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.     

Analytical and numerical approach to prediction of forming limit in tube hydroforming

Analytical and numerical analyses of forming limit in tube hydroforming under combined internal pressure and independent axial feeding are discussed in this paper. To predict the initiation of necking, Swift's criterion for diffuse plastic instability is adopted based on Hill's general theory for the uniqueness to the boundary value problem. In addition, in order to predict fracture initiation, Oyane's ductile fracture criterion is introduced and evaluated from the histories of stress and strain calculated by means of finite element analysis. From the comparison with a series of tube bulge tests, the prediction of the bursting failure based on the plastic instability and the ductile fracture criterion demonstrates to be reasonable so that these approaches can be extended to a wide range of practical tube hydroforming processes.     

A prediction of bursting failure in tube hydroforming process based on plastic instability

Based on plastic instability, an analytical prediction of bursting failure on tube hydroforming processes under combined internal pressure and independent axial feeding is carried out. Bursting is an irrecoverable phenomenon due to local instability under excessive tensile stresses. In order to predict the bursting failure, three different classical necking criteria – diffuse necking criteria for a sheet, and a tube, and a local necking criterion for a sheet – are introduced. The incremental theory of plasticity for an anisotropic material is adopted and the hydroforming limit, as well as a diagram of bursting failure with respect to axial feeding and hydraulic pressure are presented. In addition, the influences of material properties such as anisotropy parameter, strain hardening exponent and strength coefficient on plastic instability and bursting pressure are investigated. As a result of the above approach, the hydroforming limit with respect to bursting failure is verified with experimental results.     

Plastic instability in dual-pressure tube-hydroforming process

The tube-hydroforming process has become an indispensable manufacturing technique in recent years. Successful tube hydroforming requires bulging to take place without causing any type of instability such as bursting, wrinkling or buckling. The dual-pressure tube-hydroforming process was introduced to achieve a favorable tri-axial stress state in the deformation process. In this paper, the effect of applying counter pressure on plastic instability of thin-walled tubes is analyzed. It is concluded that in dual-pressure tube hydroforming, the onset of plastic instability is delayed and the ductility of the metal is increased.     

Hydroforming of aluminum extrusion tubes for automotive applications. Part II: process window diagram

Based on the mathematical formulations for predicting forming limits induced by buckling, wrinkling and bursting of free-expansion tube hydroforming, a theoretical “Process Window Diagram” (PWD) is proposed and established in this paper. The theory developed in the first part of the present work was formulated within the context of free-expansion tube hydroforming with both combined internal pressure and end feeding. The PWD is designed to provide a quick assessment of part producibility for tube hydroforming. The predicted PWD is validated against experimental results conducted for 6260-T4 60×2×320 (mm) aluminum tubes. An optimal loading path is also proposed in the PWD with an attempt to define the ideal forming process for aluminum tube hydroforming. Parametric studies show that the PWD has a strong dependency on tube geometry, material property and process parameters. To the authors’ knowledge, this is the first attempt that a PWD is being formulated theoretically. Such a concept can be advantageous in deriving design solutions and determining optimal process parameters for tube hydroforming processes.     

Analytical approach to bursting in tube hydroforming using diffuse plastic instability

Analytical studies on onset of bursting failure in tube hydroforming under combined internal pressure and independent axial feeding are carried out. Bursting is irrecoverable phenomenon due to local instability under excessive tensile stress. In this paper, in order to predict the bursting failure diffuse plastic instability based on the Hill's quadratic plastic potential is introduced. The incremental theory of plasticity for anisotropic material is adopted and then the hydroforming limit and bursting failure diagram with respect to axial feeding and hydraulic pressure are presented. The influences of the plastic anisotropy on plastic instability, the limit stress and the bursting pressure are also investigated. Finally, the stress-based hydroforming limit diagram obtained from the above approach is verified with experimental results.     

Numerical analysis and design for tubular hydroforming

To get an optimum deformation path for tubular hydroforming, the hydroforming limit of isotropic and anisotropic tubes subjected to internal hydraulic pressure, independent axial load or torque is firstly proposed based on the Hill's general theory for the uniqueness to the boundary value problem and compared with those of the conventional sheet forming. The influences of the deformation path, the material properties and the active length–diameter ratio on the nucleation and the development of wrinkling during the free tubular hydroforming are also investigated. The above theory is used as a criterion and implemented with some new functions in our ITAS3D, an in-house finite element code for simulating the sheet forming, to control the materials flow and to prevent the final failure modes from occurring. Finally, the tubular hydroforming of an automobile differential gear box is taken as an example to show the efficiency and usefulness of the algorithm.     

Investigation of deformation and collapse mechanism for magnesium tube in axial crushing test

This paper demonstrates the quasi-static axial compression and high-speed axial compression tests of extruded magnesium alloy circular tubes for evaluating the crash and fracture behavior of mg parts. To capture the buckling and fracture behavior of Mg tube during the axial compression tests, FE simulation adopts different types of flow curves depending on the deformation mode such as tension and compression with LS-DYNA software. The Mg tube undergoes compressive plastic strain prior to buckling while according to the model based on Hill yield criterion only bulging along the radial direction is predicted. Due to the tension-compression asymmetry of Mg alloys, diameter of Mg tube expands largely at the initial plastic strain before having bulging or folding while only a bulging mode typical for materials with cubic crystal structure can be predicted. Simulation results such as punch load and deformation mode are compared with experimental results in the axial crushing test with AZ61 alloy.     

Influence of end-conditions during tube hydroforming of aluminum extrusions

Tube hydroforming experiments were conducted to develop the forming limit diagram of AA6082-T4 by utilizing three types of end-conditions: (i) “free-end”, (ii) “pinched-end” or “fixed-end” and (iii) “forced-end”. It was found that “free-end” hydroforming gives the lowest forming limits followed by “pinched-end” and “forced-end” hydroforming. It was noticed that the tube failure occurs within 5° to the extrusion weld in the “free-end” experiments, within 7° in the “pinched-end” condition and extended up to 10° in the “forced-end” hydroforming experiments. Finite element simulations were carried out to capture the effects of the weld geometry, weld mechanical properties and the end-conditions of the extruded tube on the maximum induced stress and location of the maximum von Mises stress. It was found that the anisotropy of the weld material and the end-condition used during hydroforming experiments have the largest influence on the failure location with respect to the weld center.     

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 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”.     

Ductile fracture and lifetime prediction of structural materials and components under high-temperature creep conditions

The problem of long-term strength prediction of structural materials under uniaxial and biaxial creep conditions, which give rise to ductile fracture, is considered. A new approach to long-term fracture modelling based on constitutive equations of the “isochronous creep theory” and criteria of ductile fracture is suggested. The change in the momentary tangential modulus specifies the materials rheological behaviour and a value tending to zero is assumed as the ductile fracture criterion. Using the model the times to fracture of rods, beams and thin-walled tubes made of high-temperature and heat-resistant materials are estimated and they show good agreement with experimental data.     

内高压成形机理与关键技术

在中国国家杰出青年科学基金资助项目“镁合金热态液力成形技术”、中国国家自然科学基金资助项目“轻体件高内压液力成形机理的研究”、“管材热态内压成形新方法及其机理研究”和“激光拼焊管内高压成形机理”、以及中国教育部高等学校博士学科点专项科研基金资助项目“镁合金热态内高压成形机理研究”共同资助下,开展内高压成形机理及关键技术研究,在内高压成形塑性变形规律、起皱和破裂等失稳行为、提高成形极限和降低成形压力方法,以及液力胀接、热态内压成形和拼焊管内高压成形等方面取得重要进展,并在汽车和航天等领域实现内高压成形技术产业化应用,报告上述研究的理论和工程体系。 根据塑性变形特点,将内高压成形分为变径管内高压成形(IHPF of TPVD)、弯曲轴线管内高压成形(IHPF of TPCA)和多通管内高压成形(IHPF of TPB/BT)等3类,提出IHPF of TPVD由充填、成形、整形等步骤组成,IHPF of TPCA由弯曲、预成形、内高压成形等步骤组成,IHPF of TPB/BT由胀形、补料、整形等步骤组成。以此为出发点,通过实验和理论分析,研究IHPF塑性变形规律与失稳行为。     

An analysis of localized necking in aluminium alloy tubes during hydroforming using a continuum damage model

In this work, localized necking in aluminium alloy tubes subjected to free hydroforming is analyzed. The main objective is to study the influence of loading conditions, such as prescribed fluid pressure or volume flow rate in conjunction with axial end feed, on the nature of the forming limit curve (FLC). To this end, the strain histories experienced at the tube mid-length, which were computed in an earlier investigation [14] [Varma NSP, Narasimhan R. A numerical study of the effect of loading conditions on tubular hydroforming, Journal of Materials Processing Technology 2005; [Submitted for publication]], are analyzed using the Marciniak–Kuczynski (M–K) method along with an anisotropic version of the Gurson model. The Gurson constitutive parameters are determined following an inverse approach using the sheet FLC for the chosen alloy. The predicted FLC for combined pressure and axial contraction corroborates well with the experimental data obtained in [12] [Kulkarni A, Biswas P, Narasimhan R, Luo A, Stoughton T, Mishra R, Sachdev AK. An experimental and numerical study of necking initiation in aluminium alloy tubes during hydroforming. International Journal of Mechanical Sciences 46:2004;1727–46] and is almost flat, whereas it is akin to the sheet FLC and increases with negative minor strain when fluid volume is specified. The forming limit strains for loading with specified fluid volume are in general higher when compared to those with prescribed fluid pressure. Finally, it is demonstrated that a transition from axial to circumferential necking occurs when high ratios of axial extension to volume flow rate are applied to the tube.     

Plastic buckling of tubes under axial compression and internal pressure

The plastic buckling and collapse of long cylinders under combined internal pressure and axial compression was investigated through a combination of experiments and analysis. Stainless-steel cylinders with diameter-to-thickness values of 28.3 and 39.8 were compressed to failure at fixed values of internal pressure up to values 75% of the yield pressure. The first effect of internal pressure is a lowering of the axial stress–strain response. In addition, at some plastic strain level, the cylinder develops uniform axisymmetric wrinkling. Under continued compression, the wrinkles grow stably, gradually reducing the axial rigidity of the structure and eventually lead to a limit load instability. All pressurized cylinders remained axisymmetric until the end of the test past the limit load.The critical stress and wavelength were established using classical plastic bifurcation theory based on the deformation theory of plasticity. The evolution of wrinkling, and the resultant limit state, were established by modeling a periodic domain that is one half of the critical wavelength long. The domain was assigned an initial imperfection corresponding to the axisymmetric buckling mode calculated through the bifurcation check. The inelastic material behavior was modeled through the flow theory of plasticity with isotropic hardening. The variations of the axial response and of the limit strain with pressure observed in the experiments were reproduced well by the model. Inclusion of Hill-type anisotropic yielding in all constitutive models was required for good agreement between predictions and experiments.     

Analytical and numerical study on plastic instabilities for axisymmetric tube bulging

In this paper, plastic instabilities of elasto–plastic tubes subject to internal pressure are discussed. For diffuse necking prediction, the classical intrinsic criteria for diffuse necking are accurate for long cylindrical tubes. However, for short tubes, geometric changes are important, and the intrinsic criteria become insufficient. For this purpose, a new diffuse necking criteria is proposed including geometric effects in the prediction.On the other hand, for the local necking prediction, the Hill's criterion is not accurate for short tubes, due to the biaxial stretching. As an alternative, a local necking criterion based on a modified Hill's assumption for localized necking is proposed. The numerical calculations carried out for different tube dimensions, explains the geometrical effects on the localization of deformations for pressurized tubes, and improves the accuracy of the proposed criteria.     

Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure

The dynamic plastic collapse of energy-absorbing structures is more difficult to understand than the corresponding quasi-static collapse, on account of two effects which may be described as the “strain-rate factor” and the “inertia factor” respectively. The first of these is a material property whereby the yield stress is raised, while the second can affect the collapse mode, etc. It has recently been discovered [6,7]that structures whose load-deflection curve falls sharply after an initial “peak” are much more “velocity sensitive” than structures whose load-deflection curve is “flat-topped” (Fig. 1a); that is, when a given amount of energy is delivered by a moving mass, the final deflection depends more strongly on the impact velocity. In this paper we investigate strain-rate and inertia effects in these two types of structure by means of some simple experiments performed in a “drop hammer” testing machine, together with some simple analysis which enables us to give a satisfactory account of the experimental observations. The work is motivated partly by difficulties which occur in small-scale model testing of energy-absorbing structures, on account of the fact that the “strain-rate” and “inertia” factors not only scale differently in general, but also affect the two distinct types of structure differently.     

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.     

Bursting failure prediction in tube hydroforming using FLSD

In tube hydroforming, circular components are hydrobulged or hydroformed from tubular blanks with internal pressure and simultaneous axial loading. Thus the tube can be fed into the deformation zone during the bulge operation allowing more expansion and less thinning without any defects such as wrinkling, buckling, and bursting. By contrast with the buckling and the wrinkling, the bursting is generally classified as an irrecoverable failure mode. Hence in order to obtain the sound hydroformed products, it is necessary to predict the bursting behavior and to analyze the effects of process parameters on this failure condition in hydroforming processes. In this study, a forming limit stress diagram (FLSD) is constructed by plotting the calculated principal stresses based on the local necking criterion. Using the theoretical FLSD, we carry out the numerical prediction of bursting failure in a hydroforming process, which usually has non-linear strain path. Finite element analyses are carried out to find out the state of stresses during simple hydroforming operation, in which the FLSD is utilized as the forming limit criterion for assessment of the initiation of necking, and influences of the material parameters on the formability are investigated. In addition, the numerical results obtained from the FEM combined with the FLSD are confirmed with a series of bulge tests in view of bursting pressure and show a good agreement. Consequently, it is shown that the theoretical and numerical approach to bursting failure prediction proposed in this paper will provide a feasible method to satisfy the increasing practical demands for assessment of the forming severity in 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.