METALS AND MATERIALS, Vol. 5, No. 1 (1999), pp. 17-23

Analysis of Texture Evolution of Cubic Metals by Isotropic and Anisotropic 、Tiscoplastic Self-Consistent Models

Kyu Hwan Oh

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

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

The texture evolutions of cubic metals during tension, compression and rolling have been analyzed by isotropic and anisotropic self-consistent models with the large deformation theory of viscoplastic material. The final tεxturε components prεdicted from an initial random texture were the same in both isotropic and anisotropic self-consistent models. The effect of rate sensitivity on the rolling texture component was analyzed. π1ε intεnsity of the major Copper component of the roIling texture increased with the decreasing of rate sensitivity and the increasing of rolling strain.

Key words : texture. self-consistεnt modcl, tension, rolling, compression

1. INTRODUCTION

In polycrystal plasticity, two extreme points of view have been used, i.e., Taylor and Sachs approaches. The Taylor approach requires a complete strain compatibility between single crystals and results in a uniform plastic strain throughout single crystals and the p이ycrystal.

This assumption has been successfully applied to modeling of weakly anisotropic materials, but hardly appJied to strongly textured materials such as hexagonal ones. Thε Sachs approach requires a stress equiJibrium condition and imposes a constant stress in each singJe crystal and the poJycrystal, which leads to a )ow number of active slip systems, typically a single slip. The Sachs theory has not been very successful for polycrystalline metals [1]. The self-consistent approach lies between the Sachs and the Taylor models and takes into account both the compatibility and equilibrium conditions.

A self-consistent scheme has been formulated for small deformation by Kroner [2], Berveiller [3] and Hill [4], for large deformation of elasto-plastic material by Iwakuma [5] and for large deformation of isotropic viscoplastic material by Molinari [6] and the anisotropic viscoplastic material by Lebensohn [7]. The application of the equilibrium and compatibility equations gives an integral equation in the self-consistent schemes, where a single crystal in the polycrystal is considered as an inclusion embedded in a homogenous equivalent medium

(HEM). The interaction law between a single αystal and HEM and the constitutive equations should be solved together, which req띠res so-called inner and outer iterations in the Newton Raphson method [6,7]. πle viscoplastic self­consistent model has been used to analyze the te었ure

evolution at various deformation modes [6,7], multipha"e material [8,9] and damage modeling [10].

In this study, the viscoplastic self-consistent models by Molinari [6] and Lebensohn [7] were used for the analysis of texture evolution in fcc metals during tension, compression and rolling. The texture evolutions from the two models were compared. During rolling, the effect of rate sensitivity on the texture evolution was aJso discussed in both models. The calculated results were compared with reported experimental data.

2. FORMULATION

2.1. Stress-strain relation The shear rate of a slip system s, ys is related to the

resolved shear stress, 1:', by a power law relation, which is called the 、riscoplastic law [11].

훨 = sgn<y')[쩔r where m is the rate sensitivity. 10' and 강 are the reference shear rate and reference critical resolved shear stress, respectively. The microscopic hardening law of

18 Kyu Hwan Oh

singJ e cηstals 때 be taken into account by changing the critical resolved shear stress 캠. The strain rate can be calculated as follows:

Dl=쭈R;(縣1 摩 I ,j,-]

where D and d are the strain rate and the deviatoric Cauchy stress tensor, respectively. The Schmid tensor, R , is associated with the slip systems as follows:

” ] / n ‘ “

, , J

A U

+ s j

n A U

1? = j ” R

where b and n are the slip direction and the slip plane normal direction, respectively. The above stress and strain rate relation can be inverted. First order Taylor expansion and a secant expansion of the stress-strain relation give the following Eqs. 4a and 4b, respectively.

ð;j=Aijk1 :Dk1 +(500 , and Dij = Mijk1 : 따+DI; (4a)

(5ij=A i써 :DId , and Dη =Mi쩌 : 따 (4b)

where A and M are the tangent stiffness and compliance tensor, respectively, which are dependent on strain rate D and stress s. The superscripts 0 and s denote the back extrapolating tensor and the secant tensor, respectively. The equations, 4a and 4b are called the tangent and secant formulations, respectively. These formulations are the bases of the self-consistent models and are assumed for the singlε crystals and for a whole p이ycrystal as follows:

급j=Aijk1 :Dk1 +육lyO and jlJ =MIlkl :마'+Di~ (5a)

i;=A짧 : Dk1 and Dij = M;씹 :급kl (5b)

The bar over the tensor property denotes the macros­copic property over polycrystal. When the rate sensitivity, m, is the same for all single crystals, the overall secant and tangent stiffness and compliance are line ,:trly related as follows:

A;샘 = nlAljkl and k쇄 =mMijk1

2.2. Self-Consistent Model [12] In the self-consistent model, the p이ycrystalline

material is continuous in space, but the properties are not uniform. The self-consistent condition can be represented by

<(5(j> = (50 (7a)

(7b) <Dij> =Dij

(2)

the p이ycrystal. Eqs. 7a and 7b represent the stress equilibrium and compatibility conditions, respectively. Using the Eshelby formalism, the polycrystalline material is regarded as a homogeneous equivalent matrix (HEM) having an unknown viscoplastic stiffness or compliance tensor, which should be solved from the self­consistent model. A single crystal in a p이yσystal can be regarded as an inhomogeneity embedded in HEM. Within this approach, an inhomogeneity or a single crystal is replaced by an ’equivalent inclusion' having the same modulus of the polycrystal and undergoing a fictitious transformation strain rate or body force. The equilibrium and compatibility conditions lead to a set of differential equations expressed by the strain rate or stress state of single crystal. Knowing the macroscopic velocity gradient, L at the boundary, the set of the differential equation can be solved by means of thε Green function method. During the analysis, the stress and strain can be decomposed into the space independent part and the space dependent part, which are related to the macroscopic property and the single crystal property, respectively.

(3)

σ'，= <r :，+급'， and D" =δ +D 'J - lJ -'J ---- - 'J - lJ . - 'J (8)

(6)

The tilde over tensor property denotes the space dependent part. 2.2. 1. Isotropic Self-Consistent Model [6,13]

Molinari derived the general interaction equation between a single crystal and HEM after proper derivation and symmetrization of the set of differential equations. Finally, the strain rate at any location of the polycrystalline material was obtained as follows.

D~ =Dij + J많(싫n :Dmn +삼n)따 (r’) (9)

The influence tensor r is associated with the interaction of strain rates between two points r and r’. Assuming a uniform stress and strain rate in the single crystals and ignoring the interaction between single crystals, the above equation can be expressed as follows, which is called the l-site formulation.

σll/-δtl/=Oll/-Al써n :Dmn

=a{_T;j짧g+A쇄):(Dμ -D，μ )

α=0 : Sachs Model, a=m : Tangent Model α:=1 : Secant Model, α=∞: Taylor Model

The meaning of parameter a is the degree of

interaction between a single σystal and matrix (HEM).

The Sachs model has no interaction and the Taylor

m / , , ‘ 、

where < > means thε arithmetic volume average over model has an infinite interaction. The degree of

Analysis o[ Texture Evolution o[ Cubic Metals by Isotropic 19

interaction of the self -consistent model is located between the 싱achs and Taylor models. The recommended value of inten따ion parameter, α is as follows:

a==1-0.4(m-l)2 (11)

The above equation can give the macroscopic secant modulus .:4(잉 in a numerical way. In case of a plastically isotropic material, the property of HEM is isotropic and the interaction tensor .:4(') can be given as follows:

A폐=띠l}씨 2 σJ

μ== -:::;mτ-j Deq

(12)

where 1 is the fourth order Kroneck delta. α~ and Deq are the effective stress and the effective strain rate of the HEM, respectively. μ is the plastic modulus of HEM. 2.2.2. Anisotropic Self-Consistent Model [7]

The problem of viscoplastic inhomogeneity embedded in a homogeneous viscoplastic matrix being acted upon by a uniform stress at infinity can be solved by the inhomogeneity problem of the Eshelby formalism [12] At a single crystal, the following equation can be obtained.

D" - D/f = M"" : (J/,; ij -n~ijkl'Vkl (13)

where Dt is a fictitious transformation strain rate due to the replacement of the inhomogeneity as an equivalent inclusion. At the uniform strain rate in the inclusion, the fictitious strain rate can be given by

D i ; = S;jkl : Dk1* (14)

where S is the viscoplastic (instead of elastic) Eshelby tensor which is a function of the shape of the polycrystal and the macroscopic tangent viscoplastic modulus. Using thε above equation, the solution of the 、'iscoplastic inhomogeneity problem leads to the mtenchon equatton

D" = - M",., : 0',.; ijkl . Vkl (15)

where the interaction tensor of a single crystal, M , is defined as follows:

M'jkl = (/;jmn -Sijmnt1 : Smnpq : Mpqkl

= 승(I;jmn -S;jmn r 1 : 쇄q . 짧 (16)

Using Eqs. 2, 13 and 14, M，꾀， the secant viscoplastic modulus of a polycrystal can be obtained from the following transcendental equation in a numerical way.

M，씹 =<M싸 :(M，싸q + Mmnpq t 1 : (Ã짧'( + Mpqk1 ) > (17)

까(s) represents the anisotropic behavior of HEM. Using the obtained M') and Eqs. 2, 15 and 16, the stress state of each single crystal can be solved. AlI of the iterations continue to satisfy Eq. 7 over the polycrystal.

2.3. Calculation of Rotation Rate [13] After a convergence of the isotropic and anisotropic

self-consistent solutions, the lattice rotation rate of a single crystal, Q , can be given as follows:

!2ij = f2; j +n，옆1: 며짧 : (Dmn -Dmn)

-잃(binj -bjni )’ ys (18)

where .Q is the maσoscopic rotation rate. The tensor n can be obtained by integrating the antisymmetric paπ of the second derivative of the Green function. The second term is the reorientation of the associated single crystal due to the difference between plastic strain rates of a single crystal and a polycrystal. 까le last term is the antisymmetric component of the plastic rotation rate. From the obtained lattice rotation rate of a single crystal , Q , a new orientation of single crystals can be calculated from the rotation rate of the Euler angle given as follows:

ψ 2 = (.Q닝2sin φ ]-.Q13COS 망 l )lsinφ

φ = μ2COS P 1 + .Q13sin φ1 φ 1 =~1- P2COSφ. (19)

2.4. Calculation of an Ideal Component During De­formation

To obtain the ODF value of a single component during deformation, the following procedures were used. Single orientations in the texture data during deformation were converted into the orientation distribution function around a single ideal component, called the "standard function" [15 ,16]. πle "Gaussian standard function" can bε given by

[(S , w)=N(S)éCOSω , N(S)= 1/(/o(S)-/ 1(S)) , 1 π

μ (x) = ~ I ex cost cos kt dt n" 0

S =ln2/[2sin2(b/4)] for b~ 2껴드S 츠 Inδ) (20)

where f(S, w) expresses the relative densities of a orientation rotated through an orientation and distance &띠gkÜ， g) from the single ideal orientation gko

. b and Ik (x) mean the full width at half-maximum of the bell­shaped curve and modified Bessel function, respectively. A b of 12S was used in this study. The obtained ODF from all of the single crystals was normalized to have

20 Kyu Hwan Oh

the ìntegrated value of 1 for the whole Euler space. During calculation of the texture evolution by the two self-consistent models, the major components are calculated for each calculation step using Eq. 20.

3. CALCULATION PROCEDURE

The texture evolution of cubic metals is calculated in a model material with no strain hardening. The reference critical resolved shear stress τs of any slip system is taken as constant. The slip system is {111 }<1 1O> and the shape of a single crystal is assumed to be spherical during deformation. 까uee deformation modes are considered: tension, compression and rolling. For each deformation mode, the macroscopic velocity gradient L is given. For example, in the case of rolling, macroscopic velocity gradient L is set to be

11 0 0 1

Lij = 비 o 0 0 I Ðij =Lij qj =0 (21)

100-11

where 11 is the increment of the velocity gradient, typically set to be 0.01.

Fig. 1 shows the (111) pole figure of an initial random texture having 200 grains.

The convergence of interactìon laws, Eqs. 10 and 15 is considered to be achieved during the two-level iterations of self-consistent calculations, when

1 <Ðii >-Ðij 1 max 'J 'J' < 0.01 and νeq

|<qI/ > -q/|max 'J 'J • < 0.01 (22)

,. .... , -eq

Fig. 1. (111 )pole figure of thε random texture.

4. RESULT AND DISCUSSION

Figs. 2(a) and (b) show the calculated (111) pole figures from the isotropic and anisotropic self-consistent models, respectively, after a tensile strain of 1.0. The two pole figures are very similar and show two texture components of (100) and (111) fiber components. Figs. 3(a) and (b) show the calculated inverse pole figures along the tension axis from the isotropic and anisotropic self-consistent models, respectively. The reported texture components of most fcc metals after tension were minor (100) and m매or (111) fiber components. The ratio of (111) to (100) fiber components is dependent on the stacking fault energy [17). The reported {1l 1} to {100} fiber ratio is about 2 in pure Copper. πle {111} to {100} fiber ratio from the isotropic and anisotropic self­consistent schemes are 2.5 and 1.1, respectively. The ratio from the isotropic self-consistent model is c10ser to the experimentally measured ratio than that from the anisotropic self-consistent models.

Figs. 4(a) and (b) show the calculated (111) pole figures from isotropic and anisotropic self-consistent

Maximum 19.9 Max imμm 17.2

RD RD

• ( , 1 2) I 57 o? - ~71 • ( ‘ 1 1) [ ~7 " -,,’ • ( , 1 2) f - 57 - 57 ~7 1 • ( 1 ‘ 2) [- 57 -57 57)

w (~

Fig. 2. (111 )pole figures after tensile strain of 1.0 by self-con­sistent models. The contour values are 1, 2, 5, 10, and 15. a)iso­tropic model, b )anisotropic model

mox . 20.0 m ax . 17. 3

1 0。 ’ 10 10 0 (8) (b)

\1 0

Fig.3. πle Inverse pole figure along the t，εnsile axis from a) iso­tropic rnodel, b) anisotropic model. πle contour values are 1, 2, 5, 10, and 15

Analysis of Texture Evolution of Cubic Metals by lsotropic 21

Moximum 4.6 Moximum 4.6 RD RD

. ' ( , 0 이 1 0 , ‘J • ( , 0 이 1 0 , ’l (a) (b)

Fig. 4. (l11)pole figures after a compression strain of 1.0 by self­consistent models. ηle contour values are 1, 2, 3, and 4. a) iso­troplc model, b) anisotropic model

‘ 1 0。

mox. 17.8 max. 17 .4

110 100 11 。

(a) (b)

Fig.5. 꺼le inverse pole figure along compression 앓is from a) isotrppic model, b) anisotropic model. The contour values are 1, 2, 5, 10, and 15.

models, respectively, after a compression strain of 1.0. Figs. 5(a) and (b) show the calculated inverse pole figures along the compression axis from the isotropic and anisotropic self-consistent models, respectively. The reported texture component of fcc metals after compression was the (11이 fiber component. The calculated results from the two self-consistent models

Moximum 11.0 RD

Moximum 12.2 RD

show that the m매or component is the (110) component. The maximum intensity of the two inverse pole figures is the same, which represents the same texture evolution of the two self-consistent models.

Figs. 6(a), (b), (c) and (d) show the calculated (111) p이e figures from the isotropic self-consistent model after a rolling strain of 1 with the rate sensitivities of O. 05, 0.1 , 0.2 and 0.5, respectively. At the low rate sensitivity of 0.05, the m떼or Copper component {112} <11 1> and minor Brass component {1 1O}<111> are shown in Fig. 6(a). The Copper and Brass components decrease with increasing rate sensitivity. Figs. 7( a), (b), (c), and (d) show the calculated (111) pole figures from the anisotropic self-consistent model after a rolling strain of 1 with a rate sensitivity of 0.05, 0.1, 0.2 and 0.5, respecti vely . πle m에or Copper and minor Brass components develop at the low rate sensitivity and intensity of the components decreases with the increasing of rate sensitivity. The behavior of texture evolution from the isotropic and anisotropic self­consistent models shows almost the s없ne pole figures as shown in Figs. 6 and 7.

With the increasing of the rate sensitivity, the random component of the obtained texture increases and at rate sensitivity of m=l , a random texture was obtained. With the increasing of the rate sensitivity, the rotation rate of all orientation decreases and at the rate sensitivity of m= 1, the rotation rate of any orientation becomes zero. Thus the calculated texture is the same as the original texture, i.e., the random texture as shown in fig. 1.

In the self-consistent models, due to the low interaction between single crystals, the m매or texture component was the Copper component [7]. In full constraint Taylor material, the degree of interaction between single crystals is infinite as shown in Eq. 10

Moximum 10.7 Moximum 4.7 RD RD

• ( 1 1 2) [ ~ ~7-~7J • ( 1 1 2) ( 57 "-5끼 • ‘ 1 1 2) [ 57 ~7-~71 • ( 1 1 이 [ 5 ’ -~I -뻐l • ( 1 1 2) [-57-57 ~끼 • ( 1 1 2) [ -57-57 57) • ( 1 1 2) [ - 57- 57 5기 . ( , 1 이 [ -51 5’- M)

(a) (b) (c) (d) Fig. 6. (1l1)pole figures from the isotropic self-consistent model after a rolling strain of 1.0 at various rate sensitivities. The contour values arε 1, 2, 5, and 10 in a), b), c) and 1, 2, 3, 4 in d). a) m=0.05, b) m=O.l, c) m=0.2, and d) m=O.5

Kyu Hwan Oh 22

4.5 Moximum RD

9 .9 Moximum RD

Moximum 12.6 RD

Moximum 13.5 RD

• ( 1 1 2) ( 51 07-5기 • ( 1 1 2) ( 57 57-57J • ( 1 1 0) [ 51 -51-... J

• ( 1 1 2) (-57- 57 57) • ( \ 1 2) (-57- 57 57J • ( 1 1 0) [-51 51-08J

(b) ~) (d) Fig. ~'. (111)pole figures from the anisotropic self-consistent model after a rolling sσain of 1.0 at various rate sensitivities. The contour values are 1, 2, 5, and 10 in a), b), c) and 1, 2, 3, 4 in d). a) m=O.05, b) m=O.l , c) m=0.2, and d) m=O.5

. ( 1 ’ 지[ 57 5’‘ 57)

• ( \ t 2) <-57-57 57J

(a)

the Copper and Brass components increase almost linearly with the increasing of the rolling strain in the two seJf-consistent models. The applied strain causes the plastic shear in the slip system, which is the intemal shear strain. The symmetric part of the intemal shear strain is equivalent to the applied strain and the anti­symmetric part of the intemal strain is εquivalent to the lattice rotation rate in the case of rolling [14]. ln Eq. 18, the lattice rotation rate is composed of the macroscopic rotation rate, the reorientation of the single crystal and the plastic rotation rate. In the case of rolling, the macroscopic rotation rate is zero as shown in Eq. 21 and

and the stable texture component is the D component. With the decreasing of the interaction, the stable component changed from the D component to the Copper component. With the increasing of the rate sensitivity, the ratio of major Copper to minor Brass components decreased [13].

Figs. 8 and 9 show the calculated ODF values of the Copper and Brass components, respectively, during their rolling from the isotropic and anisotropic self-consistent models. ln the initial stage of rolling, the ODF intensity of the Copper and Brass ∞mponents are 1 because the initial random texture was used. The ODF intensities of

: m-0.05 : m=0.10 : m=0.20 : m=O. 50

Isotrople Anlsotrople

--- ‘ -

-“--‘,,_

30

25

20

15

10

s

(@)i

-

ιQ。

: m=0.05 : m=0.10 : m=0.20 : m-O.50

Isoν'op1e Atil.,otropic

---‘--• -

-‘,.... _,_ -‘0-

~

30

25

20

(

g ;;:- 15 u.. 。

。

10

5

1.0

Fig. 9. The variation of the ODF value of the Brass component at various rate sεnsitivities from isotropic and anisotropic models

0.8 0.4 0.6 Strain. &

0.2 1 .。

Fig. 8. The variation of thε ODF value of the Copper component at various ratc scnsitivitiεs from isotropic and anisotropiι models.

0.8 0.4 0.6 Strain. &

0.2 O 0.0

Analysis 01 Texture Evolution 01 Cubic Metals by Isotropic 23

the reorientation of the single crystal is zero because the tensor II is zero in the sphe더cal shape. Thus the linear increase of the applied strain causes the linear increase of the internal shear strain and consequently the linear increase of the lattice rotation rate. The linear increase of the lattice rotation rate leads to the almost linear development of the major Copper component in the range of rolling strain 1 as shown in Fig. 8. πle intensity of the Copper component in thε two self­

consistent models is almost the same at the same rate sensitivity. The same ODF value of the major component during deformation shows that the texture evolution mechanism is similar in both self-consistent models. The similar texture evolution in the isotropic and anisotropic self-consistent models shows that the anisotropy of the viscoplastic secant matrix of the fcc polycrystal in Eq. 17, 까(，\ is low and sirnilar to the inverse of the plastic sti뼈less given in Eq. 12. The texture evolutions in highly anisotropic material such as hexagonal and orthogonal materials are differεnt from those in isotropic materials [7] because the viscoplastic secant matrix is a highly anisotropic secant matrix due the low symmetry of the slip system. But in the cubic metals, the anisotropy is not high, because the {111} <110> slip system has a cubic symmetη. The high symmetry of the slip system gives rise to the high symmetry of the viscoplastic secant matrix, leading to the almost isotropic behavior of the texture evolution in fcc metals.

At thε low rate sensitivity of m=0.05 and 0.1 , the intensity of the major Copper component is almost the same in both self-consistent models. Below the rate sensitivity of m=0.15, the texture evolution of rate sensitive material with a full constraint Taylor model shows the same behavior [14]. With the increasing of the rate sensitivity, the intensities of Copper and Brass components decrease as shown in pole figures in Figs. 6 and 7.

5. CONCLUSIONS

The texture evolution of fcc metals during tension, compression and rolling has been analyzed by isotropic and anisotropic self-consistent models. The final textures

of tension, compression and rolling showed almost the same texture components in both isotropic and anisotropic self-consistent models. The effect of rate sensitivity on the rolling texture component was analyzed. With the in다'easing of the rate sensitivity, the rate of texture evolution decreases. The major Copper component increases almost linearly with the increasing of the rolling strain. The texture evolution is the same in both self-consistent models due to the high symmetry of the slip system at various rate sensitivities.

ACKNOWLEDGEMENT

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

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