Finite-element modelling of the stretch forming of coated steels

Journ~ of

ELSEVIER

Materials Processing Technology

Journal of Matcriah l'rocc,~xingTechnology 6~g (1997) 71 75

Finite-element modelling of the stretch forming of coated steels A.G. Mamalis *, D.E. Manolakos, A.K. Baldoukas DcT~ttrtmc'ltt o/ ,%h,chmlical /:nghwcring. Nt:ti, mal Tc~tmt(al I.mrcr.~ttv (,1 ,.Ithcn.~. 42. 2gth OctoBer .4re., lob Y,2 ,,Itht'm, Grcc('c

Received 20 September 1995

Abstract The simulation of stretch forming using the explicil non-linear FE-('ode DYNA 3D is reported. FE models concerning forming-limit diagrams and the physical modelling of the process from a macroscopic point of xiew hase been constructed, the theoretical results obtained being compared with the resuh~ of experiments carried out on the stretching of four coated sheet steels. For these materials, modelled as isotropic elastic-phtstic work-hardening, good agreement between modellirg and experiment has been obtained, indicating, therefore, that the proposed FE model predicts FLDs satisfactorily, c 1997 Elsevier

Science S.A. Key,ords: Finite element method: Simulation: Sheet metal ft)rming: Stretch forming: Coated steels

I. Introduction

In sheet-metal Iorming processes the end of useful straining is defined by the onset of Iocalised necking and fracture. Forming-limit diagrams (FLDs), which express the formability of sheet metals, are well known and much work has been done to determine them. both theoretically and experimentally [1-7]. The first theoretical approach was made by Hill [2] and was based on the mathematical estimation of necking, whilst the most widely used theoretical model is that proposed by Marciniak and Kuczynski [3,4] which imposes an inhomogeneity in the sheet metal, introducing a reduced thickness. The m-yield criterion proposed by Hill [5] was used subsequently for modifying the FLD theoretical model, the latter being validated experimentally, see Refs. [6,7]. Finiteelement methods were used recently in the modelling of sheet metal forming [8-10]. In the present work, a numerical model for constructing the FLDs of coated (galvanised) sheet steels is proposed, the explicit FE-Code DYNA 3D being employed. The "necking' region of the deforming sheet is determined by estimating the time step when the value of the strain rate, & of the 'necking' element reaches a particular limit. The results obtained are in good agreement with the experi*

Corresponding author. Fax: + 30 I 7723689.

0924-0136/97/517J)0 ~'~ 1997 Elsevier Science S.A. All rights reserved. P i l S0924-0136(96)02543-5

mental results. The material properties were obtained by standard tensile testing, anisoiropy not being taken into account.

2. Finite element modelling

The explicit FE-Code DYNA 3D was employed to simulate the stretch forming of sheets. As an explicit code, DYNA 3D uses the central-difference method in order to solve the equations of motion; see Refs. [8- 10] for detail~. The punch stretch forming, as described in Ref. [1], was modelled using the 8-node 'brick' element for the simulation of the blank and the punch, as this represents the macroscopic mesh distortion in a better way and is the only element than can provide the value

j-.-Btonk

li| ,: ....

C~,

• ,~tlt=

(ol

II

1.1

•

; V ~

'Punch-I !~ }-

I

BLonkhctder

;~ 100--I

" I Die

•

~ 120

-I

(hi T

Fig. I. Stretch forming of sheet: (a) FE modelling: (b) experimental set-up.

A.G. Mamalis et al. / Journal of Materials Processing Technology 68 0997) 71 75

72 Table 1 Material properties Material

E (N/mm-') Sire.

Et (Ntmm-') Sire.

Y ( N / m m 2) Exp. Sire.

UTS (N/ram-') Exp.

R-value Exp. Sire.

/~ Exp.

Sire.

0° 45 ° 90 °

21

482

180.1 192.4 190.5

286

307.5 310.5

2.15 1.78

I

2.04

I

303.2

2.43

Galvo 2

0° 45 ° 90 °

21 x l0 ~

535

312.2 322.0 330.7

357

383.9 384.2 384.3

1.13 1.35 1.40

I

1.31

I

Galvo 3

0° 45 ° 90 °

21 x 10~

578

313.6 304.8 319.4

360

394.1 386.0 405. I

0.92 1.35 1.27

I

1.22

I

Galvo 4

0° 45 ° 90 o

21 x i03

630

210.1 184.3 197.7

291

318.0 325.7 311.2

2.08 1.73 2.31

l

1.96

1

Galvo 1

Orientation (0)

x

10~

Table 2 Chemical composition o f the steel specimens used Material

Coating thick. (pm)

Chemical composition (10~%l C

Galvo Galvo Galvo Galvo

I 2 3 4

7 8 l0 7.5

Mn

Si

8

152

12

64 39

337 209

9 l0

P

S

AI

N

Nb

V

Ti

B

Cu

Cr

Ni

7

4

29

3,1

<2

4

70

0.1

3

16

33

12 7

7 8

34 53

3.9 5.1

19 22

2 I

I l

8 0.1

l0 ll

20 17

22 24

of the strain rate, ~ (the change of the equivalent strain, d~ in the corresponding time unit, dt, i.e. ~ = dE/d0, at each time step. A schematic diagram of the FE simulation of the stretch forming operation is shown in Fig. I(a). Due to shape symmetry and in order to minimise the CPU time cost, only one quarter of the blank was considered in the calculations, the required nodes therefore being constrained in the appropriate directions. Four galvanised steels were used, two hot-dip galvanised (Galvo 1 and Galvo 2) and two ¢lectro-galvanised (Galvo 3 and Galvo 4), their mechanical properties as obtained from standard tensile tests and their chemical composition being presented in Table 1Table 2, respectively, Each steel was modelled as isotropic elastic-plastic work-hardening material, selected as being the most efficient and simple to fit the experimental tensile curves; see for example Fig. 2 for Galvo I. Note that anisotropy is not taken into account, although the coated steels tested showed some degree of anisotropy (their R-values were 2.04, 1.31, 1.22 and 1.96 for Galvo 1, Galvo 2, Galvo 3 and Galvo 4, respectively). The hemispherical punch was simulated using 8nodes 'brick' elements also. The punch velocity was taken to be about 100 times greater than the actual velocity of the hydraulic press (8 ram/s), in order to minimise the CPU time cost, see also Refs. [8-10]. The

interface phenomena at the contact zone between the punch and the workpiece were analysed using the 'sliding with separation and friction' interface model as defined in the sliding interface DYNA 3D Code library, estimating a Coulomb friction coefficient of about # = 0.2. Note that simulation of the blank-holder and the die, which provide the non-sliding of the sheet as shown in

0.00"

0.10 0.20 True strain

0,30

Fig. 2. Tensile stress-strain curve o f hol-dip galvanised steel (Galvo I).

A.G. ~lamali.~ et e l . . Jooroal ol Materials Proce.~'.~ittg Te~ Imolog.v fig ( 199 7) 71 - 75

® J

Constructthe FE model ] (lleumttry, materials, elemenl~. boundar~,io.nditlons) [ Runninll completed I

I

! Select Iroup of elements i I Plot the tulSa.rates, l(tO I

I at each time step st for [. the seletted elements

I No

Sel~ lmothef F

gro of e!ements group

I

These elements ate the "necking" elemems I

Estimate the strain-rate I i(t,) at, the time st~ tl J ! [ Estimate the overall I

I

strain-rate ~o(t,) at time step tt

Fig. i(b), is not required in the proposed FE model: lbr simplicity sliding of the sheet is prevented by constrainmg the relz.tive nodcs of the sheet mesh il~.all directions. The modelled FLD is constructed by calculating the m~jor, e~, and the minor, e 2, principal strains of the "necking elements, th, latter being determined as the el,',n~ents wlaere ti~e maximum strata rate occurs as the deformation proceeds, compared with their neighbouring etemenr.s: greater strain rates indicate that on these elements a strain concentration occurs, which is similar to "necking' of the steel obtained at instability in tensile testing. Note that "necking' is well predicted by the simulation, whilst the material model used does not account for fracturing. The critical time step, t,,, at which the "necking" elemcnts were deformed providing the useful strains E) and ~, for the construction of the F L D , is estimated, based on experimental observations, as the time step at which their strain rate reaches 0.9 and I.I of the overall strain rate, ~,), the latter being defined as thc ratio of the punch velocity and the punch travel at the critical time step. The flow chart of the simulation procedure is shown in Fig. 3. The effect of strain rate sensitivity on the limit strain curves is discussed by Marciniak et al. [4].

3. Experimental validation In Fig. lib) a schematic diagram of the experimental set-up for the stretch forming of coated steels is presented. All tests were performed on a CNC SMG hydraulic press of I MN capacity, fully equipped with the necessary measuring devices. The punch speed was about 8 mm,'s and the blank-holder force was 2.4 kN, preventing the sheet from sliding. The blanks were initially cut at 0, 45 and 90 ° to the rolling direction of the sheet, a grid, consisting of 2.5 mm diameter circles, being printed on the initial surface of each specimen; see Fig. 4. Tests were performed under 'dry' conditions and under lubricated conditions using oil and a 0.1 mm

NO

[

7-

Estimateshe cl(t,) and al(t,) J strains of the "necking = elements

Sek,ct u m b e r

~

I

i Iroup of q~'m~ts [

-:. ~ Fig. 3. Flow chart of the FE simulation procedure for FLD construction.

. . . . .

p'.

"¢,

acceptable

..

?)~i~-~

10 mm

t

I

Fig. 4. Deformed region of a 25 mm width stretched blank of coated steel.

74

A.G. Mamalis et al./Journal of Materials Processing Technology 68 (1997) 71.75

10

u

1.0

u

Galvo 2

Oalvo 1 '

:_:

_

E.per. FEbl

:;

E oer. FEU

°~ 0.6 m

02f "-0.4

-0.2 00 Minor strain

0.2 ~.~

0_%,

0.41,

'

-d~

' Minor

(e) 1o

C.II

1.0 [

o'~ F.2

u

,

'

'

0.4

u

~.o.[8

Galvo

I

Exoer.

:''''

¢

'

strain (b)

i / I Galvo 3

I

o.o

___

!

4

FEM

"L~ 0.6 :

.9.o, -

.~--..04

o 02'

~

T

4

4-

0.'I

-0

i

i

i

-02

1

i

0~"

M:mor

I2 O.

s',~;m

i

0.00.4

0.4

i

I --02

,

~:,

l O0

,

0.4 J ,

Fig. 5. Experimental and FE-predicted FLDs for coated (gah, aniscd) punch-stretched shcel steels.

~Omr9

Is;

25mm

50ram

75ram (b)

lOOmm

120rnm

Fig. 6. Presenting: (a) experimental; (b) FE-predicted; stretched forms for hot-dip galvanised steel (Galvo I).

thick polyethylene sheet as the lubricants between the punch and the sheet. In order to obtain limit strains in the first and second quadrants of the yield surface, specimens of each steel

with 25, 50, 75, 100 and 150 mm width were tested to failure without load interruption until fracture occurred, the punch travel varying from about 26 mm for narrow strips to 12 mm for the blank. The deformed

A.G. Mamafi.s t'l al. ,hmrnal "4 Matcrials Pr+,ccs.sing li,chmd,gy (J,'~(1997! 71 75

circles at the 'necking' area were classified as "acceptable' if there was not any 'necking" locus on the circle surface, as 'necking" if Iocalised thinning was present inside its surface and as "fracture' if a fracture occurred inside the circle surface; see also Fig. 4.. The major and minor strains, Ej and E2, were estimated by mcasuring the distorted circles. Thc forming-It,nit diagram of each material was drawn by interpolating the "neck,ing" points using a 3rd-order polynomial fitting curve.

75

posed FE model, using the expiicit FE Code DYNA 3D. predicts satisfactorily the FLDs of the tested galvanised steels. Material anisotropy and fracture at failure were not taken into account, but local tensile instability, i.e. necking, was considered The estimated CPU time in constructing the FLDs is reasonable.

References 4. Results and discussion

The predicted FLDs, constructed using the FE procedure as described previously, are plotted togetl.er with the related experimentally obtained FLDs in Fig. 5 !bt each material. With the assumptions ntade and within the limitations of the theoretical and experiment,,| procedure, good agreement between the theoretical predictions and the experimental measurements was obtained. in Fig. 6 photographs of stretch-formed specimens are presented, together with their corresponding simulations. Although material anisotropy as well as the occurrence of fracture were not taken into account. macroscopically the numerical simulation seems to well predict the material plastic flow. It may be noted that the total FE sim,:lation results in a reasonable CPU time, estimated to bc about 15 h for each material, for a SUN Sparcstation I ~

5. Conclusions

Summarising the main features of the results reported, related to the theoretical modelling of the stretch forming of coated sheet steels and the construction of the FLDs, it may be concluded that the pro-

[I] S.P. Keeler. l)clcrrnhmtioa of (bwming limit.,, m a m o n m m c stampings, S.A.E., 650535 (1965) I -9. [2] R. Hill. Discontinuous plaslic stoics. ~silh special reference to localised necking in thin sheets, J. :lh,ch. Phrs. Solid.s. I (19521 19. 131 Z Martin|ok and K. Kuczynski. Limit ,,trains m the proccs.,, of stretch-l,,_,rmiilg shcct-melal. Int..I. ,~Ic~ h. Set.. 9 (106"7) 3(]9. [4] Z. Marciniak. K. Kuczynski and T. Pokora. Inlluence of the plas!ic properlies of a material on Ihe forming limil diagram for sheet metal in tern,ion. Ittt. J..~h'ch. Sci.. 15 (1973~ 789. [5] R. Hill. Theoretical plasticity of texture aggregate,,..~lath Pr,,. ("ulnh. PhHm. Sin.. 79 (1979) t79. [6] A.G. Mamalis. L.P. Hatzikonstanfis. A.J. Zavaliangos. G.C. Vosniakos and D.E. Manolakos, Evaluation of the m-yield criterion of anisotropic sheet metal in predicting limit strains, in S.K. Ghosh and A. Niku-Lar~ (eds.L ('ADC'A..t/ am/ F E M m Metal I.t'orkmg. 1988. p. 121. [7] A.G. Mamalis. A.P. Karat|Ills and N.M. Vaxe~.anidis. Prediction of the limit strains of steel thermall3 mad rncchanicall.~ v.orkcd in rclalion to surface integrity changes: a theoretical model. J. Mater. Prm'c.~.~. Teclmol.. 25 (1991} 15. [8] A.G. Mamalis. D.E. Manolakos and A.K. Baldot, ka,,. Applicl,lion of an explicit finite element model in deep-dra~ing of cylindrical cups of coated steels, in I.J.P. Singh. B. Shirvani and It.J.J. Kals (eds. l. P r m . . 5 , 1 hu. ( " h i . Shccl .~h,tal. 1994. p. 153. [91 A.G. Mama'.",. D.E. Manolakos and A K . Baldouka~,. On the tinite element modelling of the deep-drawing of ,,quare seclioll~. of coated steels. 2n,! ,.|.~'l~l Pa('I/IC ('on/..~lltlh'rt~tl.~ PreJcc~.~'ltle..1. Mulct. Prmc.~.~. Fcchno/., (m pres%

[I0] R.G. Whirley, B.E. Engelman and R W .

Logan, Numerical

methods for simulation of industrial metal forming processes, CED 5 . A M D {1992) 156.