Finite-element simulation of the lateral compression of aluminium tube between rigid plates

The lateral compression of aluminium and clad tubes owing to a large deformation is examined by an incremental elasto-plastic finite-element method based on an updated Lagrangian formulation in which a sliding-sticking friction mode is specially considered. It is mainly expected to predict the buckling process and load–deflection curves for energy dissipation capacity during the design stage, before trials. The high non-linearity of the process due to geometric changes, the inelastic constitutive behavior, and the deformation-dependent boundary conditions are taken into account in an incremental manner. A static explicit approach to the solution is applied, tangent stiffness matrix equation is solved without iteration and the r_min technique is employed to limit the step size to linear relation. The simulated load–deflection curve agrees with a published experimental result. The predicted geometries of the compressed tube clearly demonstrate the processes of the formation of buckling until unloading. The effects of various parameters of the process, such as elastic modulus, strain hardening exponent, tube thickness, friction coefficient and configurations of the clad tube, on the occurrence of buckling of tube are discussed and interpreted in simulation. The present work may be expected to improve the understanding of the buckling mechanism of lateral compression.     

Collapse mode transitions of thin tubes with wall thickness, end condition and shape eccentricity

Transition of deformation mode shapes of round aluminum tubes from axisymmetric concertina to non-axisymmetric diamond mode have been studied with varying tube wall thickness, boundary conditions and tube shape eccentricities. Quasi-static axial compression experiments were carried out on as received aluminum tubes and tubes with wall thickness eccentricity, incorporated by off center machining. Tubes were of D/t=29 and L/D=1.4. The numerical simulation of the collapse phenomenon has been undertaken using a static non-linear finite element analysis in ANSYS with a fine mesh discretization of the tube domain and small incremental displacements as load steps. Convergence studies for the finite element model with respect to load step size and mesh density have also been established. The numerical results are found to compare well with the experimental load compression and energy absorption responses both for the axisymmetric concertina and non-axisymmetric diamond collapse modes. Having validated the numerical model with experiments, it has been used to undertake a systematic study of the load–deformation characteristics, energy absorption response and collapse mode transition of the tubes in varying configurations of wall thickness, shape and inplane boundary condition eccentricities. Dependence of tube collapse characteristics and collapse mode transitions on such eccentricities have been discussed.     

Plastic buckling of circular tubes under axial compression—part II: Analysis

In the second part of this study, the evolution of uniform axisymmetric wrinkling in axially compressed cylinders is modeled using the principle of virtual work. A version of this formulation also allows localization of wrinkling. The model domain is assigned an initial axisymmetric imperfection of a chosen amplitude and the wavelength yielded by the first bifurcation check. The solution correctly simulates the growth of wrinkles and results in a limit load instability. The limit strain is influenced by the amplitude of the imperfection. Beyond the limit load, wrinkling tends to localize, eventually leading to local folding.The possibility of bifurcation of the axisymmetric solution to non-axisymmetric buckling modes is examined by using a dedicated bifurcation check. The bifurcation check was found to yield such buckling modes correctly. The evolution of such buckling modes is simulated by a separate non-axisymmetric model assigned imperfections with axisymmetric and nonaxisymmetric components. The domain analyzed is one characteristic wavelength long (2λ_C). Initially, compression activates mainly axisymmetric deformation. In the neighborhood of the bifurcation point, non-axisymmetric deformation starts to develop, eventually leading to a limit load instability. Experimental responses were simulated with accuracy by assigning appropriate values to the two imperfection amplitudes. Prediction of the limit strains for the whole range of diameter-to-thickness ratios (D/t) considered in the experiments was achieved by making the amplitude of the non-axisymmetric imperfection proportional to (D/t)²/m³ (m is the circumferential wavenumber). Matching all aspects of the experiments required inclusion of the anisotropy measured in the tubes tested through Hill's yield criterion in all models.     

Hydroforming of aluminum extrusion tubes for automotive applications. Part I: buckling, wrinkling and bursting analyses of aluminum tubes

Three possible failure modes have been identified in tube hydroforming: buckling, wrinkling and bursting. A general theoretical framework is proposed for analyzing these failure modes as an elastoplastic bifurcation problem. This framework enables advanced yield criteria and various strain-hardening laws to be readily incorporated into the analysis. The effect of plastic deformation on the geometric instability in tube hydroforming, such as global buckling, axisymmetric wrinkling and asymmetric wrinkling, is precisely treated by using the exact plane stress moduli tensor. A mathematical formulation for predicting the localized condition for bursting failure is established herein. Furthermore, the critical conditions governing the onset of buckling, axisymmetric wrinkling and asymmetric wrinkling are derived in closed-form expressions for the critical axial compressive stresses. Closed-form solutions for the critical stress are developed based on Neale–Hutchinson's constitutive equation and an assumed deformation theory of plasticity. It is demonstrated that the onset of asymmetric wrinkling always requires a higher critical axial compressive stress than the axisymmetric one under the context of tube hydroforming with applied internal pressure and hence the asymmetric wrinkling mode can be excluded in the analysis of tube hydroforming. Parametric studies show that buckling and axisymmetric wrinkling are strongly dependent on geometric parameters such as t₀/r₀ and r₀/ℓ₀, and that axisymmetric wrinkling is the predominant mode for short tubes while global buckling occurs for long slender tubes.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.     

A study of the plastic buckling of axially compressed cylindrical shells with a thick-shell theory

A thick shell theory is used to calculate the critical load of plastic buckling of axially compressed cylindrical shells. The buckling equations are derived with the principle of virtual work on the basis of a transverse shear deformable displacement field. The deformation theory of plasticity is used for constitutive equations. To fit the uniaxial stress–strain curve, the Ramberg–Osgood equation is used. In the numerical examples special attention is paid to the dependence of the buckling mode on the ratios of radius to thickness R/h and length to radius L/R. This dependence divides the (R/h,L/R)-plane into simply connected regions each of which corresponds to a buckling mode. These regions form a “buckling mode map”.     

管轴压成形失稳起皱力学过程的研究

管轴压成形的先进性、优势及成形过程的成功实施受到变形区开裂和传力区起皱等失稳因素的制约,为此采用理论解析与试验相结合的方法,对失稳起皱的力学过程、影响因素及影响规律进行了研究,并给出了失稳判据,以实现稳定成形时变形参数的取值范围的确定。研究表明:成形过程的失稳起皱受到几何参数、材料参数及摩擦边界条件的共同作用。其中,模具圆角曲率半径是决定性因素,并且存在一个最大和最小圆角半径区域使成形过程顺利进行而不发生失稳起皱;增大管坯材料硬化指数,减小相对厚度及接触面的摩擦,有利于减少失稳的发生。对牌号为5052,原始管平均直径d_0=31.0mm,41.0mm铝管的应用表明所提出的判据是可靠实用的。     

Plastic buckling of circular tubes under axial compression—part I: Experiments

Elastic buckling of cylindrical shells due to axial compression results in sudden and catastrophic failure. By contrast, for thicker shells that buckle in the plastic range, failure is preceded by a cascade of events, where the first instability and failure can be separated by strains of 1–5%. The first instability is uniform axisymmetric wrinkling that is typically treated as a plastic bifurcation. The wrinkle amplitude gradually grows and, in the process, reduces the axial rigidity of the shell. This eventually leads to a limit load instability, beyond which the cylinder fails by localized collapse. For some combinations of geometric and material characteristics, this limit load can be preceded by a second bifurcation that involves a non-axisymmetric mode of deformation. Again, this buckling mode localizes resulting in failure.The problem is revisited using a combination of experiments and analysis. In Part I, we present the results of an experimental study involving stainless steel specimens with diameter-to-thickness ratios between 23 and 52. Fifteen specimens were designed and machined to achieve uniform loading conditions in the test section. They were subsequently compressed to failure under displacement control. Along the way, the evolution of wrinkles was monitored using a special surface-scanning device. Bifurcation buckling based on the J₂ deformation theory of plasticity was used to establish the onset of wrinkling. Comparison of measured and calculated results revealed that the wrinkle wavelength was significantly overpredicted. The cause of the discrepancy is shown to be anisotropy present in the tubes used. Modeling of the postbuckling response and the prediction of the limit load instability follows in Part II.     

A prediction of bursting failure in tube hydroforming processes based on ductile fracture criterion

Bursting is an irrecoverable failure mode in tube hydroforming, in contrast with buckling and wrinkling. To predict bursting failure in the hydroforming processes, Oyane's ductile fracture criterion is introduced and evaluated from the results of stress and strain productions obtained from finite element analysis. The region of fracture initiation and the bursting pressures are predicted and compared with a series of experimental results. It is shown that the material parameters used in the criterion can be obtained from the forming limit diagram. From the simulation results of tube bulging, the prediction of the bursting failure based on the ductile fracture criterion was demonstrated to be reasonable. This approach can be extended to a wide range of practical tube hydroforming processes.     

Axial crushing of wood-filled square metal tubes

The behaviour of thin tubes made of sheet metal and not so thin extruded tubes filled with wood and subjected to axial crushing is studied. Experiments show that the mode of elastic buckling is changed by the presence of the wood filler. The plastic crushing of thin tubes resulted in Euler-type buckling, while a considerable enhancement in the load carrying capacity and energy absorption was seen in the case of thicker walled tubes which were examined both under quasi-static and dynamic loading conditions. A new idealized deformation mechanism for the progressive crushing of the wood-filled tube is suggested and analysed. The results obtained for the mean loads agree reasonably with experimental observations. An alternative method employed to predict the mean crushing load of empty tubes and using a thickness that provides an equivalent stiffness to that of a filled tube is also seen to produce reasonable agreement with the experiment.     

Finite element analysis of tube flaring process with a conical tool

The extended r_min technique is incorporated into an incrementally updated Lagrangian formulation (ULF) of an elasto-plastic finite element computer code in order to handle contact boundary conditions to analyse the axisymmetric tube flaring process with a conical tool. A modified Coulomb friction law was adopted to calculate the influence of the friction coefficient on the tube flaring process. The effects of size and mechanical properties of tubes, as well as lubricants and tool semi-angle on flaring load were studied. It was found that good lubrication is effective in reducing flaring load. An optimum tool semi-angle (tool load is lower) in flaring is determined by frictional and bending activity at the tool inlet. In addition, the forming behaviour of the tube end is investigated to understand whether outward curling takes place and how much influence the tool semi-angle and tube size (initial thickness/initial mean diameter of tube) have on outward curling mode.     

An elasto-plastic finite element analysis of the sheet metal stretch flanging process

The incremental updated Lagrangian elasto-plastic finite element method (FEM) was employed in this study to analyse the stretch flanging of circular plates with a pre-determined smaller hole at the centre of the sheet metal. An extended r _min technique was employed such that each incremental step size is determined not only by the yielding of an element Gaussian point, but also by the change in the boundary condition along the tool-sheet interface. The experimental results, using a low-carbon (BA-CQ₂) sheet plate with a thickness of 1.0 mm, have been obtained and compared with the corresponding theoretical results. It was found that the flange thickness does not always decrease monotonically from the die shoulder to the flange edge. Reducing the punch diameter and increasing the flange height significantly reduced the flange thickness. Web width does not influence the thickness distribution of the flange. The tendency of flange thickness to thin decreases as punch diameter increases. The reduction of thickness at the die shoulder depends on the die shoulder radius. Simulation results of punch load of stretch flanging, the deformed geometry, and the distribution of thickness are compared with experimental data and found to satisfactorily agree.     

Some mixed boundary value problems for the semi-infinite elastic solid subjected to tangential surface tractions

A method is developed for representing the tangential displacements and tractions at the surface of the semi-infinite solid in terms of potential functions. In this form, a mathematical analogy is revealed between corresponding mixed boundary value problems involving tangential and normal surface displacements respectively. This analogy enables a general solution to be obtained to the problem in which the surface tangential displacements are specified axisymmetric functions inside the circle ar0 and the tangential surface traction is zero outside this circle. The method can also be used for certain non-axisymmetric problems, but it fails if the indentation analogue has a stress singularity at the boundary of the stressed area.     

Thermoelastic stability of duplex heat exchanger tubes

When duplex tubes are used for heat exchange purposes, thermal distortion affects the contact pressure and, hence, the thermal contact resistance at the interface. The resulting coupling between the thermal and thermoelastic fields for the system can lead to instability and non-uniqueness of the steady-state solution and, hence, to erratic heat transfer performance.Stability of the system is investigated by determining the conditions under which a small (axisymmetric or non-axisymmetric) perturbation on the steady state can grow exponentially in time. Substitution into the governing heat conduction and thermoelastic equations enables the unknown functions to be determined except for a set of arbitrary constants, which are then determined from the thermal and mechanical boundary conditions, including a statement of the relation between thermal contact resistance and pressure, linearized for small perturbations about the steady state.Results are presented for a range of material combinations and for both directions of steady state heat flux. It is shown that unstable axisymmetric perturbations can only occur for inward heat flow and, hence, the stability boundary for outward heat flow is always associated with a non-axisymmetric mode. In the latter case, the circumferential wavelength of the critical mode is generally about twice the tube thickness and a good approximation to the stability boundary can be obtained using previously published results for the stability of two plane layers in thermoelastic contact.For inward heat flow, the critical heat flux for axisymmetric perturbations is independent of the mean contact resistance, whereas that for non-axisymmetric modes increases with contact resistance. Thus, the stability boundary in this case is determined by non-axisymmetric modes for small values of contact resistance and by axisymmetric modes for large contact resistance.     

Coupled buckling and plastic instability for tube hydroforming

In this paper, the hydroforming limit of isotropic tubes subjected to internal hydraulic pressure and independent axial load is discussed.Swift's criterion is often used in this case for the prediction of diffuse plastic instability. Here, we first highlight the existence of two different Swift's criteria (for sheets and for tubes).Then, we recall that these types of approaches do not take into account buckling induced by axial loading. In fact, buckling may obviously occur before plastic instability; consequently, Swift's criteria must not be used alone to predict instability in the case of tube hydroforming.Numerical simulation was used to confirm these points and to analyse both the buckling and striction phenomena together. The two types of instability must be treated together in a reasonable approach to the hydroforming process.In this paper, the material verifies a “J2-flow” constitutive rate constitutive law. Jaumann's derivative was chosen and the Prandtl–Reuss equations with von Mises’ yield criterion and the associated flow rule were used. Isotropic hardening was taken into account.     

工艺参数对三维非轴对称管件缩径旋压成形的影响

以6061T1(挤压态及退火态)铝合金为研究对象,探讨工艺参数对三维非轴对称管件缩径旋压成形的影响。结果表明,当材料状态及其他工艺参数相同时,与轴对称零件相比,偏心及倾斜类零件的极限名义缩径量要小得多;轴对称件所能采取的进给比下限最低,而倾斜件的进给比下限最高;当旋转圆角半径过小时,会使工件表面粗糙度值变大,甚至出现破裂现象;旋轮直径对极限名义缩径量的影响不大。当工艺参数选择不合理时,工件口部或起旋点附近易产生破裂现象,为危险截面所在位置。     

Influence of a ring flat zone in the point contact surface on thermal elastohydrodynamic lubrication

This paper describes a geometrical profile, an elastohydrodynamically lubricated point contact surface with a ring flat zone, aimed at building up local line contact elastohydrodynamic lubrication (EHL) in point contact conjunctions to reduce the influence of side-leakage on the central film thickness. Effects of the ring flat zone on the thermal EHL characteristics are studied. A dimensionless coefficient, r_W, is defined to represent the relative half width of the ring flat zone in a point contact EHL surface. Thermal EHL numerical simulations have been performed to investigate the influence of r_W on the film thickness as well as pressure, temperature and friction coefficients under different operating conditions. In the range of 0≤r_W≤1.0 results show that the minimum film thickness decreases with increasing r_W and the central film thickness increases with increasing r_W, and the influence of r_W on the film thickness is more pronounced than those on the maximum pressure, the maximum temperature and the friction coefficients. It is revealed that the proposed ring flat zone with appropriate width is beneficial to the thermal lubrication.     

Crashworthiness study of a keel beam structure

In this paper, crashworthiness of the energy-absorbing keel beam that could be used for sub-floor of an aircraft was numerically studied using finite element software (ABAQUS/EXPLICIT). Effects of various parameters, i.e. web thickness, radius of everting stringer, and friction coefficient, were investigated and the conditions under which the keel beam could produce progressively crushing deformation were explored. The results show that web thickness, t, and radius of everting stringer, r, have great influence on the performance of the keel beam. When the ratio of r/t is in the range of 4-6, a low peak force (peak force reduction of 50%), stable and relative high stress plateau, and progressively crushing deformation mode can be produced from the energy-absorbing keel beam. By selecting appropriates parameters, energy absorption behavior has been significantly improved comparing with the traditional keel beam. Finally, a new energy-absorbing keel beam with variable web thickness was proposed. It has been demonstrated that the keel beam with variable web thickness is an ideal crashworthiness structure. It could not only significantly reduce the peak force, but also remain the stable and effective force plateau; meanwhile avoid web buckling.     

Influence of Punch Radius and Angle on the Outward Curling Process of Tubes

Using the theory of updated Lagrangian formulation, this study adopted the elasto-plastic finite-element method and extended the increment determination method, the r_min method, to include the element’s yielding, nodal contact with or separation from the tool, maximum strain and limit of rotation increment. The computer code for a finite-element method is established using the modified Coulomb’s friction law. Conical punches with different radii and angles are used in the forming simulation of hard copper and brass tube ends. The effects of various elements including the half-apex angle of punch (α) and its radius (R), the ratio of the thickness of the tube wall to the mean diameter of tube, mechanical properties, and lubrication on the tube’s outward curling, are investigated. Simulation findings indicate that when the bending radius at the punch inlet (ρ) satisfies the condition of ρ≤ρ _c , curling is present at the tube end. On the other hand, if ρ≥ρ _c , the tube end experiences flaring. The variable ρ _c is called the critical bending radius. The value of ρ _c increases as the value of α increases. Furthermore, the findings also show that ρ _c is neither correlated with tube material nor lubrication.     

Buckling of elastic—plastic square tubes under bending

This paper addresses the response, buckling and collapse of long, thin-walled, seamless steel square tubes under pure bending using a combined experimental and analytical approach. All tubes tested had nominal cross-sections with height equal to 1 in. (25.4 mm) and ranged in height-to-thickness ratios (h/t) from 15.4 to 28.6. The experiments were conducted under curvature control. It was found that the deformation of the cross-section that accompanied bending was uniform along the tubes for low values of curvature. At higher values, periodic ripples with wavelengths approximately equal to the width of the cross-section appeared on the compression flange. These ripples increased in amplitude with further bending. For tubes with higher h∼ the increase was more pronounced. Tubes with lower h/t showed more moderate increases in ripple amplitude but developed regions spanning several ripples in which the cross-section deformation was more pronounced. In all cases, collapse occurred when a kink formed on the compression flange of the tube.Rayleigh—Ritz type formulations based on the principle of virtual work were developed to predict the response and buckling of the tubes. Results include predictions of the response considering the effect of uniform cross-section deformation and predictions of the critical curvature at which the ripples appear. The numerical results are in good agreement with experimental observations.