earing in deep-drawing - hindawi publishing...

Texture,1974, ol. 1, pp. 173-182.

Gordon and Breach Science Publishers Ltd.Printed in the United Kingdom

EARING IN DEEP-DRAWING STEELS

G. J. DAVIES, D. J. GOODWILL and J. S. KALLEND

Department of Metallurgy and Materials Science, University of CambridgePembroke Street, Cambridge, U.K.

(Received March 20, 1972)

The variations of the fourth-order coefficients of the crystallite orientation distribution function, with rollingreduction have been determined after cold-rolling and annealing for a deep-drawing quality rimming steel and analuminium-killed steel. These coefficients influence drawability and earing behaviour and by the manipulation ofthe coefficients in the distribution function of a 60o cold-rolled and annealed rimming steel, a hypothetical non-earing sheet texture has been derived. By comparison with the actual sheet texture those textural components whichmost affect earing behaviour are identified.

INTRODUCTION

The effects of normal and planar anisotropy ofmechanical properties on the deep-drawing per-formance of sheet metals have been variouslyreported. 1-s In particular, the relationship betweenanisotropy and the appearance of "earing" hasattracted considerable attention. The eating pheno-menon has been reviewed by Wright6 and byGrewen. 7 Although earing in sheet metal is knownto be determined by mechanical anisotropy attribu-table to the existence of crystallographic preferredorientation (texture) in the sheet, there are fewquantitative treatments available. 5,8,9 One of themost important aspects of texture study is thedevelopment of methods for changing the texturetowards the ideal. For control ofeating this requiresclear identification of the textural components con-tributing most strongly to the mechanical aniso-tropy. Although empirical methods have beendeveloped with some success, 5’7 fundamentalanalyses are still needed.The application of spherical harmonic analysis

methods in texture researcht o-t has yielded quan-titative descriptions of texture, in the form ofcrystallite orientation distribution functions, whichcan be used to predict elastic and plastic anisotropywith some accuracy. The object ofthe present paperis to show how the crystallite orientation distribu-tion function can be used to study earing in sheetsteels of deep-drawing quality. The proceduresdescribed are ofmore general applicability.

173

ANALYTICAL BASIS

Pursey and Cox15 in their analysis of elasticanisotropy in cubic crystals showed that only fourindependent coefficients of the erystallite orienta-tion distribution function enter into the expressionfor the polyerystalline elastic constants. The usualexpression for the distribution function, is

W(I, , ) I=0 m=--l n---- WlmnZlmn () exp(-img,)

exp(- in)

where g,, 0 (= cos-) and are the Euler anglesdescribing the crystallite orientation, W. are thecoefficients of the generalised spherical harmonicseries and Z. (0 is a generalisation of the asso-ciated Legendre function (the so-called augmentedJacobi polynominalsX2). In this function it isthe independent coefficients, Wooo, W.oo, W.2o andW4,0 (W4.m4. being linearly dependent on Wo)which determine elastic anisotropy.

Similarly, it has been shown 6 that these fourthorder coefficients largely determine the plasticanisotropy of cubic metals. This is a consequenceof the observation that when the anisotropy ofplasticity of a cubic single crystal deforming bypolyslip according to the Bishop and Hilltheory 7, 8 is expressed as a series of generalisedspherical harmonics the coefficients, Gt,,,,(q) ofhigher than fourth order are an order of magnitude

174 G.J. DAVIES, D. J. GOODWILL AND J. S. KALLEND

smaller than those for the zerotWf or fourth order(see Table I).

TABLE I

Mean absolute values of the coefficients Gtmn(r) for variousorders of the expansion of the function describing flow stressanisotropy according to the Bishop and Hill theory, with

contraction ratio r

Order Mean absolute value of Gmn(r)

0 4.4702 04 0.1916 0.0128 0.02210 0.00312 0.00514 0.003

In making the analysis of plastic anisotropy16

the series coefficients, Glmn(q) are evaluated forsuccessively different values of an assumed con-traction ratio, q, given by

dyy r

a%+a +r

where d%y and dzz are strains in the width andthickness directions, respectively, and r is the con-ventional strain ratio (width strain/thicknessstrain).

In deriving Table I the values of Gmn(q) have beencalculated for contraction ratios q 0, 0.1,0.2... 1.0 for different orders of 1 up to andincluding the fourteenth. In predicting the plasticanisotropy of polycrystals 14, 6 this information iscombined with the crystallite orientation distribu-tion data. If the lower-order texture coefficients,W’tmn, are of the same order of magnitude (this isnormally the case), the contribution of productterms of order greater than four can be neglected.Cross terms are not present because of the ortho-gonal properties of generalised spherical harmonicfunctions.The analysis of drawability can be undertaken

in a parallel way. Hosford and Backofen2 showedthat good drawability in sheets of cubic metals isobtained when the parameter, r, given by the ratio

of the yield strengths under the plane-strain condi-tions existing in the cup wall and flange respectively,is large.: This parameter can easily be related to thecrystallographic texture. 2 Kallend 9 showed thatfor the coefficients of the crystallite orientationdistribution function a large negative W4oo gives alarge fl value and thus good drawability. Further-more, a positive W420 controls the tendency toform two ears at 90 and 270 from the rollingdirection and a positive W44o leads to four ears at0, 90, 180 and 270 to the rolling direction.Correspondingly, a negative W42o gives two ears at0 and 180 and a negative W44o gives four earsat 45 135 225 and 315 to the rolling direction.The earing tendency is decreased as the values of1/r420 and W440 tend to zero.The present work involves firstly the manipula-

tion of the fourth-order coefficients of the cry-stallite orientation distribution function obtainedexperimentally for a commercial rimming steeland the correlation of changes in the drawabilityand earing tendency with changes in the texturalcomponents of the distribution. Subsequently theresults are applied to a commercial killed steel.The initial texture analysis was made using rimmingsteel because the orientation distribution functionfor this steel was such as to allow the ready identifi-cation of the different texture components. Thecrystallite orientation distribution function for thekilled steel obscured small changes in intensity dueto its higher overall severity.

EXPERIMENTAL

The steels considered were commercial deep-drawing quality rimming steel and killed steel whosecompositions are given in Table II. Samples of thesteels were cold rolled to different reductions usingan oil lubricant. Rolling was followed by a simu-lated commercial annealing treatment, viz." slowheat (25C per hour) to 700C, hold for 20 hours,slow cool to room temperature (25C per hour),in an atmosphere of argon at reduced pressure.Texture measurements were made on samplesprepared by the composite specimen method ofElias and Heckler2 using a Siemens texture gonio-meter with filtered MoK radiation. {110}, {200}

"]’The zeroth order coefficients give rise to orientationindependent properties.

:l:It should be noted that the ]/-parameter is difficult tomeasure experimentally. For this reason the plastic strainratio, r, is more frequently used as a measure of drawability.

EARING IN DEEP-DRAWING STEELS 175

TABLE II

Composition and hot process treatment of rimming andkilled steels

Element Rimming steel wt. Killed steel wt. oC 0.020 0.035S 0.022 0.022P 0.006 0.008Mn 0.27 0.24Cu 0.090 0.030Ni 0.060 0.025Sn 0.020 0.010Si 0.01Cr 0.070N 0.0028 0.0074

Total A1 0.030 0.047Sol AI 0.01 0.040

FinishingTemperature 898-937C 882-904C

(mean 920C) (mean 893C)CoilingTemperature 653-620C 598-549C

The section used for experimentation was from the centreof the coil.

and {211} pole figures were used to determine thecoefficients ofthe crystallite orientation distributionfunctions for each pre-annealing reduction.

Constant b sections of the crystallite orientationdistribution function were plotted automaticallyusing a curve plotter controlled by a contouringprogram. Specific orientations were indexed byreference to the charts published previously. 21

Table III lists orientations of interest in terms of thecorresponding Euler angles 0, b and k.Using the experimental distribution function

for a sample of rimming steel given a treatmentclose to that adopted commercially (viz. 60o coldreduction prior to annealing) the effect of settingW420 and Wo to zero both separately and togetherwas evaluated. In each case constant-ff sectionswere reconstituted and comparison made with theoriginal plots. In this way those textural componentsinfluencing earing could be identified.The crystallite orientation distribution function

data were also used to predict the planar anisotropyof the yield stress and of the plastic strain ratiousing the method of Bunge and Roberts1 andKallend and Davies. a’z

TABLE III

Euler angles for given ideal orientations

Idealorientation

hkl uvw

100 011

110 001

320 001

100 001

111 uvw

110

100 uvw

o ,90 45 090 45 900 (+,)=45

45 0 9045 90 9090 45 0

56.3 0 9090 33.7 033.7 0 9090 56.3 0

0 +=0, 9090 0 090 0 9090 90 090 90 9054.7 45 any

45 0 any45 90 any90 45 any

0 (+,)=any90 0 any90 90 any

RESULTS AND DISCUSSION

Fourth Order Texture CoefficientsFigure 1 shows the variation in the W4mo coefficientwith increasing true strain from cold rolling priorto annealing for the rimming steel. As mentionedabove, a large negative W4oo gives rise to a larget-value and good drawability. This is achieved byincreasing the rolling reduction prior to annealingalthough the rate of change decreases significantlyabove reductions ofabout 60 o.

W,2o is always negative and nearly independentof reduction. Thus a pair of ears at 0 and 180 areexpected after all reductions. Wo, however, isnegative below 309/0 reduction and above 88oreduction. The positive values reach a maximumat 60o reduction. Thus this coefficient shouldgive rise to ears at 45 135 225 and 315 outsidethe range 30-88 reduction and at 0, 90, 180 and270 within this range. Experimental observationsconfirm these predictions. This is also confirmed byresults presented in Figure Tf which shows thevariation in ear height with rolling reduction for a

Data supplied by R. C. Hudd, British Steel Corporation,Strip Mills Division.


3

2

1

0

-1

W4m0x103-2

-3

-4

-5

-6

-7

g|,,F \<o,nn r2W440 I-- t TRUEgSTRAINO | ’(.5 1.0 1’.5’ /’0 2|5

! !2"]):4b 6"0 libROLLING REDUCTION (PER CENT)

FIGURE 2 The variation in ear height with rollingreduction for a rimming steel similar to the one used toderive Fig. 1.

Sample from the front end of the coil.(C) Sample 2 from the back ofthe coil.

0.5 1.0 1.5 2.0 2.5TRUE STRAIN

2"o ioREDUCTION (PER CENT)

FIGURE The variation of the W+.o texture coefficientswith rolling reduction prior to annealing for the rimmed steel.

rimming steel similar to the one used to deriveFigure 1.The best earing behaviour is obtained by mini-

mising the deviation ofthe top ofthe cup from beingflat. In some eases this can be achieved by usingbalanced textures. The procedures developed em-piricallyTM for cubic metals to minimise earing bycombining 0/90 and 45 earing, effectively mini-mise W++o. In the present case since W+2o isindependent of reduction optimisation wouldinvolve minimising Wo to as large a degree aspossible consistent with increasing drawability.Notwithstanding this, it should be noted that theW2o and Wo coefficients are not crystallo-graphically independent since textural componentscontributing to W2o can also contribute toW440.

In commercial practice the cold reductionadopted depends on the final thickness and prop-erties required of the sheet. Figure 1 shows thatWoo does not change much beyond 60% reduc-tion. For good drawability and in applications whereearing is not disadvantageous, reductions of about60% are commonly employed. Where non-eatingproperties are particularly desirable, the coldreduction will be closer to either 30 or 88 o andthis may result in some loss of drawability.2

In Figures 3, 4 and 5 only the series coefficientsWooo and W+oo, Wooo and W+2o, and Wooo andWo together with the linearly dependent coeffi-cients have been used, respectively. W0o0, the tex-ture independent eoettieient, effectively sets thecontours for random texture to be everywhereunity. Examination of these figures with referenceto Table III and the indexing charts2 shows thatWoo is controlled mainly by the amount of orienta-tions with {111}-planes parallel to the plane of thesheet. This was expected, sine it is well known that alarge proportion of {111}-oriented planes in atextured material gives good drawing properties, x

and, as stated above, the overall level of any para-meter used to quantify drawability is dependent on

4oo-

From Figure 4 it can be seen that the orientationswhich most affect W42o are those close to {110}

EARING D,,I DEEP-DRAWInG STEELS 177

I

( =0

I1.2\

..!’ ,,( 25

0,5 1.3 ".2II

45 55

FIGURE 3 The crystallite orientation distribution func-tionforthe kVooo and W,oo coefficients alone, forrimmed steelrolled 60 and annealed.

( =0

( 25.

])= 45

@: 35

FIGURE 4 The crystallite orientation distribution functionfor the IVooo and I442o coefficients alone, for rimmed steelrolled 60% and annealed.

(001). These orientations, if present, will causethe "two ear" phenomenon. Figure 5 shows thatthe major orientations contributing to the "fourear" phenomenon are those with (001) parallelto the rolling direction, in particular the cubetexture, {100} (010). Since these include {110}(001), and cube texture is not present to any greatextent in the steel, it is clear that the amount oforientations close to {110} (001) is going to controlthe extent to which earing occurs on deep drawingin rimming steel.The levels of contours for the peak orientations

in Figure 3 should be made as large as possible forW,oo to be large and if good drawability is to beachieved. This will be a result of normal anisotropy

and hence will involve the preferred alignment ofplane normals. {111}-planes are most conduciveto favourable drawability. {ll0}-planes give gooddrawability while {100}-planes will give the mini-mum drawing performance. From Figures 4 and 5it can be seen that IV,2o and W**o should be mini-mised as far as possible to avoid eating. This in-volves planar anisotropy the extent of which will bedetermined by the degree of alignment of crystallo-graphic directions. Orientations with (100) in therolling direction should be eliminated as far aspossible. In addition the amount of the {100}(011) component should be increased in order toreduce IV**o and get a reduction in ear height, butthis would decrease the value of IV,oo.


25

(= 45

1.4

1.4

( 20

o.

( 35

(p 55

FIGURE 5 The crystallite orientation distribution functionfor the Wooo and W44o coefficients alone, for Yimmed steelrolled 60 and annealed.

Variations in the Cold-Rolled Texture of RimmingSteel Produced by Manipulation of the TextureCoefficientsConstant- sections of the erystallite orientationdistribution function for the sample ofrimmed steelcold rolled 60% and annealed involving all coeffi-cients up to twentieth order are presented in Figure6. Setting W42o to zero produces the new set ofconstant @sections given in Figure 7. Likewise,W44o is set to zero in Figure 8, and both W42o andW4o are zero in Figure 9.Removing W2o measurably affects orientations

close to {110} (001) (including {320} (001)), asdoes the removal of W4o. This latter coefficientaffects all orientations ofthe form {hkl} <001 ).

Sorting both W2o and Wo to zero has an evenmore marked effect. Figure 9 represents the hypo-

(=0

,05 -( 25

2.0

=45

=35

=55

HGURE 6 The crystallite orientation distribution functionfor rimmed steel cold rolled 60 and annealed.

thetical texture of the rimmed steel which wouldproduce the same level of deep drawability as theexperimental sample (same average plastic strainratio) but without a significant earing tendency.It is worth noting that an improvement effected bymaking W4o zero by slightly increasing the pro-portion of {100} (01 I) would lower the overalllevel of performance by reducing r. Some compro-mise would therefore seem to be inevitable inpractice.

Variations in Plastic Anisotropy of Rimming SteelProducedby Manipulation ofthe Texture CoefficientsFigure 10 shows the predicted variation of the plas-tic strain ratio, r, with angle from the rolling direc-tion using coefficients up to the fourteenth order,


(= 25

(I) 45

20

35

55

FIGURE 7 The crystallite orientation distribution functionfor rimmed steel cold rolled 60% and annealed with theW,2o coefficient set to zero.

2.0

:00.

25

:45

1.5

0.5

(: 20

35

) :55

FIGURE 8 The crystallite orientation distribution functionfor rimmed steel cold rolled 60 and annealed with theI4/44o coefficient set to zero.

2.0

2.0

for the various cases considered above. The un-modified curve exhibits considerable planar aniso-tropy, the r-value in the transverse direction beinggreater than that in the rolling direction. The resultsagree fairly well with the expected behaviour fromsimilar steels.2 Removing W420 means that W440alone controls the eating behaviour. This correlatedwell with the observed form of the predicted planaranisotropy. Wo is seen to have the greatest in-fluence on r. When both W20 and Wo are set tozero almost no planar anisotropy is predicted. Thislatter observation is further confirmation of thepredominance of the fourth order coefficients indetermining the plastic anisotropy.The data of this and of the preceding section can

readily be combined to define the developments to

be aimed for to produce steels of high drawabilitywith low earing tendency.

Applications to Killed Steel

Figure 11 shows the variation in the W,oo, ’r420and W,40 coefficients for the killed steel reducedby varying amounts and annealed. Comparison withFigure 1 shows that the overall variations are quitesimilar in rimmed and killed steels. The much largerW400 values, however, reflect the superior draw-ability of killed steel. The variation of W400 withcold reduction agreed well with the variation ofaverage strain ratio obtained by Held25 for akilled steel given a series of reductions prior toannealing under similar rolling conditions, viz.:


0.5

25

.!4.50

10.5

35

55

\0

\

-2

W4moxlO3

FIGURE 9 The crystallite orientation distribution functionfor rimmed steel cold rolled60 and annealed with both theW,2o and W,,o coefficients set to zero.

-4W400

-6

-8

-t0

-12

-14I.

0.5 1.0 1,5 2.0 2.5TRUE STRAIN

" 4o s’oREDUCTION ( PER CENT)

FIGURE 11 The variation of the W4mo texture coefficientswith rolling reduction prior to annealing for the killed steel.

2.5

2.0

z

I--. 0.50 30 60 90

ANGLE FROM ROLLING DIRECTION

FIGURE 10 The predicted angular variation of the plasticstrain ratio for the texture data of Figures 6, 7, 8, and 9.

predicted from experimental texture data.

as with W,,o 0./ as with Wo 0.(C) as with W2o and W,,o 0.


good lubrication and heavy reductions on eachpass.

Figure 12 shows the variation in ear height withrolling reduction for an aluminium stabilised steelsimilar to the one used to derive Figure 11. This issimilar to Figure 2 for the rimmed steel. In generalaluminium stabilised steel shows a greater degree ofeating than rimming steel but this effect is not shownbetween the curves presented in Figures 2 and12. 24 Usually the lower critical cold reduction forzero eating is less for the killed steel than therimmed, and the upper critical reduction higher.

TRUE STRAIN

ROLLING REDUCTION (PER CENT)

FIGURE 12 The variation in ear height with rollingreduction for an aluminium stabilised steel similar to theone used to derive Fig. 11.

Examination of the erystallite orientation dis-tribution function for killed steel reduced by 40and annealed (Figure 13) shows that the compo-nents described above which contribute to W42oand W440 are present. A specimen of killed steelcold rolled 40 prior to annealing was chosensince this treatment produced a similar overalllevel of texture severity as that of the rimmed steelcold rolled 60 and annealed (Figure 6). It canthus be deduced that the steps necessary to reducethe earing tendency in this steel are the same asthose for the rimming steel.

CONCLUSIONS

1) W4oo is the single texture coefficient obtainedfrom spherical harmonic analysis of texture data

fData supplied by R. C. Hudd, British Steel Corporation,Strip Mills Division.

25

v

(]) 20

( 35-(J) 55

FIGURE 13 The crystallite orientation distribution func-tion for killed steel cold-rolled 40% and annealed.

for commercial steels which most affects the overallnormal anisotropy and drawing performance.W2o and W4o control the planar anisotropy andearing behaviour.2) Components with(111}-planesinthe planeofthesheet will produce the optimum normal anisotropyand associated maximum drawability. { 110} (uvw)will have a deleterious effect.3) The magnitudes of W42o and W44o are decidedlargely by the proportion of {hkl} (001) compo-nents present.4) To improve the earing behaviour some compro-mise with the overall drawability is necessary.Increasing the proportion of {100} (011) willimprove earing behaviour but reduce the drawa-bility. However, reducing components near {110}


(001) in steels should not significantly affect thedrawability but is expected to reduce earing con-siderably.

ACKNOWLEDGEMENTS

The authors would like to thank the Science Research Coun-cil and the British Steel Corporation for financial supportduring the course of the research from which this paperevolved. They would also like to thank Mr. R. C. Hudd ofthe British Steel Corporation Strip Mills Division for supply-ing the steel and for stimulating discussions.

REFERENCES

1. R. L. Whiteley and D. E. Wise, in Flat Rolled ProductsIII (Interscience, New York, 1962), pp. 47-63.

2. W. F. Hosford and W. A. Backofen, in Fundamentals ofDeformation Processing (Syracuse University Press,New York, 1964), pp. 259-292.

3. D. V. Wilson, J. Inst. Metals 94, 84 (1966).4. D. J. Blickwede, Trans. ASM61, 653 (1968).5. D. V. Wilson, Met. Reviews 14, 175 (1969).6. J. C. Wright, Sheet Metal Ind. 42, 814 (1965).7. J. Grewen, Proc. Polish Acad. Sci. (in press).

8. G. E. G. Tucker, Acta Met. 9, 275 (1961).9. R. W. Vieth and R. L. Whiteley, see Refs. 5 and 7.

10. R. J. Roe, J. AppL Phys. 36, 2024 (1965).11. H. J. Bunge and W. T. Roberts, J. AppL Cryst. 2, 116

(1969).12. P. R. Morris and A. J. Heckler, in Advances in X-ray

Analysis (Plenum Press, New York, 1968), Vol. 11, pp.454-472.

13. H. J. Bunge, Krist. u. Technik 5, 145 (1970).14. J. S. Kallend and G. J. Davies, J. Inst. Metals 99, 257

(1971).15. H. Pursey and H. L. Cox, Phil. Mag. 45, 295 (1954).16. G. J. Davies, D. J. Goodwill and J. S. Kallend, Met.

Trans. 3, 1627 (1972).17. J. F. W. Bishop and R. Hill, Phil. Mag. 42, 414 (1951).18. J. F. W. Bishop and R. Hill, Phil. Mag. 42, 1298 (1951).19. J. S. Kallend, Ph.D. Thesis, University of Cambridge

(1970).20. J. A. Elias and A. J. Heckler, Trans. Met. Soc. AIME

239, 1237 (1967).21. G. J. Davies, D. J. Goodwill and J. S. Kallend, J. AppL

Cryst. 4, 67 (1971).22. J. S. Kallend and G. J. Davies, J. Inst. Metals 98, 242

(1970).23. R. S. Burns and R. H. Heyer, Sheet Metal Ind. 35, 261

(1958).24. R. C. Hudd, Private Communication.25. J. F. Held, Trans. Met. Soc. AIME 239, 573 (1967).

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