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Materials Science and Engineering A 464 (2007) 101–109

Analysis of crystallographic twinning and slip in fcccrystals under plane strain compression

Zhiyong Chen a,b,∗, Hongnian Cai a, Saiyi Li c, Xinming Zhang b,Fuchi Wang a, Chengwen Tan a

a School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, PR Chinab School of Materials Science and Engineering, Central South University, Changsha 410083, PR China

c School of Mechanical Engineering, South China University of Technology, Guangzhou 510640, PR China

Received 4 July 2006; received in revised form 20 January 2007; accepted 19 March 2007

bstract

The Bishop–Hill maximum work principle was applied to analyze the contribution of {1 1 1}〈1 1 0〉 slip and {1 1 1}〈1 1 2〉 twinning to theeformation of face-centered cubic crystals under plane strain compression. The influence of ξ, the ratio of critical resolved shear stress forwinning to that for slip, on the yield stress states and the selection of active systems for main ideal orientations of interest was investi-ated. The results showed that twinning is difficult for the {1 0 0}〈0 0 1〉, {1 1 0}〈0 0 1〉 and {1 1 2}〈1 1 0〉 orientations, and becomes easieror the {1 1 0}〈1 1 2〉, {1 1 2}〈1 1 1〉, {1 2 3}〈6 3 4〉, {1 1 0}〈1 1 1〉 and {1 1 0}〈1 1 0〉 orientations. Compared with pure slip, the yield strengthnisotropy decreases with the introduction of twinning. A factor based on the Taylor factors was introduced to describe the twinning ability of

ifferent orientations. The analytical results are in qualitative agreement with the experimental rolling textures in a Cu5%Zn alloy. The effectf the ξ on the rolling texture development at the polycrystalline level was also illustrated by simulations using the full constraints Taylorodel. 2007 Elsevier B.V. All rights reserved.

press

mawaaatit

(s

eywords: Twinning; Slip; Texture; Deformation mechanism; Plane strain com

. Introduction

Both slip and twinning can take place for face-centered cubicfcc) metals in many cases, especially for those with mediumo low stacking fault energy and deformation at low tempera-ures and high strain rates [1–6]. In such cases, investigationsnto the relative contributions of slip and twinning to the plasticeformation are of importance in understanding the deforma-ion mechanism under a given deformation conditions. However,nvestigations about slip and twinning deformation mechanismn fcc crystals are often conducted separately and twinningas received less attention than slip [7]. Although extensive

tudies have been carried out to evaluate the influence of twin-ing on the texture evolution during plastic deformation forolycrystalline aggregates [2,3,8,9], there exist relatively few

∗ Corresponding author at: School of Materials Science and Engineering, Cen-ral South University, Changsha 410083, PR China. Tel.: +86 731 8830489;ax: +86 731 8830489.

E-mail address: czysh@netease.com (Z. Chen).

(itcmmlt

921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2007.03.081

ion

odeling and simulations results for polycrystalline materi-ls that deform by slip and twinning simultaneously comparedith those for polycrystalline materials that deform by slip

lone [10]. There are even fewer investigations on the slipnd twinning co-deformation and the corresponding active slipnd twinning systems for single crystals with arbitrary orienta-ions, although such studies can provide useful information innterpreting experimental and simulation results for polycrys-als.

In a recent investigation [11], the single crystal yield surfacesSCYS) and their characteristics for fcc crystals were studiedystematically when ξ, the ratio of critical resolved shear stressCRSS) for {1 1 1}〈1 1 2〉 twinning to that of {1 1 1}〈1 1 0〉 slip,s different. All possible types of yield stress states for slip and/orwinning were tabulated and subdivided into crystallographi-ally non-equivalent groups. As an extension of Ref. [11], the

aximum work principle of Bishop–Hill [12,13], which wasainly applied to slip deformation, has been employed to ana-

yze plastic behavior of fcc metals for crystallographic slip andwinning with different ξ. The present study focuses on the influ-

1 nd En

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δ

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wid

diticIatare shown in Table 1. The influence of ξ on the yield strengthanisotropy and rolling texture developments has also been ana-lyzed.

Table 1The main ideal orientations in fcc metals considered in this study

Notation Miller index {h k l}〈u v w〉 Euler angles (◦)

ϕ1 φ ϕ2

W {1 0 0}〈0 0 1〉 0 90 90G {1 1 0}〈0 0 1〉 90 90 45B {1 1 0}〈1 1 2〉 54.7 90 45C {1 1 2}〈1 1 1〉 90 35.3 45

02 Z. Chen et al. / Materials Science a

nce of ξ on the yield stress states, twinning ability and yieldtrength anisotropy for selected orientations of interest in fccrystals under plane strain compression (PSC) deformation. Thenalytical results will be discussed in comparison with modelingnd experimental rolling textures in a model fcc material and au5%Zn alloy, respectively.

. Theory

.1. Co-yield stress states for slip and twinning

For plastic deformation of a single crystal, it obeys thechmid law. When a single crystal is deformed in uniaxial ten-ion, only one shear is required to fulfill this tensile deformationince the lateral dimensions can be deformed freely. In this case,he shear system with the maximum resolved shear stress oper-tes. On the other hand, for the deformation of an arbitrarilyrientated crystal in a polycrystalline aggregates, five indepen-ent shears are required to accommodate five independent strainomponents. It has been confirmed [11] that the SCYS variesith ξ and there exist four basic types of yield stress states. Only

lip occurs when ξ > 2/√

3, and there are 56 yield stress stateshat can be subdivided into five groups according to the crystalymmetry. Only twinning happens when ξ < 1/

√3, and there

re 25 yield stress states that may be classified into 4 groups. For/√

3 ≤ ξ ≤ 2/√

3, both slip and twinning take place and therere only two types of yield stress states, depending on whetheris greater or smaller than

√3/2: one type consists of 259

tress states when√

3/2 < ξ < 2/√

3, which can be divided into1 groups; another 259 ones as well when 1/

√3 < ξ <

√3/2,

hich can be subdivided into 19 groups. Between the two typesf the stress states, 139 ones are common and 120 ones areifferent.

Therefore, just the four kinds of cases need to be consideredhen ξ is within the different ranges. According to all the possi-le slip and/or twinning yield stress states, the maximum workrinciple of Bishop–Hill [12,13] has been applied to analyzedeal PSC deformation of fcc metals for {1 1 1}〈1 1 0〉 slip and1 1 1}〈1 1 2〉 twinning.

.2. Application of Bishop–Hill maximum work principle tolane strain compression

The plastic deformation under ideal PSC can be described byn elongation in x1-direction and a contraction in x3-direction,r in the form of strain increment tensor:

εskl =

⎡⎢⎣

δε11 0 0

0 0 0

0 0 −δε11

⎤⎥⎦ (1)

here the superscript ‘s’ refers to the sample coordinate system.he absolute value of the deformation is assumed to be a small

uantity. Using the fully constraints (FC) assumption of Tay-or [14], all grains experience the same deformation as that ofolycrystalline aggregate. Since the macroscopic strain tensors defined respected to the sample coordinate system, it must be

SPrA

gineering A 464 (2007) 101–109

ransformed into the crystal coordinate system in order to applyhe maximum work principle of Bishop–Hill. According to theensor transformation rule:

εcij = aikajlδε

skl (i, j, k, l = 1, 2, 3) (2)

here the superscript ‘c’ stands for the crystal coordinate sys-em, and aik is the orientation matrix of the crystal coordinateystem with respect to the sample coordinate system. For anrbitrarily oriented crystal, since all possible yield stress statesincluding slip and/or twinning yield stresses) are known, thectual stress state (and hence active systems) can be selectedccording to the maximum work principle of Bishop–Hill12,13]:

w = σijδεcij = σkδε

ck = δwmax (3)

That is, the stress state giving the maximum value of δw, i.e.wmax, will actually motivate the strain. Here, σij and σk, δεij

nd δεk are the yield stress tensor and vector, strain tensor andector, respectively. As a measure of the relative yield strengthf the crystal orientation, the so-called Taylor factor (M) can bealculated by

= δwmax

2/√

3δεs11τcs

(4)

here τcs is the CRSS for {1 1 1}〈1 1 0〉 slip. Then, the M-factors proportional to the work needed to bring about the sameeformation in crystals with different orientations.

Using the maximum work principle of Bishop–Hill, the five-imensional yield stress states (the definition of notations is seenn Ref. [15]), active slip and/or twinning systems (the defini-ion of notations is seen in Ref. [11]) and M-factors of maindeal orientations in fcc crystals under ideal PSC have beenalculated for four representative cases with ξ > 2/

√3 (Case

),√

3/2 < ξ < 2/√

3 (Case II), 1/√

3 < ξ <√

3/2 (Case III)nd ξ < 1/

√3 (Case IV), respectively. The orientation nota-

ions, Miller indexes and Euler angles of the studied orientations

{1 2 3}〈6 3 4〉 59.0 36.7 63.4{1 1 0}〈1 1 1〉 35.3 90 45

ot.G {1 1 0}〈1 1 0〉 0 90 45{1 1 2}〈1 1 0〉 0 35.3 45

Z. Chen et al. / Materials Science and Engineering A 464 (2007) 101–109 103

Table 2The Taylor factor values, yield stress states, numbers of sets of five independent systems and active systems for the ideal orientations in fcc crystals under planestrain compression (Case I: ξ > 2/

√3)

Notation Taylor factor Yield stress states(multiplied by

√6τcs)

Sets of fiveindependent systems

Active slip systems Common active slip systems

W 3√

2/2 (−1/2, −1/2, 0, 0, 0) 32 s1 s4 s7 s10 s15 s18 s21 s24 s15 s18 s21 s24

3√

2/2 (−1/2, 1/2, 0, 0, 0) 32 s2 s5 s8 s11 s15 s18 s21 s24

3√

2/2 (−1/2, 0, 0, 0, −1/2) 36 s2 s5 s7 s10 s15 s18 s21 s24

3√

2/2 (−1/2, 0, 0, 0, 1/2) 36 s1 s4 s8 s11 s15 s18 s21 s24

G 3√

2/2 (0, 1, 0, 0, 0) 32 s2 s5 s8 s11 s13 s16 s19 s22 s2 s5 s13 s16

3√

2/2 (0, 0, 0, 0, −1) 32 s2 s5 s7 s10 s13 s16 s20 s23

B 2√

2 (0, 0, 0, 0, −1) 32 s2 s5 s7 s10 s13 s16 s20 s23 s5 s7 s13 s20

2√

2 (0, 0, −1/2, 1/2, −1/2) 6 s3 s5 s7 s13 s18 s20

C 9√

2/4 (0, −1/2, −1/2, −1/2, 0) 4 s1 s9 s10 s14 s20 s24 s1 s9 s14 s24

9√

2/4 (0, 0, −1/2, −1/2, 1/2) 6 s1 s9 s11 s14 s19 s24

S 3.02 (0, −1/2, −1/2, −1/2, 0) 4 s1 s9 s10 s14 s20 s24 s1 s9 s10 s14 s20 s24

P 5√

2/2 (0, 0, 0, 0, −1) 32 s2 s5 s7 s10 s13 s16 s20 s23 s2 s5 s7 s10 s13 s16 s20 s23

rot.G 3√

2 (0, 0, 0, 0, −1) 32 s2 s5 s7 s10 s13 s16 s20 s23 s2 s5 s7 s10 s13 s16 s20 s23

A

3

3s

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2√

2 (0, 0, −1/2, −1/2, −1/2) 62√

2 (0, 0, 0, 0, −1) 32

. Results and discussion

.1. The influence of ξ on the yield stress states and activeystems

Generally, the yield stress states and active systems vary withhe crystal orientation and the ξ value. In Case I (ξ > 2/

√3,

able 2), only slip occurs. The orientations W, G, B, C and A arectivated by more than one yield stress state and there are fourommon active slip systems associated with each orientation.or orientation W, the first two yield stress states and the last

wo belong to the same group according to the crystal symme-ry. For orientations B and A, the yield stress states that activateight slip systems are common and the stress states that activateix systems belong to the same group of stress states. The orien-ations S, P and rot.G are activated by only one yield stress state.or orientations P and rot.G, the yield stress state is the same as

he one activating eight systems for orientations G, B and A.In Case II (

√3/2 < ξ < 2/

√3, Table 3), slip and twinning

an take place simultaneously. For orientations W, G, B, C and A,hey are also activated by more than one yield stress state com-are with pure slip. For orientations W, G and A, the stress stateshose active systems are all slip systems belong to the yield

tress states of Bishop–Hill slip. They have four common activelip systems, which are the same as those when only slip occurs.or orientation W, the stresses that activate six slip systems and

wo twinning systems belong to the same basic group of yieldtress states. For orientations B and C, there are six stress stateshat can fulfill the yield condition and their common active slipystems include two slip systems and one twinning systems. Forrientation S, the stress state activates four slip systems and one

winning system. For orientations P and rot.G, they are activatedy the same yield stress state.

In Case III (1/√

3 < ξ <√

3/2, Table 4), slip and twinningan also occur together. As far as orientations W, G, S, P, rot.G

sc

s5 s9 s10 s16 s20 s24 s5 s10 s16 s20

s2 s5 s7 s10 s13 s16 s20 s23

nd A are concerned, the analytical expressions of yield stresstates and corresponding active systems are the same as those of3/2 < ξ < 2/

√3 (Table 3). Particularly, it should be noted

hat for orientations W, G, S, P, rot.G and A, the analyticalxpressions of yield stress states can fulfill the yield conditionithin the whole range of 1/

√3 ≤ ξ ≤ 2/

√3. This means that

hey belong to the 139 common yield stress states. However,or orientations B and C, only four stress states fulfill the yieldondition in the case of 1/

√3 < ξ <

√3/2 while there are six

tress states for√

3/2 < ξ < 2/√

3. In other word, varying the ξ

alue from Cases II to III shows the most significant influencesor the B and C orientations.

In Case IV (ξ < 1/√

3, Table 5), only twinning occurs. Onlyrientation G is activated by more than one stress state. The yieldtress state is the same in orientations C and S, and the same in, P and rot.G.

Obviously, the orientations P and rot.G are activated by theame yield stress states when ξ is located within the differentanges. Furthermore, for the orientations S, P and rot.G, theyre activated by only one yield stress states while the other fiverientations are activated by more than one for Cases I–IV.

It is noted that, the yield stress states at the boundariesetween the four cases stated above can be calculated by sub-tituting ξ = 1/

√3,

√3/2 and ξ = 2/

√3 into the analytical

xpressions. In light of space, no detail discussions are madeere. Generally, the number of yield stress states at the bound-ries will be smaller than those in the ranges of

√3/2 < ξ <

/√

3 and 1/√

3 < ξ <√

3/2.

.2. The influence of ξ on the M-factors and yield strengthnisotropy

As can be seen from Tables 2–5, the M-facors for all thetudied orientations vary continuously with ξ and no abrutpthang occurs at the boundaries. The orientation position for the

104 Z. Chen et al. / Materials Science and Engineering A 464 (2007) 101–109

Table 3The Taylor factor values, yield stress states, numbers of sets of five independent systems and active systems for ideal orientations in fcc metals under plane straincompression (Case II:

√3/2 < ξ < 2/

√3)

Notation Taylor factor Yield stress states (multiplied by√

6τcs) Sets of fiveindependent systems

Active systems Common activesystems

W 3√

2/2 (−1/2,√

3ξ/2 − 1, 0, 0,√

3ξ/2 − 1/2) 40 s1 s4 s15 s18 s21 s24 t7 t10 s15 s18 s21 s24

3√

2/2 (−1/2,√

3ξ/2 − 1, 0, 0, 1/2 − √3ξ/2) 40 s7 s10 s15 s18 s21 s24 t1 t4

3√

2/2 (−1/2, −1/2, 0, 0, 0) 32 s1 s4 s7 s10 s15 s18 s21 s24

3√

2/2 (−1/2,√

3ξ − 3/2, 0, 0, 0) 44 s15 s18 s21 s24 t1 t4 t7 t10

G 3√

2/2 (0, 1, 0, 0, 0) 32 s2 s5 s8 s11 s13 s16 s19 s22 s2 s5 s13 s16

3√

2/2 (0, 1/2 − √3ξ/4, 0, 0, −1/2 − √

3ξ/4) 6 s2 s5 s13 s16 t9 t12

B 3√

2/2 + √6ξ/4 (−1/2 + √

3ξ/4, 1/2 − √3ξ/4, 0, 1 −√

3ξ/2, −√3ξ/2)

1 s2 s5 s7 s13 t9 s5 s13 t9

3√

2/2 + √6ξ/4 (0, 1/2 − √

3ξ/4, 0, 0, −1/2 − √3ξ/4) 6 s2 s5 s13 s16 t9 t12

3√

2/2 + √6ξ/4 (1/2 − √

3ξ/4, 1/2 − √3ξ/4, 3/2 −√

3ξ,√

3ξ/2 − 1/2,√

3ξ/2 − 3/2)1 s5 s7 s13 t4 t9

3√

2/2 + √6ξ/4 (1/2 − √

3ξ/4, 1/2 − √3ξ/4,

√3ξ/2 −

1, 0, −√3ξ/2)

1 s5 s13 s16 s20 t9

3√

2/2 + √6ξ/4 (1/2 − √

3ξ/4, 1/2 − √3ξ/4, 1/2 −√

3ξ/2,√

3ξ − 3/2,√

3ξ/2 − 3/2)1 s5 s13 s20 t2 t9

3√

2/2 + √6ξ/4 (0, 1/2 − √

3ξ/4, 1/2 − √3ξ/2,

√3ξ/2 −

1/2,√

3ξ/4 − 1)1 s5 s13 t2 t4 t9

C 7√

2/4 + √6ξ/4 (0, 1/2 − √

3ξ/2, 1/2 − √3ξ/2,

√3ξ/2 −

3/2, 0)1 s1 s9 s10 s24 t3 s9 s24 t3

7√

2/4 + √6ξ/4 (0,

√3ξ/2 − 1, 1/2 − √

3ξ/2,√

3ξ/2 −3/2,

√3ξ − 3/2)

1 s1 s9 s24 t3 t8

7√

2/4 + √6ξ/4 (0, 1/2 − √

3ξ/2,√

3ξ/2 − 3/2, 1/2 −√3ξ/2, 0)

1 s9 s14 s20 s24 t3

7√

2/4 + √6ξ/4 (0, −√

3ξ/4, −1/2, −1/2,√

3ξ/4 − 1/2) 1 s9 s10 s20 s24 t37√

2/4 + √6ξ/4 (0,

√3ξ/2 − 1,

√3ξ/2 − 3/2, 1/2 −√

3ξ/2,√

3ξ − 3/2)1 s9 s14 s24 t3 t10

7√

2/4 + √6ξ/4 (0,

√3ξ/4 − 1/2, −1/2, −1/2, 3

√3ξ/4 −

1)1 s9 s24 t3 t8 t10

S (1/61 +1/56)15

√6ξ/2 +

(18/61 + 11/14)3√

2/2

(0, −√3ξ/4, −1/2, −1/2,

√3ξ/4 − 1/2) 1 s9 s10 s20 s24 t3 s9 s10 s20 s24 t3

P 3√

2/2 + √6ξ/2 (0, 1/2 − √

3ξ/4, 0, 0, −1/2 − √3ξ/4) 6 s2 s5 s13 s16 t9 t12 s2 s5 s13 s16 t9 t12

rot.G 3√

2/2 + 3√

6ξ/4 (0, 1/2 − √3ξ/4, 0, 0, −1/2 − √

3ξ/4) 6 s2 s5 s13 s16 t9 t12 s2 s5 s13 s16 t9 t12

A 2√

2 (0, 0, −1/2, −1/2, −1/2) 6 s5 s9 s10 s16 s20 s24 s5 s10 s16 s20√ √ √ √2)

mttt

uFotitttt

ξ

o

aadsiawtwrdnT

2 2 (0, 0, 3ξ/2 − 1, 3ξ/2 − 1, − 3ξ/

aximum value of M-factor is always located at the rot.G orien-ation, regardless of the ξ value. For some orientations, thoughhe stresses are the same, the M-factor values are different sincehe orientation matrices are different.

For ξ < 1/√

3 (Table 5), only twinning takes place; the val-es of M-factor for all orientations should be multiplied by

√3ξ.

or ξ > 1/√

3 (see Tables 2–4), the M-factors for the W, G and Arientations do not change with the ξ value; for the other orienta-ions (i.e. B, C, S, P and rot.G), their M-factors are constant onlyn the range of ξ > 2/

√3. It is interest to note that, in the cases

hat slip and twinning occur simultaneously (see Tables 3 and 4),he active systems and analytical expressions of M-factors arehe same though their yield stress states belong to two different

ypes of yield surfaces.

For a more comprehensive understanding of the influence ofon the M-factor and yield strength anisotropy, the M-factors

f ideal plane strain compression for the four different slip

toit

6 s5 s10 s16 s20 t9 t12

nd/or twinning cases were explicitly calculated. The resultsre shown in Fig. 1 as a function of (ϕ1, φ, ϕ2) in the three-imensional Euler space using Bunge’s notation. It can beeen that both the maximum and minimum values of M-factorncrease with the ξ value in all cases studied. The reductionmount of the maximum values is higher than the minimum’sith the decrease of the ξ value. Further analysis showed that

he orientation position for the maximum value of M-factorithin whole Euler space is always located at the orientation

ot.G. Meanwhile, the M-factor varies by more than two forifferent crystal orientations in the four cases, indicating sig-ificant variations of the yield strength with crystal orientation.he change of the M-factor with crystal orientations appears

o be the largest when only slip occurs and be the least whennly twinning occurs, indicating a general tendency of decreas-ng anisotropy with the decrease of ξ or the introduction ofwinning.

Z. Chen et al. / Materials Science and Engineering A 464 (2007) 101–109 105

Table 4The Taylor factor values, yield stress states, numbers of sets of five independent systems and active systems for ideal orientations in fcc metals under plane straincompression (Case III: 1

√3 < ξ <

√3/2)

Notation Taylor factor Yield stress states (multiplied by√

6τcs) Sets of fiveindependent systems

Active systems Common activesystems

W 3√

2/2 (−1/2,√

3ξ/2 − 1, 0, 0,√

3ξ/2 − 1/2) 40 s1 s4 s15 s18 s21 s24 t7 t10 s15 s18 s21 s24

3√

2/2 (−1/2,√

3ξ/2 − 1, 0, 0, 1/2 − √3ξ/2) 40 s7 s10 s15 s18 s21 s24 t1 t4

3√

2/2 (−1/2, −1/2, 0, 0, 0) 32 s1 s4 s7 s10 s15 s18 s21 s24

3√

2/2 (−1/2,√

3ξ − 3/2, 0, 0, 0) 44 s15 s18 s21 s24 t1 t4 t7 t10

G 3√

2/2 (0, 1, 0, 0, 0) 32 s2 s5 s8 s11 s13 s16 s19 s22 s2 s5 s13 s16

3√

2/2 (0, 1/2 − √3ξ/4, 0, 0, −1/2 − √

3ξ/4) 6 s2 s5 s13 s16 t9 t12

B 3√

2/2 + √6ξ/4 (0, 1/2 − √

3ξ/4, 0, 0, −1/2 − √3ξ/4) 6 s2 s5 s13 s16 t9 t12 s5 s13 t9

3√

2/2 + √6ξ/4 (1/4 − √

3ξ/4, 1/2 −√3ξ/4, 0,

√3ξ/2 − 1/2, −3/4)

1 s2 s5 s13 t4 t9

3√

2/2 + √6ξ/4 (

√3ξ/4 − 1/4, 1/2 − √

3ξ/4, 1/2 −√3ξ/2, 0, −3/4)

1 s5 s13 s16 t2 t9

3√

2/2 + √6ξ/4 (0, 1/2 − √

3ξ/4, 1/2 −√3ξ/2,

√3ξ/2 − 1/2,

√3ξ/4 − 1)

1 s5 s13 t2 t4 t9

C 7√

2/4 + √6ξ/4 (0, −√

3ξ/4, −1/2, −1/2,√

3ξ/4 −1/2)

1 s9 s10 s20 s24 t3 s9 s24 t3

7√

2/4 + √6ξ/4 (0, −1/4,

√3ξ/2 −

1, −√3ξ/2,

√3ξ/2 − 3/4)

1 s9 s20 s24 t3 t8

7√

2/4 + √6ξ/4 (0, −1/4, −√

3ξ/2,√

3ξ/2 −1,

√3ξ/2 − 3/4)

1 s9 s20 s24 t3 t10

7√

2/4 + √6ξ/4 (0,

√3ξ/4 −

1/2, −1/2, −1/2, 3√

3ξ/4 − 1)1 s9 s24 t3 t8 t10

S (1/61 +1/56)15

√6ξ/2 +

(18/61 + 11/14)3√

2/2

(0, −√3ξ/4, −1/2, −1/2,

√3ξ/4 −

1/2)1 s9 s10 s20 s24 t3 s9 s10 s20 s24 t3

P 3√

2/2 + √6ξ/2 (0, 1/2 − √

3ξ/4, 0, 0, −1/2 − √3ξ/4) 6 s2 s5 s13 s16 t9 t12 s2 s5 s13 s16 t9 t12

rot.G 3√

2/2 + 3√

6ξ/4 (0, 1/2 − √3ξ/4, 0, 0, −1/2 − √

3ξ/4) 6 s2 s5 s13 s16 t9 t12 s2 s5 s13 s16 t9 t12

A√

ξ/2)

3t

tmtIa

dabfo

TTc

N

W

G

BCSPrA

2 2 (0, 0, −1/2, −1/2, −1/2)2√

2 (0, 0,√

3ξ/2 − 1,√

3ξ/2 − 1, −√3

.3. The influence of ξ on the twinning ability and rollingextures

In addition to the influence on the yield strength anisotropy,he induction of twinning has obvious effect on the develop-

ent of deformation textures. The above results showed that inhe case that slip and twinning occur simultaneously in CasesI and III, for orientations W, G and A, all of the commonctive systems are slip systems, which indicates twinning is

uξ

ws

able 5he Taylor factor values, yield stress states, numbers of sets of five independent sysompression (Case IV: ξ < 1/

√3)

otation Taylor factor (multiplied by√

3ξ) Yield stress states (multiplied b

3√

2/2 (−1/2, −1/2, 0, 0, 0)

3√

2/2 (0, 1, 0, 0, 0)3√

2/2 (0, 1/4, 0, 0, −3/4)

7√

2/4 (0, 1/4, 0, 0, −3/4)2√

2 (0, −1/4, −1/2, −1/2, −1/4)2.66 (0, −1/4, −1/2, −1/2, −1/4)2√

2 (0, 1/4, 0, 0, −3/4)ot.G 9

√2/4 (0, 1/4, 0, 0, −3/4)

2√

2 (0, 0, −1/2, −1/2, −1/2)

6 s5 s9 s10 s16 s20 s24 s5 s10 s16 s20

6 s5 s10 s16 s20 t9 t12

ifficult for the three orientations. In contrast, the commonctive systems for orientations B, C, S, P and rot.G includeoth slip and twinning systems and hence, twinning is pre-erred for the five orientations, that is, they have the trendf twining. The analysis of M-factors showed that the val-√

es for orientations W, G and A do not change when 1/ 3 ≤≤ 2/

√3 while the values for the other five orientation varies

ith ξ, which is consistent with the analysis of common activeystems.

tems and active systems for ideal orientations in fcc metals under plane strain

y√

6ξτcs) Sets of five independent systems Active twinning systems

56 t1 t3 t4 t6 t7 t9 t10 t12

56 t1 t2 t4 t5 t7 t8 t10 t11

6 t1 t2 t4 t5 t9 t12

6 t1 t2 t4 t5 t9 t12

1 t3 t8 t9 t10 t12

1 t3 t8 t9 t10 t12

6 t1 t2 t4 t5 t9 t12

6 t1 t2 t4 t5 t9 t12

6 t4 t5 t8 t9 t10 t12

106 Z. Chen et al. / Materials Science and Engineering A 464 (2007) 101–109

F r plan√

√hould

tb

μ

wffsctaataaiP

sto

tofsslt

sey

ig. 1. The distribution of M-factors in the Euler spaces for fcc metals unde3/2 < ξ < 2/

√3 (ξ = 1); (c) 1/

√3 < ξ < 2/

√3 (ξ = 3/4); (d) ξ < 1/

√3. M s

In order to describe the twinning ability for a crystal orienta-ion, the concept of twinning ability factor μ can be introducedased on the M-factor values, i.e.

= Ms − Mt

Ms= 1 − Mt

Ms(5)

here Ms and Mt are the M-factors of the orientation calculatedor ξ = 2/

√3 and ξ = 1/

√3, respectively. The analysis of M-

actors showed that the values for some orientations in Eulerpace decrease with ξ and some do not change. Obviously, for arystals with an arbitrary orientation, its Ms value is not smallerhan the Mt. If Ms = Mt (or μ = 0), such as for orientations W, Gnd A, twinning is difficult since all the common active systemsre slip systems. In fact, further analysis shows that for orien-ations with a high symmetrical multiplicity, e.g. 96, they are

ctivated by only one yield stress state and all the active systemsre slip systems in this case. For these orientations, twinnings difficult. If Ms > Mt (μ > 0), such as for orientations B, C, S,

and rot.G, twinning becomes easier since the common active

tsta

e strain compression deformation with different ξ values: (a) ξ > 2/ 3; (b)be multiplied by

√3ξ.

ystems include both slip and twinning systems. The lower ishe Mt (or the larger is the μ), the easier is the twinning for therientation.

Fig. 2 shows the distribution of the twinning ability factor inhe Euler space. The twinning is preferred for the orientationsutside the shadow regions (μ > 0) while the twinning is difficultor the orientation within the shadow regions (μ = 0). It can beeen that for orientations W, G and A, they are located within thehadow regions. The other five orientations B, C, S, P and rot.Gocated outside the shadow regions. Certainly, only twinningakes place if ξ < 1/

√3.

It also should be noted that the Euler space in Fig. 2 can beubdivided into some certain regions. The crystals for differ-nt orientations within these regions are activated by a specificield stress state, so active slip systems are the same. Some of

hese regions contain just five active systems, and some containix or eight active systems. For an arbitrarily orientated crys-al, only five independent slip systems are necessary in order toccomplish an arbitrary shape change. Since the rotation for an

Z. Chen et al. / Materials Science and

Fm

actatlsmO

tc

aHaacrfitceahiotint

T

dm

θ

ig. 2. The distribution of twinning ability factor μ in the Euler space for fccetals under plane strain compression.

rbitrarily orientated crystal during deformation depends on theombination of active systems, for the orientation that only con-ains five active systems, there is no ambiguity in selecting thective systems. For orientations that contain six or eight systems,here is uncertainty in the selection of active systems, which

eads to an uncertainty in predicting texture development. Inuch cases, a certain rule in selecting the active systems [16,17]ust be followed when deformation textures are simulated.nce the active systems are determined for every orientation,

wta

Fig. 3. Orientation line analysis of textures in a Cu5%Zn alloy after

Engineering A 464 (2007) 101–109 107

he orientation changes or the development of texture can bealculated.

The present results can be applied to assess the formationnd development of rolling textures in polycrystalline materials.ere we consider first the experimental textures in a Cu5%Zn

lloy after various rolling reductions from 60% to 99%, whichre shown in Fig. 3 [18]. The rolling deformation in such casesan be expressed approximately by ideal PSC. At 60% rollingeduction, it can be seen that crystallites accumulate to the �-ber with preferences at the orientations from {0 1 1}〈1 1 1〉

o {2 1 1}〈1 1 1〉 (Fig. 3a), and the �-fiber with an apparent Component (Fig. 3b). With increase of rolling strain, the ori-ntation concentrations around �-fiber become more apparentt the expenses of �-fiber. Because of the influence of workardening, twinning becomes more and more obviously withncreased rolling deformation. In this case, twinning and slipccur simultaneously. To illustrate the possible contributions ofwinning to the development of deformation textures, it is ofnterest to analyze the orientation changes associated with twin-ing. According to the relationship between the non-twined andwinned parts of a crystal:

t = ΩT (6)

where T is the orientation matrix of the non-twinned part, Ttescribes the orientation of the twinned part, and Ω is a 3 × 3atrix, the elements of which are

ij = 2vivj − δij, δij

{1, i = j

0, i �= j(7)

here v1, v2 and v3 are the direction cosines of the normal onhe twinning plane. For orientation C (1 1 2)[1 1 1], the commonctive twinning system for Cases II and III (Tables 3 and 4) is t3

various rolling reductions: (a) �-fiber; (b) �-fiber; (c) �-fiber.

1 nd En

(

b

T

Ich(aett

−2/3 2/3 −1/3√ √ √

Fξ

08 Z. Chen et al. / Materials Science a

1 1 1)[1 1 2]. Hence, we have

Ω =

⎡⎢⎣

−1/3 2/3 2/3

2/3 −1/3 2/3

2/3 2/3 −1/3

⎤⎥⎦ and

T =

⎡⎢⎣

−1/√

3 1/√

2 1/√

6

−1/√

3 −1/√

2 1/√

6

1/√

3 0 2/√

6

⎤⎥⎦ (8)

According to Eq. (6), the orientation of the twinned part cane calculated as

t = ΩT =

⎡⎢⎣

1/3√

3 1/√

2 5/3√

6

1/3√

3 −1/√

2 5/3√

6

−5/3√

3 0 2/3√

6

⎤⎥⎦ (9)

ig. 4. ODFs of textures simulated for a model fcc materials after 86% reduction wit< 2/

√3 (ξ = 3/4); (d) ξ < 1/

√3.

gineering A 464 (2007) 101–109

t can be worked out that the orientation of the twinnedrystal is (5 5 2)[1 1 5]. The experimental results (Fig. 3c)ave demonstrated that there is an accumulation nearby the5 5 2)[1 1 5] orientation, which is a symmetrically equiv-lent orientation of (5 5 2)[1 1 5] orientation. As anotherxample, for orientation B (1 1 0)[1 1 2], the common activewinning system is t9 (1 1 1)[1 1 2] (Tables 3 and 4) andherefore

Ω =

⎡⎢⎣

−1/3 −2/3 −2/3

−2/3 −1/3 2/3

⎤⎥⎦ and

T =

⎡⎢⎣

1/ 6 1/ 3 2/2

−1/√

6 −1/√

3√

2/2

2/√

6 −1/√

3 0

⎤⎥⎦ (10)

h different ξ values: (a) ξ > 2/√

3; (b)√

3/2 < ξ < 2/√

3 (ξ = 1); (c) 1/√

3 <

and

T

otope

ttdwo(√ξ

eot

o“aew4wttr(eaLtbddft

4

{

ortwtitidaeTaCmr

A

SP

R

[

[

[[[[

[[

(2004) 2845.

Z. Chen et al. / Materials Science

The orientation of the twinned part can be determined as

t = ΩT =

⎡⎢⎣

−1/√

6 1/√

3 −√2/2

1/√

6 −1/√

3 −√2/2

−2/√

6 −1/√

3 0

⎤⎥⎦ (11)

r (1 1 0)[1 1 2], which is also a symmetrically equivalent orien-ation of B. Therefore, twinned orientations from the B and Crientations might partly contribute to the weakening of com-onent C and the strengthening of component B found in thexperimental textures for the Cu5%Zn.

Although the present analysis focuses on the effect of ξ onhe deformation behavior of single crystals, it is of interesto illustrate its effect at a polycrystal level. For this purpose,eformation texture simulations for polycrystalline aggregatesith a starting randomly distributed texture has been carriedut using the FC-Taylor model, at various rolling reductionsfrom 63% to 95%) for four cases with ξ > 2/

√3 (ξ = 1.25),

3/1 < ξ < 2/√

3 (ξ = 1), 1/√

3 < ξ <√

3/2 (ξ = 0.75) and< 1/

√3 (ξ = 0.5), respectively. These results confirm the gen-

ral tendency of rolling textures with the rolling strain. In lightf space, Fig. 4 shows only the results for 86% rolling reductiono illustrate the effect of ξ.

Fig. 4 reveals four types of textures. In the case thatnly slip occurs (ξ = 1.25, Fig. 4a), the simulated texture is acopper-type” rolling texture. The majority of the orientationsccumulate along the �-fiber, which runs from the stable ori-ntation (4 4 1 1)[1 1 1 1 8] or (90◦, 27◦, 45◦) to (30◦, 45◦, 90◦),hich is not located at the ideal orientation B (0 1 1)[2 1 1] (35◦,5◦, 90◦). The orientation densities along the �-fiber are mucheaker than those along �-fiber. For ξ = 1 (Fig. 4b), slip and

winning take place simultaneously. A transition texture withhe stronger �-fiber than the �-fiber is obtained. It is compa-able to those for the Cu5%Zn alloy in Fig. 3. For ξ = 0.75Fig. 4c), a typical “brass-type” texture with predominant ori-ntation distributions along the �-fiber is obtained. The resultsre in qualitative agreement with those measured by Hirsch anducke [2] for Cu30%Zn. For the case of ξ = 0.5 (Fig. 4d), only

winning occurs, the obtained texture, similar to that obtainedy Van Houtte [8], is hard to be related to any known rollingeformation textures in practice. According to these results, theifferent ξ values may be applied to match to textures results incc metals with different stacking fault energy and thus twinningendencies.

. Conclusions

The plane strain compression deformation in fcc crystals for1 1 1}〈1 1 0〉 slip and {1 1 1}〈1 1 2〉 twinning has been analyzed

[

Engineering A 464 (2007) 101–109 109

n the basis of the maximum work principle of Bishop–Hill. Theesults showed that all the common active systems for orienta-ions W, G and A are slip systems and the twinning is difficulthen ξ > 1/

√3. The twinning becomes easier for the orienta-

ions B, C, S, P and rot.G since the common active systemsnclude both slip and twinning systems. As an indication ofhe yield strength anisotropy, the difference between the max-mum and the minimum values of M-factor decreases with theecreases of ξ due to the introduction of twinning. The twinningbility factor based on the M-factors can describe the differ-nt preferences of twinning to slip for different orientations.he analytical results are in reasonable agreement with the vari-tion of general tendencies in the texture development for theu5%Zn alloy. The polycrystal simulations using the FC-Taylorodel revealed four types of characteristic rolling textures cor-

esponding to the four different ranges of ξ.

cknowledgements

This work was financially supported by the National Naturalcience Foundation Council (Project 50301016) and the Chinaostdoctoral Science Foundation (Project 2005037003).

eferences

[1] G.Y. Chin, W.F. Hosford, D.R. Mendorf, Proc. R. Soc. A 309 (1969) 433.[2] J. Hirsch, K. Lucke, Acta Metall. 36 (1988) 2863.[3] J. Hirsch, K. Lucke, M. Hatherly, Acta Metall. 36 (1988) 2905.[4] E. El-Danaf, S.R. Kalidindi, R.A. Doherty, Metall. Mater. Trans. 30A

(1999) 1223.[5] E. El-Danaf, S.R. Kalidindi, R.A. Doherty, Int. J. Plast. 17 (2001) 1245.[6] M.S. Szczerba, T. Bajor, T. Tokarski, Philos. Mag. 84 (2004) 481.[7] M.A. Meyers, O. Voheringer, V.A. Lubarda, Acta Mater. 49 (2001) 4025.[8] P. Van Houtte, Acta Metall. 26 (1978) 591.[9] Z.Y. Chen, X.M. Zhang, C.M. Liu, S.Y. Li, Z.P. Zhou, Y. Yang, Acta Metall.

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12] J.F.W. Bishop, R. Hill, Philos. Mag. 42 (1951) 414.13] J.F.W. Bishop, R. Hill, Philos. Mag. 42 (1951) 1298.14] G.I. Taylor, J. Inst. Met. 62 (1938) 307.15] Z.Y. Chen, X.M. Zhang, Z.P. Zhou, S. Li, C.M. Liu, Metall. Mater. Trans.

A 31 (2000) 2449.16] J. Hirsch, K. Lucke, Acta Metall. 36 (1988) 2883.17] Z.Y. Chen, X.M. Zhang, C.M. Liu, H.N. Cai, F.C. Wang, Mater. Trans. 45

18] X.M. Zhang, Entstehung und Entwicklung der Walz- und Rekristallisation-stextur in Cu-Zn und Cu-P Legierungen in Abhangigkeit von der Walz-und Gluhtemperatur sowie von der Ausgangskorngroesse, Dissertation,RWTH-Aachen, 1987, p. 229.