METALS AND MATERIALS, VoL 4, No. 3 (1998), pp. 489-497

Stability of Initial Texture Components during Deep Drawing of FCC Polycrystals

Shi-Hoon Choi, Kyu Hwan Oh, Kwansoo Chung* and Frederic Barlat**

Division of Materials Science and Engineering, Seoul National University San 56-1 Shinrim-dong, Kwanak-ku, Seoul 151-742, Korea

*Department of Fiber and Polymer Science, Seoul National University 56-1 Shinrim-dong, Kwanak-ku, Seoul 151-742, Korea

**Aluminum Company of America, Alcoa Technical Center, 100 Technical Drive, Alcoa, U.S.A.

Stability of ideal orientations and texture formation during deep drawing were investigated for a hot rolled AA 1050. Lattice rotation fields around ideal orientations were nmnericlly predicted using a rate sensitive poly- crystal model with full constraint boundary conditions. In order to evaluate the strain path during deep drawing of an AA1050, simulations using a finite element analysis were carried out. The stability of orientations and tex- ture formation were examined at the sequential strain paths such as flange deformation and w._~l deformation. During flange deformation, Goss and Pf {0 1 1} <21 32 32> (shifted form P {0 1 1} <8 11 11>) sa__tisfied the stability condition of texture formati_on, while during wall deformation, only Bw {0 1 1} <17 13 13> (shifted from Brass) and Dw {5 5 12} <6 6 5> (shifted from Dillamore) satisfied the stability condition of orientation.

Key words : deep drawing, rate sensitive polycrystal model, stability condition, finite element analysis

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

The evolution of anisotropic properties and for- mability during deformation has been regarded as an im- portant subject in metal forming processes. In particular, the crystallographic texture evolution by plastic de- formation such as tension [1, 2], rolling [3-7], deep draw- ing [8, 11] and torsion [12, 13] were experimentally in- vestigated and theoretically predicted both in mi- croscopic and macroscopic scales. Recently, Savoie et al.

[10] experimentally investigated the texture evolution of aluminum alloys in deep drawing by considering non- orthorhombic sample symmetry. Zhou et al. [11] also in- vestigated the stability with strain of initial texture com- ponents of FCC sheet metals during deep drawing. However, these investigations were performed under the plane strain condition, even though deformation path of the deep drawing does not completely comply with the plane strain condition, and depends on the location (flange or wall). In this study, therefore, the non-plane strain effect of a sequential strain path on the stability of orientations is accounted for. In order to investigate more realistic strain paths, the deep drawing for an AA1050 sheet having rolling texture was simulated by finite element analysis using Baflat's anisotropic yield

function for the material description. From these deep drawing simulations, sequential strain paths were iden- tified and the stability of ideal orientations and texture formation for hot rolled AA1050 along these strain paths were studied with a rate sensitive polycrystal model with full constraint boundary conditions. Three-dimensional lattice rotation fields around the ideal orientations were numerically calculated.

2. A N A L Y S I S

2.1. Rate sensitive analysis The deformation of rate sensitive polycrystals is usu-

ally modeled by a power law relationship between the shear rate Ys and the resolved shear stress ,~s on a slip sys- tem s.

m m - l

' = (1) = Zo sgn 0~) j'o yo ~'o

where m is the rate sensitivity parameter, to is a ref- erence shear stress and ~/o is a reference shear rate. The sign term in Eq. 1 means that the shear rate has the same sign with the resolved shear stress. The resolved shear stress is related to the Cauchy stress tensor ~ij of

3,90 Shi-Hoon Choi et al.

the crystal through the following relation.

= mi~ ~j (2)

where the Schmid tensor mi~ (=b~ n~) is defined with the component of the unit vector n~ which is normal to the slip lane and the unit vector b~ which is parallel to the slip direction of the slip systems. When the elastic de- formation is ignored, the vectors n~ and b~ are orthogonal. The component of the strain rate tensor D~j associated to the given stress tensor (Yij is

Dij=~-, 1 . s s- ~{.mij + ~ i ) j's "7"

1 (3) I s = 3o (mi~ + mj~) m u % m~q O'pq

~o ~

It should be noted that the strain rate is deduced form the following stress potential [14]

m ~+1 F(f f i j )= ( m ~ I S s ~ [ l (mi~+mJ~)~ i J

_ m (1 + m~ W (crij) (4)

a F ( q j ) and Dij = 3o.i j

where W(ff~j) is the rate of plastic work according to the prescribed strain rate D~j in the Taylor-Bishop-Hill analysis. The stress state which satisfies the above equation for a given strain rate can be numerically ob- tained by the Newton-Raphson method [15, 16]. The solution of Eq. 3 always converges regardless of the in- itial guess.

The lattice rotation rate ~ j with respect to the la- boratory is given as follows [15]

~'~2iJ = L i i - E mi~ ys (5) s

The lattice rotation rate can be obtained from the pres- cribed velocity gradient ~j and the calculated shear rate ~,~. The Euler angles (01, ~ , ~2) of the individual orien- tations should be updated according to the lattice ro- tation rate as [17, 18].

~1 : (~'~3 sin q~2-- ~1 COS ~2)/sin cO

: ~ 3 cos #2- ~1 sin r (6)

= ~ 2 - ~'~ cos r

Table 1 shows the Miller indices and Euler angles of the initial orientations and the main orientations developed dur- ing deep drawing for the hot rolled AA1050 [19].

Table 1. Euler angles and Miller indices of the ideal orientations

Normal Drawing 01 ~ ~2 direction direction

G 0 45 90 1 0 1 0 -1 0 B 35.25 45 90 1 0 1 -1 -2 1 Bw 47.24 45 90 1 0 1 -13 -17 13 S 58.98 36.70 63.43 2 1 3 -3 -6 4 C 90 35.26 45 1 1 2 -1 -1 1 D 90 27.37 45 4 4 11 -11 -11 8 Dw 90 30.51 45 5 5 12 -6 -6 5 P 62.63 45 90 1 0 1 -11 -8 11 Pf 65.11 45 90 1 0 1 -32 -21 32

To examine the stability of ideal orientations during de- formation, a parameter to describe orientation change at the Euler space is required. The behaviors of orientation change at a given Euler space can be expressed by g=(01, ~, +2), the gradients and divergence from the following

div i = -~1 + 3q~ 3r 2 (7)

Negative div g implies that, around g, more orien- tations rotate toward g that those which rotate away form g. The behaviors of orientation change at a given Euler space can also be expressed by the ODF (Orientation Distribution Function), f(g). In order to des- cribe the change of the ODF during deformation, the continuity equation at a fixed point of the Euler space can be derived as follows [7, 11, 17, 18]

(f/f)g + r div g + g grad (In f) = 0 (8)

Assuming the fourth term in Eq. 8 g grad (In f ) is negli- gible near the ideal orientations, (t'/f)g can be calculated numerically [7].

2.2. Finite element analysis of deep drawing

In Barlat's anisotropic yield criterion for three di- mensional deformation, a yield function, suitable for alu- minum alloys, is defined as [20, 21]

q~= IS3-S2IM+ IS3-SIIM+ I S I - S 2 I M = 2 ~ M (9)

where ~ is the effective stress and Si are the principal values of a symmetric matrix S,~ defined with respect to the components of the Cauchy stress as

C 3 (O'xx - (7yy) - C 2 (r - O'xx) Sxx -

3 C 1 (O'yy - r - C 3 (O'xx - O'yy)

Syy - 3

Stability of Initial Texture Components During Deep Drawing of FCC Polycrystals 49]

C 2 (O'zz - O'xx) - C 1 (O'yy - O'zz) Szz =

3 Syz -- C 40"yz

Szx = C 50"zx

Sxy = C 60"xy

(10)

where x, y and z refer to the mutually orthogonal axes of the orthotropic symmetry. The material coefficients C~ represent anisotropic properties. S,~ reduces to the matrix of deviatoric stress when C~=I.0, that is, when a material is isotropic (particularly, Tresca yield condition for M=I and Von Mises yield criterion for M=2 or 4). The ex- ponent m is mainly associated with the crystal structure [20]. For FCC metals, M=8 is recommended [20].

The coefficients of the yield function were obtained from three R values measured at 0 ~ 45 ~ and 90 ~ from the rolling direction of the sheet. After the yield function was implemented into ABAQUS using the user sub- routine UMAT [22], deep drawing simulations for the hot rolled AA1050 sheet were conducted. Fig. 1 shows the yield surface of the yield function together with the crystallographic yield surface obtained from the rate sen- sitive polycrystal model. The yield surface from the yield function is in good agreement with the yield sur- face computed with the rate sensitive polycrystal model. Fig. 2 shows R values calculated with the rate sensitive polycrystal model and the yield function and those de- termied experimentally. The R values predicted from the

Fig. 2. R values for the AA1050 sheet predicted from the rate sensitive model (RS) and Barlat's yield function (yld91).

rate sensitive polycrystal model shows slight differences with the R values predicted from the yield function and obtained experimentally, but the trend is similar. Fig. 3 shows the normalized yield stresses calculated with the yield function and the rate sensitive polycrystal model and those determined experimentally. The yield function underestimates the normalized yield stresses in the entire range of directions and exhibits a trend different from the experimental data.

In the deep drawing simulation, blanks of initial thick- ness 2.5 mm and initial diameter of 24 mm were drawn into 180 mm diameter cup (drawig ratio is 1.8 : 1). Fig. 4

Fig. 1. Yield surfaces calculated from the rate sensitive model (RS) and Barlat's yield function (yld91) for the AA1050 sheet. Yld91 coefficients: m=8, C1=0.921, C2= 1.095, C3=0.898 and C6=1.031.

Fig. 3. Normalized yield stresses for the AA1050 sheet predicted from the rate sensitive model (RS) and Barlat's yield function (yld91).

zt92 Shi-Hoon Choi et aL

Fig. 4. Finite element meshes for the deep drawing analysis.

shows 937 elements on the initial blank with one layer of elements. In order to account for the planar an- isotropy of the material, FEM simulation was performed with three-dimensional, eight node brick elements, type C3D8H [22]. The full section of the cup was analyzed to obtain strain path. In order to evaluate deformation in the flange of half processed cup (after 51 mm of punch travel), the strain components of an element located at the center of the flange area in the rolling direction were traced. In order to evaluate deformation in the wall area, strain components of elements at 1/2 and 2/3 height from the bottom of fully drawn cup after punch stroke of 105 mm were traced. The elements at 1/2 and 2/3 height undergo different strain path and different amount of deformation during deep drawing. The circumferential deformation at 2/3 height is larger than that at 1/2 height. In Fig. 4, the elements of the flange area, 1/2 height and 2/3 height of the wall area are denoted by 1, 2 and 3, respectively. The input parameters used in the analysis are:

Stress strain characteristics: = 532 - 376 exp (-2.5 e) (MPa)

Anisotropic material data: M=8, C1=0.921, C2=1.095, C3=0.898 , C6=1.031

Punch radius: 90 mm Punch profile radius: 13 mm Die opening radius: 13 mm Blank radius: 162 mm Blank thickness: 2.5 mm Blank holder force: 100 kN Coefficient of friction:

(blank/punch): 0.1, (blank/die): 0.1, (blank/blank holder): 0.1

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

3.1. Strain path during deep drawing Fig. 5 shows the deformed shape obtained at the final

stage of drawing. It shows the elements denoted by 2 and 3 located at 1/2 height and 2/3 height from the cup bottom and aligned with the rolling direction, RD. Con- sidering the geometrical symmetry of the set-up, the strain rate tensor D of these three elements (1, 2 and 3) during deep drawing becomes

[D 00 [i01 D= Doo 0 =D~r a (11)

0 Dff 0

where r, 0 and t denote the three principal directions in the drawing (radial), circumferential and thickness direc- tions of the cup, respectively. The calculated de- formation path during deep drawn cup was different from the plane strain condition. In the flange area, the thickness strain component was not zero and in the wall area, the circumferential strain was not zero. Fig. 6(a) represents the ratio of the circumferential strain rate to the radial strain rate components calculated from the deep drawing simulations. In order to evaluate the sta- bility of ideal orientations, these strain components should be taken into account in the polycrystal simu- lations. Fig. 6(b) represents the ratio of the thickness strain rate to the radial strain rate components calculated from deep drawing. Again, this depends on the initial po- sition in the blank. During the flange deformation, the element (1) in the outer position from the center exhibits the highest strain component ratio; i.e., higher thick-

Fig. 5. Deformed finite element meshes at the final stage after deep drawing simulation (punch stroke: 105 mm).

Stability of Initial Texture Components During Deep Drawing of FCC Polycrystals 493

wall area was

li o o

D = D~r -0.23 0

0 --0.77

(13)

3.2. Rotation fields around the ideal orientations Rotation rate maps for the flange and wall de-

formations were calculated from Eq. 12 and 13 using the rate sensitive polycrystal model. Fig. 7(a) and Fig. 7(b) show ~2=45 ~ and ~2=90 ~ ODF sections representing the rotation rate maps for the flange deformation. The direction of the arrows represents the orientation change and the length represents the total rotation rate. The

Fig. 6. (a) Ratio of radial strain rate and circumferential strai rate components calculated from deep drawing simulation. (b) Ratio of radial strain rate and thickness strain rate components calculated from deep drawing simulation.

ening. In the flange area, the non-vanishing I] value represents the deviation from the plane strain state. The calculated average value of ~ was 0.23. As a result, the average strain rate in the flange area was

fi ~ a l D = Drr -1.23 0 0.23

(12)

After the flange deformation, elements in the blank un- dergo bending deformation which is transient between the flange and wall deformation in Fig. 6. In this study, the bending strain effect was neglected for the cal- culation of stable orientations, assuming that the amount of bending strain is very small. After the bending de- formation, elements undergo tension deformation in the wall area. The non-vanishing ct represents the deviation from the plane strain state. The calculated average value of t~ was -0.23. As a result, the average strain rate in the

Fig. 7. Lattice rotation rates in the (a) t~2=45 ~ and (b) 02 =90 ~ ODF section for flange deformation along RD.

4 9 A Shi-Hoon Choi et al.

stable orientation are those for which all direction of the arrows converge. In this figure, only the Pf {0 1 1} <21 32 32>, (0~, ~, ~)=(65.11 ~ 45 ~ 90 ~ component can be stable orientation. The orientation P represents the stable orientation in the case of plane strain deformation (thickness strain is zero) in the flange area [11]. The stable orientation was shifted from the P to the Pe by the thickening of the flange during deep drawing.

An orientation g=(~,, ~, 02) formation if and only if the ditions are satisfied [7]

O, 9=0, _<0,

remains stable during de- following stability con-

a,b 04)

These stability conditions for orientation determine whether or not orientations around g rotate away from g during deformation. However, these conditions cannot determine whether or not orientation density around g in- creases during deformation.

In the stability for texture formation, a texture com- ponent at a given g is stable during deformation as long as

= = 0 , (i 0g > 0 (15)

These stability conditions represent that, at the stable orientation, zero rotation rates occur and orientation den- sity around g increases during deformation. Therefore, these conditions only describe the stability of texture for- mation around g, but cannot determine whether or not orientations around g rotate away from g. Thus, Eq. 15 is the criterion for the stability of texture formation rath- er than for orientation stability during deformation.

In order to investigate the stability of the initial tex- ture components during flange deformation, the rotation rate g =(+b ~, (~2), (0'1/31~i), div g and relative ODF in- tensity changes (f/f)g were calculated at some ideal orien- tations using the rate sensitivity polycrystal model.

Table 2 shows the calculated results for the strain rate at the flange deformation as shown in Eq. 12. No orien- tations satisfy the stability condition of orientation shown in Eq. 14. Among the seven ideal orientations, only the G and Pe components partially satisfy the sta- bility condition, because they show zero rotation rate. However, the C and P orientations exhibit only two di- mensional convergence with negative div g and positive (f/f)g. The graphical behaviors of the three-dimensional lattice rotation fields around these ideal orientations were also investigated for the flange deformation. The lattice rotations of orientations in the range 20~215 20~215 20 ~ around the given orientation in the Euler space were

Table 2. Rates of change at, gradients 3+~/bd~, diver- gence div g and relative rate of change of ODF intensity (f/f)g for the ideal orientation in the flange deformation

G B S C D P Pf

+1 0 0.36 1.13 0 0 0.16 0 dO 0 0 -0.38 (I.19 0.2 0 0 +2 0 0 -0.58 0 0 0 0

~+1/~)1 -5.17 3.74 1 .11 -5.22 -5.35 -3.88 -3.87 0dO/3qb-0.98 2.57 -0.18-0.17 0 . 0 6 - 2 . 0 6 - 2 . 1 1 ~+r 0.24 0.17 0.07 1 .73 2.19 0.26 0.16 div g -5.9 6.48 1 .01 -3.66 -3 .1 -5.68 -5.81 (f/f)g 5.9 -6.48 -0.5 3.39 2.71 5.68 5.81

calculated for a strain increment of AD,=0.01. Selected cross-sections of these fields are illustrated in Fig. 8 and 9. Fig. 8 shows the rotation fields around the G, B, S and C orientations. The orientations around G converge in the ~1 and �9 directions and diverge in the ~)2 direction. The rotation fields around B, S and C diverge from these three orientations. The rotation rates around C is re- latively low. The rotation fields around the D, P and Pf orientations are shown in Fig. 9. The rotation fields

Fig. 8. Lattice rotation fields around the G, B, S and C orientations during flange deformation.

Stability of Initial Texture Components During Deep Drawing of FCC Polycrystals 495

condition of orientation is satisfied at B and the sta- bility condition of texture formation is satisfied at D

Fig. 9. Lattice rotation fields around the D, P and Pf orientations during flange deformation.

around D and C are similar. The orientations around P, which is the stable end orientation for plane strain draw- ing (thicknes strain is zero), are relatively stable and ro- tate slowly. The rotation fields around Pf converge in the 01 and qb directions and diverge in the 02 direction. However, according to the stability condition of texture formation shown in Eq. 15, the orientation density around G and Pe should increase during the flange de- formation, because (f/f)g at these orientations has the highest value among the seven ideal orientations.

Figs. 10(a) and (b) respectively show the 02=45 ~ and 02 =90 ~ ODF sections representing the rotation rate maps for the deformation in the wall. Stable orientations can be seen as Bw {0 1 1} <17 13 13>, (01, * , 02)=(47.24 ~ 45 ~ , 90 ~ ) and Dw {5 5 12} <6 6 5>, (01, ~, 02)=( 900 , 30.51 ~ , 45 ~ ) components . For the plane strain de- formation (circumferential strain is zero), the stability

Fig. 10. Lattice rotation rates in the (a) 02=45 ~ and (b) 02=90 ~ ODF section for wall deformation along RD.

Table 3. Rates of change +i, gradients ~+~/~+~, divergence div g and relative rate of change of ODF intensity (f/f)g for the ideal orientation in the wall deformation

G B Bw S C D Dw P Pt +1 0 0.33 0 0.02 0 0 0 -0.48 -0.48 r 0 0 0 -0.19 -0.25 0.16 0 0 0 +2 0 0 0 0.46 0 0 0 0 0

3+1/b#1 0.05 -1.19 -1.81 0.56 -1.04 -1.15 -1.03 0.02 0.24 ~r -0.28 -1.13 -1.35 -2.43 -2.96 -3.08 -3.05 -0.68 -0.57 ~+2j/~)2 -3.62 -2.83 -1.75 -3.22 -0.81 -1.38 -1.21 -1.42 -1.3 div g -3.85 -5.15 -4.91 -5.1 -4.81 -5.6 -5.29 -2.08 -1.63 (f/f)~ 3.85 5.15 4.91 5.35 5.17 5.29 5.29 2.08 1.63

4 9 6 Shi-Hoon Choi et al.

[7]. The two stable orientations, B and D, are shifted to the Bw and Dw by the circumferential shrinking of thd cup wall. The associated shifting angles appear to be a function of the ratio of the circumferential strain component to radial strain component of the cup wall; i.e., the ratio cc

Table 3 shows the calculated results for the strain rate at the wall deformation as shown in Eq. 14. The stability conditions of orientation and texture formation are simultaneously satisfied at Bw and Dw. The stability condition is only partially satisfied at G, because of the zero rotation rate. The B, C and D orientations satisfy partially one stability condition, because they have ne- gative rotation rate gradient. For all the ideal orien- tations, the values of div g are negative and the values of (f/f)g are positive. This indicates that these orien- tations can be preferred orientations of texture during the deformation of the wall. The orientation density around Bw and Dw increases during the deformation, be- cause (f/f)g at these orientations has the highest values. Fig. 11 shows the rotation fields around the G, B, S, Bw and S orientations during the deformation of the wall. The orientations around G converge in the 02 and

directions and diverge in the 01 direction. The rotation field around B moves away along the direction 01. The rotation field around Bw exhibits three dimensional con- vegence. The orientations around S rotate quickly in the 02 direction. The rotation fields around the C, D, Dw, P and Pf orientations are shown in Fig. 12. The rotation rate fields around the C and D orientations are re- latively stable, but show the strong movement of orien- tation in one direction. The negative ~ value of C and the positive �9 value of D mean that the orientations around C and D will rotate toward �9 < 35.26 ~ and �9 > 27.3 ~ respectively. The rotation fields around Dw ex- hibits three dimensional convergence. The rotation fields around P and Pf move away along direction 01. The negative ~1 values for P and Pf indicate that the orientations around P and Pe rotate toward 01 < 62-63~ and 01 < 65.11~

Fig. 11. Lattice rotation fields around the G, B, Bw and S orientations during wall deformation.

Fig. 12. Lattice rotation fields around the C, D, Dw, P and Pf orientations during wall deformation.

Stability of Initial Texture Components During Deep Drawing of FCC Polycrystals 497

5. CONCLUSIONS

In order to estimate the strain path during deep draw- ing, FEM simulation was conducted. It was shown that the material elements underwent the sequential strain paths of the flange, bending and wall deformations. As- suming bending deformation is negligible, a rate sen- sitive model was utilized to understand the effect of the flange thickening and wall shrinking on the stability of initial texture components and additional texture com- ponents by simulating three dimensional lattice rotation fields around the orientations. For deformation in the flange, no orientation satisfied the stability condition of orientation, but G and Pf (shifted form P) orientations sa- tisfied the stability of texture formation. The orientation density around the G and Pf orientations increased dur- ing the deformation of the flange. For the deformation of the wall, only Bw (shifted from B) and Dw (shifted from D) orientations satisfy the stability condition of orientation and the G, Bw and Dw orientations satisfied the stability condition of texture formation.

ACKNOWDLEGEMENTS

This work was financially supported by the Korean Ministry of Education through the Advanced Materials Research Program in 1998.

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