A note on the forming limit curve

Ramaekers, J.A.H.; Bongaerts, P.C.P.

Published: 01/01/1994

Document VersionPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differencesbetween the submitted version and the official published version of record. People interested in the research are advised to contact theauthor for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

Citation for published version (APA):Ramaekers, J. A. H., & Bongaerts, P. C. P. (1994). A note on the forming limit curve. (TU Eindhoven. Fac.Werktuigbouwkunde, Vakgroep WPA : rapporten). Eindhoven: Technische Universiteit Eindhoven.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ?

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 19. May. 2018

Einhoven University of TechnologyFaculty of Mechanical EngineeringDepartment of Production Engineering and Automation (WpA)

A note on the forming limit curvePaper SheMet '94

J.A.H. Ramaekersp.e.p. Bongaerts

January 1994WPA reportnr.: 120003

A note on the Forming Limit Curve.

Forming Limit Curves, theory and application.

I.A.H. Ramaekers, P.C.P. BongaertsEindhoven University of TechnologyLaboratory for Forming Technology

Abstract

A new theory to estimate the Forming Limit Curve ( FLC ) is presented in this paper.This theory is verified by practical results. Some materials show a disagreement to thepresented theory. Several items like inhomogeneity, ductility, strainrate sensitivity andlogarithmic strain hardening are discussed in this paper to give an explanation for thisdisagreement.

Introduction

SClf.

FlIil FlIil

Sofe20

40

80

....

-30 -20 -10 0 10 20 30

£ 1%1

Figure 1 The original Keeler-Goodwin FW(Pearce).

Usually, the FLC is measured experimen­tally, but because of the importance of theFLC, many attempts have been made inorder to describe the FLC mathematically.

100

In sheet metal working, FLC'S, serve as one of the guidelines to estimate the formingcharacteristics of sheet metal (Keeler1 Goodwin2).

The FLC is the combination of loci of thestrains with necking initiation. This can bepresented in an EcE.z or e1 - e2 diagram(see fig. 1).

First theories, Hi114, and SwiftS have beenbased on the mathematical estimation ofnecking in sheet metal working. Otherauthors, like MarciniaJc6·7, need to assumean inhomogeneity in the sheet metal,which is introduced as a smaller thickness.

The above-mentioned theories are complex and do not satisfy available data in the firstquadrant of the FLD, see fig. 1.

The theory presented in this paper professes to contribute to the understanding of thephenomenon and to supply a relatively simple theory for the FLC, without claimingcompleteness.

Basics of anisotropic material behaviour

For the formulation of an instability criterion, certain material behaviour has to beassumed. Here, a simplified formulation, based on Hills, Backofen and Hosford9

, is used.Only the essential equations are presented here.

(1)

", =In(~]

"2 = In(~]

"0 = In (~]

Starting points of the analysis are :o Shear strains are negligible.o The deformation can be described by the

principal strains :

Figure 2 The basic deformationmodel.

A straight strain path is assumed :

(2)e2 .- = J ,j « 1) = constant.e1

o The stress perpendicular to the sheet is negligible :

~=O ~

This implies together with (1) and (2) that a straight stress path also exists:

(4)°2 . .- =/ , 1 « 1) = constant.°1

o The material exhibits normal anisotropy, Le. the planar anisotropy i.\R is zero.Anisotropy then can simply and solely be described with the R factor as measuredin the tensile test :

(5)

Taking the above assumptions into account, the following equations can be obtained :o The effective strain :

R+1 (1 +R+2jR+(R+1)j2)2R+1

o The equivalent or Von Mises stress :

a=IRC11

1 - 2R -2-1--+/R+1

(6)

(7)

o The stress strain relations according to Levy-Von Mises then become :

t:1 = ~(C11----.!!...-C12)C1 R+1

e 1e3 = --=--(01+Cl2)

o R+1

o The material shows exponential strain hardening behaviour:

Clf = C(e+ ear- Clf the flow stress.

- C characteristic deformation resistance,- n strain hardening exponent,

- eo pre-strain.

o The yield criterion remains :

(8)

(9)

(10)CI = Of

By means of (2), (4) and (8), i and j can be expressed in each other:

j = (R+1)i-R and i = (R+1)j+R (11)R+1-iR R+1 +jR

See figure 3. This means that a straight stress path (i), implies an R dependent, straightstrain path and vice versa. Since a straight strain path implies a minimum of plastic work,the assumption of a straight strain path is likely to be correct in most sheet metal formingprocesses. We are dealing with processes like bending and deep drawing.

The local necking in the tensile test

In a tensile test necking is related to the maximum of the force in the force-path diagram :

concentration ofstrain • necking

dF = 0 (12)

Figure 3 Relation between strain path j Figure 4 The initiation ofnecking causedand stress path i with the normal by the weakest section.anisotropy R as a parameter.

The hypothesis that necking occurs when the force reaches its maximum is plausible, withthe condition that there is always a weakest cross-section in the test specimen. As isshown in fig. 4 a weakest spot implies, in the case of a decreasing force, that there willbe a strain concentration, called necking, at the weakest section. By the decrease in force,the stronger cross-sections will lose part of their elastic deformation (spring-back).

From figure 2 it follows :

~ = (J1 bc

and with (1), (6), (7), (9) and (10) it yields:

~ = C (J~1+Eo)ne-e1 bosoIRWith the instability criterion :

a~d~=-de1 = 0ae1

It then yields :

(13)

EOE1 cr = n- - (14)

JR

This is the well known critical strain in a tensile test, but actually only for q=O correctwith respect to the experimentally obtained FLCs.

The local necking in the formed sheet

Experimentally it is found that at the moment of instabilitythe strain q perpendicular to the major strain G1 remainsstationary (Veerman~. This can be explained by the factthat the rapid local increase of the strain in 2-directionq as observed in the tensile test will be suppressed by theneighbouring material, which is relatively stable comparedto the weakest spot in a relatively large sheet. Figure 5shows the strain path as has been found experimentally.

From this, a second condition for instability can beobtained, in the case of a relatively large sheet :

o paintof InItabIIy

Figure S The initiation ofstrain localisation in 1direction.

= 0 (15)

So, as an instability criterion it is proposed that :

dt=.. = 0 and dE2 = 0 (16)

Relation (13) can be rewritten in several ways as a function of the strains. Using theinstability criterion (16), three different formulations for the critical strain G1er areobtained.

First formulation

(17)

and:

dF..

Then the instability criterion becomes :

a~ = 0aeS

and the critical major strain becomes:

eoe1rr = n-JR

which conforms to the former result.

Second Formulation

and :

a~ a~d~ = - de2+- des

ae2 a£sThen the instability criterion becomes :

a~ = 0aes

and the critical major strain becomes:

Third Formulation

and :

(18)

(19)

(20)

(21)

Then the instability criterion becomes :

at=, = 0a€3

and the critical major strain becomes :

(22)

The Forming Limit Curve

With the three previously described formulations (18), (20) and (22), the FLC can bedrawn (see figure 6).

In sheet metal forming it is obvious that the highest values of the critical strain are closestto the experiments, in the case of relatively large sheets. However, if the instability occursnear an edge of the sheet, then the second condition dq=O is not fulfilled (Veerman1o.>,and the critical strain moves downward towards line I.

s

II

\.

s

-1·1

Figure 6 The FLC in absence ofprestrain,with line I according to (18), line IIaccording to (22) and line III according to(20).

Figure 7 The influence ofprestrain andanisotropy on the FLC.

Figure 7 shows the influence of the prestrain £0 and anisotropy R. In most cases the pre­strain is so small that it can be neglected.

Experimental results

From Painterll experimental results of several materials are used to verify the theorypresented in this paper. The leftside values are obtained by tensile tests; grooved test­strips (to produce different strain ratio) were pulled to failure.The rightside values are obtained using the hydraulic bulging technique.In figure 8, 9, 10 and 11 the theoretical lines (according to (18), (20) and (22» andexperimental values for steel, aluminium and brass are plotted.

groovedepth

+ O.OOnvn 0 O.12mmo 0.24 mm * 0.38 rnm

+

o 0

o

o 0.1

E1ILl

~00q; 0>p-

,,-0 ~

~Iy,/

OJ

n-o.~EO- ~8R-1 ~

.0.4 ,(.2 OJ! cu

Figure 8 CRrsteel tensile test. Figure 9 FLC for CR1-steel.

0-0.336EO -0.255R-O.37

-0.4 -0.2 o 0.2

"-

~V

o?,.. 0.8 /r '\. /

0

0.4 R> 0 o~

0.2

n-O eEO- ~.oeR-C f85

,(.4 .0.2 0 0.2 0.4

Figure 10 FLCfor aluminium. Figure 11 FLC for brass.

Factors influencing the FLC

Quite a number of measurements deviate substantially from the theoretical curves. Fourcauses will be discussed here :

o Inhomogeneity of the material.o Ductility.o Strain rate sensitive material.o Logarithmic strain hardening.

A : Inhomogeneity of the material

Inhomogeneity of the material causes stress-concentration in the workpiece andconsequently early crack initiation, even before the necking point is achieved(Ramaekers12

). This fracturing is often interpreted as fracture precededby necking instability. In fact, the material shows lacking ductility.

B : Ductility

4

recycled AI:o 20·C+ 100·CD 150·Cx 200·CA 250·C• 300·C

320.0 +-~-e+..,tHI ............-~tr---.-----o~~--l

o

<5 0.2 .....

~:2~I~

:::c 0.6 • A1CuBiPb : "

~ • A1MgSil : "~ ,,:,,:bE 0.4 \ ~ \ ; .

...- ., .. ~ .~ :,-,:" o 350·C" .

Strain at fracture iF [-]

Figure 12 Ductility according to Bolt andSillekens.

Ductility is defined as the strain untillcrack initiation in the workpiece. As isshown by Boltl3, and Sillekens14, the strainto fracture depends on the material(homogeneity (A), structure, grainsize)and the state of stress, in particular thehydrostatic stress (see fig. 12).For the hydrostatic stress the followingholds:

Om 1° = 3(01 + 02 + 03) (23)

it then follows for the current analysis :

1

Om 1 (1 1\(1 "2R "2)-2-=- = - +'J -1--+/o 3 R+1

and so with (11) :

(24)

Om = 1(1 +j+R* J 1-2 j+R* +( j+R* J2 -j° 3 1+J"R* .1 1+J"R*J+-

R*

(25)

in this:

R*= RR+1

, ~~-.0"" ft-! .- /lClJISlfJlt-IfDDlFIED

HSLA SI'IEl.lJ If' .ft-,

~"III ~HSl.A5TIn

ft- I,".

AI "'fNrhHw -.......~,

~~...J- .

-1,1 , ',3 4' -1,1 , uIt

Figure 13 The hydrostatic stress as afunction ofthe strain path.

Figure 14 Em for an HSLA Steel,also one inclusion-modified.

As can be seen from fig. 13, early crack initiation will preferably occur at the right partof the FLC (j >0). This is also illustrated by Pearce3 (see fig. 14).

C : Strain rate sensitivity.

Strain rate sensitive material shows a delay inthe occurrence of necking. With a constantvelocity (v) of the moving head of the tensiletester, it yields the following for the strain rateduring uniform deformation (see fig. 15):

. V .e 1 = L ,L the length of the test piece.

During necking this becomes :

. Ve 1 = L ,~length of the necking zone.

N

Figure 15 Strain rate in the tensile testwith a constant velocity.

Thus with~ < < L, the strain rate will increase substantially in the necking zone, so thestrain resistance or yield stress will increase, and consequently a more uniformdeformation will appear.

Generally, it is assumed that carbon steel is strain rate sensitive, contrastingly, aluminiumis not. Experimental curves of C-steel show some delay in the occurrence of necking,while aluminium mostly does not show this delay. This is in agreement with thestatements made.

D : Logarithmic strain hardening.

This behaviour is experimentally found for copper and brass (KMS 63) (Ramaekers14).

Fig. 12 shows the results of the tensile test.

In (E+Eo)

Figure 16 The strain hardeningbehaviour ofcopper and brass.

For e ~ es the well-known exponential

strain hardening behaviour is found and for

e ~ es the so-called logarithmic strain

hardening:

Of = B +q Lrt...e +eo) (26)

A possible explanation for this behaviour is theappearance of cross-slip after a certain amountof plastic work. With (26), (13) becomes:

(27)~ = B~Co(1 +q* ~J#l+£O))exp-"R

From a limited number of experiments on copper and brass (Ramaekers14), it follows

that:

* q nq = - =-B 2

(28)

Analogue to exponential strain hardening material, the critical strain e1cr is derived. Fig 17shows the result of the calculations.

-1 .& 0.&

The n-value of formula (28) is the real value

obtained from the tensile test for e < es .

Normally this deviant material behaviour is nottaken into account, and one will find anaverage value of the strain hardening exponent,which is smaller than the one above-mentioned

for e < es .

The theoretical FLC is in fairly good agreementwith measurements for e2<0 (fig. 17). Fore2> 0 the ductility (B) will influence themeasurements.

Figure 17 The FLCfor brass.

Conclusions

A plausible theory for the FLC is derived. For aluminium and carbon steel, theory andexperiment correspond very well. For some materials, for example low carbon steels, thestrain rate sensitivity is a possible explanation for the delay in the occurrence of necking.Some materials, i.e. brass, show a large deviation, for 82>0, from the theory, which canbe explained by the moderate ductility of the material. In certain conditions the materialtends to fracture before localised necking.For brass the different strain hardening behaviour also gives a possible explanation for thedeviation.

References

[1] Keeler S.P., Determination of Forming Limits in Automotive Stampings,Society of Automotive Enggs., No. 650535, 1965.

[2] Goodwin G.M., Application of Strain Analysis to Sheet Metal Forming Problems inthe Press shop, Society of Automotive Enggs., No 680093, 1968.

[3] Pearce R, Sheet Metal Forming, 1991, ISBN 0-7503-0101-5, pp. 145 and pp. 202.[4] Hill R, Discontinuous Plastic States, with special reference to Localized necking in

thin sheets, Journal of the Mechanics and Physics of Solids, 1952, vol. 1, pp. 19-30.[5] Swift H.W., Plastic Instability under Plane Stress,

Journal of the Mechanics and Physics of Solids, 1952, vol. 1, pp. 1-18.[6] Marciniak: Z, Kuczynski K, Limit Strains in Stretch Forming Sheet Metal, Int. J.

Mech. Sci., 1967, vol. 9, pp. 609-620.[7] Marciniak: Z, Kuczynski K, Influence of the plastic properties of a material on the

forming limit diagram for sheet metal in tension, Int. J. Mech. Sci., 1973, vol. 15,pp. 789-805.

[8] Hill R., The mathematical theory of plasticity, 1950, Oxford Clarendon Press.[9] Hosford W.F., Backofen W.A., Burke 1.1., "Texture Hardening" Transactions of the

ASM, 1962, Vol. 55, pp. 264-267.[10] Veerman C.Chr., Groot de M., Hartman L., Peels J.J., "Bepaling en toepassing van

de grensvervormingskromme voor een begin van de zichtbare insnoering", poly­technisch weekblad 1974, 23 - 1, pp. 35-44.

[11] Painter M.J., Pearce R, Instability and fracture in sheet metal, Papers IDDRGAmsterdam, 1972, 1.1-1.5, l.1.b-1.14.b.

[12] Ramaekers J.A.H., "Harte und Verformung metallischer Werkstoffe", Ph.D. thesis,1970.

[13] Bolt P.J., "Prediction of Ductile Failure", Ph.D. thesis, Eindhoven University ofTechnology, 1989.

[14] Sillekens W.H., "Backward Can Extrusion and Materials Behaviour", Ph.D. thesis,1992.