ISIJ [nternational, Vol, 31 (1991 ). No. 7, pp. 696-705

UItrason ic Prediction of r-value In DeepDrawing Steels

D. DANIEL.Kei SAKATA1)and J. J. JONASDepartment of Metai[urgical Engineering, McGill University, 3450 University Street. Montreal PO. H3A2A7. Canada.1)Technical Research Division, KawasakiSteel Corporation, Kawasaki-dori. Mizushima, Kurashiki, Okayama-ken,71 2Japan.

(Received on January 7. l991; accepted in final form on March22. 1991)

The textures of five types of deepdrawing steels were measuredand analyzed using the series expansionmethod. Modul-r and electromagnetic acoustic (EMAT)techniques were employedto determine the elastic

anisotropy in terms of the angular variation of Young's modulusand the ultrasonic velocities, respectively.

The plastic anisotTopy wasassessed by measuring r-values as a function of inclination with respect to therolling direction. The series expansion forma]ism was employed for predicting the elastic and plastic

anisotropies from the initial texture data. Comparison with the experimental measurementsof Young'smodulus indicates that the so-called elastic energy methodcan accurately reproduce the elastic anisotropyif the single crystal elastic constants are appropriately chosenwithin their ranges of uncertainty. Theangularvariation of r-value in the rolling p]ane wascalculated from the ODFcoefficients by meansof a relaxedconstraint model (pancake version). The best quantitative agreement is obtained whenthe CRSSratio for

glide on the {1 12} 11>and {1 10} 11>slip systems is I .O. 0.95 and 0,90 for the IF2, IFI and AKDagrades, respectively. The ODFcoefficients of order greater than 4 were evaluated and calculatednon-destructive/y from the anisotropy of the ultrasonic velocities of the lowest order symmetrical Lamb(S.)

and shear horizontal (SH.) wavespropagating in the rolling plane. The calculated pole figures based onthe ODFcoefficients obtained in this wayare similar to those derived from complete X-ray data. It is shownthat the plastic properties of commercial deep drawing steels are predicted moreaccurately whenthe 4thand 6th order ODFcoefficients are employedthan whenonly the 4th order ones are used,

KEYWORDS: texture, r-value, ultrasonic velocities, on- Iine evaluation, elastic anisotropy, plastic anisotropy.

1. Introduction

The existing techniques of texture identification (e.g.

the determination of pole figures using X-ray diffracto-

metry) are not particularly well suited to on-1ine ap-plications for reasons of lengthy experimentation timeand intricate sampling. Methodsof measuring the textureon-line are therefore of great practical interest. Althoughactive research is being carried out on rapid methodsofpartial X-ray analysis,1) an alternative solution to this

problem involves deducing the required texture infor-

mation through the anisotropy of a suitable physical

property. The latter must be chosenjudiciously; it mustbe readily measurable, and its anisotropy must be easily

related to the distribution of crystallographic orienta-tions. Of the manypossible choices, the elastic properties,

determined by meansof ultrasonic measurements,satisfy

both of the aboveconditions. However, the basic textureinformation contained in the anisotropy of the elastic

parameters does not describe the crystallographic ani-

sotropy in sufficient detail to permit the accurate pre-diction of the plastic properties. In this work, attemptsare made to extract further information from theultrasonic anisotropy and to demonstrate how suchadditional texture coefficients can be employed for

improving the prediction of plastic properties. Thus, therelation between texture and the resulting elastic and

plastic anisotropy is the main concern of this paper,centered specifically on the case of deep drawing steels.

2. Experimental WorkThirty one samples of low-carbon steel sheets were

collected from various producers in Canada, the UnitedStates and Japan. A11 these steels were cold rolled andannealed (recrystallized). They were divided into five

types: (1) batch-annealed Al-killed drawing quality

(AKDQ) steel, (2) commercial-grade rimmed steel

(denoted as RIM below), high-strength low-alloy

(HSLA)steel, and (4) and (5) two types of interstitial-free

(IF) extra-10w-carbon steels. The latter distinction wasmadebecause of the different textures observed, induc-ing different plastic anisotropies. In the first type ofinterstitial-free steel (IF1), the r~-values fell in the rangefrom 1.5 to 1.9, and the Ar's of 0.2 to 0.4 were belowthose of the AKDQsteels in the samef range. In the

second type (IF2), the r~-values were above 2.0, and the

Ar's were still smaller, ranging from -0.2 to 0.2. Thethickness range of the sheets studied was0.7 to I.6mm.Anexampleof the chemistry of each type of steel is givenin Table 1.

2.1. Mechanical Testing

The lowest order symmetrical Lamb(So) and shearhorizontal (SHo) ultrasonic wave velocities were mea-

C 1991 ISIJ 696

IS]J International. Vol. 31 (1991), No. 7

Table l. Chemical composit{ons of some of the deepdrawing steels studied.

Composition (wt%)Grade

C Mn P S Si Cu Cr Al N Ti Nb

IF2 .005 .18 .OIO ,007 012 ,049 .014 ,060 080

IFl .002 ,09 OIO ,004 .O11 .Ol ,02 .032 .04

AKDQ .O1 .22 ,006 .009 .008 .O1 .Ol .034 .Ol

RlM .07 .33 OIO .O11 .Ol .O1

HSLA .07 .73 .006 .O11 ,020 .020 020 .034 046

o

30

~60

90

IF2

90o

30

~~

60

90

90 oo

T: EMATTransmitterRl, R2: EMATReceivers

EMATCONTROL

rigidspacer R2

Rl '7

T ~~ 4-~~20crnsoundL~L~'~e~ ~ Propagation

RD

SCOPE

l]

30

~~

60

90

IF1

30 ci 60 90o

30

~60

90

tllO]

(112)[110]

(OO1) I110]

[1101 [4~2] EIZ2](223)

Ilizl

[1TO] [l~l] [Oi 1](1 1l)

(332)[lTol (554)t225]

[1 1O] (110) [OOIl

PCANALYSIS

sheet metal

~~~Frg. l. Schematic diagram of the experimental ~MATap-

paratus,

sured along three directions of propagation (O', 45'and 90') for So and two directions (O', 45') for SHOin

the rolling plane using 500kHZelectrornagnetic acous-tic trnasducers (EMAT'S). The experimental device is

illustrated in Fig. l. The speed of sound is determinedby measuring the time-of-flight of an acoustic pulsebetween the two EMATreeceivers. The So modevelocities were extrapolated to zero frequency to correctfor the dispersion effect asspciated with finite platethickness. The errors associated with the wavevelocity

measurementsare estimated to be ~5m/s. For com-parison purposes, Young's modulus was also mea-sured at every 15' in the rolling plane using Modul-Requipment. This induces a resonant frequency in thespecimen, from which Young's moduluscan be calculat-

ed directly. According to Mould and Johnson,2) theuncertainty in the Modul-R frequency measurementsis

O.070/0, which corresponds to O.140/0 for Young'smodulus. Thus the uncertainties associated with the twomethodsof measuring the elastic anisotropy are similar.

Tensile tests for r-value measurementwere carried outon 200mmby 20mmtapered specimens cut from thesheets at several orientations with respect to the rolling

direction, i,e. every 15', 22.5' or 45'. The r-values wereevaluated at various strain levels from the distortion of

a grid printed on the specimens, which were tapered from20.0 to 19.3mmwithin the gage length of 100mm.Further details of this procedure and of the results

3)obtained are presented in an earlier paper.

2.2. Texture Analysis

The texture of each steel sheet was determined by

means of X-ray measurements, using standard tech-

AKDQ c2 =450

Fig. 2. ip2=45' ODFsections oflF2, IFI and AKDQsteels

(1evels: 1, 2, 3, • ••).

niques. Quantitative analysis of the pole figures led tothe C~" coefficients of the ODF.4)Typical ip2 =45' sec-tions of the ODFfor three of the deep drawing steels

are presented in Fig. 2. The particularity of deep draw-ing steels is caused by the distribution of the grainorientations along the ND-fibre (i.e. grains whose 111>directions are parallel to the normal direction). Thenon-uniformity of the texture along that tube inducesthe planar anisotropy of properties such as r-value andthe ultrasonic velocities. Another characteristics of thesesteels is the earing behaviour displayed during thedrawing of cylindrical cups. Drawncups of the AKDQand IFI type steels have four ears located at O', 90', 180'and 270' from the rolling direction (RD). Theother typeof phenomenonis observed in the IF2 steels, which havesix ears at O', 60', 120', 180', 240' and 300'. The latter

behaviour can be attributed to the sharpness of the IF2texture and to the strong six-fold symmetryof the plastic

properties of 111>single crystals. In the case of the

AKDQand IFI steels, the six-fold symmetry of theRD-fibre (i.e, grains whose I0> directions are parallel

to the rolling direction) is obscured by the larger

dispersion around the fibre axis and by the presenceof secondary componentsalong the RD-fibre, such as{332} I0> and {223} I I0>,

The Cu1" coefficients of the ODFcharacterize in acomplexmathematical waythe symmetryof the texture.

As shown below, they can be used to describe theproperties of the polycrystal since the ODFacts as aweighting function in the averaging procedure based onthe behaviour of individual grains. It is clear that thefourth order ones cannot describe the six-fold symmetryof the plastic properties. Consequently, whenthe earingbehaviour displays six-fold symmetry, the higher order

ODFcoefficients are required for a full description, thesixth order ones being the most important.

697 C 1991 ISIJ

ISIJ International, Vdl. 31 (1991), No. 7

3. Elastic Behaviour of Polycrystalline Aggregates

3.1. First Order Voigt, Reussand Hill Approximations:Conventional Method

Theclassical crystal-elasticity models are based on: 1)

the Voigt hypothesis5) of an identical strain state in all

the crystallites, 2) the Reuss hypothesis6) of constantstress, and 3) the Hill approximation,7) which involvesthe use of an arithmetic average of the above upper(Voigt) and lower (Reuss) Iimits. The Hill formulationis widely accepted as a reasonably accurate descriptionof the elastic anisotropy.

In the Voigt framework,4) the polycrystalline stiffness

tensor ~iVj is the average of the elastic stiffness tensorthrough the entire aggregate*:

~iVj

= c,?j +c.(agl- tij +a 11(U)C11+a~2(U)C~2

+a~3(lj)C~3)......

..........(1)

and

c cll cl2-c~6""

"""""(2)

Here c,?j are the single crystal elastic constants, all - tij

and a~" are the coefficients tabulated in Ref. 4), and C~"are the fourth order ODFcoefficients. Similar expressionshave been derived for the Reuss assumption,4) whichrequires the averaging of the elastic compliance tensorsv' It is clear that, for sheets of cubic materials, thepolycrystal elastic constants calculated from the first

order Voigt, Reussand Hill aipproximations are solely

functions of the single crystal onesandof the three fourthorder ODFcoefficients C'l".

3.2. Second Order Voigt, Reuss and Hill Approxima-tions: Elastic Energy Method

This method, originally proposedby Bunge,9) involvesthe introduction of perturbation terms into the procedurefor averaging the single crystal constants. Theperturba-tion terms are adjusted so as to minimize the meanelastic

energy, which leads to the following expression for theelastic stiffnesses in the Voigt formulation:

cij-cij 81(c~)-lc~ a~"(ik)al'at)F1:v v ) (3)_ -v.,.' = o(2)

with

8 lF(v, v')= ~ ~ {4, 4, 1 1Il l}

l =021+ I ~=.+.'o* l~-"I

l C~"C~"'. . . . . . . . .

.(4)x {4, 4, v, v' l, n}.hC11~_81

'

l lThe Clebsch-Gordan coefficients, {4, 4, 1, I l, 1}, forcubic symmetryand {4, 4, v, v' l, n}.h for orthorhombicsymmetry, are all numerical constants available in theoriginal paper.9) Similarly, by substituting the com-pliances sij for cij, the effective elastic constants s.. can'Jbe estimated. In the elastic energy method, the stiffnesses

and compliances are "corrected" by a secondorder termcontaining the function F(v, v') in which the fourth, sixth

and eighth orders appear.

As in the conventional method, the effective elastic

constants c. , s.. and their mathematical averages lead to'J' 'J

the planar distributions of Young'smodulusor ultrasonicvelocity pertaining to the respective Voigt, Reuss andHill type solutions. It is a characteristic of the elastic

energy methodthat 12 independent ODFcoefficients (3

for !=4, 4for l=6 and 5for l= 8) are necessary for thecalculation of Young's modulus, instead of only 3according to the conventional approach.

4. Prediction of r-value from Texture Data

4.1. Description of the Theoretical Mode]lingPlasticity calculations on bcc materials involve ques-

tions regarding the slip systems that are operative andthe critical resolved shear stresses (CRSS's) associatedwith each of these systems. Both experimental and theo-retical studies support the view that the crystallograph-ic description of glide on {1 10} and {1 12} planes alongthe 111>directions is sufficient for the description ofplastic flow. Another feature of bcc deformation is theexistence of asymmetric slip on the {I12} planes. Inthat description, the CRSSratios ocs = 1;t~i**i*g/T{110}

and OCH=T f characterize single crystal{112}**ti-t~i~~i*g/ T{ 11o}plastic deformation. Experimental values reported in

the literature include cc ;~~ I Oand oc ~~ I 04 to 114 fors ' H '

decarburized Fe single crystals.lo) From experiments

on Fe-3wtoloSi single crystals, Orlans-Joliet et a/.1 1) ob-tained values of ocs~;0.93 and o(H ~~0.96. Dueto the un-certainties regarding these values in the case of low car-bon steels, several hypothetical sets of ocs and oeH Wereused in this study.

The following grain interaction models were testedsystematically: the Taylor (FC), Sachs-Kochend6rfer(SK) and relaxed constraints (RC) models. Four RCmodels were evaluated: Iath (~i3 relaxed), RC4(~

23relaxed), RC3(~l

3and~23 relaxed, i.e. pancake) andRC2(~l2, ~l3 and ~23 relaxed). The 1, 2and 3 indices refer,

respectively, to the longitudinal, width and thicknessdirections of the tensile sample. The application of theseries expansion methodto the prediction of r-values is

described in Refs. 3), 4), 12) and 13) and will not bereviewed further here.

In addition to the existing grain interaction assump-tions, a new model,13'14) referred to as LWfor least

work, wasalso tested. It involves combining the FCandRC3deformation modesin proportions which dependon the grain orientation g. Analysis of the single crystalplastic work functions* for different CRSSratios andvarious r-values shows that the average value of thenormalized difference Abetween M.(g)RC= and M.(g)Fc(A (M.(g)Fc ~M.(g)Rc.)/M.(g)Fc) is about II o/o and thatl/3 of the grain orientations in Euler space haveAvaluesless than 6o/o

.Theabovetwo values weretherefore chosen

to define a least work criterion in which l/3 of the grainorientations deform according to the FCmodel, 50 olo

by the RC3deformation modeand the remaining onesselect the FCor RC3modewith a probability depending

on the A-value.

* The matrix notation (two suffixes) for the elastic constants is used here instead of the tensor one (four suffixes).8)

C 1991 ISIJ 698

ISIJ lnternational, Vol, 31 (1991), No. 7

~d

4

3

2

1o 30 e 60 90

o

30

~60

90

~

o

30

~60

o

r=0

30 c1 60

90

7

5

3

1O 9030 e 60

3. Influence ofCRSSratio on the values ofr(e) obtainedfrom the RC3modelfor anAKDQandan IF2 steel,

Experimental r-values are indicated by (A). O ocs'

ccH>1,155; ~) c(s=0cH=1.05; (~ ccs=0eH=1.0; ~)o(s =oeH=0,95; (~) ocs' o;H 0.866; Cc(s = I.O, CCH= I , I ,

90

90

r=1c1

o

30

~60

90

Fig.

4.2. Modelling Results

Results for fifteen of the current steels were presentedin an earlier publication.3) Sixteen additional steels wereinvestigated in the present study, and the methods ofcalculation were further improved and refined byemploying the LWcriterion. With the increase in the

numberof steels studied, the following tendencies havebecomeapparent. As shownin Fig. 3, it appears that anoc-value of I .O is the most suitable for r-value predictionin the IF2 steels. Avalue of 0.95 is preferred for the IFlgrades (not shownin Fig. 3), whereas restricted slip onthe {I12} planes or an oc-value of 0.9 gives relatively goodagreementwith experimental results in the AKDQsteels.

These trends confirm the experimental evidence accord-ing to which the presence of interstitial elements in

solution makesslip moredifficult on the {I lO} planes. i s)

The previous analysis indicated that the RC3modelgives the best overall agreement with experimental

measurements. The more recent calculations revealedthat the RC3and LWcalculations lead to similar results

for a given CRSSratio. This reflects the fact that, for

most of the texture componentsencountered in deepdrawing steels, the RC3deformation modesatisfies theleast work criterion defined above. Mapsof the preferred

deformation modesfor orientations belonging to the

c2 =45' section of Euler space are presented in Fig. 4.

It can be seen that for r ranging from Oto 2.5, only the

orientations near {1 1l} 12> (such as {554} and{223})

amongthe main texture componentsnear

Fig. 4.

r=1.5 r=2.3

Map illustrating the preferred deformation modescalled for by the LWmodel: RC3([]), FC (~~) ormixed FC-RC3(EEI).

Thegrain orientations being considered are located in

the ip2 =45' section and have been subjected to vari-

ous strain paths characterized by their r-values. Theline within the mixed FC-RC3areas indicates the

50o/oFC-500/0RC3Iimit.

the ND-fibre deform according to the FCmodeor to amixture of FCandRC3.This result provides anargumentin favour of the use of the RC3model, which is en-ergetically the most favourable deformation mode. TheLWcriterion, which justifies the use of this deformationmode,seemsespecially satisfactory whenthe deformationof an approximately equiaxed polycrystal is concerned(i,e, it explains why the 'pancake' model, originally

proposed for heavily cold rolled materials, can also beemployedeven whenthe grains are equiaxed).

5. Theory of Acoustoelastic Texture Analysis

5.1. Relation betweenUltrasonic Velocities and TextureData

The general theory linking sound velocity to textureis based on the Voigt constant strain assumption and

wasintroduced by Sayersl6) andThompsonet al, 17) The"plane wave" solutions for an unboundedmediumin

the unstressed state can be modified so as to apply to

wave propagation in relatively thin plates.18) Therelationship between wavevelocity and the polycrystal

elastic constants can be derived from the Christoffel

equaiton. 19) If the direction cosines are expressed in termsof the angle ebetweenthe measurementand rolling (RD)directions, the velocities of the lowest order symmetrical

LambSo(Vs.) and shear horizontal SHo(VsH.) modespropagating in the rolling plane can be obtained from20):

* Theplastic work function M.(g) specifies the work dissipated in a given grain gin order to accommodatethe macroscopic deformation associated

with a given ,'-value in the framework of the grain interaction model being considered.

699 C 1991 ISIJ

ISIJ International, Vol. 31

p+fll4~V~0(e)~2p ""~""'

"""~"(5)

p-~T1~~VsHo(O)-

2p ~"""'"""""(6)

where Pand Qare defined by:

P=c66 +611 cos2e+i22 sin20....

..........(7)

Q= (iiIcos2e+c66 sin2e)(c66 cos20+~22sin2e)

-(c66 +~l2)2 sin2ecos2e .............................(8)

and

,Jcij-ci3cj3/c33

.......,,......(9)

Here ~ij are the effective elastic constants applicable tothin sheet.

5.2. Effect ofDispersion onSoLambModePropagation

The value of Vso required by the theory in Eq. (5) is

the long wavelength (10w frequency) Iimit of the velocity.

Dueto dispersion, this cannot be determined directly at

any finite measurementfrequency. However, since thedispersion is small at wavelengths that are large withrespect to the plate thickness, accurate corrections canstill be made. Strictly speaking, the correction should bebased on the theory of a Lambwavepropagating in anorthotropic plate, and the dispersion correction shoulddepend on the ODFcoefficie, nts. Such a complex cal-

culation has not yet been carried out. Moreover, whenthe analysis concerns commercial textures and thedispersion is small (long wavelength with respect to thethickness), it is believed that the effect of planaranisotropy on the correction is negligible with respect tothe errors associated with the experimental measure-ments. Thecorrections can therefore be madeinstead onthe basis of the muchmoredeveloped theory of Lambwavesin isotropic plates.19) Accordingly, it can be writ-

ten that:

TV'sot.[1 1( A Irft

-)(2 2J1/2

(lO)Vis*t*f'l-

'

so ~"~ so T~l~+2,4 Vo

where t is the plate thickness; Aand u are the elastic

constants of the isotropic polycrystal (the Lam6constants); f is the frequency (500 kHZ in the presentwork); and Vis~0't* is the fundamental So Lambmodevelocity in the absenceof texture. (The latter is calculatedfrom Eq. (5), the elastic stiffnesses being determined bythe choice of the grain interaction model and the singlecrystal constants.)

ThemeasuredVso(t) can be considered as the sumofViss:o' and a velocity term induced by the texture effect.

As the latter can be assumedto be unaffected by thedispersion, the Vso defined in Eq. (5) can be obtainedafter the following correction of the experimentalmeasurements(in the conventional Hill approximation):

isot* isot*Vs0=Vso(t)+ Vso ~ Vso (t)= Vso(t)+13t2 ......(11)

The sheet thicknesses of the steels studied ranged from0.7 to 1.6 mm;thus the correction for dispersion induces

a change of 0.2 to 0.30/0 of the measuredvalue. This

C 1991 ISIJ 700

(1991 ). No, 7

cannot be neglected with respect to an experimental errorof O.13 "/o. As the horizontally polarized shear velocity,

VsH., is frequency independentl9) and exactly equal tothe velocity of a plane shear wave, Eq. (6) does not needto be corrected along the lines of Eqs. (lO) and (1 l) for

Vs.'

5.3. Principle of the Determination of Texture Data fromUltrasonic Measurements

5.3.1. EstimationofthcFourthOrderODFCoefficientsby the Conventional Method

Onthe basis of single crystal values from the literature,

and employing the Hill averaging procedure, the planardistribution of ultrasonic velocity can be expressed as afunction of the three fourth order ODFcoefficients:

C~1, C~2and C13. Conversely, by measuring three in-

dependentultrasonic quantities, these coefficients can beestimated, from which a partial description of the textute

can be obtained. This procedure for determination of thefourth order ODFcoefficients can be employedwith anyset of elastic data, such as Young's modulus measure-ments, the angular variation of which dependson threeindependent elastic parameters (i,e. on sll, s22, ands66 + l/2si2)'

Thetechnique employedhere differs from the previousones20,22-24) in that it retains the original expressionsfor the ultrasonic quantities as functions of the texturecoefficients without any simplifications. Due to thecomplexity of these expressions, a numerical method is

required, which can nevertheless be carried through tocompletion in about 0.5 son amicrocomputer. It employsandoperates iteratively on a systemof equations (at least

three independent ones) with three unknowns.If the inputis a set of three independent ultrasonic quantities, thesolution is exact. Alternatively, if more than three sets

of elastic data are available, the methodleads to a "bestfit" (least squares type) solution. Comparison of theresults obtained from this methodwith those from theapproximate analytical one indicates a difference of less

than 2o/.. A similar approach was used for the de-termination of the texture coefficients from Young'smodulusdata,2s)

The SHOvelocity measurementsare redundant in-

formation with regard to the calculation of the ODFcoefficients. However, it was found that the planaranisotropy of VsH, is moresensitive to the texture thanthat of the SoLambmodevelocity; thus its use improvesthe accuracy of determination of the C~3coefficient. Inthe first step of the present calculation, five ultrasonicvelocity measurementswere employed to determine thethree fourth-order ODFcoefficients through a least

squares fitting. The major problem encountered in thedetermination of texture coefficients from ultasonic mea-surements arises from the inaccuracies associated withthe C11 coefficient, which is calculated from the absolutevalues of the velocities; by contrast, C~2and C~3are de-termined mainly from the variations in the velocities.26)

5.3.2. Estimation ofthe Fourth andHigher Order ODFCoefficients Using the Elastic Energy Method

Theaim of the approachdescribed below is to improve

ISIJ International, Vol.

on the conventional method for determination of thecharacteristics of the texture from ultrasonic measure-ments, which is limited to deriving the fourth order ODFcoefficients. It can be shownthat at least the sixth ordercoefficients are needed for accurate prediction of theplastic anisotropy. The description that follows is asummaryof this method; further details can be found in

Refs. 28) and 29).

Using the effective polycrystal constants calculated bythe elastic energy method, the planar distribution of theultrasonic velocities or of Young's modulus can beexpressed in terms of the three l=4, four l= 6and five

l= 8ODFcoefficients. It is, however, impossible to invertthe calculation and to estimate the above twelve ODFcoefficients directly from such elastic data because theangular variation of Young's modulusor of the ultason-ic velocities depends on three independent elastic pa-rameters. Nevertheless, in addition to the fourth ordercoefficients, only the sixth order coefficient with thehighest absolute value (Cll) plays a significant role in

the direct calculation of the polycrystalline elastic

constants for the deepdrawing textures. Thus, if the C11coefiicient can be estimated from other (non-elastic)information, through iteration of the elastic energy cal-

culation, moreaccurate values of Ci I and of the fourthorder coefficients can be obtained.

For the present purpose, the additional information

comesfrom the decomposition of the texture into its

principal preferred orientations and their representa-tion in terms of gaussian spreads, as explained in de-tail in Ref. 29). In addition to the {111}, {110} and

{100} components, other preferred orientations, such asthe {223} I0> and {554} (or the neighbouring{223}),

found in cold rolled and annealed sheets(the latter often appearing in interstitial free (IF) steels)

are in turn considered in the decomposition. Thesecancontribute significantly to C~l, as well as to the valuesof the higher order coefficients.29)

5.4. Influence of the Sing]e Crystal Elastic Constants

Twoproblems are involved in the calculation of thepolycrystal constants: the limitations of the averagingmodel, and the somewhatarbitrary choice of the single

crystal constants to be used. As mentioned earlier, for agiven set of single crystal constants, the conventional Hill

assumption gives results similar to those obtained froma more rigorous calculation, especially whencommonindustrial textures are being considered. As shownbythe present authors, the single crystal elastic constantshavea strong effect on the predictions,25) as do the CRSSratios on the plastic anisotropy.3) A comprehensivecompilation of the elastic constants of pure iron wascarried out by Ledbetter et a/.,30) whosuggested that,

according to the literature data, c~l, c 12 and c~4 Iie

respectively in the ranges 229~9, 136~9 and I15~3GPa. Someof these values are listed in Table 2. Theseuncertainties are related to the experimental methodsofmeasurementand to the varying chemical compositionsof the crystals tested. Also, in the present study dealingwith low carbon steels, it is expected that these valuesdiffer not only from those of pure iron, but also from

31 (1991),

Table 2.

No. 7

Single crystal stiffnesses c,'j (GPa units) of iron(from Ref. 30, where the detailed references canbe found).

Investigator (year) Cllo C12 c4

(:) Markham (1957) 233 139.2 ll6.2

C Rayneet al. (1961) 233.1 135.4 ll7.8

R Lord et al. (i965) 228 132 116.5

C Rotter et al. (1966) 231.4 134.6 ll6.4

C Leeseet al. (1968) 226 140 ll6

@ Guinanet al. (1968) 230.l 134.6 ll6.7

@ Kimura (1984) 241 146 ll2

C Goens (1931) 237 141 116

225

220

~C) 215

~i!

210

205

225

220

_ 215

~C;

~il 210

205

200

Fig. 5.

o

IF1

@

Q)

HSLA

@

~)

X

30e

X

60

X

error : l

error : I

90

O 80 60 90e

Infiuence of the choice of the single crystal elastic

constants on the edependenceof Young's moduluspredicted by the conventional Hill assumption.Curves Oto ~) correspond to the elastic constantslisted in Table 2, Theexperimental data are indicatedby crosses.

grade to grade (1ike the CRSSratios).

Figure 5illustrates the effect of various sets of c,?j, takenfrom the literature, on the e dependenceof Young'smoduluspredicted from the ODFcoefficients accordingto the conventional assumptions. It can be seen that theshapesof the E(e) curves are not affected by the different

sets. However, the magnitudes of the Young's m6duli arevery sensitive to the ~alues of ci'j, just as the r-values

were shownto dependstrongly on the CRSSratios. Thesametendencies are observed when the elastic energymethod is used. Consequently, since the average E is

70~ C 1991 ISIJ

ISIJ International, Vol.

mainly related to the C11 ODFcoefficient, the calcula-

tion of the latter from experimental data can only lead

to accurate values if suitable c,?j's are employed. Dis-crepancies observed by various authors23,26) jn the de-

termination of this particular coefficient can be cor-rected in this way. After investigation of the most re-

cent data reported by Ledbetter et al.,30) the use of the

reduced ranges: c11=230~5GPa,c12= 136~5GPaand

c44= I16.5 ~2.5 GPafor steels of commercial purity is

suggested here,

On the basis of the Ist and 2nd order Hill ap-proximations, the present calculation was "calibrated"

by varying the single crystal data in the above ranges.Their values were adjusted so that the measuredangulardistribution of Young's modulus agreed well with that

predicted from the X-ray measurements,again using the

two Hill assumptions. Dueto the high sensitivity of the

Young's moduluspredictions to these constants, andalso

to the limited accuracy of the experimentally determined

ODFcoefficients, the sameset of c,?j were assumedfor

the five types of steel studied. By modifying these

constants within their intervals of reliability, it wasfoundthat good overall agreement between the predicted

Young's moduli and the experimental measurementswasobtained with the following set: cll =236GPa,cl2=

31 (1991), No. 7

141 GPaand c~4=118.5GPawhenusing the conven-tional Hill hypothesis. The 2nd order Hil] assumptionrequired a slightly different set of values: c11=233GPa,cl2= 141 GPaand c~4=117.5GPa. The results of the

reverse calculation, presented below, showthat these sets

also lead to good agreement between the experimentaland predicted ultasonic velocities, pointing out the

coherency between these two approaches to elastic

anisotropy.

6. Results of the Acoustoelastic Texture Analysis ofDeepDrawing Steels

6.1. Comparisonbetween the Acoustoelastic and X-rayTexture Analyses

As indicated above, measuredultrasonic velocities orYoung's moduli can be employed to calculate ODFcoefficients with the aid of the elastic energy method.The fourth order coefficients derived in this way arecomparedwith those calculated from Young's modulusandX-ray data in Figs. 6(a) and6(b). Direct comparisonsbetween the coefficients deduced from the Young'smodulus data and from the X-ray measurementswerepresented elsewhere.29)

There is significant scatter between the ultrasonic and

~)

~

o

-l

-2

-3

-4

,,, ,

,,,

,

, ,

,

-4 --23

C411(X-ray)

-1 O-4 -3 -2 -1 OC411(Young's modulus)

O

:;~o ~2

~~ -4

,,,

~-6

o,,

8'

..eo

,,

o.2

.4

.6

.8

,,

,

3

-6

~)

o,

,

,

,

,, ,,

-4 -2

C611(X-['ay)

1,

• e

,,,

1 -.8 -.6 -.4

C412(X-ray)

-.2 O-1 -.8 -.6 -.4 -.2 OC412(Young's modulus)

O-6 -4 O-2

C611(Young's modulus)

oe~2u'

~)

q,1

O

o

,

, ,

,,lce

, ,

,,

e

, ,,

l

.5

O

.5

-1

,

d'

2v'~ 1~O;~

~-l

O _2

-3

o

~)

o,

,

,l

-l-. O .5 1-1 1-.5 O .55

C413(x-ray) C413(Young's modulus)

Comparison between the fourth order ODFco-efficients derived from ultrasonic velocity, Young'smodulusand X-ray diffraction measurements.

1 2C612(X-ray)

30 2 31C612(Young's modulus)

,,,

,

,,

,,

.e .

Fig. 6(a).

e' r ,

,

@1991 ISIJ 702

-3 -2- I O 1 2-3 -2 -1 O 1 2C614(X-ray) C614(Young's modulus)

Fig. 6(b). Comparisonbetween the sixth order ODFcoeffi-

cients derived from ultrasonic velocity, Young'smodulusand X-ray diffraction measurements.

ISIJ International, Vol. 31 (1991), No. 7

X-ray results. Theexperimental errors of ~7m/s in theultrasonic velocities and in the X-ray data contribute tothe observed dispersion. The C~3 coefficients obtainedfrom the ultrasonic measurementsare slightly higher thanthose deducedfrom either the Young's modulusdata orthe pole figures, although the effects of such discrepancies

on the r-value predictions are negligible (see Sec. 6.2.2below). This deviation seerns to be caused by the Hill

approximation, which slightly overestimates the planarelastic anisotropy, as can be seen from Fig. 6(a). Similarresults were obtained with the conventional Hill

assumption (with the appropriate set of single crystalelastic constants) and were reported by the author in

Ref. 25). Nevertheless, the elastic energy method is

preferred, since it leads to estimates of the higher order

ODFcoefficients.

With respect to the sixth order coefficients, shownin

Fig. 6(b), there is considerable scatter whenthe ultrasonic

parameters are compared with the X-ray ones. Bycontrast, the dispersion is much reduced when theYoung's modulus and ultrasonically based coefficients

are contrasted, as expected. This latter observation points

out the connection between these two types of elastic

anisotropy, andespecially justifies the assumptions madeconcerning the correction of the SoLambwavevelocities

for dispersion (see Sec. 5.2). For the C~3coefficient, thecalculated ultrasonic and Young's modulus values didnot agree at all well with the X-ray based results. Thesedata are therefore not presented here. The lack ofagreement probably arises because the {100} and

{100} 1>components, which involve different valuesof C~3, are represented here by only one of these twofiber components.However,since the magnitudes of C~3in steel sheets are rather small (-O.5 ~0.2), these wereassumedto be -0.5 in the present work, which did notlead to significant changes in the r-value predictions.

In the averaging of the local plastic work rate Mq(g)from which the r-values are predicted, the l-th ordercoefficients are weighted by a l/(21+ l) term.3'12) Thus,the low order coefficients on their own can fully

characterize the infiuence of texture on the macroscopicproperties of the polycrystal. Regarding the sixth order

ones, the X-ray data for the deep drawing steels studiedshowthat mainly C~l (ranging from - 6to - 2), followedin importance by C~2 (from I to 2), and then by C~4(- I only in the IF2 steels) are of interest.

The eighth order coefficients determined from theX-ray measurementsare normally all smaller (their

absolute values are generally less than l) than those ofthe fourth and sixth orders for the textures studied. Thus,the effect of the eighth order coefficients on r-value is

not particularly significant. As reported in Ref. 29), thesimplified texture decomposition permits reasonableestimates to be madeof the eighth order and of the Cl~,Cl~, and C~~coefficients. Although Cl~ is nearly zero,the others are of importance for deep drawing steel

textures since their values are related to the sharpness ofthe ND-fibre (they depend solely on the Miller indices{hkl} parallel to the rolling plane4)). Cl~ and C~~reachvalues of 3to 4 in the sharp textures present in the IF2steels.

6.2. Applications of the Partial Texture Analysis

6.2.1, Ultrasonic Pole FiguresVarious authors have demonstrated the practical

interest of the ultrasonic pole figures calculated from thefourth order ODFcoefficients.24,26) Although they donot agree at all well with the X-ray ones, they can beof use for the control of sheet processing by indicatingthe presence or absence of certain componentsof thetexture. This is because ultrasonic pole figures aresensitive to the evolution of texture during rolling andannealing, Nowthat additional texture information canbe determined by the newmethodpresented above, it is

of interest to analyze how this can improve suchultrasonic pole figures (and inverse pole figures) com-pared to the X-ray ones.

(200) pole figures are presented in Fig. 7for an AKDQsteel. The results derived from the ODFcoefficients upto l=6 are comparedwith those obtained solely fromthe fourth order coefficients and with those calculatedfrom the X-ray basedcoefficients up to I=22. Thefourthorder coefficients alone lead to a muchweaker {I I l}

fiber and to a nearly planar isotropic texture. Bycontrast,employmentof the elastically derived ODFcoefficients

up to l= 6determined in this study leads to considerablycloser agreementwith the X-ray based results.

The above results suggest that acoustoelastic tech-

niques are useful in the non-destructive characteriza-tion of sheet metal textures. With the increasing use ofcontinuous annealing lines in steel processing, on=1inecontrol of the recrystallization process by such fast andrelatively inexpensive methodsis of considerable interest.

6.2.2. The NonDestructive Prediction of Plastic An-isotropy

Planar r-value distributions were calculated with the

levels : .5, 1, 1.5,... levels : .5, 1, 1.5,

...

levels : .5, l, 1.5,...

Fig. 7. Comparisonof the (200) pole figures for an AKDQsteel calculated using only the fourth order coefficients,

the coefficients up to !=6 derived from the ultrasonic

measurements, and the coefficients up to !=22 ob-tained from X-ray analysis.

703 C 1991 ISIJ

ISIJ International, Vol. 31 (1991), No. 7

~t

3

2

l

3

o 30e

60 90

~~

2

lO 9030

e60

Fig. 8. r(e) dependencescalculated from the RC3model (andthe appropriate CRSSratio).

The predictions using the ODFcoefficients of fourth

(-) andsixth (-) order determined from the ultrasonic

measurementsare compared with those measuredexperimentally (x ).

aid of the ODFcoefficients up to l= 12 obtained in this

study. Thecalculations showedthat the estimated eighth

and twelfth order coefficients did not improve thepredictions. Instead, it is the sixth order coefficients, C11

and C~2, which are the most important for the ODF'sstudied.

The r(e)'s deduced from the ODFcoefficients up to

/= 6with the aid of the RC3model (associated with the

appropriate CRSSratio) are compared with thosepredicted by the samemodel but employing only the

fourth order coefficients in Fig. 8. Thesecalculations arecontrasted with the experimental results for one of eachof the five different types of steel. Whenthe r-values arerelatively low, as in the case of the rimmedand HSLAsteels, the predictions using the ODFcoefficients up to

l=4 and 6showgood agreementwith the experimentaldata. By contrast, whenthe r-values are higher, i.e, for

sharper textures, as in the AKDQIFI and IF2 steels,

the values obtained from the fourth order coefficients lie

below the experimental oncs, especially in the 90'

direction. Thepredictions basedon the ODFcoefficients

up to l= 6, on the other hand, Iead to very good quan-titative agreement with the experimental values. Thusit is clear that the accuracy of r-value predictions is

improved considerably whenthe sixth order coefficients

are taken into consideration.

The agreement between the ultrasonic predictions

using the present method(with ODFcoefficients up to

l=6) and the experimental measurementsis as good asthat observed for the calculations from the completeX-ray data. The reason for the good agreementcan befurther appreciated from Fig. 9, iri which a yield surface

section computed from the complete X-ray data is

comparedwith sections derived from the fourth andsixth

@1991 ISIJ 704

l=22 l=6 l=4022

Fig. 9. Yield surface sections calculated for one of the IFlsteels by the RC3model (c(s=c(H=1) plotted with

respect to the rolling reference frame.

The X-ray based prediction (!=22) is comparedwiththe ultrasonic onesbasedon the fourth and sixth order

ODFcoefficients.

all

order ODFcoefficients obtained ultrasonically (the

calculation method is decribed in Ref. 3)). As for ther(e) prediction, the use of the ODFcoefficients up to I=6leads to results almost identical to those obtained fromthe complete ODFinformation. This indicates that

forming operations can be readily and accurately

simulated from the acoustoelastic characterization ofsteel sheet.

7. Conclusions

The modelling of the elastic and plastic anisotropyof deepdrawing steels wascalibrated after testing variousgrain interaction modelsandby adjusting the CRSSratio.

Theelastic anisotropy wasbest predicted by the so-called

'elastic energy' methodassociated with the appropriate

set of single crystal elastic constants (cI I =233, c12 = 141

and c~4= I17.5 GPa). The calculation of plastic anisot-

ropy from texture data shows that a CRSSratio of I .O

is suitable for IF2 steels, while values of 0.95 and 0.9 arepreferred for, respectively, the IFI and AKDQgrades;

this tendency is in agreementwith observations reportedin the literature. The good predictions obtained by the

RC3relaxed constraint model for deepdrawing textures

were justified by 'least work' considerations,

The above results point out the high accuracy andreliability of the texture information obtained fromacoustoelastic measurements when the sixth ordercoefficients are employed. Comparedto conventionalanalyses, which generally lead to the empirical prediction

of f and Ar, the method developed in this study for

characterizing the behaviour of deepdrawing steels leads

to aconsiderably morecomplete description of the plastic

anisotropy, which is highly accurate when a suitable

crystal plasticity model is used. TheRC3modeland the

appropriate CRSSratios should be employed for this

purpose. Being potentially usable for on-1ine measure-ments, this analysis represents a significant improvementover the empirical Modul-r technique for characterizing

the formability of steel sheets.

ISIJ International, Vol. 31 (1991), No. 7

Acknowledgments

The authors are indebted to Dr. J. F. Bussi~re ofNRCC-IMRI,Boucherville, PQfor providing ultrasonicdata and to Prof. J. Szpunar of McGill University forassisting with the texture measurements. They aregrateful to the KawasakiSteel Corporation, Stelco SteelInc., Dofasco Inc. and the AlgomaSteel Corporationfor supplying steel sheets. They acknowledge withgratitude the financial support received from theCanadian Steel Industry Research Association, theNatural Sciences and Engineering Research Council ofCanada, and the Ministry of Education of Quebec(FCARprogram).

l)

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4)

5)

6)

7)

8)

9)

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13)

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705 C 1991 ISIJ