International Journal of Mechanical Sciences 105 (2016) 266–272

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International Journal of Mechanical Sciences

http://d0020-74

n CorrE-m

journal homepage: www.elsevier.com/locate/ijmecsci

Prediction of springback in air-bending of Advanced High Strengthsteel (DP780) considering Young's modulus variationand with a piecewise hardening function

Xi Yang n, Changhyok Choi, Nimet Kardes Sever, Taylan AltanCenter for Precision Forming (CPF, www.cpforming.edu), The Ohio State University, 339 Baker Systems, 1971 Neil Avenue, Columbus, OH 43210, USA

a r t i c l e i n f o

Article history:Received 21 June 2015Received in revised form16 November 2015Accepted 19 November 2015Available online 2 December 2015

Keywords:V-die bendingSpringbackAHSSAnalytical methodFinite element method

x.doi.org/10.1016/j.ijmecsci.2015.11.02803/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ1 614 620 3002.ail address: yang.1418@osu.edu (X. Yang).

a b s t r a c t

Advanced High-Strength steels (AHSS) are increasingly used in industry. Thus, the springback predictionin bending of AHSS is important to maintain close geometric tolerances in deformed parts. The uniqueproperties of AHSS: a non-constant Young's modulus and a stress–strain relation which does not follow asimple flow stress equation make the prediction of springback very difficult. In this study, an analyticalmodel was developed to predict the springback in air-bending of AHSS, considering the special prop-erties of these materials. A computer code which is developed based on classical bending theory wasupdated based on the new model. The finite element simulation which includes the properties of theAHSS is also presented in the paper. The comparison between the experimental results and predictionsindicates that the detailed consideration of the properties of AHSS affects the accuracy of the springbackprediction with the analytical method. It is hoped that this new analysis can be incorporated into thecontroller of a press brake to adjust the machine for compensating springback during bending.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Springback is a common phenomenon in sheet metal forming.It refers to the dimensional change of the deformed part afterunloading due to the elastic recovery. The prediction of springbackin bending of Advanced High-Strength steels (AHSS) is becomingincreasingly important because these materials are now widelyused in industry. Springback can be estimated by Finite ElementAnalysis (FEA). Selections of the appropriate element, solver type,definition of the tool model and boundary conditions are crucial toobtain an accurate solution in reasonable amount of computationtime. Furthermore, it is very important to input into the FE modelthe correct material data such as hardening model, hardening lawand Young's modulus.

FEA requires a commercial code and computer time. On theother hand, a simple computer code, which can predict thespringback fast and accurately, can be incorporated into a pressbrake controller. Thus, the bending operation can be improved tocompensate for springback. Certain properties of AHSS make thespringback prediction difficult in using both FEM and analyticalmethods. First, Young's modulus of AHSS is not constant duringthe forming process. Previous researchers have reported that the

unloading and loading Young's moduli of AHSS are not the sameand both of them decrease with strain. Fei and Hodgson [5]observed the Young's modulus variation in simple tensile test.Stoughton et al. [14] pointed the difficulties of constitutive mod-eling of metal forming considering the modulus variation. Levyet al. [9] studied the unloading Young's modulus effect on auto-motive metals. Cobo et al. [4] specifically investigated the decreaseof Young’s modulus of Advanced High Strength steel and how itaffects the springback in forming. Researchers at Center for Pre-cision Forming (CPF) Palaniswamy and Altan [10] and Kardes et al.[6] also observed the decreasing Young's modulus of AdvancedHigh Strength steel in tensile tests and studied the effect ofspringback in V-bending. Most of the methods, used to predictspringback, assume a constant Young's modulus during deforma-tion. To predict springback accurately, modulus variation has to beconsidered. Considerable research has been conducted for esti-mating plane strain bending and springback using analyticaltechniques. For instance, Buranathiti and Cao [3] use analyticalmodel to predict the springback in straight flanging. Zhang et al.[16] developed a model for springback as well as side wall curlprediction in U-bending. Alexandrov et al. developed an analyticalmodel to calculate the large strain bending of incompressible sheetunder tension in his series publications (2009, 2011) [1,2]. Someother studies developed semi-analytical approach that combinesthe analytical model with a set of simple FEM problems, whichsignificantly reduces the computational cost of a full Finite

1

2

1

2

3

4

X. Yang et al. / International Journal of Mechanical Sciences 105 (2016) 266–272 267

Element simulation. Examples include the 2D draw bead spring-back prediction by Lee et al. [8] and U-channel bending undertension study by Quilliec et al. [12].

Other than these research, an analytical model, which is pro-posed by Raghupathi et al. [13], for springback prediction in airbending, is further improved by Wang et al. [15] to developpractical design aids and adaptive control of a press brake. Centerof Precision Forming (CPF) at The Ohio State University developeda FORTRAN code names “BEND” based on this model which cangive an accurate springback prediction in bending of low carbonsteels [7]. Two important assumptions were made in this model:

) Swift's material model ½σ ¼ K ϵ0þ ϵ� �n� was used to describe the

sheet metal's property.) The Young's modulus was assumed to be constant during theloading and unloading process.

However, AHSS does not fully satisfy these two assumptions.Constant strength coefficient, K, and strain hardening exponent, n,values are not sufficient to describe the material's behavior inplastic state. Thus, a new analytical model is developed, whichdescribes the material's property (flow stress) piecewisely andconsiders Young's modulus variation. The computer code is nowupdated based on the new method. In addition, the FEM simula-tions that consider the unique properties of AHSS are also pre-sented as comparison. This study is an extension of the previouswork by Kim et al. [7].

True strain0.0 0.1 0.2 0.3 0.4

True

Stre

ss(M

Pa)

0

200

400

600

800

1000

1200

1400K=1465MPa, n=0.14

Flow stress from experiment

Fig. 2. Flow stress data from measurement bulge test and tensile test vs. recreatedstress–strain relation using constant K and n values (Material: DP780) [6].

2. Analytical model

2.1. Material properties

The unique properties of an AHSS (DP780) are clearly shown inFigs. 1 and 2. Fig. 1(a) shows the stress–strain curves obtained fromload-unload tensile test. Fig. 1(b) plots the corresponding Young'smodulus at each load-unload cycle [6]. Notice here that there is asmall deviation between the unloading and reloading curves. Theplotted Young's moduli are the average of unloading and reloadingvalues at each strain level. Apparently, Young's modulus of DP780decreases nonlinearly as the strain increases. When the strain levelis greater than 0.12, Young's modulus is assumed constant.

Fig. 2 illustrates that stress–strain relation, expressed by Swift'smodel with constant K and n, gives a flow stress curve whichdeviates from the experimental data. These two features of DP780cause the existing analytical method to be inaccurate in predictingspringback.

Fig. 1. (a) Stress–strain curve in load and unload tensile tes

One effective way to improve the methodology for predictingspringback in bending of AHSS is to use the Young’s modulus, aswell as K and n values that vary with strain. Therefore, instead ofusing traditional hardening rule, a piecewise hardening equationas a function of strain could be used. Young's modulus could betreated piecewisely as a function of strain as well. The experi-mental tabular data from tensile test or bulge test are required torepresent K, n and Young's modulus values. The following steps arenecessary:

) Tabular data of Young's modulus vs. strain will be obtained fromload–unload tensile test, and flow stress curve will be obtainedfrom the bulge test [6]. The bugle test was conducted at TheOhio State University, Center of Precision Forming.

) The flow stress curve will be divided into many small regionsaccording to the recorded strain data points. Ideally, the straininterval should be as small as possible to capture the material'sbehavior. Based on a parametric study, the maximum straininterval should be no larger than 0.005 to ensure accuracy. Inthis study, a strain interval of 0.002 was selected.

) Young's modulus will be linearly interpolated at the boundariesof each region. Within the small region, the modulus is assumeconstant and with the average value of the boundary points.

) K and n values will be calculated by fitting the equation ½ σ ¼ Kϵ0þ ϵ� �n� with the two data points at the boundaries of eachregion. A demonstration of the interpolation method of Young'smodulus, K and n values are shown in Fig. 3.

True strain

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Elas

tic m

odul

us (G

Pa)

140

150

160

170

180

190

200

210

220

t; (b) variation of the unloading modulus of DP780 [6].

5

6

Modulus from test at 0.03

Modulus from test at 0.02

Linear interpolated moduli(average of boundary values)

Fitted (K1, n1), (K2, n2), (K3, n3)…at each region using the two data points at boundaries.

One small region with strain interval 0.002

Fig. 3. A zoom-in stress–strain curve between 0.02–0.03. Strain interval of each small region is 0.002. The modulus of each point can be linearly interpolated using the testdata at strain of 0.02 and 0.03. Constant modulus is assume within in each region with the value of the average of boundary point. For each small region, (K, n) will be fittedwith equation ½ σ ¼K ϵ0þ ϵ

� �n� using the two data points at the boundaries of that region.

RoutRin

O

Fig. 4. Dividing thickness into small regions with respect to tabular strain data andusing constant K, n and E values corresponding to strain values at different regions.

X. Yang et al. / International Journal of Mechanical Sciences 105 (2016) 266–272268

) New two sets of tabular data will be obtained: Young's modulusvs. strain, and K and n vs. strain.

) Instead of integrating over the entire domain with one set ofconstant parameters, the maximum moment, elastic stiffnessand springback will be calculated by integrating over each smallregion with different K, n and Young's modulus values, infunction of strain. The final result is obtained by summing up allthe integrals in each region.

2.2. Loading moment in the plastic zone

The loading moment in the plastic zone, ML, needs to be deter-mined first. The strain values along the thickness change from “0” atthe un-stretched fiber to maximum at the inner and outer fibers(Fig. 4). The calculation of the location of the un-stretched fiber isdiscussed in [11]. Due to the properties of the AHSS, the stress–strainrelation will follow different K and n values as strain changes, andYoung's modulus changes as well. So the thickness of the sheet metalis divided into several small regions, such that the strain values at theboundaries of each region correspond to the experimental data points.Then, within each region, the stress–strain relation can be expressed

by Swift's model with a pair of constant K and n. Such division ofstress–strain curves will only assure C0 continuity, which means thestress–strain curve are continues at the boundaries of each region, butthe first derivative of the fitted function for each interval may not becontinues. This is because at each interval, the fitted K and n values aredifferent. Young's modulus can be assumed to be constant within eachregion since the change of strain is small enough (Fig. 3).

As a result, the loading moment in the plastic zone MLð Þ can becalculated by using Eq. (1):

ML ¼wZ y1

0σxydyþ

Z y2

y1

σxydyþ…þZ rout

ym

σxydy

!

þwZ y1

0σxydyþ

Z y2

y1

σxydyþ…þZ rin

yk

σxydy

!ð1Þ

where y is coordinate at the boundary of each region, σx is cir-cumferential stress within each region, rin and rout are radii at theinner and outer fibers, respectively (Fig. 4).

By using the flow stress relation, Eq. (1) can be modified toEq. (2):

ML ¼wZ y1

0FK1 ε0þFεxð Þn1ydy þ

Z y2

y1

FK2 ε0þFεxð Þn2ydyþ…

þZ rout

ym

FKn�1 ε0þFεxð Þnmydy

!

þwZ y1

0FK1 ε0þFεxð Þn1ydyþ

Z y2

y1

FK2 ε0þFεxð Þn2ydyþ…

þZ rin

yk

FKn�1 ε0þFεxð Þnkydy

!ð2Þ

where F is the index to account for anisotropy, F ¼ 2 1þ R� �� � 1

M=2½1þ ð1þ2 RÞ 1

1� M �ðM �1Þ=M , M¼ (1þ R) is the anisotropy index, εx isthe circumferential strain at each region, pre-strain ε0 is calculatedaccording to the initial yield of the material, K1,…, Kn, n1,…, nm arethe strength coefficients and strain hardening exponents in dif-ferent regions. Once the loading moment in plastic region, ML, isobtained, the moment distribution along the length of the bentsheet, MðSÞ, can be written as a linear function of arc length S, asdiscussed in [15].

Fig. 5. Stiffness variation along the length of the bent sheet. In elasto-plastic zone,the arc is divided into small regions and the stiffness in each region is assumed tobe constant. (Refer to [15] for the detail calculation of the arc length OA, AE and EB).

Table 1Properties and dimensions of DP780 sheetsamples.

Strain hardening exponent (n) 0.14Strength coefficient (K) 1465 MPaInitial yield stress (σ0) 540 MPaInitial Young's modulus 207 GPaVariable Young’s modulus Fig. 1(b)Poisson’s ratio (ν) 0.3Initial thickness (t0) 1 mmSheet width (w) 60 mmFriction coefficient (μ) 0.12Anisotropy (R) 0.896

X. Yang et al. / International Journal of Mechanical Sciences 105 (2016) 266–272 269

2.3. Stiffness variation of the bent sheet

For low carbon steels, since Young's modulus is constant, thestiffness of the bent sheet can be calculated as E0I and remainsconstant through the thickness, where E0 is the Young's modulusin plane strain condition (E0 ¼ E=ð1�ν2Þ). Based on the classicelastic bending theory, the moment could be expressed in Eq. (3):

M¼ 1r�1r0

� �E0I ð3Þ

Where r and r’ are the radius of the curvature of the sheet metalbefore and after the springback.

However, the stiffness of the bent sheet has to be re-calculatedto include the effect of Young's modulus variation for AHSS. Wefollow the procedure described in Section 2.2 to calculate theunloading moment in the plastic region, MU, as given in Eq. (4):

MU ¼w1r�1r0

� � Z y1

0E01y

2dyþZ y2

y1

E02y2dyþ…

þZ rout

ym

E0my2dyþ

Z y1

0E01y

2dyþZ y2

y1

E02y2dyþ…

þZ rin

yk

E0ky2þ dy

!

¼ 1r�1r0

� �E0I� �

P ð4Þ

where E0I� �

P is stiffness of the sheet in the plastic zone. In thebending operation, the amplitude of loading moment is equal tothe unloading moment (|ML|¼ |MU|). Thus, the springback can becomputed using Eq. (4) if E0I

� �P is calculated first. A more detailed

explanation of the moment calculation can be found in [7].Another issue to consider is that the stiffness along the length

of the bent sheet is not constant. Fig. 5 illustrates that the strain atouter and inner fibers of the sheet decreases from maximum atplastic zone to zero at the die corner, where at point E which is theboundary of elastic zone, the strain reaches the elastic limit. Thecalculation of arc length OA, AE and EB is explained carefully inWang et al. [15]. Due to the length of the paper, the detailedequations are not discussed here. Since there is only elasticdeformation in the elastic zone (E–B), the stiffness in this portioncan be calculated by using E0I, where E0 is the maximum Young's

modulus. So the equation to calculate springback in the elasticregion remains the same:

θe ¼Z B

E

M Sð ÞE0I

dS ð5Þ

where θe is springback in the elastic region, M(S) is the momentdistribution in elastic zone. In the elasto-plastic zone, the strains atinner and outer fibers along the length of the bent sheet are dif-ferent. As a result the stiffness at different cross sections is notconstant any more. To solve this problem, the arc length is dividedinto several small regions (1, 2, 3, …, n), as seen in Fig. 5. Thestiffness of each small section is assumed to be constant and thespringback of one region can be calculated by using equa-tionθn

S ¼R

ME0 Ið ÞndS, where E0I

� �n is the stiffness of the given section.

The total springback angle (θS) for the bent sheet is obtained bysumming up the springbacks of all the small regions in the elasto-plastic zone and also in the elastic zone as given in Eq. (6):

θS ¼Z

M Sð ÞE0I

dS¼Zð1Þ

M Sð ÞE0I� �

1

dSþZ

2ð Þ

M Sð ÞE0I� �

2

dSþ…

þZ

nð Þ

M Sð ÞE0I� �

n

dSþZ B

E

M Sð ÞE0I� �dS

" #ð6Þ

3. Results obtained from the mathematical model

V-bending tests were conducted at Cincinnati Inc. In V-bendingtest, the sheet metal did not touch the die bottom, thus the resultscan be used to compare with air bending. The dimensions andproperties of the tested samples assuming constant K and n valuesare listed in Table 1. No lubrication is used during the test. Thecoefficient of friction (0.12) between tool and sheet is calibratedaccording to the FEA simulation [6]. Table 2 is the list of the tooldimensions used in V-bending test as well as in the computer pro-gram for calculation. The die opening is calculated by the geometricrelations between V-bending and air-bending, as shown in Fig. 6. Inthe test, flat cut DP780 samples were bent with different press brakestroke depth. The stroke was selected to bend the sample to 30°, 45°,60°, 70°, 80°, 90°, based on the geometrical relation of the tool.Digital photos were taken when the sample was under loading aswell as after springback, Fig. 7. Springback angles were calculated bymeasuring the bending angles of the samples before and afterspringback on the digital pictures with a small Matlab code.

The computer code based on classical bending theory (BEND)was modified according to the improved mathematical model totest the effectiveness of the new method. Experimental springbackmeasurements of AHSS DP780 were compared with the predic-tions made with the analytical model.

The modulus variation from tensile test and flow stress datafrom bulge test were input for calculation. The evaluation of theprogram was conducted as follows:

Table 2Dimensions used in V-bending test and Air-bending test.

Punch radius,Rp (mm)

Die cornerradius, R (mm)

Die opening(mm)

V-Bending 4.75 3.81 36.77Air-Bending(Calculated)

4.75 3.81 32.91

Fig. 6. V-die opening (V) and the equivalent air-bending opening (DL).DL ¼ V�2R 1� tan π�α

4

� �� �, where α is the V-die opening angle (α¼75°), R is the die

corner radius.

Fig. 7. Pictures of bent DP780 samples under loading (a) and after springback (b).

Stroke (mm)0 2 4 6 8 10 12 14 16

Spr

ingb

ack

(deg

ree)

0

5

10

15

20

25

MeasurementsE=207GPaE=153GPaE as a fuction of average largest strain of inner and outer fibervariable K, n and E

Fig. 8. Predicted springback using Young's modulus value corresponding to thelargest average strain in bending.

X. Yang et al. / International Journal of Mechanical Sciences 105 (2016) 266–272270

(1) Input the highest and the lowest Young's moduli fromtensile test while keeping K an n constant. The result from thiscase can show the influence of Young's modulus in springbackprediction; (2) use the Young's modulus value in function of thestrain in the plastic zone for a given bending angle; (3) considerthe Young’s modulus variation and non-constant K and n inmoment and stiffness calculation.

3.1. Springback prediction with young's modulus in function of thelargest strain

To explore the influence of Young's modulus variation, first thehighest Young's modulus (207 GPa) when strain is zero and thelowest Young's modulus (153 GPa) when strain is around 0.1 areused as input to the program, Fig. 1. The results are shown in Fig. 8.When punch stroke is small (e.g. less than 2 mm), the program didnot calculate the springback. This is because when calculating thespringback, we assumed there is a pure plastic region near thepunch tip which does not contribute to springback. However whenthe punch stroke is small, the elastic recovery of this region cannotbe ignored. Therefore the springback estimation is not accurate. It isobserved that the springback angles when punch strokes are largerthan 3 mm are located between the boundary values predictedusing highest and lowest Young's moduli. Therefore, the averageunloading Young's modulus vs. strain (Fig. 1) was used as the inputto the program next, while K and n are kept constant. For a givenbending angle, we first calculate the maximum strains along thelength of the bent sheet at inner fiber εin and outer fiber εout . Thelargest strain is estimated by taking the average of these two values,εavg ¼ 1

2 εin þ εout jÞ����������� . Young's modulus used for this angle is

interpolated from the input Young's modulus vs. strain data basedon this largest strain value. The predicted results with variableYoung's modulus are also shown in Fig. 8. The results show that thepredicted springback with Young's modulus as a function of strain isvery close to the average measured springback angles.

3.2. Springback prediction with variable K, n and Young's modulus

In this case, the springback calculation will consider variable K, nand Young's modulus values in function of strain. Flow stress dataobtained from the biaxial VPB test [6] (Fig. 9) are input to the com-puter program. The loading moment in plastic zone was calculatedfirst using Eq. (2). As discussed previously, the moment distributionin elasto-plastic and elastic zones can be written as a linear functionof the arc length S. However instead of using Eq. (5) to calculatespringback, the stiffness in elasto-plastic zone was also assumed tobe distributed linearly as a function of arc length, E0IðSÞ, Fig. 10. Sincethe stiffness in elastic zone is constant, the springback angle (θS) ofthe whole bent sheet can be calculated as given in Eq. (7):

θS ¼Z E

A

MðSÞE0IðSÞdSþ

Z B

E

MðSÞE0I

dS ð7Þ

The predicted results are also plotted in Fig. 8. The plot indicatesthat, using variable K, n and E values, the springback predictions are

0

200

400

600

800

1000

1200

1400

0.0 0.1 0.2 0.3 0.4

Tru

e St

ress

(MPa

)

True Strain (mm/mm)

VPB test data

Tensile test data

Fig. 9. Flow stress obtained by tensile and viscous pressure bulge (VPB) tests [6].Data from viscous pressure bulge (VPB) test starts from 0.01 true strain and goes upto 0.35 true strain. Yield strength was found from tensile test.

Fig. 10. Stiffness variation E’I along the length of the bent sheet. AE: elasto-plasticzone; EB: elastic zone (Fig. 5).

Table 3FE simulation parameters for AHSS (DP 780).

Simulation parameters Descriptions

Material type Sheet-elastic plastic, V-die and punch-rigidElement Type Solid (2D),Young's modulus Constant: 207 GPa and 153 GPa (max. and min.

values in Fig. 1)Variable (See Fig. 4)

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16Sp

ring

back

ang

le (d

eg)

Punch stroke (mm)

FEA - cons E=153 GPa

FEA - cons E=207 GPa

FEA - var E

measured springback-rolling direction

Fig. 11. Comparison of the springback angles obtained by FEA (by using FE codeDEFORM 2D) and experiments (using constant and variable Young's modulus).

X. Yang et al. / International Journal of Mechanical Sciences 105 (2016) 266–272 271

slightly smaller than the calculations using E as a function of thelargest strain. This is because if the Young's modulus is the functionof the largest strain, it is the smallest E value within the part (Fig. 1).As a result, the springback calculated using this E value will belarger. While the variable E method considers strain variationwithin the part, the E value is slight larger than the previousmethod. Therefore, the estimated springback is smaller. The invol-vement of variable K and n in this calculation does not show sig-nificant influence on the results. The reason is that although theSwift's flow stress model cannot describe the material's property ofAHSS precisely, the deviation between the mathematical model andthe experimental data is not that large (Fig. 2). Therefore in thiscase, variable K and n did not considerably improve the estimationaccuracy. However, the springback prediction may be improved fordifferent sheet materials and tooling setups.

1

2

3

4. Prediction of results from finite element simulations

Finite element method is another common way to predictspringback, especially in cases where sheet, tool geometry andloading conditions are difficult to model analytically. Here in thispaper, finite element simulation of the V-die bending test discussedabove is presented here as a comparison with analytical method.

Depending on bending geometry, 2D or 3D simulations have tobe considered. In most practical bending operations, however,when no flanging around a curved line is involved, 2D simulationsgive reasonably accurate results. The simulation parameters thatwere used for FE analyses are given in Table 3. A half model wasutilized for all FE simulations to save computation time. The flowstress data (Fig. 9) is used in tabular form in FEM. In order to bettercompare with the analytical model, Bauschinger effect and ani-sotropy was also neglected in FE simulation to keep consistency.Isotropic hardening model (von-Mises) was used in the FE simu-lations. The FE simulation was conducted using commercial

software DEFORMTM. A user subroutine was utilized to input theunloading modulus as a variable (Fig. 1). The flow stress was inputas tabular data from Fig. 9. Comparison of springback anglesobtained from FEA and experiments is given in Fig. 11.

It can be seen from Fig. 11 that FEA predicted much larger spring-back with minimum Young's modulus (153 GPa) as an input, which isconsistent with the analytical model. With E¼207 GPa, FEA's predic-tion is still smaller but is very close to the test data. This is becausetabular flow stress data input has the advantage of variable K and n andFEA simulation handles friction much better than the analytical model.By including decreasing Young's modulus, FEA predicted largerspringback than the test data, which is an improvement of the pre-diction with constant Young's modulus. However, with the currentsimple model without considering the Bauschinger effect and aniso-tropy, the FEA overestimated the springback with variable modulus.This will be improved by considering more complex material model.The study is still in process will be discussed in future publications.

5. Summary and conclusions

) The analytical model to predict the springback in bending oflow carbon steel has been well developed and proved effective.This method requires that the material has constant Young'smodulus and its property can be described by Swift's model½ σ ¼ K ϵ0þ ϵ

� �n�with constant K and n values.) The challenges in predicting the springback in bending of AHSSare: (1) Young's modulus is not constant in bending; (2) swift’smodel is not able to describe accurately the material propertyof AHSS.

) An improved analytical model that describes hardening rule andYoung's modulus with a piecewise function was proposed.Instead of using constant K, n and E, the strain distribution wasdivided into small regions. Different Young's modulus, K and n

4

5

6

7

8

X. Yang et al. / International Journal of Mechanical Sciences 105 (2016) 266–272272

values, obtained from the experimental data, are assigned foreach small region.

) The equations used to calculate the moment and stiffness wereincorporated in the computer code developed by Center ofPrecision Forming: The moment and stiffness in plastic zonewere computed by summing up the integrals over the smalldivided regions along the thickness.

) In addition, the stiffness along the length of the bent sheet is notconstant in the elasto-plastic zone. The total springback is thesum of the springback values in elasto-plastic and elastic zones.

) The springback predictions, based on the new model, werecompared with the experimental measurements. The multi-stepverification study indicates that the Young's modulus has asignificant influence on the springback estimation. The accuracyof the estimation can be considerably improved when usingYoung's modulus as a function of the largest strain. Whenincluding all the K, n and E variations in the computation, thepredicted springback angles are slightly smaller than using E asa function of the largest strain. The level of improvement in thiscase is not significant because although the Swift's model is notaccurate, but it is still very close to the experimental data.However considering different sheet materials and thickness,the variable K, n and E may show a great potential of improvingthe estimation accuracy.

) Friction is not considered in the analytical model. The friction forcewill add additional tension during bending. The accuracy could beimproved by including the friction/tension in the analytical model.The effect of friction will be investigated in future work.

) Compared to the analytical method, FE simulations with max-imum value of unloading modulus show better agreement withexperiments than the predictions with minimum value ofunloading modulus. This is because FEA uses flow stress astabular data is more accurate than the K and n values in theanalytical model. When the variation of unloading modulus wasconsidered in FE simulations, the prediction was improved aslarger springback angle were calculated. Although currently theprediction overestimates the springback, the results can beimproved by including more complex material model such asBauschinger effect and anisotropy.

Acknowledgments

We would like to thank Edison Welding Institute for conduct-ing load–unload tensile tests to obtain unloading modulus

variation and Cincinnati Inc. for allowing us to run V-die bendingtests in their facility. We would like to thank Mr. Gautham Suku-maran for his help in the experiments.

References

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