the elastic regime the moment initially traced a somewhat ragged plateau. At the beginning of the plateau Lüders bands appeared on the tension and compression sides of the cross section and simultaneously the curvature localized in one or two short zones while the rest of the tube maintained a much lower curvature. As the rotation of the ends was increased, one of the higher curvature zones spread at a nearly steady rate, affecting an increasingly larger part of the tube. When the whole tube was deformed to the higher curvature, the moment started to gradually increase while the tube deformed uniformly. A moment maximum was eventually attained and the structure failed by localized diffuse ovalization without any apparent effect from the initial Lüders bands-induced propagating instability. The problem was analyzed using 3D finite elements with a fine mesh. The material was modeled as an elastic–plastic solid with an up–down–up response over the extent of the Lüders strain, followed by hardening. The calculated response reproduced all major structural events observed experimentally including the initiation of the Lüders deformation, the moment plateau that followed, its extent, and the curvature localization and propagation associated with it. As in the experiments, once the high curvature extended over the whole tube length, the response of the tube became stable and the curvature uniform. With further bending the increasing ovalization induced a limit moment at a very high curvature.     

Initiation of aneurysms as a mechanical bifurcation phenomenon

Recent studies on localized bulging in inflated membrane tubes have shown that the initiation pressure for the onset of localization is determined through a bifurcation condition. This kind of localization has also been shown to be much more sensitive to geometrical and material imperfections than classical sub-critical bifurcation into periodic patterns. We use these results to show that the initial formation of aneurysms in human arteries may also be modeled as a bifurcation phenomenon. This bifurcation interpretation could provide a theoretical framework under which different mechanisms leading to, or reducing the risk of, aneurysm formation can be assessed in a systematic manner. In particular, this could potentially help in assessing the integrity of aneurysm repairs.