Texture evolution of FCC sheet metals during deep drawing process

Transcription

1 International Journal of Mechanical Sciences 42 (2000) 1571}1592 Texture evolution of FCC sheet metals during deep drawing process Shi-Hoon Choi, Jae-Hyung Cho, Kyu Hwan Oh*, Kwansoo Chung, FreH deh ric Barlat School of Materials Science and Engineering, Seoul National University, Research Institute of Advanced Materials, San 56-1 Shinrim-dong, Kwanak-ku, Seoul , South Korea Department of Fiber and Polymer Science, Seoul National University, 56-1 Shinrim-dong, Kwanak-ku, Seoul , South Korea Aluminum Company of America, Alcoa Technical Center, 100 Technical Drive, Alcoa, USA Received 27 June 1999 Abstract The stability of ideal orientations and texture evolution was investigated for FCC sheet metals during deep drawing. Lattice rotation "elds around ideal orientations were numerically predicted using a rate-sensitive polycrystal model with full constraint boundary conditions. In order to evaluate the strain path during deep drawing of an AA1050, simulations using a "nite element analysis were carried out. The stability of orientations and texture formation was examined at sequential paths such as #ange deformation, transition and wall deformation. Depending on the initial location in the blank, the deviation from the plane strain state in the #ange deformation path decreased the orientation density around P and shifted the "nal stable end orientation from P to Y near The texture evolution in AA1050 sheet metals during deep drawing was experimentally investigated. The change of orientation density around ideal orientations in the RD and TD samples was in good agreement with the rate-sensitive polycrystal model Elsevier Science Ltd. All rights reserved. Keywords: Texture; Deep drawing; FEM; Rate-sensitive polycrystal mode 1. Introduction The evolution of anisotropic properties and formability during deformation has been regarded as an important subject in metal forming processes. In particular, the crystallographic texture * Corresponding author. address: chunsoo@snu.ac.kr (K.H. Oh) /00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 1572 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Nomenclature γ shear rate on a slip system s τ resolved shear stress on a slip system s m rate sensitivity parameter τ reference shear stress γ reference shear rate m Schmid tensor σ Cauchy stress tensor D strain rate tensor F(σ ) stress potential =Q (σ ) rate of plastic work ΩQ lattice rotation rate prescribed velocity gradient φ, Φ, φ Euler angles of the individual orientations φ, Φ, φ lattice rotation rate g(φ, Φ, φ ) orientation at a given Euler space g (φ, Φ, φ ) orientation change at a given Euler space f (g) orientation distribution function (ODF) fq (g) change of the ODF σ e!ective stress C material coe$cients of Barlat's anisotropic yield function M exponent used in Barlat's anisotropic yield function r, θ, t three principal directions in the drawing (radial), circumferential and thickness directions of the cup ρ ratio of the circumferential strain rate to the radial strain rate component κ ratio of the thickness strain rate to the radial strain rate component evolution by plastic deformations such as tension [1,2], rolling [3}7], deep drawing [8,11] and torsion [12,13], was experimentally investigated and theoretically predicted both in microscopic and macroscopic scales. Recently, Savoie et al. [10] experimentally investigated the texture evolution of aluminum alloys in deep drawing, considering non-orthorhombic sample symmetry. Zhou et al. [11] also investigated the stability of initial texture components of FCC sheet metals during deep drawing. However, these investigations were performed assuming plane strain conditions, even though the deformation path in the deep drawing is not completely plane strain, and depends on the location (#ange, bending area and wall). In this study, in order to investigate actual strain paths, the deep drawing of an AA1050 sheet exhibiting rolling texture was simulated by "nite element analysis using Barlat's anisotropic yield function for the material description. From these deep drawing simulations, sequential deformation paths were identi"ed 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.

3 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Three-dimensional lattice rotation "elds around the ideal orientations were numerically calculated. In order to verify the predicted results, deep drawing tests were conducted and the change of initial texture in the rolling direction (RD) and transverse direction (TD) samples was experimentally investigated. 2. Analysis 2.1. Rate-sensitive analysis The deformation of rate-sensitive polycrystal is usually modeled by a power-law relationship between the shear rate γ and the resolved shear stress τ on a slip system s: τ"τ sgn(γ ) γ γ γ "τ γ γ γ. (1) where m is the rate sensitivity parameter, τ is the reference shear stress and γ is the reference shear rate. The value of γ does not a!ect the texture evolution, but the value of τ generally depends on the microscopic hardening such as self and latent hardening. In the present study, the microscopic hardening is not considered and reference values are assumed to be constant during deformation. The sign term in Eq. (1) means that the shear rate has the same sign as the resolved shear stress. The resolved shear stress is related to the Cauchy stress tensor σ of the crystal by the following relation: τ"m σ. (2) where the Schmid tensor m ("b n )isde"ned with the component of the unit vector n which is normal to the slip plane and the unit vector b which is parallel to the slip direction of the slip system s. When the elastic deformation is ignored, the vectors n and b are orthogonal. The component of the strain rate tensor D associated with the given stress tensor σ is 1 D " 2 (m #m )γ " γ 1 τ 2 (m #m )m σ m σ. (3) It should be noted that the strain rate is deduced from the following stress potential [14]: F(σ )" m (m#1) 1 2 (m #m )σ " m (1#m) =Q (σ ), D " F(σ ) σ. (4) where =Q (σ ) is the rate of plastic work according to the prescribed strain rate D. The stress state which satis"es the above equation for a given strain rate can be numerically obtained by the Newton}Raphson method [15,16]. The solution of Eq. (3) always converges regardless of the initial guess.

4 1574 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Table 1 Euler angles and Miller indices of the ideal orientations φ Φ φ Normal direction Drawing direction G !1 0 B !1!2 1 S !3!6 4 C !1!1 1 D !11!11 8 P !11!8 11 P !32!21 32 Y !1!1 2 Y !16!16 45 The lattice rotation rate Ω with respect to the laboratory is given as follows [15]: Ω "! m γ. (5) The lattice rotation rate can be obtained from the prescribed velocity gradient and the calculated shear rate γ. The Euler angles (φ, Φ, φ ) of the individual orientations should be updated according to the lattice rotation rate g (φ, ΦQ, φ ) as [17,18] φ "(Ω sin φ #Ω cos φ )/sin Φ, ΦQ "Ω cos φ!ω sin φ, φ "Ω!φ cos Φ. (6) Table 1 shows the Miller indices and Euler angles of the initial orientations and the main orientations developed during deep drawing for the FCC sheet metals [19]. To examine the stability of ideal orientations during deformation, a parameter that describes orientation change in the Euler space is required. The behavior of orientation change can be expressed by the lattice rotation rate g "(φ, ΦQ, φ ), divergence of g, div g as div g " φ # Φ φ Φ #φ. (7) φ Negative div g implies that, more orientations around g rotate towards g. The behaviors of orientation change at a given Euler space can also be expressed by the orientation distribution function (ODF), f (g). In order to describe the change of the ODF during deformation, the continuity equation at a "xed point of the Euler space can be derived from published work [7,11,17,18]: ( fq /f ) #Φ cot Φ#div g #g grad(ln f )"0. (8) Assuming that the fourth term g grad(ln f ), in Eq. (8) is negligible near the ideal orientations, ( fq /f ) can be calculated numerically [7].

5 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Finite element analysis of deep drawing In Barlat's anisotropic yield criterion for three-dimensional deformation, a yield function, suitable for aluminum alloys, is de"ned as [20,21] Φ"S!S #S!S #S!S "2σ, (9) where σ is the e!ective stress and S are the principal values of a symmetric matrix S de"ned with respect to the components of the Cauchy stress as S " C (σ!σ )!C (σ!σ ), 3 S "C σ, S " C (σ!σ )!C (σ!σ ), 3 S "C σ, S " C (σ!σ )!C (σ!σ ), 3 S "C σ, (10) where x, y and z refer to the mutually orthogonal axes of the orthotropic symmetry. The material coe$cients C represent anisotropic properties. S reduces to the matrix of deviatoric stress when C "1.0, when a material is isotropic (particularly, Tresca yield condition for M"1 and Von Mises yield criterion for M"2 or 4). The exponent M is mainly associated with the crystal structure [20]. For FCC metals, M"8 was recommended [20]. The coe$cients of the yield function could be obtained from three R values measured at 0, 45 and 903 from the rolling direction of the sheet. After the yield function was implemented into ABAQUS using the user subroutine UMAT [22], deep drawing simulations for the hot-rolled AA1050 sheet were conducted. Fig. 1 shows the yield surface from Eq. (9) together with the crystallographic yield surface obtained from the rate-sensitive polycrystal model using ODF. In the deep drawing simulation, blanks of thickness of 2.56 mm and diameter of 324 mm were drawn into 180 mm diameter cup (drawing ratio is 1.8 : 1). In order to account for the planar anisotropy of the material, the FEM simulation was performed using 240 elements as shown in Fig. 2 with three-dimensional, eight-node brick elements, type C3D8H [22]. The quarter section of the specimen was analyzed. In order to evaluate deformation paths at di!erent locations, the strain history of elements at 1/2, 2/3 and 6/7 of the height of the fully drawn cup (see Fig. 3) after a punch stroke of 105 mm, was obtained. These elements undergo di!erent strain paths and di!erent amounts of deformation during deep drawing. The total circumferential deformation at the 6/7 height location was the largest. The input parameters used in the analysis were Stress strain characteristics: σ "532!376 exp(!2.5ε ) (MPa) Anisotropic material data: M"8, C "0.921, C "1.095, C "0.898, C "1.031 Punch radius: 90 mm Blank holder force: 100 kn Punch pro"le radius: 13 mm Coe$cient of friction: Die opening radius: 13 mm (blank/punch): 0.1 Blank radius: 162 mm (blank/die): 0.1 Blank thickness: 2.56 mm (blank/blank holder): 0.1

6 1576 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Fig. 1. Yield surfaces calculated from the rate-sensitive model (RS) and Barlat's yield function (yld91) for the AA1050 sheet. Yld91 coe$cients: M"8, C "0.921, C "1.095, C "0.898 and C " Fig. 2. Finite element meshes for the deep drawing analysis.

7 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Fig. 3. Deformed "nite element meshes at the "nal stage after deep drawing simulation (punch stroke: 105 mm). 3. Experimentals In the actual deep drawing process, 324 mm diameter blank (initial thickness 2.56 mm) was drawn into 180 mm diameter cup using a drawing ratio of 1.8 : 1. Pole "gure measurements were carried out in Seifert D3000 PTS X-ray di!ractometer to characterize crystallographic texture. From the three (1 1 1), (2 0 0) and (2 2 0) incomplete pole "gures, the ODF of crystallite was calculated using the WIMV method [23], which used the conditional ghost correction. In order to observe the texture evolution of sheet during deep drawing deformation, samples were taken at 1/2, 2/3 and 6/7 height of the cup wall along the RD and TD directions after a punch stroke of 105 mm. The ODF was calculated from the pole "gures considering the orthorhombic sample symmetry. Such symmetry requires the elementary Euler space de"ned by 03)φ )903, 03)Φ)903 and 03)φ ) Results and discussion 4.1. Strain path during deep drawing Fig. 3 shows the deformed shape obtained at the "nal stage of deep drawing simulation. It shows three locations denoted by 1/2, 2/3 and 6/7 and located at 1/2, 2/3 and 6/7 height from the cup bottom and aligned in the RD and TD directions. Fig. 4 shows the cup obtained after actual deep drawing. The positions for measuring the pole "gure are shown along RD and TD, respectively. Fig. 5 shows the experimentally measured cup height pro"le and predicted pro"les using FEM simulation for the completely drawn cup. Even though the FEM results underestimated the actual experimental cup height, the calculated anisotropic behavior of the deep drawn cup was in relatively good agreement with the experimental one.

8 1578 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Fig. 4. Completely drawn cup by deep drawing operation (punch stroke: 105 mm). Fig. 5. Experimentally measured and predicted cup height pro"les using FEM simulation for the completely drawn cup. During deep drawing the strain rate tensor of points located on a symmetry axis (RD or TD) D is D D"D 0 D 0 0 ρ 0 ρ", κ"!(1#ρ), (11) 1 D 0 0 D 0 0 κ, "D

9 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Fig. 6. (a) Ratio of circumferential strain rate to the radial strain rate components along the RD calculated from deep drawing simulation. (b) Ratio of circumferential strain rate to the radial strain rate components along the TD calculated from deep drawing simulation. where r, θ and t denote the three principal directions in the drawing (radial), circumferential and thickness directions of the cup, respectively. ρ"!1 and κ"0 denote the plane strain condition in the #ange area. Fig. 6(a) represents the ratio of the circumferential strain rate to the radial strain rate components calculated from the deep drawing simulations along the RD. The elements at 1/2, 2/3 and 6/7 of the height undergo di!erent strain paths and di!erent strain amounts during deep drawing. The calculated deformation paths during deep drawing are di!erent from the plane strain condition (ρ"!1). Considering the change of the ratio ρ, the deep drawing deformation can be divided into three deformation paths (#ange deformation, transition deformation and wall deformation). As the punch stroke increases, the three regions in the blank undergo deformation at approximately constant ratio ρ in the #ange. This is de"ned as the #ange deformation path. After the #ange deformation path, ρ in the three regions changes continuously during the deformation. This is de"ned as the transition deformation path. After the transition deformation path, the two regions in the blank undergo deformation at constant ρ in the wall area. Fig. 6(b) represents the ratio of the circumferential strain rate to the radial strain rate components calculated from deep drawing along the TD. In the #ange deformation path, the 6/7 location in the outer position from the center exhibits the highest strain component ratio, i.e. the highest thickening. In the #ange area, the non-vanishing κ value represents the deviation from the plane strain state. The calculated average values of κ along the RD and TD were about 0.23 and 0.33, respectively. As a result, the average strain rate in the #ange area along the RD and TD was D"D 0! for RD, D"D 0! for TD. (12) Fig. 7 shows the measured and predicted thickness distributions along the RD and TD. The di!erence of κ between the RD and TD is related to the thickness pro"le of the deep drawn cup.

10 1580 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Fig. 7. Measured and calculated cup thickness pro"les along the RD and TD for the completely drawn cup. The thickness along the TD is larger than that along the RD. Except near the punch bottom and nose areas, the thickness of blank increased during deep drawing. The measured thickness pro"le shows a good agreement with the predicted one Rotation xelds around the ideal orientations In order to analyze the texture development around the ideal orientation during deep drawing, strain amounts as well as deformation paths should be considered. Fig. 8(a) shows the total D and D calculated from the FEM simulation for the three locations along the RD. It is shown that the #ange deformation path is the most dominant path among the three deformation paths during deep drawing. The total D in the wall deformation path is negligible compared to the other deformation paths. The 6/7 location undergoes only the #ange and transition deformation paths. Fig. 8(b) shows total D and D calculated from the FEM simulation for the three locations along the TD. The total circumferential deformation at the 6/7 location was the largest. The results were consistent with the RD. In this study, the #ange deformation path for RD in which the ratio κ was 0.23, was considered to examine the texture formation around the ideal orientations. Rotation rate maps for the #ange deformation path were calculated from Eq. (12) using the rate-sensitive polycrystal model. Figs. 9(a) and (b) show φ "45 and 903 ODF sections representing the lattice rotation rate maps for the #ange deformation path. The direction of the arrows represents the orientation change and the length represents the total rotation rate. The stable orientation is that for which all the directions of the arrows converge. As shown in the "gure, the Y, (φ, Φ, φ )"(903,633,453) and P , (φ, Φ, φ )"(65.113,453,903) components are stable orientations. The orientation P represents the stable orientation in the case of plane strain deformation (thickness strain is zero) in the #ange area [11]. However, due to thickening of the #ange during deep drawing, the stable orientations changed from P and Y components to P and Y components, respectively.

11 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Fig. 8. (a) Total D and D for three deformation paths along the RD calculated using FEM simulation. (b) Total D and D for three deformation paths along the TD calculated using FEM simulation. Fig. 10 shows the position of the two rolling "bers, i.e. the α "ber (G}B) and β "ber (B}S}C/D). During rolling, orientations either move directly into the β "ber, or "rst move into the α "ber and into the β "ber, then "nally rotate along the β "ber toward the stable end orientation C/D. If #ange thickening during deep drawing does not occur, the deep drawing could be considered as plane

12 1582 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Fig. 9. Lattice rotation rates in the (a) φ "453 and (b) φ "903 ODF section for #ange deformation path along RD. Fig. 10. Schematic representation of orientation development in Euler space during rolling and deep drawing. strain deformation, i.e. there is no thickness change of the sheets. The orientations, the α and β "bers in the rolling frame are equivalent to the α "ber (G}Y) and β "ber (Y}P) in the deep drawing reference frame. The positions of the two deep drawing "bers, i.e. the α and β, are shown in Fig. 10. During deep drawing process, orientations either move directly toward the β "ber, or

13 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Table 2 Rates of change φ, gradients φ /φ, divergence div g and relative rate of change of ODF intensity ( fq/f ) for the ideal orientation in the plane strain drawing G B S C D P P Y Y φ ! ΦQ 0 0! !0.12 φ 0 0! φ /φ! !5.39!5.68!3.47!3.46!4.00!3.72 ΦQ /Φ! !0.21! !2.88!2.82!1.09!0.69 φ /φ ! !0.57! div g! !3.55!2.80!6.92!6.91!4.09!3.97 ( fq /f ) 6.01!6.68! "rst move toward the α "ber and the β "ber, then "nally rotate along the β toward the stable end orientation P [9]. However, due to the #ange thickening, the two "bers are shifted and the stability conditions at the ideal orientations are changed as shown in Fig. 9. An orientation g"(φ, Φ, φ ) remains stable during deformation if and only if the following stability conditions are satis"ed [7]: g "(φ, Φ, φ )"0, φ )0, φ Φ Φ )0, φ )0. (13) φ These stability conditions of orientation determine whether or not orientations around g rotate away from g during deformation. However, these conditions (Eq. (13)) cannot determine whether or not orientation density around g increases during deformation. In the stability for texture formation, a texture component at a given g is stable during deformation as long as g "(φ, Φ, φ )"0, ( fq /f ) '0. (14) These stability conditions mean that, at the stable orientation, zero rotation rates occur and orientation density around g increases during deformation. Therefore, these conditions (Eq. (14)) only describe the stability of the texture formation around g, but cannot determine whether or not orientations around g rotate away from g. In order to investigate the stability of the initial texture components at the #ange deformation path, the rotation rate g "(φ, ΦQ, φ ), (φ /φ ), div g and relative ODF intensity changes ( fq /f ) were calculated at some ideal orientations using the rate-sensitive polycrystal model. Table 2 shows the calculated results for plane strain drawing (thickness strain is zero). Only orientation P satis"es the stability condition of orientation shown in Eq. (13) [11]. Table 3 shows the calculated results for the strain rate in the #ange deformation path as given in Eq. (12), whose thickness strain is "nite. Only orientation Y satis"es the stablility condition of orientation shown in Eq. (13). Among the nine ideal orientations, the G, P and Y components satisfy the stability condition of texture formation. However, the C, P and Y orientations exhibit only two-dimensional convergence with negative div g and positive ( fq /f ).

14 1584 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Table 3 Rates of change φ, gradients φ /φ, divergence div g and relative rate of change of ODF intensity ( fq/f ) for the ideal orientation in the plane strain drawing G B S C D P P Y Y φ ΦQ 0 0! φ 0 0! φ /φ! !5.22!5.35!3.88!3.87!4.70!4.68 ΦQ /Φ! !0.18! !2.06!2.11!0.48!0.37 φ /φ !0.03 div g! !3.66!3.10!5.68!5.81!4.70!5.08 ( fq /f ) 5.90!6.48! Fig. 11. α and β "ber orientations calculated using rate-sensitive polycrystal model along RD during the flange deformation path. Fig. 11 shows the α and β "ber orientations calculated using the rate-sensitive polycrystal model for the three locations along RD during the #ange deformation path. With the increase in the ratio ρ, the α "ber moves toward the Φ direction and the β "ber orientation moves toward φ and φ. Table 4 shows the stability of orientation and texture formation in the wall 1/2,

15 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Table 4 Stability of orientation and texture formation in wall 1/2, 2/3 and 6/7 parts along RD in deep drawn cup P (w1/2) Y (w1/2) P (w2/3) > (w2/3) P (w6/7) Y (w6/7) φ ΦQ φ φ /φ!3.56!4.21!3.77!4.57!4.17!4.92 ΦQ /Φ!2.68!0.86!0.05! !0.03 φ /φ! ! !1.67!0.87 div g!6.69!4.32!6.05!4.82!4.86!5.82 ( fq /f ) /3 and 6/7 location along RD in deep drawn cup. For the 1/2 and 2/3 locations P component satis"es the stability condition of orientation but, for the 6/7 location only Y component satis"es the stability condition of orientation. In all the three locations, orientations P and Y satisfy the stability condition of texture formation. With the increase in the height from the bottom of the cup, the relative rate of change of ODF intensity in P decreases, but the relative rate of change of ODF intensity in Y increases. Fig. 12 shows the α and β "ber orientations calculated using rate-sensitive polycrystal model for the three locations along TD during the #ange deformation path. The change of α and β "ber orientations is very similar to that of the RD case. But, P orientation moves much slower in the φ case, compared with the RD case. Table 5 shows the stability of orientation and texture formation in the wall 1/2, 2/3 and 6/7 locations along TD in the deep drawn cup. For all locations P and Y components satisfy the stability condition of texture formation and only Y component satis"es the stability condition of orientation. Similar to the RD case, with the increase in the height from the bottom of the cup, the relative rate of change of ODF intensity in P decreases, but the relative rate of change of ODF intensity in Y increases Experimental texture evolution Fig. 13 shows the φ "0 and 453 sections of ODF for RD samples in the deep drawn cup as shown Fig. 4. In order to investigate the rotations of the crystal orientations during deep drawing, the initial and deformed textures are presented together according to the height of the locations at wall 1/2, wall 2/3 and wall 6/7. The crystal orientations distributed around the D component move toward the C component. The crystal orientations distributed between the G component and the B component move toward the P and G components. Fig. 14 shows the orientation density along the α "ber for the RD samples. As predicted by the rate-sensitive polycrystal model, the orientation density around the C and G components was higher than that of the initial state. Fig. 15 shows the orientation density along β for the RD samples. Similar to the predicted results, the orientation density around P and Y are dependent on the ratio ρ. As predicted, with the increase in the ratio ρ, the orientations around Y become more stable compared to the plane strain

16 1586 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Fig. 12. α and β "ber orientations calculated using rate-sensitive polycrystal model along TD during the flange deformation path. Table 5 Stability of orientation and texture formation in wall 1/2, 2/3 and 6/7 parts along TD in deep drawn cup P (w1/2) Y (w1/2) P (w2/3) > (w2/3) P (w6/7) Y (w6/7) φ ΦQ φ φ /φ!3.86!4.67!4.00!4.78!4.04!4.87 Φ /Φ 0.16! ! !0.13 φ /φ!2.11!0.02!1.89!0.41!1.84!0.48 div g!5.81!5.06!5.41!5.39!5.26!5.48 ( fq /f ) condition. Fig. 16 shows orientations of the β "ber for the RD samples. The orientation shifted in the φ direction. This tendency is attributed to the larger circumferential deformation and the higher ratio ρ. Fig. 17 shows the φ "0 and 453 sections for TD samples in deep drawing cup. The crystal orientation distributed around a speci"c component (φ "03, Φ"27.373, φ "453) moves

17 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Fig. 13. φ "0 and 453 ODF sections of (a) initial state (b) 1/2 part (c) 2/3 part (d) 6/7 part along the RD in deep drawing cup. toward rotated cube component (φ "03, Φ"03, φ "453) and the orientation density decreases drastically. The crystal orientation distributed between the Rotated Goss (φ "03, Φ"903, φ "453) component and orientation (φ "32.253, Φ"903, φ "453) moves toward P. With the increase in the total circumferential strain, the orientation density between C and Y increases as predicted by the rate-sensitive polycrystal model. Fig. 18 shows the

18 1588 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Fig. 14. Orientation density f (g) along the α "ber for the RD samples. Fig. 15. Orientation density f (g) along the β "ber for the RD samples. Fig. 16. Orientation of β "ber for the RD samples.

19 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Fig. 17. φ "0 and 453 ODF sections of (a) initial state (b) 1/2 part (c) 2/3 part (d) 6/7 part along the TD in deep drawing cup.

20 1590 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571}1592 Fig. 18. Orientation density f (g) along the α "ber for the TD samples. Fig. 19. Orientation density f (g) along the β "ber for the TD samples. orientation density along α "ber for the TD samples. As predicted by the rate-sensitive polycrystal model, orientation density around the C and G components was higher than that of the initial state. Fig. 19 shows the orientation density along β "ber for the TD samples. With the increase in the ratio ρ, the orientations around Y become more stable as predicted by the model. Fig. 20 shows the β "ber orientation for the TD samples. The orientation shifted in the φ direction. This trend is consistent with the results obtained at the RD locations. 5. Conclusions In order to estimate the strain path during deep drawing, FEM simulation was conducted. It was shown that the blank sheets underwent sequential deformation paths of the #ange, transition and

21 S.-H. Choi et al. / International Journal of Mechanical Sciences 42 (2000) 1571} Fig. 20. Orientation of β "ber for the TD samples. wall region. The #ange deformation path was the most dominant deformation path. Assuming negligible bending deformation, the rate-sensitive model was utilized to understand the e!ect of the #ange thickening on the stability of initial texture components and additional texture components by simulating three-dimensional lattice rotation "elds around the orientations. For the #ange deformation path, the stability of orientation and texture formation around ideal orientations along RD and TD was di!erent from the initial locations in the blank. The outer position from the center of the blank exhibits the highest ratio of the circumferential strain to the radial strain rate components and the highest total circumferential deformation. The orientation of α "ber (G}Y ) and β "ber (P }Y ) in Euler space was dependent on the ratio of the circumferential strain rate to the radial strain rate components. From the texture measured in the drawn AA1050 cup, it was clear that the orientation density around Y increased with increasing the total circumferential deformation. With the increase in the ratio ρ, the orientation of β "ber shifted in the φ direction. Acknowledgements This work was "nancially supported by the Korean Ministry of Education through the Advanced Materials Research Program in References [1] Becker RC, Butler Jr. JF, Hu H, Lalli LA. Metallurgical Transactions A 1991;22:45. [2] Choi S-H, Cho JH, Barlat F, Chung K, Kwon JW, Oh KH. Metallurgical Transactions A 1999;30:377}86. [3] Hirsch J, LuK cke K. Acta Metallurgica 1988;36:2883}904. [4] Choi S-H, Kwon JW, Oh KH. Metals and Materials 1996;2:133}40. [5] Choi CH, Kwon JW, Oh KH, Lee DN. Acta Metallurgica 1997;45:5119. [6] Choi S-H, Oh KH. Metals and Materials 1997;3:252}9. [7] Zhou Y, ToH th LS, Neale KW. Acta Metallurgica Materials 1992;40:3179}93.

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