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.     

Analysis of forming limit in tube hydroforming

The automotive industry has shown increasing interest in tube hydroforming. Despite many automobile structural parts being produced from cylindrical tubes, failures frequently occur during tube hydroforming under improper forming conditions. These problems include wrinkling, buckling, folding back, and bursting.We perform analytical studies to determine forming limits in tube hydroforming and demonstrate how these forming limits are influenced by the loading path. Theoretical results for the forming limits of wrinkling and bursting are compared with experimental results for an aluminum tube.     

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.     

管材屈曲和起皱分析

研究了薄壁管在高压成形过程中的屈曲、起皱失稳现象,通过对所建立的弹塑性屈曲（全局屈曲）和起皱（局部屈曲）模型的分析计算,获得了管子屈曲、起皱的初始条件、临界载荷和临界应力计算公式。研究表明,屈曲和起皱的产生取决于几何参数,如相对半径r0/l0和相对厚度d0/r0等,且对长管而言,屈曲是主要的失稳形式,但对于短管,起皱则是主要的失稳形式,薄壁管内压对薄壁管起皱、屈曲没有影响。     

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.     

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.     

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.     

Inelastic wrinkling and collapse of tubes under combined bending and internal pressure

The problem of inelastic bending and collapse of tubes in the presence of internal pressure is investigated using experiments and analyses. The experiments involve 1.5-inch diameter, D/t=52 stainless steel tubes bent to failure at fixed values of pressure. The moment-curvature response is governed by the inelastic characteristics of the material. Bending induces some ovalization to the tube cross section while, simultaneously, the internal pressure causes the circumference to grow. Following some inelastic deformation, small amplitude axial wrinkles appear on the compressed side of the tube, and their amplitude grows stably as bending progresses. Eventually, wrinkling localizes, causing catastrophic failure usually in the form of an outward bulge. Internal pressure stabilizes the structure, it increases the wavelength of the wrinkles and can increase significantly the curvature at collapse. The onset of wrinkling is established by a custom bifurcation buckling formulation. The evolution of wrinkling and its eventual localization are simulated successfully using a FE shell model. The material is represented as an anisotropic elastic-plastic solid using the flow theory, while the models are assigned initial geometric imperfections with the wavelength of the wrinkling bifurcation mode. It is demonstrated that successful prediction of collapse requires very accurate representation of the material inelastic properties including yield anisotropies, and that as expected, the collapse curvature is sensitive to the imperfection amplitude and wavelength imposed.     

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.     

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.     

An experimental and numerical study of necking initiation in aluminium alloy tubes during hydroforming

In this paper, a combined experimental and numerical investigation of free hydroforming of aluminium alloy tubes is conducted. The tubes are subjected to different loading histories involving axial compression and internal pressure. The circumferential and axial strains experienced by the tubes are continuously recorded along with the pressure and axial load. The numerical simulations are carried out using both 2D axisymmetric and 3D finite-element formulations by applying the experimentally recorded axial load and internal pressure. In the latter, a geometric imperfection is introduced in the form of wall thickness reduction at the tube mid-length in order to trigger necking which happens after significant bulging and beyond the stage of peak pressure. The strain histories and peak pressures obtained from the simulations agree well with those determined from the experiments. Further, the forming limit curve predicted by the simulations as well as from a M–K analysis incorporating the computed strain paths corroborate well with the experimental data. The role of nonproportional straining on the mechanics of failure of the tubes due to bulging and necking is studied in detail.     

Prediction of necking of high strength steel tubes during hydroforming—multi-axial loading

This article studies tubular hydroforming of high strength low alloy (HSLA) and dual phase (DP600) straight tubes under the action of end feeding loads. Experiments demonstrate that higher end feed loads enhance the formability of the tubes and increase the internal fluid pressure for onset of necking and bursting. Because of the action of the internal pressure and the axial compressive load, the onset of localization (necking) is due to a complex three-dimensional state of stress. Using free expansion experiments, approximate upper and lower bound strain-based forming limit curves are determined for the tube materials. These limit curves, in turn, are used to derive upper and lower bound extended stress-based forming limit curves [Simha et al., Prediction of necking in tubular hydroforming using an extended stress-based FLC. Transactions of the ASME Journal of Engineering Materials and Technology 2007;129(1): 36-47]. In conjunction with finite element computations that use solid elements to model the tube, these stress-based limit curves are used to predict upper and lower bound necking pressures under the action of end feed loading. These predictions of necking pressures, when an appropriate coefficient of tube-die friction is used, are found to bracket the experimentally measured necking pressures. Computations using plane stress shell elements to model the tubes are shown to give erroneous results, since the plane stress approximation is not valid when tubes are hydroformed in a die.     

Non-axisymmetric buckling behaviour of elastic-plastic circular tubes subjected to a nosing operation

Theoretical investigations have been carried out to predict the non-axisymmetric buckling seen in the throat of circular elastic-plastic tubes subjected to a nosing operation along a frictionless conically shaped die. The buckling points and associated modes are determined by Hill's bifurcation theory in conjunction with a non-axisymmetric buckling mode. The results show that increasing the die semi-angle and the tube thickness or decreasing the material work-hardening rate are always beneficial to improve the buckling limit. The associated critical buckling mode number generally decreases as the tube thickness increases and does not depend drastically on the die semiangle nor on the hardening rate. The effect of the average r value is substantial and the buckling limit is improved when the r value reduces. Part of the present results predict well the results observed in existing experimental work.     

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.     

Bursting failure prediction in tube hydroforming processes by using rigid–plastic FEM combined with ductile fracture criterion

The most common failure in tube hydroforming is the bursting failure due to excessive thinning of large deformation. To evaluate the forming limit of hydroforming processes, the Oyane's ductile fracture integral I was introduced and calculated from the histories of stress and strain according to every element by using the rigid–plastic finite element method. The region of fracture initiation and the forming limit for three hydroforming processes, such as a tee extrusion, an automobile rear axle housing, and a lower arm under different forming conditions are predicted in this study. Also it is shown that the material parameters used in the ductile failure can be obtained from the experimental forming limit diagram. From the results, the prediction of the bursting failure and the plastic deformation for the three hydroforming examples demonstrates to be reasonable so that this approach can be extended to a wide range of practical tube hydroforming processes.     

Die shape design for T-shape tube hydroforming

In this paper, two design methods for T-shape tube hydroforming dies are proposed, namely, the extrusion-cutting-fillet method (ECFM) and the intersection-fillet method (IFM). Simulations on hydraulic expansion and axial feeding of T-shape tube hydroforming with two dies using the program DEFORM-3D were performed. The influence of the two dies on workpiece formability of T-shape tube hydroforming was examined. Experiments were carried out with SUS304 stainless steel tube at room temperature. A qualified product of T-shape tube, without wrinkling or bursting, was obtained using the die designed by the IFM method.     

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.     

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.     

Analysis and design of hydroformed thin-walled tubes using enhanced one-step method

This paper deals with the analysis and design of tube hydroforming parameters in order to reduce defects which may occur at the end of the forming process, such as necking and wrinkling. We propose a specific methodology based on the coupling between an enhanced one-step method for the rapid simulation of tube hydroforming process and a surrogate model based on a metamodeling technique. The basic formulation of the one-step method has been modified and adapted for the modeling of 3D tube hydroforming problems in which the initial geometry is a circular tube expanded by internal pressure and submitted to axial feeding. In the surrogate model, approximate responses are built using moving least squares method and constructed within a moving region of interest which moves across a predefined discrete grid of authorized experimental designs. Two applications of tube hydroforming of aluminum alloy 6061-T6 have been utilized to validate our methodology. The final design is validated using experiments together with the classical explicit dynamic incremental approach using ABAQUS? commercial code.     

On the analysis of sheet metal wrinkling

An analysis for the prediction of wrinkling in curved sheets during metal forming is presented. Using a local approach, similar to that employed for conventional forming limit diagram representations, we construct “wrinkling limit curves” (WLCs) which represent the combinations of the critical principal stresses for wrinkling in curved sheet elements. Wrinkling limit curves are first determined using a bifurcation analysis for plastic buckling in short-wavelength shallow modes. A study of the effects of material properties and sheet geometry on the critical conditions for wrinkling is carried out. We then analyse the effects of geometric imperfections on wrinkling. This analysis is based on the implementation of a finite element scheme. The influence of nonproportional loading is also investigated. In our analysis the material is assumed to be isotropic, elastic-plastic with the plastic part modelled using both J₂ deformation theory and J₂ flow theory of plasticity.