Simulation of earing in textured materials

Materials Science and Engineering, A 131 ( 1991 ) 161 - 170

161

Simulation of Earing in Textured Materials D. W. LIN*, D. DANIEL and J. J. JONAS

Department of Metallurgical Engineering, McGill University, Montreal, PQ H3A 2A 7 (Canada) (Received April 27, 1990; in revised form June 22, 1990)

Abstract

The cup drawing of anisotropic materials was analyzed to determine the relation between earing behavior and mechanical anisotropy. Two factors referred to here as the radial-to-circurhferential strain ratio and the normalized circumferential stress were shown to play important roles in ear formation. Ear profiles were estimated from texture data by a tension-compression model with the aid of the continuum mechanics of textured polycrystals method. The treatment was extended to ear evolution during drawing by means of the elemental upper bound technique. The predicted cup profiles showed good agreement with experimental results. I. Introduction

The earing phenomenon, which occurs during deep drawing owing to inhomogeneous deformation, can either be analyzed by continuum approaches or in terms of crystal plasticity. In the former, the earing behavior is related directly to the planar anisotropy of the mechanical properties. For example, a correlation was noted in early experimental work [1, 2] between ear position and the local maximum in R, the ratio of width-to-thickness strain in tensile testing. Continuum descriptions of ear formation have also been published [3-6]. When using physical methods, however, the crystallographic texture must first be known, since this is ultimately responsible for producing earing. This relationship can be confirmed empirically for both single crystals [7] and polycrystals [8-11]. Models have also been proposed based on the mechanical anisotropies which can be deduced from the *Present address: Department of Metallurgical Engineering, East China Institute of Metallurgy, Ma'anshan, Anhui (China). 0921-5093/91/$3.50

orientation distribution functions (ODFs) of textured materials [12-14]. Alternatively, the occurrence of earing can be linked to the presence of certain texture components by employing the continuum mechanics of textured polycrystals (CMTP) method, which provides a connection between the preferred orientations present in polycrystals and the plastic anisotropy of the material [15, 16]. In the present work, the relationship between crystallographic texture, plastic anisotropy and earing was investigated with the aid of this technique. The deformation in the flange of a blank was analyzed in this way and two explicit factors were shown to influence ear formation. These two factors can be evaluated using tensile yield stresses. Simulations were carried out, by which ear evolution during drawing is described. The calculated cup heights were compared with experimental results for a commercial IF steel, as well as for single crystals and polycrystals taken from the literature. 2. A method for estimating the ear profile

During the deep drawing of a circular blank, the radial stresses are tensile and the circumferential stresses are compressive at all points in the flange, as illustrated in Fig. 1. In anisotropic materials, after an increment of drawing, individual elements in the flange are elongated radially and compressed circumferentially to different degrees. Ears are produced by the accumulated effects of the different radial strains during drawing. In this section, a simple model is presented to describe the variation in the radial strain at different locations. For this purpose, the shear strains between adjoining segments are neglected. Then the final cup height is predicted, on the assumption that the total elongation of the flange at different locations can be directly © Elsevier Sequoia/Printed in The Netherland,

162

initial orthorhombic textures than from the evolving textures, which are largely monoclinic and much less well known. Furthermore, predictions based on the initial texture are in reasonable agreement with observations, at least for foureared cups [7, 10-14]. This is probably because the shape of the yield surface does not change as rapidly as the texture does during straining. The radial strain taking place in a given element during an increment of drawing can be expressed as the product of two factors:

(a)

(b)

m (e)

Fig. 1. (a) Cross-section of a deep drawing operation. (b) During drawing of the flange into the die, the flange is subjected to tension along the radial direction and compression along the circumferential direction. (c) Owing to the lack of cylindrical symmetry in the texture of the blank (which is orthotropic), ears are formed along the rim of the cup.

d~:ri

d~:rl dtr2

dtrm-I d~:rm

d~ei~ dtotdto.i 1,

where Q is the ratio of the radial to the circumferential strain increment, deJdeo, and the circumferential strain increment de0 will be shown below to be linked to the normalized circumferential stress. These two factors will be shown to influence the extent of ear formation. In order to apply the Taylor model to the present problem, the Cauchy (i.e. true) stress and strain increment tensors are defined as follows:

a = dtri (a)

derldtr2

dtrm-I derm

related to the radial strains of the individual elements in the model. 2.1. Deformation model The present model is based on the series of elements shown in Fig. 2. Each element is subjected to a compressive strain de0 along the simulated circumferential direction and a tensile strain d g r along the simulated radial direction. For simplicity, the two corresponding stresses, or and Cro, are taken to be numericall3~ equal, although the real ratio of the tensile-to-compressive stress varies along the radius from zero at the rim of the flange to about two near the edge of the die [7]. For the present purpose, it is also assumed that the progressive changes in mechanical anisotropy which accompany the changes in texture during drawing [17] can be neglected. This can be justified from a practical point of view in that it is more useful to be able to predict earing from the

O0r O0

0

0

(2)

[ der/de0

(b)

Fig. 2. A tension-compression model of flange deformation during cupping: (a) an element of the model (ori = -aoi); (b) groups of elements combined.

(1)

der=Qdeo

de=de0[

0

1

0 det/deo

where the subscripts r, 0 and t refer to the radial, circumferential and thickness directions. The strain increment ratio Q and the stress components normalized by the critical resolved shear stress r c can be found by applying the' normality rule to the CMTP yield locus at the point where ar= - Oo. This model is similar to that proposed by van Houtte [14], as the ratio of radial to circumferential strain during cup drawing is taken as constant for a given radial direction in both models. By contrast, instead of assuming the circumferential strain increment to be constant around the flange, the circumferential strain increment is taken to vary inversely with the circumferential stress in the present model, as given by deo

Ool

deol

(70

(3)

where deol and oro1 are the circumferential strain increment and stress in the element which corre-

163

sponds to the rolling direction. The above relation is based on the observation that the local deformations are usually larger in regions where the material yield stresses are lower. Now the two factors can be combined into the parameter P=

de, deo deo deo~

ao. ao

O--

(4)

It should be noted that work hardening must be taken into account when the stress ratio in eqn. (4) is being evaluated. This can be done on the basis of power law hardening

(5) where ou and eu are the flow stress and strain measured in a uniaxial tensile test, and k and n are constants. If work hardening is solely a function of the amount of plastic work per unit volume, the equivalent strain increment can be defined by 1 dgcq = - - ( o r der + Ou

_46 (St der+So de0)

(6)

(7)

where M is the Taylor factor for the uniaxial tensile test from which the flow stress in eqn. (5) is obtained, and S r and S o are the normalized deviator stresses. Eqn. (3) is now rewritten to give de0

_ rclS01 -

r~So

OeqlSol

(8)

aeqSo

Sol 1 - Qjl "/l'+lj

d601-So

1~-

As will be profiles can be together with circumferential circumferential

shown in the next section, ear readily estimated using eqn. (10) the values of the radial-tostrain ratio and the normalized stress.

2.2. Final cup height In order to predict the final cup height from the present analysis, the radial strain increment is first expressed as der = -Pdeol

(11)

The final cup height h is now composed of two parts

]/

(12)

where h 0 is the cup height which would be obtained if the radial strain were zero, and h r is the elongation of the flange. According to ref. 7, the former is given by h 0 =I{4(R b - Rp) + (4 - :r)(2 rp + t)}

(13)

where R b and Rp are the radii of the blank and the punch, rp is the radius of the punch profile, and t is the thickness of the blank. The flange elongation can be determined by introducing a constant Cr=

hr

(14)

dgr

where dgr is the average radial strain increment, and /i r is the average elongation of the flange. These are given in turn by

°'

yg dgr = 2 m d g r i

/ ~

(15)

i=1

where r~ and r~ are the critical resolved shear stresses and a~q and a~q, are the equivalent stresses in the element of interest and in the element corresponding to the rolling direction respectively. Using eqn. (8) together with eqns. (5)-(7), the normalized circumferential strain increment can be expressed as

deo

Sol 1 - Q j ] '/~" +l>

P=-O_ so T2 I

h = h 0 +h r

ao de0)

For the present stress state, the deviator stress is identical to the corresponding normal stress, so that, after normalizing by , ~ re, eqn. (6) becomes

deol

The expression for P is now obtained by substituting eqn. (9) into eqn. (4):

(9)

where m is the number of elements, and 2

hr = Rb" - R p +t-ho+O.43rp 2Rp

(16)

This formula is based on the assumption that the surface area of the cup is approximately the same as that of the blank [18]. On the assumption that the final elongation of the flange at different locations is proportional to

164

the local radial strain der, eqn. (14) can be developed into

der

1

I

I

I

\X

(I

(17) 0

So the final cup height can be calculated from h =h0 q-cr der

I

h (in.)

hr

cr = -

I

1.5

0

I

i

i

1

J

60

90

120

150

180

,

i

,

,

i

I

I

I

I

1

30

60

90

120

150

i

i

e

(18)

2.3. Comparison between the present predictions and experimental results The cup heights predicted by the present method for some common ideal orientations are exhibited in Fig. 3. A single-exponent Hill-type CMTP yield function was used in conjunction with the Taylor model for all the calculations. The values of the work hardening exponent n and the geometry of the cupping dies for the ideal orientations were chosen from ref. 7. The predicted cup heights for the {001}(100), {011}(100) and {012}(100) orientations are in good agreement with experimental results for the corresponding single crystals (see Fig. 3). By contrast, the predictions for the {111}(110), {112}(110) and {122}(011) orientations all show divergence. The predicted ear positions for the {111}(110) polycrystalline ideal orientation are in fact opposite to that found by experiment for the single crystal. This is essentially because the single crystal behavior investigated by Tucker and displayed in Fig. 3 can only be simulated by a Sachs model, whereas the Taylor model was employed in the present polycrystal calculations. Nevertheless, as indicated in ref. 19, the plastic anisotropies calculated for ideal orientations such as {100}(001), {011}(100) and {012}(100) are less affected by the choice of grain interaction model. Accordingly, the present ear profile predictions for these ideal orientations are consistent with the observations from the corresponding single crystals. It will be seen that if the Sachs-Kochend/Srfer interaction model is used in conjunction with the method described below for the {111}(110), {112}(110) and {122}(011) orientations, whose calculated plastic anisotropies are more sensitive to the interaction model, better agreement with the experimental results is obtained. The simple model is also used to predict the ear profiles for some polycrystalline copper alloys. The average values of n are chosen from ref. 20. Reasonable agreement for the polycrystalline materials is indicated in Fig. 4.

i 30

1.5

h (in.) .5 {011}< 100 > 0

0

180

I!

1.5

i

~

~

!

I

I

I

I

30

60

90

120

150

I

I

I

I

I

30

60

90

120

150

180

,

~

i

,

I

1

h (in.) .5 {111}<110> 0

180

6 1.5

h (in.) .5 0 0

0

1.5 I

h (in.) .5

{I 0

0

!

I

I

I

90

120

150

180

30

60

~

~

i

I

I

I

l

l

30

60

90

120

1S0

8 1.5

~

h (in.) .S {122}<011 > 0 / 0

180

0

Fig. 3. C o m p a r i s o n of calculated c u p h e i g h t s ( ) for s o m e c o m m o n ideal o r i e n t a t i o n s with e x p e r i m e n t a l results for single crystals (. . . . ) t a k e n f r o m ref. 7.

165

I6

[

.4

h(%) ho

-

I

"6

"~'~

.4

J

[

I

~

I

Q

A0"

h(%) .2

.2

1.5 A30

0 0

l

I

I

30

60

90

ho 0

Q0 0

30

60

e .6

~

0

e i

[

0

I

30

60

I

110 0 I

90

0

I

L

60

.,o/j h(%) . i

-2

~

'

'

.6

90

.6

.4 h(%)

h(%) .2

.4 - __

•

.2

h0 0

ho 0 30

60

i

60

90

0

I

I

30

60 e

BO -1

-.5

0

90

i .

Fig. 5. Partial yield loci of two elements in a polycrystalline blank containing the {111}(110) ideal orientation: point A o, Or= Cru°, o0=0; point B 0, o r = 0 , o0= -ou°+~°; point C 0,

Or = -- O0. _

h(%) 2

CuS%Zn-F

0

30 e

~

,

o0/v'g~c 0

J

j~

-1.5

11o o

0

~

.4 ~ ' ~

.6

_

I

30

Or/V~c

.5

.6 i i ~ , , . ~

0 .6

1

'

90

,,,,:. 17::,,z3ho

I

I

Cu20%Zn-F

ho 0 90

0

i

i

30

60

90

e

Fig. 4. Comparison of calculated cup heights ( ) for some polycrystalline copper alloys with the experimental results (. . . . ) taken from ref. 20.

2.4. Estimation of P values using crystal plasticity and tensile yield stresses The earing behavior of a sheet is determined by its set of yield loci in the vicinity of pure shear. In order to predict this behavior, the corresponding radial-to-circumferential strain ratio and the normalized circumferential yield stress must be evaluated for every element in Fig. 2. For this purpose, each yield surface is referred to the current radial, circumferential and through-thickness directions. Two partial yield loci pertaining to the second quadrant are shown in Fig. 5. These correspond to the elements aligned with the 0 ° and 30 ° directions and to the {111}(110) ideal orientation. The points A 0 and B0 represent the yield stresses o~° and Ou°+9° developed when the elements aligned with the 0 and 0 + 90 ° directions are uniaxially stretched in the absence of Bauschinger effects. The two Co points represent the approximate stress states applicable to the flange during drawing. Here the two values of Q are obtained from the normals to the yield loci at Co. As the angle between the rolling direction and the direction of interest is increased from 0 °, both the shape and the position of the yield locus deviate from those corresponding to the 0 ° direction.

For example, it can be seen from Fig. 5 that the partial yield locus for 30 ° is rotated clockwise from that pertaining to the rolling direction. Both the Q and a 0 values are affected by the changes in the positions of points A 0 and B 0. These two values can be estimated as follows. If the yield locus in the tension-compression region is nearly linear, the Q value is approximately given by (7u

Q=

0 + 9(I

Ou

0

(19)

On the same assumption, the following relations are applicable to the right triangle AoOB0 in Fig. 5:

(AoCo+ B0C0) 2 = ( A o O F &Co-

BoCo -

+(B00) 2

(20)

(21)

sin q~

(22)

COS

where ~ = tan-l( - Q). Using eqns. (20)-(22), the local circumferential yield stress can be evaluated from: _

°°=

/ i!ou o l_., + ( % o+.o,21,,=

[(sec ~

~

]

(23)

The expected 0 dependencies of the uniaxial yield stress in sheets of {111}(110), {112}(110) and {122}(011) ideal orientations were also calculated by the series expansion method in conjunction with the Sachs-Kochend6rfer grain

166

interaction model described in ref. 19. The calculated yield stresses were used to estimate Q and o 0 using eqns. (19) and (23). The P values obtained from eqn. (10) in this way are plotted against the angle 0 in Fig. 6. All three predicted ear profiles are in better agreement with the experimental results than the Taylor predictions of Fig. 3. (It should be noted that, from eqns. ( 11 ) and (14), the flange elongations are proportional to P.) Thus the use of the radial-to-circumferential strain ratio and the normalized circumferential stress seems to be useful in the context of a tension-compression model. 3. A model for simulating ear evolution during drawing During an actual drawing process, ears are formed progressively in the flange, which is subjected to a complex stress state. This kind of inhomogeneous deformation can be modeled by the finite element method, which leads to the 1.5

I

i

I

i

i

1.5

current flange contour at each stage of punch travel. However, such calculations are difficult to perform on materials of complex anisotropy and are time consuming and expensive. One alternative is to calculate the deformation within the flange with the aid of the CMTP method employed in conjunction with the elemental upper bound technique.

3.1. Assumptions The assumptions associated with the simulation model, some of which were used in the previous section, are listed below. ( 1 ) The material is rigid plastic. (2) The effects of bending, straightening and ironing on ear formation can be ignored. (3) The blank-holder pressure is considered to be negligible. (4) The ideal orientations and their volume fractions remain unchanged during drawing. (5) Radial fines on the flange remain radial during deformation. (6) Work hardening is only a function of the amount of plastic work per unit volume and is independent of strain path. (7) The work dissipated at velocity discontinuities can be neglected.

3.2. Calculation procedure .5 I 0

I

I

I

I

I

30

60

90

120

150

I

I

I

i

I0 180

e

1.5

P

1 l-

I

~ /

1.5 I

h (in.) .S

.S

0

0

30

60

,

i

90

120

1SO 180

o

1.5

P

.sl ~ ~

~"

,

~-~"

,

~-~"

1.5

,

"~""

1

0'5

Owing to the orthorhombic symmetry of the initial texture, only a quarter of the flange need be analyzed. Such a quarter-flange is first divided into sectoral elements, as illustrated in Fig. 7. After an increment of drawing, the three elements shown are deformed to different degrees because the radial-to-circumferential strain ratio and the normalized circumferential yield stress depend on position within the flange. The deformed elements are characterized by azimuthal angles tXl, t~2 and ct3 and radial lengths rfl , rf2 and rf3 respectively. These two groups of unknown geometric variables can be evaluated by the following two-step calculation.

h(in.)

". "~.

0

30

60

90

120

150

QOI

180

0

Fig. 6. Comparison of P values ( ) predicted by the series expansion method using the Sachs-Kochend6rfer grain interaction model of ref. 19 with the observed cup heights (. . . . ) taken from ref. 7.

rf03

a3

a03 (a)

(b)

Fig. 7. A quarter-flange consisting of three sectoral elements: (a) before drawing; (b) after a drawing step.

167

First, the flange radius re of each element is determined by minimizing the work dissipated during the plastic deformation of the element by means of the upper bound principle. The strain state along the bisector of the element is adopted as representative during numerical integration along the radial direction to compute the work, which can be expressed as W = kao J [

de~q(e0 + de~q)ntr dr

(24)

where rf0 and a 0 are the initial flange radius and subtended angle of the element of interest prior to the drawing step. All the lengths in eqn. (24) and the equations that follow are normalized by the die radius R d. In order to account for different amounts of work hardening in different parts of the flange, every element is divided into eight sub-elements along its radius. Bearing in mind that the subtended angle of a specific element is fixed during this stage of the calculation, and that the stress components can be found by employing the stress and strain increment tensors of eqn. (2), the key point is to assume an appropriate radially admissible displacement field. In isotropic materials, the radial strain increment increases progressively from the flange rim to the die edge during drawing. If the material is anisotropic, the radial strain increment distribution within a given radial segment is still expected to have a similar shape. Thus the following function is suggested for the displacement field du

= cl(rt.

r) +c 2 In r +c 3

0 -

(25)

where du is the radial displacement during a drawing step. The radial strain increment distribution is obtained by deriving eqn. (25) C,

d 6 = - c, + -= r

(26)

which is the simplest function which can describe the desired relationship between radial strain increment and radius. The constants Cl, c 2 and c 3 are determined by three boundary conditions. Two of these are displacement conditions at the inner and outer points of the element (or near the edge of the die and the rim of the flange), as given by dulr= 1= - A r

(27)

dul,. ,.., =rf.-r,-o

(28)

where A r is the amount of draw corresponding to one step, and is equal to the displacement of the inner point. The third boundary condition is the strain condition at the outermost point derl ....... = d t r f

(29)

where dgrf is the radial strain increment at the edge of the flange. The third condition above is adopted here to enable the strain increment field to reproduce the true field as closely as possible. The circumferential strain increment at the flange edge can be determined from the assumed flange radial length and subtended angle after the drawing step as follows: rfCt

deof = In - -

(30)

rf 0 o~o

The pure compressive deformation at the flange edge can be characterized by the following stress and strain increment tensors

O" =

O'or f

(701"

0

0

{

dea/de01 0

de=de01 / \

0 0

0

1 0 0 de,f/d,%f

(31)

The radial-to-circumferential strain ratio, d£rf / dt0f, is now determined by the CMTP method as was done for the calculation of R values [16]. In this way, the radial strain increment derf can be found from d~'rf

dtrf = deof de01-

(32)

From eqns. (27)-(32), the three constants are given by cl -

re - rfo + A r -

rfo In rm d e .

rm In rf0 - rf0 + 1

(33)

c2 = rfo derf + rfocj

(34)

c3= -Ar-cl(r

(35)

m-l)

At the beginning of each calculation, the flange radius rf after the previous drawing step provides an initial value. Then the strain state at any point along the bisector of the element is determined

168

from eqns, (27)-(35). When the work expressed by eqn. (24) is a minimum and the change in the value of rf is defined, the first step of the calculation for a single element is completed. The same procedure is followed in every element in turn until the complete set of flange radii is available. In the second step of the calculation, the values of flange radii obtained from the first step are fixed while the values of the subtended angles are varied until the plastic work dissipated in the whole flange is minimized. After the set of optimized subtended angles is obtained, the calculation returns to the first step and the procedure is repeated until the required accuracy is attained. 3.3. Work correction to compensate for ear spread It should be noted that as soon as ears begin to form at the circumference of the flange, there is a reduction in the local lateral constraint, as discussed in ref. 7. As a result of this phenomenon, the true circumferential and radial strains at the edge of an ear are less than those calculated by the simulation model described above. In other words, the plastic work is overestimated, especially during the later stages of drawing when the ears constitute a substantial part of the flange. In order to take this effect into account, a correction coefficient cw was employed to reduce the calculated work dissipated in the ear portion. For this purpose, each element of the flange was divided into two parts, as shown in Fig. 8. The radius of the principal part of each element is the minimum flange radius. The volume of this part is defined as Vpi.The remainder or ear portion is referred to as V~i.The work associated with the deformation of an element is given by

Wti = Wpi "~-Cw Wei

(36)

where VCpiand W¢~are the calculated plastic work in the principal part of the flange and in the ear

Vel

Ve3 =0

"-..

Fig. 8. Division of the flange elements into a principal part Vpi and an ear part V~i.

portion. The value of Cw, as explained above, is less than unity. In the calculation, the work W'i is determined as follows. First, the work dissipated in each of the two parts is represented by

Wpi=l~iVpi

(37)

and

(38) where ~ is the average plastic work of per unit volume and is given by w, ~i - - v~, + v~,

(39)

and IVi is the uncorrected work in the element. Using eqns. (36)-(39), the corrected plastic work is determined from

(40) Now the correction coefficient cw is arbitrarily set equal to 0.5 and the calculation is stopped when an element passes over the inside edge of the die. 3.4. Predicted ear evolution and cup profile In the present calculation, the flange was divided into ten sectoral elements. One step of drawing was taken to be a strain of 0.06 in die radius units. The flange shapes and cup profiles predicted for a sample composed of the {011 }(100) ideal orientation are presented in Fig. 9 and those pertaining to a C u - 5 % Z n - P polycrystal in Fig. 10. Such predicted profiles are in good agreement with experimental results. The program was also employed to calculate the final cup heights expected in a sheet of interstitial-free extra-low-carbon steel. Six ears are frequently formed when cups are produced from this material. The main texture components and their volume fractions were determined from the O D F for this type of steel given in ref. 21 (see Table 1). The strain hardening exponent n was taken to be 0.25. The drawing conditions are listed in Table 2. The calculated and observed cup heights are shown in Fig. 11. In this case, the predicted mean cup height is less than expected. This difference can be explained in terms of the effect of wall ironing during drawing, which is not taken into account in the present analysis. According to eqn. 7.2 in ref. 22, the gap between the die and the punch should be no less than 1.06

169 TABLE 1 Volume fractions of the main texture components observed in an interstitial-free extra-low-carbon steel, after ref. 21

i

0 (a)

I

Texture component

Volume fraction (%)

{111}(110) {111}(112) {554}(225) {223}{472) {332}{110)

18.2 16.5 17.1 32.8 5.5

2

rf (Transverse direction)

1.5

I

TABLE 2 Details of the cupping geometry employed in the tests of ref. 21

i

Blank diameter Punch diameter Die diameter Punch profile radius Sheet thickness Drawing ratio

h (in.) .5

0 0

I

I

30

60

(b)

90

80

~

i 15 mm 50 mm 51.7 mm 7 mm 0.7 mm 2.3

~

~

~

0

Fig. 9. Predicted (a) ear evolution and (b) final cup profile ( ) in a sample containing the {011}(100) ideal orientation. The measurements in (b) (. . . . ) refer to the corresponding single crystal taken from ref. 7.

60

h (ram) 40

20 0

I

I

I

[

I

30

60

90

120

150

180

0

Fig. 11. Comparison of cup profiles in interstitial-free steel: experimental data (. . . . ); simple model predictions (. . . . . ); and simulation predictions ( ).

0

1 (a)

2

rf ( T r a n s v e r s e

direction)

.6

.4

h (%)

.2

ho

0 0

(b)

i

I

30

60

if n o ironing is to occur, w h e r e a s the true clearance was 0.85. It is evident f r o m the o t h e r e x p e r i m e n t a l w o r k q u o t e d h e r e [10, 11] that the a v e r a g e cup height can be satisfactorily p r e d i c t e d on the a s s u m p t i o n that t h e r e is no thickness c h a n g e during drawing. T h e c u p p i n g g e o m e t r y used in the a b o v e cases involved a larger clearance, so a little ironing o c c u r r e d in these experiments. It is of interest that the simple m o d e l d e s c r i b e d in Section 2 also leads to satisfactory predictions but takes m u c h less c o m p u t i n g time than that r e q u i r e d by the simulation m o d e l of Section 3.

90

0

Fig. 10. Predicted (a) ear evolution and (b) final cup profile ( ) for a Cu-5%Zn-P polycrystal. The measurements in (b) (. . . . ) are taken from ref. 20.

4.

Conclusions

Earing b e h a v i o r during d e e p drawing can be d e s c r i b e d by a simple t e n s i o n - c o m p r e s s i o n

170

deformation model. Two factors referred to here as the radial-to-circumferential strain ratio and the normalized circumferential stess are found to play important roles in ear formation. These factors can be estimated from the tensile yield stresses along different orientations. The cup heights can be predicted satisfactorily from texture data either by the simple model or by the simulation model. The latter is also capable of tracing ear evolution during drawing. Although no changes in the texture were taken into account in the present calculation, the simulation model can be used in conjunction with information regarding the evolution of the texture. The present analysis demonstrates that the CMTP method is useful for establishing the relationship between the crystallographic texture and the deformation behavior in metal forming.

Acknowledgments The authors are grateful to Drs. K. Neale and M. Darrieulat for helpful discussions and to Dr. K. Sakata for providing the cup test specimens and associated information. They also acknowledge with gratitude the financial support received from the Canadian Steel Industry Research Association and the Natural Sciences and Engineering Research Council of Canada.

References 1 W.M. Baldwin, T. S. Howald and A. W. Ross, Trans. Am. Inst. Min. Metall. Pet. Eng., 166 (1946) 6. 2 L. Bourne and R. Hill, Philos. Mag., 41 (1950) 671. 3 C. H. Lee and S. Kobayashi, Trans. ASME, Ser. B, 95 (1973) 865.. 4 R. Sowerby and W. Johnson, J. Strain Anal., 9 (1974) 102. 5 M. Gotoh and F. Ishise, Int. J. Mech. Sci., 20 (1978) 423. 6 R.W. Logan, Ph.D. Thesis, University of Michigan, 1985. 7 G . E . G . Tucker, Acta Metall., 9(1961) 275. 8 K.T. Aust and F. R. Morral, J. Metals, 5 (1953) 431. 9 W.T. Roberts, Texture Control Sheet Met. Ind., 43 (1966) 237. 10 N. Kanetake, Y. Tozawa and T. Otani, Int. J. Mech. Sci., 25 (1983) 337. 11 N. Kanetake, Y. Tozawa and T. Otani, Int. J. Mech. Sci., 27(1985) 249. 12 C. S. Da Costa Viana, G. J. Davies and J. S. Kallend, Proc. 1COTOM 5, Vol. 2, Springer, Berlin, 1978, p. 447. 13 P. M. B. Rodrigues and P. S. Bate, Texture in Non-Ferrous Metals andAlloys, AIME, Warrendale, PA, 1985, p. 173. 14 P. van Houtte, Mater. Sci. Eng., 95(1987) 115. 15 F. Montheillet, P. Gilormini and J. J. Jonas, Acta Metall., 33 (1985) 705. 16 Ph. Lequeu and J. J. Jonas, Metall. Trans. A, 19 (1988) 105. 17 W. Dabrowski, J. Karp and H. J. Bunge, Arch. Eisenhiittenwes., 9 (1982) 361. 18 D. F. Eary and E. A. Reed, Techniques of Pressworking Sheet Metal, Prentice Hall, Englewood Cliffs, N J, 1974. 19 D. Daniel and J. J. Jonas, Metall. Trans. A, 21 (1990) 331. 20 J. Hirsch, R. Musick and K. Lucke, Proc. ICOTOM 5, Vol. 2, Springer, Berlin, 1978, p. 437. 21 D. Daniel, Ph.D. Thesis, McGili University, 1990. 22 P. W. Lee and H. A. Kuhn, in G. E. Dieter (ed.), Workability Testing Techniques, American Society for Metals, Metals Park, OH, 1984.