springback in plane strain stretch/draw sheet forming

Pergamon

0020-7403(94) E0011-7

Int. J. Mech. Sci. Vol. 36, No. 3, pp. 327-341, 1995 Elsevier Sciettce Ltd

Printed in Great Britain 0020 7403/95 $9.50 + 0.00

SPRINGBACK IN PLANE STRAIN STRETCH/DRAW SHEET FORMING

FARHANG POURBOGHRAT and EDMUND CHU Alcoa Laboratories, Alcoa Center, PA 15069, U.S.A.

(Received 4 January 1994; and in revised form 6 July 1994)

Abstract--Accurate prediction of springback is essential for the design of tools used in automotive sheet stamping operations. The plane strain stretch/draw operation presents a complex form of springbaek occurring in sheet metal forming since the sheet undergoes stretching, bending and unbending deformations. The two-dimensional draw bending is an example of such an operation in which the complex stress-strain states in the sheet cause the formation of side wall curls after the sheet is allowed to unload. Accurate prediction of the side wall curl requires using finite element shell models which can account for curvature and through-thickness stresses caused by bending and unbending of the sheet. Since such models are generally slow and expensive to use, an alternative and efficient method of predicting side wall curls will be desirable. This paper describes a novel and robust method for predicting springback in general and side wall curls in the two-dimensional draw bending operation as a special case, using moment-curvature relationships derived for sheets undergoing plane strain stretching, bending and unbending deformations. This model modifies a membrane finite element solution to calculate through-thickness strains and stresses and springback. Accuracy of this model's predictions are verified by comparisons with finite element (ABAQUS) and experimental results.

x,y Z

C

R k n

K t

W

0, g (7

v

r

E T

M

Subscripts C

n B

UB 0

T mem

1 U

N O T A T I O N

components of Cartesian coordinate system an axis with origin at the centerline of sheet neutral axis shift from the centerline due to tension radius of curvature centerline curvature strain hardening exponent strength coefficient thickness of element width of the sheet effective stress and strain stress true and engineering strain Poisson ratio normal anisotropy parameter modulus of elasticity tension bending moment

centerline fiber neutral fiber bent element unbent element original material tangential component membrane component loaded sheet unloaded sheet

1. I N T R O D U C T I O N

The most complex form of springback in sheet forming occurs when the sheet undergoes both bending and unbending deformations. This phenomenon takes place when the sheet

327

328 F. Pourboghrat and E. Chu

first makes contact with the tool surface (bending) and then leaves the tool surface and partially or completely loses its curvature (unbending). Numerical simulation of sheet metal forming operations are currently performed by finite element analysis codes. To capture the effects of bending and stretching on the large deformation of the sheet and springback, finite element codes based on non-linear shell theory are utilized [14] . These models are usually slow due to their computational intensity. To obtain a faster turnaround time, the membrane formulation is used instead to simulate stamping operations. Deformation of the sheet in membrane solutions is by stretching alone, bending effects are neglected. When sharp tools are used in stamping operations, neglecting bending effects could result in gross underestimation of maximum strains and wall loads [5]. In addition, without the inclusion of bending effects, calculation of such important phenomena as wrinkling and springback of the sheet will be impossible [33.

Since sheet metals are most susceptible to failure under plane strain conditions (FLDo), several plane strain membrane finite element codes for sheet metal forming analysis have been developed [6-8]. These codes, in the absence of sharp tooling curvatures, are capable of predicting deformation strains very accurately, but as tooling curvatures become large enough to cause significant bending strains, these codes can no longer predict deformation strains accurately [9]. In addition, neither of these codes is capable of predicting springback after elastic unloading.

In Ref. [9], the authors presented a model, based on a mechanics approach, to calculate through-thickness strains, stresses and springback using the finite element membrane solution. Springback was calculated from predicted bending moment. By comparison with measured data [10] and other literature [11], it was found that this method would only apply to the stretch forming operation where no draw-in or unbending of the sheet takes place. In Ref. [5], the author presents a membrane model where draw-in of the sheet under plane strain conditions is considered. This model is used to study sensitivity of the solution to variations in process parameters. In this model effects of bending and unbending are again neglected.

In the following sections, development of a complete and numerically efficient sheet metal forming analysis tool to predict total strains (bending plus stretching plus unbending) as well as springback for plane strain stretch/draw operation will be described. This method is based upon linear superposition of stretching strains (obtained from the membrane FEM code) and through-thickness strains calculated separately based on a developed theory. Sections 2 and 3 describe the theory developed for calculating through-thickness strains and stresses for bending and stretching and unbending and stretching problems, respectively. For calculation of unbending strains, the kinematic hardening rule is used. Section 5 describes a simple method for calculating the unloaded shape of the sheet after springback and in Section 6 the effect of tension on the moment-curvature relationship and thinning will be discussed. Finally, in Sections 7 and 8 the accuracy of the new model is verified by application of the theory to the two-dimensional draw bending problem.

2. BENDING AND STRETCHING PROBLEM

Calculation of strains and stresses when the sheet is bent and stretched under plane strain conditions is described in detail in Ref. [93. A brief review of the assumptions used and equations derived will be presented in this section, before proceeding to discuss the draw-in problem. For the calculation of state parameters, the following assumptions were used:

(1) Plane sections remain plane after the bending (Kirchhoff-Love). (2) Transverse shear strain is neglected. (3) Strain in the width direction is neglected (ew = 0). (4) Tangential strain eT is the linear sum of bending and stretching strains. (5) Bending strains are symmetrical about the centerline. (6) Stretching strain /~mem is obtained from the membrane FEM solution. (7) Neutral axis shift is caused by tension in the sheet. (8) Normal compressive stresses aN are neglected (plane stress).

Springback in plane strain stretch/draw sheet forming 329

(9) Deformation theory of plasticity is considered. (10) Rigid plasticity is assumed.

Using the above assumptions and Fig. 1, it is possible to write the following expression for engineering tangential strain, eT, based purely on kinematic considerations:

(z + c) eT = (1)

Rn

True strain eT can then be calculated from engineering strain eT as follows.

1 /Rc+z'~ ~ : ln(1 + e~)= ntR----~_c) I2)

By neglecting higher order terms, neutral axis shift due to tension, c in Eqn (1), can be calculated for each cross-section of the sheet (element) from membrane finite element solution using the original and current thicknesses, to and t respectively, and radius of curvature, Re, as follows:

( e_ c = Rc 1 + ememJ ~o " (3)

Once neutral axis shift c is calculated from Eqn (3), tangential strain (bending plus stretching) eT can be calculated from Eqn (2). Tangential stresses aT causing bending and stretching of each element are calculated from the following constitutive equation:

= K g". (4)

For rigid plastic sheet metals, when plane stress and strain for individual fibers is assumed and Hill's 1948 yield criterion [12] is used, Eqn (2) can be rewritten for materials with normal anisotropy as

t n O'T = K ~T, (5)

where

K'=/d 7 t ~ / i61 For isotropic materials, r should be set equal to 1 in Eqn (6). The bending moment at each

cross-section can be calculated by integrating tangential stresses in Eqn (5) over the current

Rc R

~ ~s4 Z~So /

\

Fig. 1. Deformed geometry of the sheet showing various fibers.

where

thickness of each element as follows:

f tl2

M = aTzdz . (7) d - t/2

To calculate springback after elastic unloading the moment-curvature relationship for the sheet is used as shown in Fig. 2. Using this figure, it is possible to derive an expression for the unloaded centerline curvature of the sheet ku as a function of its loaded curvature k I

and bending moment M as follows:

M ku = kl (8)

t3M/Ok[e

Bending Moment,

M

~M E Wt 3

- 12(1 - v 2) (9) e

When ku, calculated from Eqn (8), is less than zero, an elastic deformation of the sheet and a return to its original flat shape after loading is assumed, i.e. ku is set equal to zero. The bending moment M in Eqn (8) is a function of the centerline curvature and tension in the sheet [see Figs 2 and 6(a)]. Centerline curvature of the sheet kl is calculated from the following expression after spatial coordinates obtained from finite element code are curve- fitted with cubic splines.

1 d2y/dx 2 R~ = kl = [1 + (dy/dx)2"] 3/2" (10)

3. UNBENDING AND STRETCHING PROBLEM

The following assumptions are made in order to calculate the induced strains and stresses in an element initially bent and stretched around the tool radius and later unbent (straightened) due to leaving the contact surface.

(1) Unbending occurs under plane strain condition. (2) Plane sections remain plane after unbending. (3) The centerline curvature of the sheet (element) will be zero after unbending. (4) Straightening of a bent element is caused by the following deformations: an elastic

unbending, AMe; a plastic unbending, AMp; and a uniform stretching, Aes. (5) Kinematic hardening governs the re-yielding in reversed loading.

ku2 kul kt Centerline Curvature,

k

Tension increasing


Fig. 2. Elastic unloading of bending moment and curvature at various tensions.


Figure 3 shows the schematic of a bent element going through elastic-plastic unbending and uniform stretching as it straightens. Consideration of the uniform stretching, in addition to unbending of the element, is necessary to assure that the sheet is in tension after straightening. It is assumed that elastic unbending AMe takes place first until the re-yielding occurs in reversed loading. Figure 4(b) shows a schematic of the moment-curvature relationship during loading (bending and stretching) and reversed loading (unbending and stretching) with the kinematic hardening rule assumed. With the kinematic hardening assumption, re-yielding occurs when the bending moment reaches twice the bending moment at initial yield, 2My, or stress reaches twice the initial yield stress, 2o- r Figure 4(b) shows that after elastic unbending AMe the centerline curvature of the sheet (element) reduces by Ake (point A).

To straighten the element, an additional and simultaneous plastic unbending AMp and stretching A~ will have to take place until the centerline curvature of the sheet (element) becomes zero (point B in Fig. 4) and tension becomes satisfied. The net change in centerline curvature during plastic unbending is equal to Akp which is equal to (k* - Ake), where k* is the centerline curvature of the sheet (element) prior to leaving the surface of contact.

4. STRAINS AND STRESSES AFTER UNBENDING AND STRETCHING

True tangential strains and stresses developing in an element of the sheet initially curved and later straightened by unbending and stretching can be computed from Figs 4 and 5. Tangential strains required for unbending about the centerline and uniformly stretching each side of the element are given by the following expression:

ev(z) = e*(z) + A~e(Z) + A,~p(Z) q- Ae~, (11)

where e*(z) are tangential strains in the curved element prior to unbending and are given by Eqn (2). Figure 5 shows engineering tangential strains after bending, unbending and stretching.

+ A M ) M r M r

F

Fig. 3. Straightening of a bent element, by elastic-plastic unbending and stretching.

(7" T

%

-O"

E M

ku A k~* b kc

(a) (b)

Fig. 4. (a) Tangential stress-strain and (b) moment-curvature relationship during the unbending. The kinematic hardening rule is assumed.


C

2 /

em~. - ( k ' - a k ) . z

+

L______

- ~ k .z

+

A e s eT, E

ml eT'F

Fig. 5. Through-thickness strains in an e lement initially curved and later straightened.

The elastic component of tangential strain caused by elastic unbending, AMe, is equal to

Ace = ln(1 - Akez). (12)

The plastic component of tangential strain caused by plastic unbending AMp is equal to:

Aep = ln[1 - (k* - Ak¢)z]. (13)

In the draw bending operation, where the sheet (element) draws over the tooling radius, the sheet further thins by At and the stretching strain Aes in Eqn (11) will be proportional to this change in thickness. The stretching strain Ae s is found such that it produces a tensile force to unbend and stretch the element. The tensile force (per unit width) needed to straighten a bent element, Tun, is equal.

TuB = TuB, mere + FB, (14)

where TUB . . . . is the unbending tensile force (per unit width) obtained from the membrane FEM solution and is given as

ZUBrnem g ' [ l n ( to / ] n (15) , = ~;UB, mem" L \ tUB,mem/- I

FB in Eqn (14) is the additional tensile force (per unit width) required to pull an element of the sheet bent around the tool with centerline curvature, k*, a distance I over the surface of the tool [131. By equating the internal work done to bend the element Wi and the external work done by the force Fe to pull it along the surface of the tool We:

W , = O d ~ d V - ~ + ~ \ - - ~ - j 2 (k , z ) .+ ldz (16)

We = F¢l (17)

force FB can be found to be equal to

FB Fe 2K' / t \ .÷2 = w - = (n + 1)(n+ 2) (k*)"+l ~ ) " (18)

Residual tangential (engineering) strains after straightening are calculated (see Fig. 5) by taking the sum of strains before unbending e*(z) and the net plastic unbending strains [ - (k* - Ake)z + Aes] as follows:

eT(Z) = e*(z) + [ -- (k* - Ake)z + Ae~]. (19)

Through-thickness tangential stresses after straightening of a bent element are calculated [-see Fig. 4(a)] by subtracting tangential stresses caused by elastic-plastic unbending and stretching from those prior to straightening, fi* (z), as follows:

= a * ( z ) T 2ay -T- { K ' [ - (k* - Ake)z + Aes]"}; ~ ! - );z and a*(z)> 0 (20) fiT(Z) + ):z and a*(z) < 0. U


Tension after straightening is found by integrating the non-symmetric portion of Eqn (20) shown in curly brackets, to be equal to:

K' r¢ = (k* - Ako)(n + 1) [(A + Aes) "+1 - (A - Aes)"+l]. (21)

After straightening, the bending moment about the centerline is found by integrating Eqn (20) through the thickness tUB to be equal to:

(A_Aes)n+2+(A+Aes)n+ 2 } i ,: , - . . . . ;, + 5 . . . .

M f = M* - 2 M y (k* - Ake) 2 Ae (A - Ae~) "+1 - (A + Aes) "+1 ' (22) +

where parameter A in Eqns (21) and (22) is defined as

A = (k* - Ak~) tub ~ - . (23)

In Eqn (22), M* is the bending moment prior to unbending and straightening, and the bending moment at which the initial yield occurs My is equal to:

My-- 2K' (tB~ n+2 (2 + n)\ 2 J k~

ky - 2(1 - v 2) ay (24)

t~ E

tub in Eqn (23), the thickness after straightening, is related to stretching strain Aes;

Ae~ = 1 tub tB (25)

and can be calculated for each element by equating Eqn (21) to Eqn (14). When only positive strains are formed through the thickness, then only the second expression (i.e. z < 0) given for stresses in Eqn (20) is used to calculate tension and bending moment after straightening.

5. UNLOADED CURVATURE OF THE SHEET AFTER UNBENDING

The unloaded centerline curvature of the element, ku, after unbending and straightening [-point C in Fig. 4(b)] is found from the following expression:

ku= - 1 2 ( i - v 2) My E tab (26)

Nodal coordinates of the unloaded elements after springback are calculated from the following expressions:

Ax(i) = As(i) cos ku(j)As ; Ay(i) = As(i)sin ku(j)As . J= j

(27) x(0) = Xo; y(0) = yo.

In Eqn (27), As(i) represents the length of the ith element.

6. EFFECT OF TENSION ON THE MOMENT-CURVATURE RELATIONSHIP AND THINNING DURING BENDING AND UNBENDING

To illustrate the effect of tension on th~ bending moment-curvature (M-k) relationship and thinning (tangential strain) during bending and unbending (straightening), Fig. 6(a, b) is provided. For this figure, the material data for mild steel given in Table 4 is used. To


(a)

1.05

1

0.95 -

6 'o'~/~.1='0.~,' I . . . . I . . . . I . . . . I . . . . 5 .[1 T I T r = 0 . 5

1 - " oo, .,,.-, ,-

0 50 100 150 200 250 300

q k ~

. . . . I . . . . I . . . . I . . . . I . . . . I . . . .

0 • 0 -

~ . : =Y-7 2 . - - o - _ _ • ~ ' ' . . . . ~ " - - ~ . . . . . . ~ - ~ E t - - ~ . , .

.....,~ "O"¢T / T r = O. O ..)¢¢.......,.,~" [] T ITy =0.5

' ~ O" T I T r = I . O ~ . . . •

, T/IT, =?.5 , , , , I , , , , I , , , , I , , , , I , , , ,

I I I I I

50 100 150 200 250 300

0 . 9 -

0.85 -

0.8 0

(b} ~[ky

Fig. 6. Effect of tension on (a) the moment-curvature relationship and (b) thinning during the bending and unbending process for the mild steel shown in Table 4.

calculate tension and bending moment, shown in Fig. 6(a, b), Eqns (2) and (5) for bending and stretching and Eqns (21) and (22) for unbending and straightening were used. Cal- culated tension, bending moment and centerline curvature were then normalized using tension at yield (Ty = art ) and moment and centerline curvature at yield given by Eqn (24). From Fig. 6(a) it can be seen that for a given centerline curvature (k/ky) an increase in tension (T/Ty) significantly reduces bending moment (M/My) during both the bending and unbending processes. Since springback is also calculated from such a moment-curvature relationship (see Fig. 2), it will also be reduced with increased tension in the sheet. Figure 6(b) shows that by increasing tension, thinning also increases for a given centerline curvature. More significantly, it shows that thinning nearly doubles as the sheet first bends to a curvature and later unbends and straightens. More verification of these results is given in Section 7.

7. V E R I F I C A T I O N

The theory described in this paper for calculating through-thickness strains for the stretch/draw operation, was implemented into SHEET-S [6] finite element membrane code to analyze the two-dimensional draw bending operation shown in Fig. 7. Using the theory developed and membrane solutions obtained from the SHEET-S code, springback and formation of side wall curls in the two-dimensional draw bending operation were predicted. To verify the accuracy of the theoretical predictions, experimental and finite element results


B l a n k ~

Die ~P

Punch

7- Yp

l Sheet

Blank Holder

Die

Fig. 7. A schematic of the two-dimensional draw bending operation.

using ABAQUS [14] were also produced. In the following sections, these results and discussion of them will be presented.

7.1. Alcoa experiment A two-dimensional rectangular hat-shaped section, shown in Fig. 7, was used to verify

the theory developed herein and predicted results were compared with experiments performed at Alcoa. The 2008-T4 aluminum alloy sheet of 1.0 mm gauge thickness as well as two test cases (a) and (b) with different punch/die radii, Rp and Rd, and clearances were used for this evaluation. Table 1 shows the material properties of the sheet and Table 2 shows the tooling geometries and forming conditions used for this experiment.

It was observed from experiments that after unloading of the hat sections, curls were formed on the sidewall of the parts as shown in Fig. 8(a, b). Figure 8(a) shows that sidewall curls are more pronounced for parts formed with a sharper die radius and tighter clearance.

The SHEET-S and ABAQUS codes were also used in this study for verification purposes. SHEET-S, a two-dimensional finite element program, was originally developed at Ohio State University using membrane line elements to simulate the stretch/draw forming operation of plane strain sections. To test the accuracy of the theory, the SHEET-S code was employed to simulate the forming of the hat sections using the appropriate forming conditions shown in Table 2. Once a membrane solution was generated using SHEET-S, the appropriate bending and unbending components were added, using the theory developed in this paper to predict strains, corrected thicknesses, springback and sidewall curls.

To employ the bending/unbending theory developed here, the following calculated information was first retrieved from the SHEET-S code: which elements made contact with either the die or the punch surface; and which elements made contact and later left the contact surface. Table 3 shows an example of such information retrieved for two elements from the SHEET-S code. A contact between the element and a tooling surface (punch or die), at a time increment, can be identified by finding a "1" in columns 2 or 3 in Table 3. Also, membrane strain, current thickness and centerline curvature of the element can be found at each time increment from columns 4-6. When the element leaves the surface of the contact, the value of"1" is replaced with a value of "0" in columns 2 or 3, and a centerline curvature of "0.0" is assigned to the element due to assumed straightening of the element (e.g. element 19, increment number 24 in Table 3). Using this generated element history information, appropriate bending and unbending corrections were made to the membrane solution for each element in post-processing.

ABAQUS was also used to model the hat sections with the aim of cross-comparing with the theoretical and experimental findings. An eight-noded, biquadratic, second-order, plane strain element with or without reduced integration was selected for this simulation. The plasticity of the hat section, including bending and unbending effects and thickness changes can be captured properly with this element. Since a higher order element was used, one would anticipate that a more accurate result could be obtained from the analysis. For the


Table 1. Material properties of the sheets used in the Alcoa experiment

K E ay Material (MPa) n (GPa) r (MPa) v

A1 2008-T4 490.0 0.260 69 0.70 138.0 0.33

Table 2. Tooling geometries used for the Alcoa experiment

Die Wall t Rp Ra opening Yp Clearance angle BHW

Case (mm) (mm) (mm) (mm) (mm) (mm) (deg) (t)

a 1.0 3.175 3.175 66.68 45.0 8.0 89.0 2.5 b 1.0 6.35 6.35 80.31 50.0 15.4 78.0 15.0

*Blank holder pressure

5 0

4 0

30

20 :S

>" 10

0

- 10

-20 1 0 0

(a)

GA 33960.1

I I I I

- 2. Abaqus

3, Theory

I I I I 120 140 160 180

X (MM) 200

6 0

5 0

4 0

3 0

1 0

0

- 1 0 -

- 2 0 120

(b~

I I I I I

D

~ l | l l l l ' l l l l l l l | l l l ~

2. A b a q u s

3. Theory

I I I I I 140 160 180 200 221) 240

X (MM)

Fig. 8. (a) Predicted and measured unloaded shape of the sheet for case a, Table 2. (b) Predicted and measured unloaded shape of the sheet for case b, Table 2.

Springback in plane strain stretch/draw sheet forming

Table 3. History of deformation obtained from membrane FEM code (SHEET-S)

Increment Punch-sheet Die-sheet t k number contact contact ~mom (mm) (mm -1)

Element 1 1 1 0 0.0000 0.7800 0.0000 2 1 0 0.0001 0.7799 0.0000 3 1 0 0.0003 0.7798 0.0000

337

50 1 0 0.0021 0.7784 0.0000 51 1 0 0.0022 0.7783 0.0000 52 1 0 0.0022 0.7783 0.0000

138 1 0 0.0059 0.7754 0.0000 139 1 0 0.0059 0.7754 0.0000 140 1 0 0.0060 0.7753 0.0000

Element 19 1 0 1 0.0000 0.7800 0.0962 2 0 1 0.0001 0.7799 0.0962 3 0 1 0.0003 0.7798 0.1854

22 0 1 0.0011 0.7791 0.1850 23 0 1 0.0012 0.7791 0.1829 24 0 0 0.0012 0.7791 0.0000 25 0 0 0.0013 0.7790 0.0000

139 0 0 0.0062 0.7752 0.0000 140 0 0 0.0062 0.7752 0.0000

ABAQUS simulation, a total of 70 elements along the arc length and one layer of element through-thickness were used to model the hat section. A friction coefficient of 0.12 was used to simulate best the sheet-tooling contact problem. ABAQUS also uses the isotropic hardening law to handle unloading and re-yielding. Results of these simulations are shown in Fig. 8(a, b). The effects of bending/unbending on simulation results and springback predictions with ABAQUS, instead of a modified membrane (superposed bending/unbending) approach, can be determined from this study.

7.2. Comparison of modified membrane code with ABAQUS A benchmark case selected by the Scientific Committee of the NUMISHEET conference

[15] has been chosen here in order to compare the predictive results generated by both the integrated SHEET-S code with the current theory and ABAQUS. A two-dimensional channel section was used for this comparison. The forming operation is essentially the same as that described previously. The conditions used for this simulation are punch and die radius of 5 mm, blankholding forces (BHFs) of 19.6 and 50kN, initial blank size of 350 × 35 mm, punch travel of 70 mm, friction coefficient of 0.144 and the material used for this application was mild steel (see Table 4) with a thickness of 0.78 mm. The predicted unloaded shapes for the NUMISHEET benchmark channel are given in Fig. 9 for comparison. Figure 10 (a, b) shows the strain distribution results predicted by the current theory for the two applied BHFs of 19.6 and 50.0kN.

7.3. Sidewall curl for different materials To examine the results of material properties on springback, the problem of two-

dimensional draw bending was simulated for the three materials (i.e. high strength steel, mild steel and aluminum) specified in the NUMISHEET benchmark test with 19.6 kN BHF. The material properties of the three sheets are shown in Table 4 and numerical simulation


Table 4. Material properties of the sheets used in numerical simulations of the N U M I S H E E T benchmark test

K E o r Case Material (MPa) n (GPa) r (MPa) v eo

a Aluminum alloy 570.0 0.347 69 0.71 137.0 0.33 0.01502 b Mild steel 567.0 0.264 206 1.77 167.0 0.30 0.00745 c High tensile steel 677.0 0.219 206 1.73 270.0 0.30 0.01129

80

6 0

4O

v

>" 2 0

I 10 =

I I I I

1. Abaqus (50.0KN) 2. Theory (50.0KN) 3. Abaqus (19.6KN) 4. Theory (19.6KN)

I I I I I 0 20 40 60 80 100 120

X (MM)

Fig. 9. Predicted unloaded shape for the N U M I S H E E T benchmark case.

conditions are shown in Table 5. A comparison of simulated results, using developed theory and SHEET-S code, with experiments shows that simulated results are in very good agreement with experimental results; see Fig. 11. More importantly, simulated results capture the observed differences in springback and sidewall curl for different materials.

8. R E S U L T S A N D D I S C U S S I O N

Figure 8(a, b) shows the superposed results of the experiment, ABAQUS and modified membrane code. Figure 8(a) shows the unloaded shape results for the case when the hat channel was formed with a punch/die radius of 3.2 mm and Fig. 8(b) shows the results for the case when it was formed with a punch/die radius of 6.4 mm. It is interesting to see that the modified membrane results match those measured experimentally and predicted by ABAQUS very accurately in both cases, with slight underestimation of the final unloaded bend angle. Accurate prediction of the shape of the unloaded sheet by the modified membrane model, for the case with sharper die radius [Fig. 8(a)] where curl is more pronounced is indicative of the accuracy of the theoretical model developed.

Figure 9 shows the unloaded shape for the NUMISHEET benchmark case predicted by both ABAQUS and the current theory for mild steel materials using BHFs of 19.6 and 50 kN. As can be seen from curves 3 and 4, for the case of the 19.6 kN BHF, the current theory slightly overestimates the amount of the curl and the final unloaded bend angle. According to ABAQUS's results (Fig. 9, curve 1), when a 50 kN BHF is applied to the sheet, the tension developed in the sheet is large enough to cause full plastic deformation along the sidewall of the channel. This causes the sheet to be stretched way beyond the elastic limit and thus eliminates sidewall curl and springback of the hat section upon unloading. The unloaded shape of the channel predicted by the theory (curve 2) had been superimposed on the ABAQUS simulation and results are shown in Fig. 9. It is apparent that, unlike


0.12

0.10

0.08 - - c

w 0 . 0 6 - - 0 P k-

0.04 --

0.02 --

0.00 -20

(a)

I I -,o- T h e o r y -o-SHEET-S

Punch contac t

I I I

Die contac t

~ 6 " J

Wall ieg Ion !

m

GA 33960.2

I

m

~(((((((((t(((ClO~l(((t'JKllOttfflmt(lQl(~((¢(~O

I I I I I I

0 20 40 60 80 100 120 Element #

_c

O @

!.-

0.18

0 .16

0 .14

0 .12

0 .10

0 .08

0 .06 - -

0 .04 -

0 .02 -

0 .00 -40

(b)

I I I I I I I | .,o. T h e o r y

-- -o-SHEET-S Wall reg ion

_ Punch. : contact ~ ! :: ~ D i e c o n t a c t -

- ! i : : . - !" - - : , - ' : - •

: ~(((((.({((~(~c(~(ut(((~(o( :

t I i I I I I

-20 0 20 40 60 80 1 O0 Element #

120

Fig. 10. (a) Predicted strain distributions by the current theory and SHEET-S for 19.6 kN BHF. (b) Predicted strain distributions by the current theory and SHEET-S for 50,0 kN BHF.

Table 5. Simulation conditions for the NUMISHEET benchmark test

t Rp Rd Yp Clearance BHF Case (mm) (ram) (mm) (mm) (ram) (kN)

a 0.81 5 5 70 1 19.6 0.162 b 0.78 5 5 70 1 19.6 0.144 c 0.74 5 5 70 1 19.6 0.129

ABAQUS, the current theory still predicts a small curl to exist after the un loading of the sheet and also predicts a final un loaded bend angle of abou t 2.5 °. This difference in predict ion between A B A Q U S and the current model is believed to be due to two factors: (1) the current theory uses the k inemat ic hardening law whereas A B A Q U S uses the


8O

6O A

40 >.

2O

0

m

Model

~T.$. ~

I I I L

20 40 60 80

X (MM)

l 100

Experiment

~ ~ s .

Fig. 11. A comparison of springback and sidewall curl for different materials. (a) Current theory, (b) experiment.

Table 6. Predicted thicknesses at various regions by different models

Model t-Punch t-Die t-Wall t-Punch t - D i e t-Wall

(ram) (mm) (mm) (mm) (mm) (mm)

BHF = 19.6 kN BHF = 50.0 kN

ABAQUS 0.7722 0.7722 0.7348 0.7410 0.7417 0.6800 THEORY 0.7743 0.7744 0.7441 0.7488 0.7494 0.6881 SHEET-S 0.7753 0.7754 0.7752 0.7526 0.7530 0.7446

isotropic hardening law; (2) calculated tensions in the sheet are different between the two models. ABAQUS calculates the restraining forces more realistically using the specified frictional condition under the blankholder, while SHEET-S uses a constant restraining force calculated simply from assumed uniform pressure and blankholder geometry. Also, it is believed that as BHF increases, the actual unbending force calculated by Eqn (14) may be underestimated, which in turn may cause more curl in the sidewall.

Figure 10(a, b) shows calculated strain distributions by the current theory at different locations along the sheet for the two cases of 19.6 and 50.0 kN BHF. For comparison purposes, the membrane strains are also superimposed to indicate the amount of strain correction due to bending and unbending. It is interesting to note that for the 19.6 kN BHF [-Fig. 10(a)], the maximum strains occur along the die and punch corners, but as BHF is increased to 50.0kN [Fig. 10(b)], the sidewalls start to carry the maximum strains and possibly become the first place for the sheet to fail by tearing [16]. To assess the accuracy of strain predictions, thickness of the sheets at punch and die radii and sidewall were obtained from the developed theory and membrane model and compared with ABAQUS. A sum- mary of the results is shown in Table 6. The results in Table 6 show that membrane solution alone could grossly underestimate the actual thickness of the sheet, while employing the bending and unbending corrections will dramatically improve these predictions. It is also obvious from the results shown in Table 6 that strains in the wall section of the sheets are slightly underestimated by the current theory when compared with ABAQUS's results. These slight underestimations of strains could explain the differences in predicted curl and springback shown in Fig. 9.

9. CONCLUSIONS

A theoretical model was presented in this paper for calculating bending and unbending strain corrections to be added to membrane strains in post-processing, in plane strain stretch/draw sheet forming problems. As a result of this theoretical work, a membrane finite element code can now also be used to predict springback. To calculate the unbending strain of a straightened element, the kinematic hardening rule was applied in reverse loading. This method, due to its post-processing nature, unlike the full shell model (ABAQUS), does not


contribute much to the overall CPU time needed to analyze the problem and therefore retains the efficiency of the membrane code. In fact, none of the examples shown here took more than 5 CPU rain, including the bending and unbending corrections, to be completed.

For comparison purposes, various results regarding the unloaded shape of the sheet, curl, strain and thickness predictions were calculated using the SHEET-S membrane code first and then corrected using the theoretical model described in this paper. These results were then compared with experimental measurements and ABAQUS's results. These comparisons indicated that the current theory is capable of predicting strains due to unbending and curl in two-dimensional draw bending very accurately. It was found that this theory, due to the use of the kinematic hardening rule, will slightly underestimate the actual springback of the sheet and tension in the sheet. Results also indicate that using the membrane code without the corrections described will grossly underestimate strain distribution and thinning in the sheet. Figure 10(b) showed the importance of including the unbending effects by showing how maximum strains are carried by the sidewalls as BHF is increased. Experi- ments have also shown that sidewalls fail by tearing in two-dimensional draw bending as BHF increases. Finally, it was shown (Fig. 11) that this theory is capable of predicting observed differences in springback and sidewall curl for different materials.

Acknowledgements--The authors wish to thank Dr K. Chung and Mr K. Shah for reviewing the transcript, Mr K. Chandorkar for running SHEET-S simulations and Dr J. Story for coordinating experimental work done on the two-dimensional channel problem.

R E F E R E N C E S

1. N. M. Wang and S. C. Tang, Analysis of bending effects in sheet forming operations. Int. J. Numer. Meth. Engng 25, 253 (1988).

2. J. K. Lee, U. Y. Cho, J. Hambrecht and S. Choudhry, Recent advances in sheet metal forming analysis, in Advances in Finite Deformation Problems in Materials Processing and Structures, Vol. 125, p. 119. ASME, WAM (1991).

3. A. P. Karafillis and M. C. Boyce, Tooling design in sheet metal forming using springback calculations. Int. J. Mech. Sci. 34, 2 (1992).

4. S. Choudhry and J. K. Lee, Dynamic plane-strain finite element simulation of industrial sheet-metal forming processes. Int. J. Mech. Sei. 36, 3 (1994).

5. M. L. Wenner, Elementary solutions and process sensitivities for plane-strain sheet-metal forming. J. Appl. Mech. 59, 23 (1992).

6. M. J. Saran and R. H. Wagoner, A user's manual and verification examples for section form (Sheet-S/version 3), section analysis of sheet stampings. Report no. ERC/NSM-S-91-17, May (1991).

7. W. H. Frey and M. L. Wenner, Development and application of a one-dimensional finite element code for sheet metal forming analysis, in Proc. Interdisciplinary Issues it~ Materials Processing and Manufacturing (edited by S. K. Samanta, R. Kommandori, R. MeMeeking M. M. Chen and A. Tseng), Vol. 1 pp. 307-320. ASME/WAM, Boston, MA (1987).

8. M. P. Sklad and J. F. Siekirk, Proe. 16th IDDRG Congress, ASM International, Bolange, Sweden, p. 295 (1990). 9. F. Pourboghrat and K. Chandorkar, Springback calculation for plane strain sheet forming using finite element

membrane solution. Numerical Methods for Simulation of Industrial Metal Forming Processes. ASME 156 (1992).

10. K. V. Chandorkar, Verification of the new version of Sheet-S by modeling plane strain sections of outer door panel of a car door, Alcoa Internal Report 1992-08-07.

11. T. B. Stoughton, Finite element modeling of 1008 AK sheet steel stretched over a rectangular punch with bending effects in Computer Modeling of Sheet Metal Forming Processes, (edited by N. M. Wang and S. C. Tang) p. 143 The Metallurgical Society (1985).

12. R. Hill, The Mathematical Theory of Plasticity. Clarendon, Oxford (1950). 13. T. B. Stoughton, Model of drawbead forces in sheet metal forming. Proc. 15th Biennial Congress, IDDRG,

Dearborne, MI, pp. 205-215 (1988). 14. ABAQUS, User's Manual, Theory Manual, version 4.7. Hibbit, Karlsson and Sorenson, Providence, Rhode

Island (1988). 15. NUMISHEET'93, Proc. 2nd International Conference, Numerical Simulation of 3-D Sheet Metal Forming

Processes - Verification of Simulation with Experiments (edited by A. Makinouchi, E. Nakamachi, E. Onate and R. H. Wagoner) Isehara, Japan, 31 August-2 September (1993).

16. R. A. Ayres, SHAPESET: a process to reduce sidewall curl springback in high-strength steel rails. J. Appl. Metalworking 3, 127 (1984).

springback in plane strain stretch/draw sheet forming

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