modelling of springback in creep forming thick aluminum sheets

Modelling of springback in creep forming thickaluminum sheets

K.C. Ho, J. Lin*, T.A. Dean

Mechanical and Manufacturing Engineering, School of Engineering, University of Birmingham,

Birmingham B15 2TT, UK

Received in final revised form 10 March 2003

Abstract

Integrated numerical techniques have been developed to predict springback in creepforming thick aluminum sheet components using physically-based unified creep constitutiveequations, which model the primary hardening, ageing, creep constrained damage and their

effects on creep deformation of an aluminum alloy. Springback effects have been studied fortwo aluminum shapes. One is a single curvature cylindrical component and the other is adoubly curved spherical component. Stress relaxation and creep deformation of the aluminum

sheet under different forming conditions are studied. The effects of forming processes and thethickness of the material sheet on springback of formed parts are investigated. The amount ofspringback on forming doubly curved and single curvature sheet components is predicted and

compared for different forming conditions.# 2003 Elsevier Ltd. All rights reserved.

Keywords: Creep forming; Numberical techniques; Sheet metal forming; Springback

1. Introduction

Creep forming is a process in which a material blank is elastically loaded onto atool surface using vacuum bagging and is held at a specific high temperature for acontrolled amount of time. During the thermal exposure, the constituents of themetal precipitate and alter the microstructure of the material. The resultant altera-tion in the internal structure of the metal can improve physical properties, such as anincrement in the yield and ultimate tensile strength (Holman, 1989). At the same

International Journal of Plasticity 20 (2004) 733–751

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0749-6419/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0749-6419(03)00078-0

* Corresponding author.

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time, stress relaxation occurs due to creep and permanent deformation of the materialtakes place. This forming process is especially used for fabricating complex shapedsheet aluminum components, such as aircraft wing parts (Pitcher and Styles, 2000).Due to the nature of the components formed by creep forming, the boundaries ofthe sheet blank are not clamped and deformation is close to pure bending, whichgenerates low plastic strains in the formed part. Thus when the vacuum force isreleased, some springback occurs since the holding time, required to achieve propermechanical properties, and the stress level, which is the driving force for creep, is notsufficient to fully fix the shape of the component. In practice, the amount ofspringback is huge in creep forming aircraft wing panels.Springback is a key problem in sheet metal forming and much research has been

carried out in the past decades (Asnafi, 2001; Carden et al., 2002; Chun et al., 2002a,b;Cleveland and Ghosh, 2002; Huang and Gerdeen, 1994; Karafillis and Boyce, 1992,1996; Kutt et al., 1999; Lee and Yang, 1998; Makinouchi, 1996; Morestin et al.,1996; Papeleux and Ponthot, 2002; Smith et al., 2003; Wagoner et al., 1997; Wu,2002; Xue et al., 2001a,b; Yao and Cao, 2002; Zhang and Lin, 1997). These researchmainly focused on the geometric and material factors that involved in the sheetmetal forming processes. In dealing with springback problem, three kinds ofapproach have been commonly used; i.e. analytical, experimental and numericalmethods.Asnafi (2001) studied the springback of doubly curved autobody pressed panels

and investigated the effects of binder force, component curvature and sheet metalthickness on springback for both steel and aluminum sheets. Carden et al. (2002)studied time-dependent springback of aluminium alloy by conducting draw-bendtests. Chun et al. (2002a,b) studied Bauschinger effect for sheet metal forming pro-cess that subjected to cyclic loading conditions. Based on membrane theory of shellsof revolution and energy method, Xue et al. (2001a,b) developed a new analyticalprocedure to predict for springback of circular and square metal sheets after adouble-curvature forming operation. Huang and Gerdeen (1994) reviewed a series ofanalytical techniques for estimating springback in forming doubly curved develop-able sheet metal components, which include small curvature bending and doublecurvature bending. In the mean time, the effects of material models on springbackprediction have been carried out for many forming processes. The material modelsused include: perfect plastic (Zhang and Lin, 1997), non-linear kinematic hardening(Morestin et al., 1996), combined isotropic–kinematic hardening (Wu, 2002),plastic–viscoplastic (Cailletaud and Sai, 1999), inelastic and cyclic loading (Chun etal., 2002a,b) models, and new material models were proposed for an aluminiumalloy and a high strength steel (Cleveland and Ghosh, 2002).For predicting springback for practical engineering components, a range of

numerical methods and finite element (FE) software packages, have been developed(Pasquinelli, 1995; Makinouchi, 1996; Lee and Yang, 1998; Yoon et al., 1999). Theanalyses mainly concentrate on rigid–plastic and elasto-plastic deformation of sheetmaterials. Analyses have indicated that, springback can be reduced by changingclamping conditions and materials properties, but can not be completely eliminateddue to the existence of residual stresses created in forming.

734 K.C. Ho et al. / International Journal of Plasticity 20 (2004) 733–751

Karafillis and Boyce (1992, 1996) developed a tool design algorithm to compensatespringback errors based on the FE simulation results of stamping processes for 2Dand 3D geometries. Nilsson et al. (1997) simulated springback in V-die bending forseveral materials neglecting friction. Their simulation results corresponded well withthe experimental results. Lee and Yang (1998) evaluated the numerical parametersthat influence the springback prediction by using FE analysis of a stamping process.From a comparison of numerical with experimental results, factors important forspringback have been assessed. The non-linear FE method was employed to inves-tigate the complicated springback behavior of doubly curved, sheet metal partsformed with reconfigurable tooling (Kutt et al., 1999). From the FE results, theyproposed the average normal distance quantity to measure springback and theirsimulation results show good agreement with the experiments conducted. Peddiesonand Buchanan (1990) carried out a preliminary simulation of creep age forming of abeam using a model based on linear viscoelasticity. Their simulations successfullycaptured many quality features of age forming, but could not accurately predictspringback. Narimetla et al. (1998) proposed a simulation procedure for panel ageforming by using FE plate model. Their model predicted similarity of the tool andpart shapes in two special cases. Narimetla et al. (2000) developed a unified ageforming model based on their previous results. Their model can be characterised bya limited number of experiments and allows for a non-iterative tool design.Dif et al. (2000) revised the development for high-formability alloys used in

aerospace industry. They studied and analysed experimental and modelling resultsconducted for various aluminum alloys with several forming conditions in order tomeasure the springback and thus improve the forming characteristics of aluminumsheet for aerospace applications. Nakamachi et al. (2002) assessed the formability ofaluminium alloys using viscoplastic FE analysis. Krempl (2001) studied relaxationbehaviour of several ductile metals and alloys by carrying out relaxation test atroom and high temperature. His results suggested that the stress relaxation ratedepends nonlinearly on the strain rate preceding the relaxation test. Numericalsimulation of the relaxation of aluminium alloy was carried out to predict thevariation of shape of sheet metal with different relaxation times (Zhu et al., 2001).Kaneko and Oyamada (2000) and Yaguchi and Takahasi (2000) carried out bothexperimental and theoretical investigation on the strain rate dependence of plasticbehaviour with ageing effect. Based on the experimental results, new constitutivemodels were developed by taking dynamics strain-ageing effect into consideration.Stoughton (2002) employed a non-associated flow rule to propose a new improvedmaterial behavior that shows good agreement with experimental data for both yieldand plastic strain ratios in uniaxial, equi-biaxial, and plane- strain tension underproportional loading for steel, aluminum and possibly other alloys.Creep forming of panel components is based on the stress relaxation phenomenon

due to creep, which occurs during the artificial ageing of a metal. Creep deformationtakes place at low stress levels and the amount of the plastic deformation is directlyrelated to ageing temperature and time. This is significantly different from the othermetal forming processes, where elastic plastic deformation of materials is dominant.The springback in creep forming panel components is dependent on creep behaviour

K.C. Ho et al. / International Journal of Plasticity 20 (2004) 733–751 735

of materials at high temperature, the thickness of sheet material and the curvature ofthe component to be formed.Creep forming is used mainly to form large aircraft wing panels, which may be

single curvature or doubly curved complex shaped components. The thickness ofcomponents varies from about 63.5 down to 2.5 mm (Holman, 1989). The thicksheets are difficult to bend and stress relaxation features through the sheet thicknessare very complicated. Experimental work on creep forming a thick cylindricalcomponent using a large radius tool shape indicates that the typical springback isabout 65–80% according to the definition given by Pitcher and Styles (2000). Theamount of springback varies non-linearly with the sheet thickness and the curvatureof the tool shape. For industrial applications, cost-efficient numerical studies onspringback in creep forming need to be available. With the aim of providing apractical technique, this paper introduces an integrated numerical procedure tosimulate springback in creep forming using the commercial FE software ABAQUS.The tool shape definition technique and the method of implementing physicallybased unified creep damage constitutive equations into ABAQUS are presented. Thestress relaxation behaviour and the effects of forming conditions and tool shapecurvature on springback are analysed.

2. Physically-based unified creep constitutive equations

2.1. Uniaxial creep constitutive equations

Continuum Damage Mechanics (CDM) with three state variables was justifiedand then used by Kowalewski (1995) to model primary creep hardening and tertiarycreep softening in aluminum alloys by; (i) hardening due to dislocation at the initialstage of creep deformation, (ii) softening due to material ageing under hightemperature, and, (iii) softening due to grain boundary cavity nucleation andgrowth. The equations describing these phenomena, set in uniaxial form, are:

":¼ A= 1� !2ð Þ

n½ �sinh B� 1� Hð Þ= 1� �ð Þ½ � ð1Þ

H:¼ h=�eð Þ"

:1� H=H�ð Þ ð2Þ

�:¼ Kc=3ð Þ 1� �ð Þ

4ð3Þ

!:2 ¼ D"

:ð4Þ

where A, B, h, H*, Kc and D are material constants, and n is given by

n ¼ B�e 1� Hð Þ= 1� �ð Þ½ �coth B�e 1� Hð Þ= 1� �ð Þ½ �

Eq. (1) describes creep behaviour for a specific aluminum alloy. The materialconstants A and B are related to secondary (steady state) creep. Eq. (2) describesprimary creep using H, which varies from 0 at the beginning of creep process to H*,


the saturation value of H at the end of primary creep and subsequently maintainsthis value until failure. More detailed descriptions of the equations have been givenby Kowalewski et al. (1995). The constants within the constitutive equations weredetermined from experimental creep data of an aluminum alloy at temperature150 �C for stresses of 241.3, 250.0, 262.0 and 275.0 MPa, and listed in Table 1.

2.2. Formulation of multiaxial constitutive equations for large deflection

First consider the creep rate Eq. (1). Without the primary hardening, ageing anddamage variables, it reduces to "

:¼ Asinh B�ð Þ. The equation can be generalized for

multiaxial conditions by assuming von-Mises behaviour and defining an energydissipation rate potential as:

¼ A=Bð Þcosh B�eð Þ ð5Þ� �1=2

where �e ¼ 3SijSij=2 is the effective stress and Sij ¼ �ij � �ij�kk=3 are deviatoricstresses. Assuming normality and the associated flow rule, the multiaxial creep strainrates are given as:

":ij ¼ @ =@Sij

� �¼ 3Sij= 2�eð ÞAsinh B�eð Þ ð6Þ

On reintroducing hardening and damage variables, H, � and !2, the effective creepstrain rate can be written as:

p:¼ A= 1� !2ð Þ

n½ �sinh B�e 1� Hð Þ= 1� �ð Þ½ � ð7Þ

The set of unified multiaxial creep constitutive equations, implemented within alarge deflection formulation, can be written as:

DPij ¼ 3Sij=2�e

� �p:

ð8Þ

H:¼ h=�eð Þp

:1� H=H�ð Þ ð9Þ

�:¼ Kc=3ð Þ 1� �ð Þ

4ð10Þ

!:2 ¼ Dp

:ð11Þ

� ij ¼ GD eij þ 2lD e

kk ð12Þ

where DPij is the rate of plastic deformation, D e

ij ¼ DTij � DP

ij is the rate of elasticdeformation, DT

ij is the rate of total deformation and � ij is the Jaumann rate of

Table 1

Values of the material constants for the constitutive equations at 150 �C

A (h�1)
B (MPa�1) h (MPa) H* (–) Kc (h�1) D (–)
4.04 10�15
0.1126 2.95 104 0.1139 1.82 10�4 2.75

Cauchy stress. G and l are the Lame elasticity constants. The multiaxial constitutiveequations have been implemented into the large deformation finite element solverABAQUS through the user-defined subroutine CREEP and used to simulate creepforming.

3. Numerical procedures for simulating springback in creep forming

FE simulations of creep forming panel components have been carried out usingtwo tool shapes. One is single curvature cylindrical tool [Fig. 1(a)], and, the other isdoubly curved spherical tool [Fig. 1(b)]. The components are made of an aluminumalloy and are creep formed at a constant temperature of 150 �C. The creep formingprocess consists of locating a flat metal sheet against the tools, the surfaces of whichform cavities in the shape required. For the convenience of locating the workpieceon the tool surfaces in the forming simulations, four soft springs, which can be

Fig. 1. FE models with (a) cylindrical and (b) spherical tool surfaces.


compressed to zero volume are used to support the weight of the workpiece, asshown in Fig. 1. Pressure is applied to the opposite side of the sheet to force it toacquire the tool shape.

3.1. Tool surface definition

In order to obtain optimal and reliable convergence, it is essential that the rigidtool surface representation is smooth. The point-based high quality surface definitionmethod (Li et al., 2001) is used to represent the tool surface for the FE simulations.This method allows tool surface reconstruction to be carried out from a cloud ofdata point generated either from a CAD system or from a digitizer. Thus the surfacedata transfer between CAD, 3D scanning and FE facilities are simplified. The basicprocedures of using the point-based technique to create high quality tool surfacesinclude:

� Selection of a sparse grid of master data points from a cloud of data set. Thesparse data points should be minimised and just sufficient to characterise theshape of a tool surface with a constraint that the minimum number of datapoints are equal or more than four in each row and column.

� Generation of dense data points based on the sparse grid of master datapoints. The density of data points is controlled by the difference of tangentdirections at adjacent data points along rows and columns (Li et al., 2001).The tool surface is approximated by small facets formed according to thedense data points.

To solve the contact problems, the orthogonal unit tangents and the rate ofchange of surface normal at contact points are computed. This ensures that if a nodeon the workpiece is in contact with the tool surface, the node can move only bysliding along the surface and further penetration is prevented. These surface definitionand contact calculation methods were implemented into ABAQUS through theuser-defined subroutine RSURFU.

3.2. Multi-step springback simulation for creep forming

Contact between the rigid tool surface and the deformable aluminum sheet isdefined by specifying (i) the tool surface model and (ii) the contact nodal set for thesheet. Friction, as a surface interaction property, is related to the contact pair byspecifying a friction coefficient of 0.3. The models, shown in Fig. 1, consist ofmaterial sheets and rigid cylindrical and spherical tool surfaces connected bysprings. Three aluminum sheets with thickness of 9, 18 and 25 mm are used in theforming simulations. The length of the aluminum sheet is 760 mm and the widthvaries from 76 to 760 mm. The three-node triangular shell element S3R, which issuitable for both thin and thick shells, is used for the analysis. The rigid tool surfaceshave the same radius of 2297 mm with a depth of 40 mm and are defined throughthe user-defined subroutine RSURFU.


The creep deformation of the aluminum sheets in creep forming is governed by theunified creep damage constitutive equations, which were implemented into ABAQUSvia the user-defined subroutine CREEP. In addition to the original constitutivemodel, the gradients, @Dp=@p and @Dp=@�e were determined accurately, so that thestep time could be controlled efficiently through the implicit integration employed byABAQUS. The primary hardening and tertiary softening variables, H, � and !2,were integrated at the end of each increment of the creep deformation computations.Multi-step numerical procedures have been developed to simulate the creep formingprocess and to evaluate the springback using ABAQUS. The procedures are brieflyas follows:

� Step 1: apply a uniform pressure normal to the top surface of the aluminumsheet to overcome the stiffness of the four springs and to deform the work-piece into complete contact with the tool surface.

� Step 2: maintain the loading pressure and hold the aluminum sheet on thetool surface for a certain period, for example 20 h. This allows creep to takeplace, thus the forming stress is relaxed.

� Step 3: remove all the pressure loading. Fully constrain the formed sheet sothat it still remains in full contact with the tool surface.

� Step 4: remove the constraints allowing springback to occur.� Step 5: load the springs until contact between the four corners of the sheetand the tool occurs.

� Step 6: measure springback; the distance between the lowest points on thesheet and tool.

3.3. Definition of springback measurement

Fig. 2 shows the definition of the springback, which is quantified by a factorR ¼ �max=�0ð Þ. �0 is the maximum vertical distance between the initial, undeformedworkpiece and the forming tool surface, �max is the maximum vertical distancebetween the formed workpiece and the forming tool surface. If R equals to 1, theworkpiece is completely springback to the original shape, while R equals to 0, thereis no springback.

4. Computational results

Fig. 3 shows the simulated maximum (surface) stresses arising in a 18 mm-thickaluminum sheet during deformation, creep and springback using a sphericallycurved tool. The total forming time is 20 h. Fig. 3(a) shows the initial loading stage,in which the springs are compressed by uniform pressure, which also deforms thematerial sheet elastically. The stress level is low at this stage. Further pressure isapplied to deform the whole workpiece into the tool shape, as shown in Fig. 3(b). At thisstage the maximum stress level reaches about 251 MPa, although stress relaxation is


allowed. With the workpiece held to the tool shape for 20 h, creep takes place, whichrelaxes the stress. The maximum stress reduces to approximately 222 MPa from 251MPa, as shown in Fig. 3(c). Springback occurs upon release of the forming pressuresince the stresses are not fully relaxed by creep. Fig. 3(d) shows the residual stressdistribution.Due to limitations on the maximum usable temperature and the forming time

dictated by microstructure requirements, it is not possible to fully relax the stresscreated in the forming process. The creep deformation is virtually in the stage ofsteady state. When the stress level drops due to creep, the creep rate reduces sharplyaccording to a power-law or sinh-law and the stresses relax in a similar way.Theoretically, an infinite time period is required to fully relax the stresses, which isnot possible in practice. Therefore, springback is unavoidable in creep forming.

4.1. Stress relaxation in creep forming

Fig. 4(a) shows the variations of longitudinal stress components through the sheetthickness at the central location of aluminum thick workpiece, which is creepformed to a cylindrical curvature for 20 h. In this forming simulation, the loadingpressure was increased immediately to deform the sheet into the tool surface con-figuration. Pure elastic deformation is assumed in this stage and the variation ofstress through the thickness is shown by the solid straight line in Fig. 4(a). Themaximum tensile stress (�490 MPa) at the bottom of the workpiece is higher thanthe maximum compressive stress (�400 MPa) at the top surface of the workpiece.This is because the large deflection theory is employed in the analysis. Based on thegoverning creep constitutive equations, the initially high stress levels yield high creepstrain rates which results in dramatic reductions of stresses. The dotted line inFig. 4(a) shows that after 20 h creep, the maximum stress drops to approximately250 MPa in magnitude. For the low magnitude initial stresses, e.g. 200 MPa, onlylittle creep take place within 20 h, thus stress relaxation is not significant. Theseresults support the fact that stress relaxation in creep forming is largely dependenton the initial stress level. After unloading, residual stresses are observed, as they are

Fig. 2. Springback is defined by a factor R ¼ �max=�0, where R=0 represents no springback; R=1

represents fully springback.


Fig. 3. Evolution of effective stresses during the forming process. The workpiece (a) deforms elastically

under the initial loading, (b) is loaded onto the tool surface, (c) deforms due to creep for 20 h; (d) is

unloaded and springback occurs.


not fully relaxed during creep forming. The residual stress profile is complicated.The presence of residual stresses causes springback.In a typical creep forming process, the deformation due to the initial loading is

assumed to be purely elastic. It can be seen from Fig. 4(a) that the stresses at theinitial elastic deformation (solid line) vary linearly across the thickness of the work-piece. If the workpiece is thicker, the induced stress will be higher and realistically,

Fig. 4. (a) Variation of stress component, �11 through the thickness at the central location of workpiece

for the cylindrically curved model, (b) variation of effective stresses, �e through the thickness at the central

location of workpiece for the spherically curved model.


plastic deformation or stress relaxation due to creep may occur during the initialloading process. To study this situation, a further creep forming simulation has beencarried out using a spherical tool surface and a 18 mm-thick square workpiece [seeFig. 1(b)]. In this simulation, the initial loading takes an hour and creep deformationis allowed to take place. This incremental loading process eliminates the extremelyhigh stress near the top and bottom surfaces of the workpiece. FE computationalresults, shown in Fig. 4(b), represent the variation of the effective stresses throughthe thickness at the central location of the workpiece. They clearly show that themaximum effective stresses at the surfaces are about 260 MPa due to the creepdeformation at the initial loading stage. If pure elastic deformation is consideredduring the initial loading, the maximum stress would reach about 1500 MPa.Referring to Fig. 4(b), the overall stress distributions are complicated because thetool has a doubly curved surface. After the creep holding period, stress relaxationtakes place. The maximum effective stress is relaxed from 260 to approximately 220MPa. Again, for low initial stress levels, e.g. below 200 MPa, the stress relaxation isnot significant. After unloading, a complicated residual stress profile through thethickness is observed.To further investigate the effect of the loading period on stress relaxation and

springback, three computations have been carried out using the procedures describedpreviously with the loading periods of 1, 2 and 5 h, followed by specified creep-holding periods. The total forming process time was constant at 20 h. The FE modelwith spherically curved tool surface and 18 mm-thick workpiece was used in thecomputations. Fig. 5 shows the maximum effective creep strains and maximumeffective stresses plotted against the forming period for all three cases. In Fig. 5, atthe beginning of the loading period, both maximum effective creep strains andstresses for all cases increase dramatically. When �e reaches above 300 MPa, it dropsimmediately even though the loading pressure continues to increase. This phenom-enon occurs because stress relaxation takes place due to creep deformation. Highstress levels lead to high creep strain rates, and thus stress relaxation is rapid. This isgoverned by the unified creep constitutive equations and the stress-strain relation-ship. However, there is a complicated interaction between the loading pressure andthe maximum effective stress level induced during this period. When workpiece iscompletely brought into contact with the rigid tool surface, the loading pressure iskept constant for a sustained time. At this stage, all effective stress levels dropgradually to about 230 MPa and become almost steady until the end of the formingperiod. In the mean time, the maximum effective strain increases slowly until abovea value of 0.75%. These features indicate that even if the total forming time isincreased, no significant changes in either maximum effective stresses or maximumeffective creep strains. Therefore, the predicted springback is not likely to be affectedby the initial loading periods after a certain creep holding period.

4.2. Effects of forming processes on springback

To assess the effect of forming period on springback, computations have beencarried out for total forming periods, t of 6, 10 and 20 h with viscoplastic loading


intervals, t2 of 1 and 5 h. The computations are carried out with both cylindricallyand spherically curved FE models as described in the Fig. 1. Fig. 6 shows thevariation of springback with different forming periods for viscoplastic loading timesof 1 and 5 h. Solid lines represent the computational results for the sphericallycurved model and the dotted line for cylindrically curved model. The amount ofspringback is dominated by the level of creep deformation. Creep is a time-dependantdeformation, the longer the forming period, more creep strains are accumulated, i.e.more elastic strains are eliminated, thus less springback occurs. With reference toFig. 6, it can be seen that overall, for a longer total forming period t, R is lower. Forexample, for spherically curved model, 20 h forming gives R=0.55 whilst for 6 hforming process, R is about 0.58. Apart from t, the loading interval t2, also influ-ences R for the cylindrically curved model but not very significant for sphericalmodel. For t2=1 h, there are 19 h of constant pressure creep-holding period. Thusmore stress relaxation can take place and therefore, more creep strains are retained

Fig. 5. The evolution of maximum effective creep strains and maximum effective stresses for 1, 2 and 5 h

creep loading periods.


by the workpiece. As a result, less springback is observed after unloading. Thepredicted springback for the single curvature tool surface and the doubly curvedtool surface are significantly different. This is dealt in latter section.

4.3. Effects of sheet thickness on springback

Referring to Fig. 3(a), the stresses vary linearly at the end of initial elastic loading.If the thickness of the workpiece increases, the stresses are expected to increaselinearly. Stress relaxation, which is the crucial factor in determining springback, ismainly dependent on initial induced stress [see Eq. (7)]. To investigate the effect ofsheet thickness on springback, computations were carried out using the two FE

Fig. 6. Variation of predicted springback with different visco-plastic loading periods and total forming

periods. Solid lines represent the computational results for the spherically curved model and dotted lines

are for the cylindrically curved model.


models as described above. Each model is investigated with 9, 18 and 25 mm-thicksquare workpieces. The computational results are shown in Fig. 7. Overall, asthe thickness of the workpiece increases, the springback, R has lower value. Forexample, the 9 mm-thick workpiece almost completely regains its original shapeafter unloading, while for the 25 mm-thick workpiece, R is around 0.7.Referring to the stress distribution in cross-section of the workpiece in Fig. 8,

during deformation, the significant creep or plastic region tends to grow from theouter surface of the workpiece, towards the center of the workpiece, where theelastic core is located. The ratio between the ‘significant creep region’ (SCR) and the‘less creep region’ (LCR) is defined as . When is zero, only pure elastic deforma-tion takes place. When has a higher value (e.g. 1), the SCR is bigger than theLCR, hence the workpiece is more plastically deformed and the induced creep strainconstrains the workpiece to springback after unloading. In creep forming, it is

Fig. 7. Effect of sheet thickness on springback with different creep loading periods. Solid lines represent

the computational results using the spherically curved model and dotted lines are the cylindrically curved

model.


almost impossible to completely eliminate the elastic core within the workpiece.However, for the same forming condition, when thicker workpiece is used, the LCRdoes not increase much while the SCR increases, i.e. increases. This explains whythicker workpiece has less springback. In addition, for a thicker workpiece, theinitial stress level induced is higher. Generally, the rate of stress relaxation is greatlyenhanced by higher initial stress level, i.e. less springback for thicker workpiece.

4.4. Effects of doubly curved tool surface on springback

From Figs. 6 and 7, the FE models consist of spherically curved tool surfacepredict lower R values compared with those using cylindrically curved tool surface.To investigate the effect of doubly curved tool surface on springback, furthercomputations were carried out using the spherically curved model with 9, 18 and 25mm-thick workpieces in different dimensions. The predicted springback is shown inFig. 9. The workpieces used in the simulation have width/length ratio w=lð Þ of 1, 0.5,0.25 and 0.01. As the w=l ratio decreases, the doubly curved surface effect decreases.Thus the spherically curved tool surface and workpiece with w=l ¼ 0:01 approx-imates to the cylindrically curved model. From Fig. 9, it can be seen that, as w=ldecreases, R increases. When w=l approaches to zero, R reaches its maximum value,which is the value obtained with the cylindrical curved workpiece.Creep forming is very similar to pure bending. For the cylindrical tool surface, the

workpiece deforms under a single moment applied at its two ends, while the otherends are free. Hence the top layer of the workpiece experiences more close touniaxial compression and the bottom layer uniaxial tension. Meanwhile, for thedoubly curved tool surface, biaxial compression and tension are observed. Creepforming a doubly curved component is similar as applying moments at four ends ofthe workpiece. The sum of these moments increases the stress level extensively andhence increases the creep strain in the bent workpiece. As the four ends are severelybent initially, more creep strains are accumulated at the outer edges of workpieceand thus constrain the workpiece from springback (i.e. the ‘hoop effect’). Comparingto the single curvature tool, during creep forming, the workpiece is only bent at two

Fig. 8. Comparison of the distribution of the SCR and LCR in the cross section of workpieces with

different thickness.


ends. The significant creep strains only accumulate at both ends. As the results, theformed workpiece has weaker constraints, hence springback occur more easily.

5. Conclusions

Integrated numerical techniques and procedures are developed to analyse spring-back behaviour in creep forming thick aluminum sheet components using FEmethods. Analyses have been carried out for a range of forming conditions and thefollowing results have been obtained.(1) The initial viscoplastic loading period has little effect on the predicted spring-

back if the ratio of creep forming time to the initial loading time is greater than 4.The creep deformation mainly takes place at the beginning of the forming processand the stress relaxation saturates quickly once the workpiece is completely in

Fig. 9. Variation of predicted springback for the spherically curved model with different dimensions.

l and w are the length and width of the workpiece respectively.


contact with the tool surface. Thus a longer holding period in creep forming doesnot reduce the amount of springback significantly.(2) Creep mainly takes place near the top and bottom surface of the workpiece,

where stress levels are high. Similar to elastic-plastic bending deformation, littleplastic (creep) deformation occurs near the neutral surface of the workpiece, wherethe stress level less than about 200 MPa during a creep forming process. Thereforethe thickness of a workpiece can be divided into two parts: one is ‘significant creepregion’ (SCR) and the other is ‘less creep region’ (LCR). The amount of springbackcan be assessed by the ratio of SCR and LCR. Thus for the same tool shapes, thethicker workpiece has less springback.(3) For the same thickness of workpiece and the same curvature of tool shape, the

doubly curved tool surface causes less springback compared to a single curvaturetool. In addition to the ‘hoop effect’, the maximum effective stress, which is thedriving force for creep, is higher in forming doubly curved sheet components. Thisresults in a higher value of the ratio of SCR and LCR.

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modelling of springback in creep forming thick aluminum sheets

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