METALS AND MATERIALS, Vol. 3, No. 4 (1997), pp. 252-259

Rate Sensitive Analysis of Texture Evolution in FCC Metals

Shi-Hoon CHOI and Kyu Hwan OH

School of Materials Science and Engineering, Seoul National University Research Institute of Advanced Materials

San 56-1 Shinrim-dong, Kwanak-ku, Seoul 151-742, Korea

The rate sensitive analysis was used to investigate the texture evolution during plane strain compression and combined deformation (plane strain compression and simple shear). The rate sensitivity of the rate sensitive model affects the orientation density distributions in Euler space and change the major orientations. As the rate sensitivity of the rate sensitive models increases, lattice rotation vectors around major components decreases.

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

The texture evolution during rolling has been studied by many researchers during last three decades [1-5]. The inhomogeneity of texture through the thickness direction is attributed to the formation of shear texture at the sur- face of sheet metals. The rolling conditions such as fric- tion, roll geometry etc., and material properties play an important role in the formation of the shear texture. These shear components have been calculated theoret- ically [6-7] using a simplified strain state and Taylor model. In case of Taylor model, the number of active slip system should be 5 in full constraint analysis or less than 5 in relaxed analysis [8-9]�9 Rate sensitive relation between resolved shear stress and shear strain rate was first proposed by Hutchinson [10]. Rate sensitive analy- sis activated all the slip systems and at the zero rate sen- sitivity gave the same result with Taylor-Bishop-Hill analysis [11]. The rate sensitive analysis was used wide- ly in the analysis of the texture evolution during rolling [12,13], biaxial stretching [14], deep drawing [15,16] and torsion [11,17,18]. Recently finite element analysis was used to analyze the texture evolution during rolling. Mathur et al. used the rate sensitive constitutive model to develop an Eulerian finite element analysis which can simulate the evolution of the crystallographic texture in the steady state bulk forming process [19[. Kalidini predicted the evolution of the texture in FCC metals with initial texture using rate sensitive model [20]. In most analyses of rolling processing which is combined with shear deformation, the sample symmetry has been orthorhombic.

In this study, texture evolution in FCC metals during plane strain compression and combined deformation (plane strain compression and simple shear) was analyz- ed by rate insensitive and rate sensitive model con- sidering non-orthorhombic sample symmetry. Rate in- sensitive analysis was carried out by the full constraint Taylor-Bishop-Hill model [21] and Renouard-Win- tenberger method [22]. Rate sensitive analysis was car- ried out by the power law relation [10] and full con- straint boundary condition. The three-dimensional lattice rotation fields around major components were det- ermined numerically using the rate insensitive and rate sensitive models.

2. A N A L Y S I S

2.1. Rate insensitive and sensitive analysis 2.1.1. Rate insensitive analysis

In the Taylor-Bishop-Hill analysis, the deformation rate of each crystallite is assumed to be identical with the prescribed deformation. The prescribed deformation

rate at crystal coordinate, k~i' can be obtained by the

transformation of deformation rate, kL~x, prescribed at the

sample coordinate.

�9 C x

~?ij =lik Ijl 8kl (1)

Using Taylor-Bishop-Hill model, tile work rate W can be given by

W(%) = ~ij eij (2)

which can be expressed as

Rate Sensitive Analysis of Texture Evolution in FCC MetaL, 253

W(~ = - Bkl' + Ak22 + 2F~23 + 2Gk31 + 2He12 (3)

A = (0-22- 0-33), B = (0-33- al 1), F= 023 , G= 0"31 , H = o-~2

where r is the crystallographic stress state. The work rate is calculated for the 56 stress state obtained by Bishop and Hill [23] and the stress states of maximum work become the yielding stress states. Thus the crys- tallographic stress state for the prescribed strain rate can be determined. From the selected stress states, the active

slip systems are determined and the shear rate ), of each slip system for the prescribed strain rate can be det- ermined according to the following equation.

1 kii = y~ -4- (mi~ + mi-,':)~

s (4)

By solving above linear equation, shear rate y, can be ob- tained. But there is an ambiguity on the selection of the active slip system for the calculation of the shear rate, because the number of active slip system at the stress state is generally larger than 5 (e.g. 6 or 8 in FCC metal). This ambiguity can be removed using "second order plastic work minimization" proposed by Renouard and Wintenberger [22]. According to the method, a small strain is applied to each possible set of slip system and crystal rotation of each set of slip system is obtained. The new orientations due to crystal rotation give the cor- responding plastic work i.e. the second order plastic work. The set of slip systems of which the second order plastic work is minimum under the applied small strain, is the active slip system. 2.1.2 Rate sensitive analysis

The deformation of rate sensitive analysis is usually modelled by a power law relationship between the shear

rate ), and the resolved shear stress r at a slip system s.

m m-I

: r,, v- (5)

m is positive and called as rate sensitivity. "co is a ref-

erence shear stress and ~, is a reference shear rate. The sign term in Eq. (5) means that the shear rate has same sign as the resolved shear stress. The resolved shear stress is related to the deviatoric stress tensor % of the

crystal through the following relation�9

r s m~i ~i (6)

s ~; q where m~i = b~ nj, is defined by the component of the unit

vector n. ~ normal to the slip plane and the unit vector bi ~ 1

parallel to the slip direction of the slip system s. Ig- noring the elastic deformation, the vectors nj ,~ and ba S can

be regarded as orthogonal. The component of the strain

tensor kii associated with the given stress tensor ~i can

be obtained as follows

k - 1 - ~ S (m~ + mig~ ij

1

gol~ , - ~ (mij + mji ) mklO'~ (7)

It should be noted that the deformation rate can be deduced from a stress potential [10]. The stress potential can be given as follows

m ~, ]1 ~,+' f(~J) (m+l ) v,, t/m ~s j T(mi~+mjg~J

_ m

(1 +m) @(~j) 0f(~j) (8)

k'J- O~j)

where W (@j) is the rate of plastic work according to the

prescribed deformation rate kii in the Taylor-Bishop-Hill

analysis. The stress state which satisfies the above e- quation at the given deformation rate can be obtained by the Newton-Raphson method [11,24]. The solution of Eq. (7) was converged regardless of initial condition. The stress and strain tensor can be reduced into the fol- lowing vectors during calculation.

0-1 = N72 (O':22--0-11)' ~ = ~ O~:~3, 0"3 : "~f2~o"23

(9) s = ~ ( s , C2 = - - E33, E'~ = x/2Cv~

~2 "4~ " --

e 4 = N[2,g31, ~c 5 = "f2E12

2.2. Calculation of the lattice rotation

~i is the rotation rate of crystal due to only slip de-

formation without rigid body rotation and elastic de-

formation. Thus &lj is called as glide rotation rate [11].

~,i = ~ 1 (mi~_ mjg~ (10)

The lattice rotation rate is defined as the rotation rate ~2ij

of the crystal lattice with respect to the laboratory. ~ij is

the rotation rate of principal interest in texture evolution and can be given as follows

25A Shi-Hoon CHOI and Kyu Hwan OH

Fig. 1. Sample coordinate system, KA and crystal coor- dinate system, KR.

S

The lattice rotation rate can be obtained from the pres-

cribed velocity gradient L~j and shear rate ~. ~, is cal-

culated from rate insensitive or rate sensitive analysis. The Euler angles (%, ~ , %) of the deformed materials should be changed according to the lattice rotation rate as follows [25].

~ l=( f223s in <p2 + {2 31cos %)/sinq~

q~=~23 c~ q~2-~231 sin <P2 (12)

2=.hi:- ~ ,cos'/'

2.3. Numerical calculation of the texture evolution The numerical procedure described in above sections

was programmed. 800 crystallite orientations were used as the initial random orientation. The prescribed deformation rates of plane strain compression, combination of plane strain and shear strain were taken in this study. The fol- lowing velocity gradient was used during calculation.

Lij = /7 0 a = L11 0

r I is the strain rate increment during plane strain com- pression and set to be 0.01 in this study, ct values of 0 and 5 were considered to investigate the effect of shear deformation on the texture evolution. The total

Fig. 2. (111) pole figure of initial random texture with 800 crystal.

plane strain of 1.0 was used. The m values of 0.05, 0.1, (/.25 and fl.5 were used in this study to see the effect of rate sensitivity on the texture evolution during de- formation. Fig. 1 shows the crystal (KB) and sample (KA) coordinate systems. In this study, the principal axes of deformation rate tensor are coaxial with those of the sam- ple coordinate system.

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

3.1. Plane strain compression (o~=0) The initial orientation distribution of FCC metals in this

study was described by 800 randomly oriented grains. Fig. 2 shows the (111) pole figure of the initial random texture. During rolling processing, the detbrrnation in a center re- gion can be characterized by plane strain compression. The major orientation of rolling textures simulated numerically by Hitch et al. [26] was cube {100} <001>, G (Goss) {110} <100>, B (brass) {110} <112% S {123} <523> or {123} <634>, C (copper) {112} <111> and D (Diilamore) {44 11} <11 11 8>. The deformation textures of FCC me- tals during rolling form ~'o fibres: the ~ (G-B) and the

Fig. 3. The calculated (111) pole figures of plane strain compression using rate insensitive and rate sensitive models.

(a) rate insensitive model (b)-(e) rate sensitive model.

Rate Sensitive Analysis of Texture Evolution in FCC Metals 255

(B-S-C/D) fibres. Fig. 3 shows the calculated (111) pole figures of

plane strain compression using rate insensitive analysis (Taylor-Bishop-Hill model and RW method) which is e- quivalent to the result of m=0 (a) and rate sensitive models for m=0.05, 0.1, 0.25 and 0.5 (b-e). The cal- culated pole figures for m=0.05 and 0.1 are similar to those of rate insensitive model.

In order to investigate texture evolution, ODF (orien- tation distribution function) was calculated using defor- med crystal orientations and filtering procedure in the Euler space [27-29]. Fig. 4 shows the calculated ODF section ((P2 = 45") for plane strain compression at various rate sensitivity. Triclinic sample symmetry was con- sidered during ODF analysis to compare with the results of the combined deformation. The sample symmetry in Fig. 4 can be characterized as orthorhombic sample sym- metry. Fig. 5(a) shows the calculated orientation density, f(g) along fibre A ((p~ = 90 '7, @ = 0'~-90 '', (2: = 45'D. As rate

C o n t o u r L i n e s : 1 2 3 5 1 0 1,5 2 0 2 5

, : : :> %

0 ~

o 9 0

3 6 0 ~

(a )

(b)

(c)

(d)

(e)

Fig. 4. The calculated ODF section (q3:=45 ~ for plane strain compression.

(a) (0 0 1) (.4 4 I I) (i I I)

[L l o ] [ ~ ~ 8~ [] I 2] 60 . . . . . . .

f(g)

40

20

.--o--- : rate insensi t ive

--,<>-- : rn=O,05

: m=O. 1

: rn=0.25

= : m=0.5 ii

~P1 = 9 0 ~

~2=-450

30

( l 0)

[0 [1

- 6"0 . . . . 90

Co) (I ~ o)

l i i o] 40

30

f(g) z0

(1 I O) (I I O) (I

[ l i l l I ~ 2 ] [0

- - o - - : rate insensit iv i ty

- - c - - m=O.05

: m=O. 1

: m=0.25

--o--- : m=0.5

r ~

q,z--45 ~

0 30 60 90

(c) (44u) (2 I 3) (] 0 1)

60 i

---o.-- : rate insensit iv i ty

- - -o- - : m=O.05

--.a,-- ; m=O.1 40

�9 : rn=0.25

- 4 - - : m=0.5

fig)

o 45 60 75 90

q~2

Fig. 5. (a) The calculated orientation density, f(g) along fibre A. (c~=D), (b) The calculated orientation density, f(g) along fibre C. (o~=0), (c) The calculated orientation den- sity, f(g) along fibre ~. (o~=0)

256 Shi-Hoon CHOI and Kyu Hwan OH

R D

�9 (4 4 11)~11 I I I-]

(a)

RD RD RD

(b) (c) (d)

Fig. 6. The calculated (111) pole figures using ODF for plane strain compression.

R D

sensitivity of rate sensitive models increases, the max- imum orientation density of D component {44 11} <11 11 8> decreases. Fig. 5(b) shows the calculated often-

oo ~so ~

-:+==~,~ . . . . . . I :.__. �9 ..... _ . . . . [I, ~1 ....... �9 �9 -- ]~`~)~U``;`~`~;``~;~:``~`~;;~5~``~-~.~:~````:```~`~;~:'~:`~.::7~:~;~`t.~ ::

- " ~ " ~ " ' " ; ; ; ; ; ; " ~ : : " ' " " ~ F~ '~ '~ '~ : :~ -~U ' ' ; ; ; ; ; ; ; ' ' ; ' ; ' ' ' ' ' ~c - ' ' ' ' l , . . . . . I . , I . . . . . - - . , ] . . . . .

(a)

: i',::..::::ii2!!!!{!!!7:::::::'::::::i',::.:::::!i!!!![!!::iT::::::i!i/

(b)

!!!~!!!!!~!f!lllll[l!!!!!!g{!!!~!!!!!!!!!!!!!t++f!lll{l!!!!!!!!!!!!!!i

(c)

iiii!!!!!iiiii}ii[!iiiiii!!! ! !!iiiiiii!! !!!iiiiiiiiiiiiiii!i!!! !!ii

(d)

!!! !j!!+ ii+ iii:iiii ((((ili ii!i!ii! i iiill iilil iiiiii iiiii iiil i+i+iiil i![

(e) �9 (+ + ~ ; ) [F i Fi B] : D

�9 (1 I o)[1 'T 2] : B

Fig. 7. Lattice rotation rates in the q02=45" section for plane strain compression.

tation density, f(g) along fibre C(q~t = 0"-90", ~ = 90", q~2 = 45+'). The maximum orientation density for rate sen- sitivity of B component {110} <112> increases with in- creasing rate sensitivity. Fig. 5(c) shows the change of orientation density along ]3 fibre at various rate sen- sitivity. With increasing rate sensitivity, the orientation density of D and S component decreases and that of B component increases. Fig. 6 shows (111) pole figures re- calculated from ODF. The predicted texture in rate in- sensitive (a) and rate sensitive models for m=0.05, 0.1 and 0.25 (b-d) can be characterized by plane strain tex- ture of {44 11} <11 11 8>. The calculated texture in rate sensitive model for m=0.5 (e) cart be characterized by {110} <112> and {44 11} <11 11 8>. To investigate the stability of a particular orientation, lattice rotation fields are calculated for plane strain deformation.

Fig. 7 shows the lattice rotation fields for the ODF section (q~2=45"). The direction of the arrows indicates the orientation change and the length indicates the total rotation rate. Most orientations around brass and D move toward the exact brass and D orientation, respec- tively. For m=0.5 (e), most orientations around brass and D move slower than for m=0.05, 0.1 and 0.25 (b-d). From the lattice rotation fields, during plane strain com- pression, orientations move first into the o~ fibre and into the ]3 fibre, then along c~ to 13. At the lower rate sen- sitivity (a-c), the total rotation rate of most orientations are large. Thus most of orientation are distributed bet- ween S and D as shown in Fig. 5(c). However, at the high- er rate sensitivity over m=0.25 (d-e), the total rotation rate of orientations around a and ]3 fibre are small. Thus most of the orientations move first into the ~z and 13 fibre and move slowly along ]3 fibre. At high value of rate sensitivity (e), the orientation density along ~ fibre shows the uniform distribution and have the maximum density at {110} <112> and {44 11} <11 11 8>, as shown in Fig. 5(c).

3.2. Combination of plane strain compression and simple shear (a=5)

During rolling of FCC metals, the deformation state in

Rate Sensitive Analysis of Texture Evolution in FCC Metals 257

Fig. 8. The calculated (111) pole figures for combined strain deformation using rate insensitive and rate sensitive models.

a surface region differs from plane strain compression due to the frictional interaction between roll and metals. The deformation state at the surface region can be characterized by the combination of plane strain com- pression and simple shear. This deformation state gives the inhomogeneous texture through thickness direction. Fig. 8 shows the calculated (111) pole figures of the combined deformation, using rate insensitive (a) and rate sensitive models for m=0.05, 0.1, 0.25 and 0.5 (b-e). The calculated (111) pole figures for m=0.05 and 0.1

Fig. 9. The calculated ODF section (q)2=45 ~ for com- bined strain deformation.

show the similar results with that of rate insensitive model. Fig. 9 shows the calculated ODF section (q02=45 ~

(a)

f(g)

O)

f(g)

(o o i) ( 4 4 I I ) ( ] II) ( l

[l 1 O] I l l ]l 8] [~ ~ 2] [o 60 i i

40

20

(1 1 l)

[ l i o] 30

20

-.-.o--- : ra te insensitive

- - o - - : m : O . 0 5 ' rn=O. 1 <P1=90~

�9 : m = 0 . 2 5 e2 = 4 5 ~

30 60

(l I l) (1 I I) (I I)

[ 1 2 I] [0 i 11 [ i ! 2

i i

: ra te i nsens i t i v i t y

- - o - - : m = O . 0 5

: m=O. 1

= : m = 0 . 2 5

�9 : m : O . 5

r ~

~2=45 ~

O)

~]

10

30 60 90

q)l

Fig. 10. (a) The calculated orientation density, f(g) a- long fibre A. (ct=5), (b) The calculated orientation den- sity, f(g) along fibre B. (~=5)

258 Shi-Hoon CHOI and Kyu Hwan OH

Fig. 11. The calculated (111) pole figures using ODF for combined strain deformation.

for the combined deformation. Triclinic sample sym- metry was considered to evaluate the ODF. The sample symmetry for combined deformation can not be charac- terized as orthorhombic or monoclinic. The major com- ponents are different from those of plane strain com- pression shown in Fig. 4. The maximum orientation den- sity around major component for m=0.5 (e) is lower

Fig. 12. Lattice rotation rates in the q02:45 ~ section for combined strain deformation.

than that of plane strain compression. The calculated orientation density shows more complicate distribution than that of plane strain compression. Fig. 10(a) shows the calculated orientation density along fibre A (q01=90 ~ q~=0%90", tp2=45'~). The maximum density orientation is between RC (rotate cube) component and D component (001) [110] and D component (44 l l ) [11 11 8]. With in- creasing rate sensitivity, the maximum density orien- tation moves toward RC component and the value of maximum density decreases. Fig. 10(b) shows the cal- culated orientation density along fibre B (q)l=0~ ~ (I)--- 55 ~ q0z=45"). The fibre B can be characterized by { 111 }// ND texture. In the values of m > 0.1, the orientation den- sity of {111}//ND texture components is negligible.

Fig. 11 shows the calculated (111) pole figures for combined deformation at various rate sensitivity. The cal- culated texture in rate insensitive (a) and rate sensitive models lot m=0.05, 0.1 (b-c) can be characterized by shear type texture components of (119) [14 143], (557) [8 11 14] and (557) [11 8 14]. For the m=0.25 (d), major component is characterized by shear type texture com- ponent, (119) [14143]. These shear type texture com- ponents (119) [14 143] and (557) [8 11 14] have 20 ~ and 15" rotational relationships with (44 11] [11 118] and (110) [001] orientations about TD, respectively. For m= 0.5 (e), intensity of pole figure has the value of about 1 and the major component can not be seen as expected from Fig. 9(c). Fig. 12 shows lattice rotation fields for combined strain deformation. During combined de- formation, orientations around shear type texture com- ponents move toward the shear type texture components.

4. C O N C L U S I O N

The texture evolution of FCC metals during plane strain compression and combined strain deformation (plane strain compression and shear strain) was analyzed by rate insensitive and rate sensitive models. During the plane strain compression, the major orientations were classified by {4411} <11 11 8>,. D component and

Rate Sensitive Analysis of Texture Evolution in FCC Metals 259

{110} <112>, B component. The D component was stable in rate insensitive and rate sensitive for m=0.05,

0.1 and 0.25. The B and D components were the major

components in rate sensitive model for m=0.5. For com-

bined deformation, shea__r type texture components ((1 19)

[14 143], (557) [8 I I 14] and (557) [11 8 14] were major orientations.

ACKNOWLEGMENT

This work was financially supported by the Korean

Ministry. of Education through the Advanced Materials Research Program in 1996.

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