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.     

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.     

Wrinkling control in aluminum sheet hydroforming

In this paper, the wrinkling behavior of 6111-T4 aluminum alloys during sheet hydroforming process was numerically and experimentally investigated. In sheet hydroforming, one or both surfaces of the sheet metal are supported with a pressurized viscous fluid, while a punch forms the part. In sheet hydroforming the use of a matching female die is not needed. The use of the pressurized fluid delays the onset of material rupture (International Journal of Mechanical Science 2003;45:1815–48) and also acts as an active blank-holding force to control wrinkling in the flange area. To form a wrinkle-free deep-drawn hemispherical cup with sheet hydroforming, a theoretical analysis based on the work of Lo et al. (Journal of Materials Processing Technology 1993;37:225–39) was initially used to predict the optimum fluid pressure profile. Simplifying geometrical assumptions and Tresca material model used in the theoretical analysis provided a fluid pressure profile that resulted in premature rupture of the sheet metal. However, an optimum fluid pressure profile generated by the finite element method, using Barlat's anisotropic yield function (Journal of Mechanical Physics and Solids 1997;45(11/12):1727–63), was successfully applied in sheet hydroforming to make the deep-drawn hemispherical cup without tearing and with minimal wrinkling in the flange area. The finite element model was also capable of accurately predicting the location of the material rupture in pure stretch, and wrinkling characteristics of the aluminum alloy sheet in the draw-in process.     

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.     

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.     

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.     

Experimental and numerical study of stamp hydroforming of sheet metals

The objectives of this research was to experimentally and numerically study the stamp hydroforming process as a means for shaping aluminum alloy sheets. In stamp hydroforming, one or both surfaces of the sheet metal are supported with a pressurized viscous fluid to assist with the stamping of the part thereby eliminating the need for a female die. The pressurized fluid serves several purposes: (1) supports the sheet metal from the start to the end of the forming process, thus yielding a better formed part, (2) delays the onset of material failure and (3) reduces wrinkle formation. This paper focuses on the experimental and numerical results of the stamp hydroforming process utilizing a fluid pressure applied to one surface of the sheet metal. The effects of applying a constant, varying and localized pressure to the surface of 3003-H14-aluminum sheet alloy were evaluated. Experiments demonstrated draw depths improvements up to 31% before the material failed. A failure prediction analysis by Hsu was also carried out to predict an optimal fluid pressure path for the varying fluid pressure case. The commercial finite element analysis code Ls-Dyna3D was used to numerically simulate the stamp hydroforming process. Both isotropic and anisotropic material models were used and their predictions compared against the experimental results. The numerical simulations utilizing Barlat's anisotropic yield function accurately predicted the location of the material failure and the wrinkling characteristics of the aluminum sheet.     

Modeling of anisotropic plastic behavior of ferritic stainless steel sheet

Modeling of anisotropic plastic behavior of ferritic stainless steel sheet (Type 409) was investigated using the three yield functions of Hill [A theory of the yielding and plastic flow of anisotropic metals. Proceedings of Royal Society of London, Series A 1948;193:281–97.], Barlat and Lian [Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions. International Journal of Plasticity 1989;5:51–66] and Barlat et al. [Plane stress yield function for aluminum alloy sheet. Part I: Theory. International Journal of Plasticity 2003;19:1297–319.] (referred to as Yld2000-2d) criteria. Mechanical behaviors were characterized based on uniaxial tension, balanced biaxial bulge, and disk compression tests. Directionalities of yield stresses and r values were predicted from the three criteria and compared with experimental results. In order to verify the modeling accuracy of the three functions under complex loading conditions, cylindrical cup drawing and limiting dome height tests were carried out numerically and experimentally. It has been demonstrated that the result from Yld2000-2d criterion exhibits good agreement with experimental data. The effects of anisotropic hardening on earing and necking were also investigated based on the different levels of plastic work.     

Forming severity concept for predicting sheet necking under complex loading histories

A new method of predicting neck formation in sheets under non-proportional loading is proposed, based on the concept of “cumulative forming severity”. This concept is borrowed from a macroscopic model of ductile fracture where the crack initiation is governed by the accumulated equivalent plastic strain modified by the stress triaxiality and the Lode angle parameter. Such an approach necessitates a representation of the forming limit diagram (FLD) in the space of the equivalent strain to neck and the Lode angle parameter.Another new factor is the assumption of the non-linear accumulation of forming severity for non-proportional and complex loading histories. A class of non-linear weighting function is proposed with only one free parameter. A starting point in the derivation is the known FLD corresponding to proportional loading. This can be determined from Hill's and Stören and Rice analytical solutions, from numerical simulation, or else taken directly from experiments. In the case of proportional loading, necking depends on the final state of stress or strain, so it does not matter if necking severity index is accumulated in a linear or non-linear way. For non-proportional loading, the unknown free parameter of the non-linear accumulation rule must be determined from a test.Experimental data on FLDs under complex strain paths for two types of material, aluminum alloy 6111-T4 [Graf A, Hosford W. The influence of strain-path changes on forming limit diagrams of A1 6111 T4. International Journal of Mechanical Sciences 1994;36(10):897–910.] and aluminum-killed sheet steel [Muschenborn W, Sonne HM. Influence of the strain path on the forming limits of sheet metal. Archiv fur das Eisenhuttenwesen 1975;46:597–602], found in the literature are revisited by the proposed model. Calibrated from only one test with non-proportional loading condition, the model is able to predict the remaining tests of complex loading paths with good accuracy.     

基于成形应力极限的管材液压成形缺陷预测

基于塑性应力应变关系及Hill79屈服准则,推导出极限应力与极限应变间转化关系,进而建立2008T4铝合金的成形应力极限图(Forming limit stress diagram,FLSD)。采用LS-DYNA软件对三通管液压胀形过程进行模拟,应用FLSD预测胀形过程中破裂的发生及成形压力极限,并与传统成形极限图(Forming limit diagram,FLD)结果进行了对比。研究表明,FLD与FLSD预测结果中破裂缺陷位置相同,但极限内压力值存在很大差别,而FLSD预测结果与物理试验结果较吻合。考虑到FLD受应变路径影响显著的因素,将FLSD作为管材液压成形等复杂应变路径下的成形极限的判据更加方便可靠。     

Prediction of sheet forming limits with Marciniak and Kuczynski analysis using combined isotropic-nonlinear kinematic hardening

The forming limit curve (FLC), a plot of the limiting principal surface strains that can be sustained by sheet metals prior to the onset of localized necking, is useful for characterizing the formability of sheet metal and assessing the forming severity of a drawing or stamping process. Both experimental and theoretical work reported in the literature has shown that the FLC is significantly strain-path dependent. In this paper, a modified Marciniak and Kuczynski (MK) approach was used to compute the FLC in conjunction with two different work-hardening models: an isotropic hardening model and a mixed isotropic-nonlinear kinematic hardening model, which is capable of describing the Bauschinger effect. Predictions of the FLC using the MK analysis have been shown to be dependent on the shape of the initial yield locus and on its evolution during work hardening; therefore the hardening model has an influence on the predicted FLC. In this investigation, published experimental FLCs of AISI-1012 low carbon steel and 2008-T4 aluminum alloy sheets that were subjected to various nonlinear loading paths were compared to predictions using both hardening models. The predicted FLCs were found to correlate quite well with experimental data and the effects of strain path changes and of the hardening model on predicted FLCs are discussed.     

Study on experimental approaches of forming limit curve for tube hydroforming

This paper proposes a set of experimental approaches to establish the forming limit curve (FLC) in different forming modes for tube hydroforming. In tension–compression strain state, analytical models are constructed to determine the linear strain paths at the pole of the hydroformed tube, and a self-designed free hydroforming apparatus with axial feeding and internal pressure are used to carry out the bulge tests. In plane strain state, the difference is that both ends of the tube are fixed with different punches. In tension–tension strain state, a novel hydroforming apparatus are designed. The novel device requires the simultaneous application of lateral compression force and internal pressure to control the material flow under tension–tension strain states. The linear strain paths for the right hand side of FLC by finite element method simulation are calculated. The linear strain paths in different strain states are verified and the FLC of roll-formed QSTE340 seamed tube is constructed through the proposed experimental approaches. Comparison between simulation and experimental results for hydroforming process of front crossmember shows that the experimental FLC is accurate and valid for tube hydroforming.     

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.     

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.     

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.     

Optimization of loading path in hydroforming T-shape using fuzzy control algorithm

The loading path is crucial to the quality of forming parts in the process of tube hydroforming, and thus the design and optimization of loading path is an important issue for tube hydroforming. Wrinkling is a catastrophic defect for thin-walled tube hydroforming. In order to avoid wrinkling, an adaptive simulation approach integrated with a fuzzy control algorithm is used to optimize the loading path of hydroforming a T-shaped tube. The tubular material used is stainless steel and has an outer diameter of 103 mm and the wall thickness of 1.5 mm. The controlled variables are the axial feeding, the counterpunch displacement, and the internal pressure. A code is developed to make the optimization automatically, which works together with LS-DYNA. Six evaluation functions are adopted for identifying geometrical shape and quality of T-shape. Failure indicators obtained from the simulation results are used as the input of the fuzzy control, and then process parameters are adjusted according to the expert experiences in the fuzzy controller. In this way, a reasonable loading path for producing a sound T-shape is obtained, and also a T-shaped product is successfully hydroformed by experiment. The result shows that the fuzzy control algorithm can provide an adequately reliable loading path for hydroforming T-shaped tubes.     

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.     

Feasibility study on optimized process conditions in warm tube hydroforming

Feasibility study has been performed to estimate the optimized process conditions in warm tube hydroforming based on the simulated annealing optimization method. Precise prediction and control of process parameters play an important role in forming at warm conditions. Optimal pressure and feed loading paths are obtained for aluminium AA6061 tubes through the simulated annealing algorithm in conjunction with finite element simulations. Numerous axisymmetric geometries are investigated and the effects of expansion ratio, corner fillet to thickness ratio, and initial diameter to thickness ratio are studied. For the feasibility estimation, warm hydroforming experiments have been conducted on aluminum AA6061 under optimal designed conditions. The results show that the optimization procedure used in this research is a reliable and feasible tool in determination of optimal process conditions for the sound warm hydroforming process.     

Effects of superimposed hydrostatic pressure on fracture in sheet metals under tension

The effect of superimposed hydrostatic pressure on fracture in sheet metals under plane strain tension is studied numerically using the finite element method based on the Gurson damage model. It is demonstrated that the superimposed hydrostatic pressure p has no noticeable effect on necking but significantly increases the fracture strain due to the fact that a superimposed pressure delays or completely eliminates the nucleation, growth and coalescence of microvoids or microcracks. The experimentally observed transition of the fracture surface, from P-type mode under atmospheric pressure to C-type mode under high pressure, is numerically reproduced.     

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.