A

tsattomofid©

K

1

fotamga

fadae

0d

Materials Science and Engineering A 432 (2006) 202–215

Investigation on cold-drawn gold bonding wire withserial and reverse-direction drawing

Jae-Hyung Cho a, A.D. Rollett b, J.-S. Cho c, Y.-J. Park c, S.-H. Park c, K.H. Oh a,∗a MSE in Seoul National University, Seoul 151-742, South Korea

b MSE in Carnegie Mellon University, Pittsburgh, PA 15213, USAc MK Electron, Pogok-Myeon, Yongin-Si, Kyunggi-Do, South Korea

Received 5 February 2006; received in revised form 31 May 2006; accepted 31 May 2006

bstract

Gold bonding wires have been manufactured through multiple drawing steps with serial and reverse-direction drawing. The texture and microstruc-ure of the gold bonding wires were characterized with X-ray diffraction and EBSD and compared with the predictions of finite element (FE)imulation. Initial 〈1 0 0〉 fiber decreases during drawing and is replaced by 〈1 1 1〉 fiber. The 〈1 0 0〉 oriented grains are concentrated in the centernd surface regions, whereas the 〈1 1 1〉 oriented grains are located throughout the cross-section of the wire. Regions near the surface often exhibithe complex textures. A simplified forward and backward drawing process was modeled by FE analysis with ABAQUS/StandardTM. The simplewo-step drawing process results in severe variation in shear strain under the surface and displays the opposite behavior in the shear componentsf the deformation gradient. The texture evolution was predicted using the deformation gradient calculated in the FE simulations together with a

odel of polycrystal plasticity. The 〈1 1 1〉 and 〈1 0 0〉 fibers are predicted to develop in the center part of the wire where homogeneous deformation

ccurs. The regions near the surface that experience repeated shear strain exhibit complex textures that deviate from the standard 〈1 1 1〉 and 〈1 0 0〉bers. The {1 1 2}〈1 1 0〉 and {1 1 1}〈1 1 2〉 components are prevalent in the higher shear strain regions. The variations of the anisotropic elasticirectional moduli with position were also calculated.

2006 Elsevier B.V. All rights reserved.

tic mo

mbg

dsditmhwp

eywords: Gold wire; FE analysis; Fiber texture; EBSD; Rate sensitivity; Elas

. Introduction

Fine bonding wires of pure Au, Cu or Al are commonly usedor interconnection in semiconductor packaging. Homogeneityf texture and microstructure of the wires is a factor that affectshe bonding wire properties. These characteristics of the wiresre related to several parameters such as the purity of the originalaterials, the drawing and annealing processes. The inhomo-

eneity of a drawn wire depends on die angle (α), reduction ofrea (r) and friction at the die/material interface [1].

High purity gold (99.999 wt.% Au) is too soft and unstableor obtaining good properties for bonding when it is drawn andnnealed. Therefore gold bonding wire commonly has various

opants at the parts per million (ppm) level in order to obtaincceptable thermal and mechanical properties [2]. Impurities,ven at these low levels, are important for controlling the final

∗ Corresponding author. Tel.: +82 2 880 8306.E-mail address: kyuhwan@snu.ac.kr (K.H. Oh).

tthast

921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2006.05.143

duli

icrostructure, texture and mechanical properties of gold wirey raising the recrystallization temperature and preventing therain growth after recrystallization [3–7].

Gold bonding wire is cold-drawn from casting bar throughiamond dies of various sizes with variable reduction in area pertep. During fabrication of fine bonding wire, large and repeatedeformation steps are required. The final wire has a diametern the range 25–30 �m, and the equivalent strain of it is greaterhan 10 without intermediate annealing. One of the most notable

echanical properties of gold is malleability. Nutting and Nuttallave pointed out that gold can accommodate plastic strain evenhen the deformed foil thickness is less than the subgrain sizeroduced by plastic deformation [8].

Axisymmetric deformation such as compression, uniaxialension or drawing, needs only one direction to fully characterizehe preferred orientation. Drawing of a bar or wire in principle

as the same deformation pattern (ε11 = −2ε22 = −2ε33) as uni-xial tension. The wire drawing textures of fcc materials, i.e.ilver, gold, copper, aluminum, and brass typically have a mix-ure of 〈1 1 1〉 and 〈1 0 0〉 fiber components [9,10]. A 〈1 1 1〉 or

and E

〈Ti[hilBipcd

aaegE〈fti〈ieecftttipdoiva

shrmarddwpdot

eoal

batsssaAHp

idudwttwiia

2

2

(5eabpsckwiet

2

wuStus0c

J.-H. Cho et al. / Materials Science

1 0 0〉 direction in each grain is aligned with the drawing axis.he textures of aluminum, copper and brass wires have been

nvestigated for “true fiber” versus “cyclic” texture component11]. True fiber component means that the material is statisticallyomogeneous such that if a large enough number of grains arencluded in the sample, cylindrical symmetry is observed regard-ess of the location of the sample within the material boundaries.y contrast, cyclic texture component means that the material

s not statistically homogeneous such that textures with sam-le symmetries lower than cylindrical (e.g. plane strain in thisase) are observed. See Kocks et al. for further references andiagrams [12].

Taylor’s theoretical analysis shows that all grains in fcc met-ls rotate so that either a 〈1 1 1〉 or 〈1 0 0〉 axis tends to becomeligned with the extension axis, depending on which axis thextension axis is closest to at zero strain [13]. A 〈1 0 1〉-orientedrain can rotate toward either the 〈1 1 1〉 or the 〈1 0 0〉 axis.nglish and Chin [14] have shown that the ratio of 〈1 1 1〉 and1 0 0〉 components varies depending on the reduced stackingault energy although Stout et al. [15] have pointed out thathe initial texture influences the outcome. Low reduced stack-ng fault energy metals, i.e. silver, typically exhibit a stronger1 0 0〉 component than 〈1 1 1〉. High and intermediate stack-ng fault energy metals, however, such as aluminum or copper,xhibit a stronger 〈1 1 1〉 component than the 〈1 0 0〉. Aernouldtt al. have reported that the textures of drawn silver wires haveyclic symmetry and is related to twinning [16]. Near the sur-ace, recrystallization can occur as a result of frictional heating inhe die. Montesin and Heizmann have reported an X-ray diffrac-ion procedure for fine wires [17]. Heizmann et al. have shownhat the strength of the cyclic texture increases as die anglencreases [18]. Rajan and Petkie measured wire textures in cop-er with automated electron back scatter diffraction (EBSD) andisplayed the results with Rodrigues-Frank [19]. The presencef inhomogeneous distribution of twins and twinning reactionsn copper wire was characterized and it was suggested that theariations in the mesotexture could contribute to mechanicalnisotropy.

Recently, numerical models have been used for better under-tanding of forming processes. Finite element method (FEM)as been applied to tube drawing to predict the distribution ofesidual stresses and temperatures with respect to process andaterials [20,21]. In order to evaluate the anisotropy, Choi et

l. used FEM combined with Barlat’s anisotropic yield crite-ion implemented in ABAQUS via a user subroutine (UMAT)uring deep drawing [22]. Cho et al. presented a simplified three-imensional tension model based on FEM with crystal plasticity,hich provides a micromechanical approach for a slip-basedlastic deformation [23]. Mathur and Dawson investigated theegree to which surface textures differ from those near the axisf the wire [24]. In particular, the effects of the tool friction onexture variations have been investigated.

The elastic modulus is one of the important mechanical prop-

rties of bonding wire. In principle, it should be possible tobtain the elastic moduli for a polycrystal from a weightedverage of the elastic behavior of all orientations of crystal-ite present in the polycrystal. Two averaging methods have

dmiE

ngineering A 432 (2006) 202–215 203

een proposed for polycrystals, namely the Voigt and Reussveraging methods [25,26]. The former is based on an assump-ion of uniform local strain and averaging the volume-weightedtiffness over all orientations. The latter assumes uniform localtress and averages the compliances over all orientations. Hillhowed that the Voigt and Reuss average correspond to uppernd lower bounds, respectively to the true behavior [27,28].lthough more sophisticated (narrower) bounds exist, such asashin–Shtrikman [29,30], we adopt the Reuss estimate for sim-licity.

In this research, the evolution of texture and microstructuren cold-drawn gold bonding wires was analyzed with X-rayiffraction and EBSD as a function of drawing strain. FEM wassed to understand the deformation during a simplified two-steprawing, i.e. forward and backward drawing. The use of FEMas confined to a mechanical analysis for material flow, and the

emperature and other internal state variables were not coupledogether. Deformation gradients extracted from the FE resultsere used to predict texture evolution using a polycrystal plastic-

ty model. The variations in strain and deformation gradient werenvestigated for the repeated drawing steps. Elastic moduli werelso calculated from both the experimental and FEM textures.

. Experimental and FE analysis

.1. Materials and sample preparation

The purity of gold used was more than 99.99% with someintentional) dopants, such as Ca and Be, that total less than0 ppm by weight. The original cast gold bar with 7 mm diam-ter was drawn down to 1000 �m diameter with a reduction inrea of approximately 98% through diamond dies in order toreak down the cast structure. The axis of the casting bar wasarallel to the subsequent drawing direction. After 1000 �m, aeries of diamond dies were used for more homogeneous andontrolled deformation. For each die, the reduction in area wasept to less than 10%. The total von Mises equivalent strainas approximately 11.3 from the casting bar to the final bond-

ng wire. Characterizations were performed on gold wires withquivalent strains of ε = 3.9, 5.2, 7.3, 8.9 and 11.3 correspondingo wire diameters of 1000, 522, 185, 81 and 25 �m, respectively.

.2. X-ray and EBSD measurement

Three incomplete pole figures (1 1 1), (2 0 0) and (2 2 0)ere measured on a cross-section of the casting gold barsing the X-ray diffraction method with Cu K� radiation on aeifert 3000 PTS diffractometer. Orientation distribution func-

ion (ODF) was calculated from the experimental pole fig-res using triclinic sample and cubic crystal symmetry. Suchymmetry requires the elementary Euler space defined by◦ ≤ ϕ1 ≤ 360◦, 0◦ ≤ Φ ≤ 90◦ and 0◦ ≤ ϕ2 ≤ 90◦. The ODF wasalculated using the WIMV method [31]. Wires with different

iameters were characterized with EBSD. The gold wires wereounted in epoxy, and then sectioned and polished. The pol-

shed specimens were cleaned with ion milling. High resolutionBSD (using a JEOL 6500F scanning electron microscope with

2 and Engineering A 432 (2006) 202–215

aaiorwsscww

2

uwolacobTl

awitdiwa

sbebmstob

Fw

04 J.-H. Cho et al. / Materials Science

n INCA/OXFORD EBSD system) was used for measurementnd the data analysis was performed using the REDS (reprocess-ng of EBSD Data in Seoul National University, 2002) [32]. Theperating voltage for EBSD system was 20 kV and the probe cur-ent was 4 nA. A rectangular grid was used and the pixel spacingas varied according to the wire diameter. A 5-�m pixel step

ize was used for the 1000 �m wire and the 25 �m wire was mea-ured with a 0.239 �m step size. EBSD maps were measured onross-section for 1000 �m wire and longitudinal section for finerires. Inverse pole figure maps (IPF) and ODFs from the EBSDere used for texture representation.

.3. Finite element analysis

For wire fabrication, die sets with several individual dies weresed to achieve further reduction in area. For convenience, theire was drawn in alternate directions through each die set inrder to avoid rewinding the wire. Each die in a die set impartedess than 10% reduction in area so that each die set results inbout 60% reduction in one drawing direction. The next die setovers another 60% reduction in the opposite direction and son and so forth. Several die sets were used to produce the finalonding wire, yielding a total strain of ε = 11.3 in experiments.he final bonding wires with 25–30 �m diameter were 10–15 km

ong.The finite element modeling was simplified in order to char-

cterize the essential features of the deformation imposed in theire drawing. A schematic diagram for the finite element draw-

ng process is shown in Fig. 1. The drawing process containswo major drawing steps, with successive forward and backwardrawing steps. FE analysis based on the ABAQUS/StandardTM

mplicit code [33] was used for modeling of this repeated for-ard and backward drawing. Fig. 2 shows the mesh before and

fter each step. The process consisted of six individual steps:

Step 1. Stabilize workpiece inside die.Step 2. Forward drawing.Step 3. Remove first die and add second die.Step 4. Second die compression for backward drawing.

sgse

ig. 2. Continuous two-step modeling for forward and backward drawings using finitas modeled.

Fig. 1. Forward and backward drawings in bonding wire fabrication.

Step 5. Backward drawing.Step 6. Remove second die.

The first die was used for the forward drawing step and theecond for the backward drawing. Each of the forward andackward drawing steps resulted in ε = 0.105 so that the totalquivalent strain was ε = 0.21. Although the total strain for theonding wire is 11.3, whereas the equivalent strain in the FEodeling is ε = 0.21, the deformation patterns are similar in each

tep. We can use the results of the two major drawing steps andheir effects on drawing because total strain is the repetitionf these two major steps. The die was assumed to be a rigidody without deformation. The deformation gradient and shear

train were taken from the mesh of the region of interest, whichave the consistent drawing results without end effects after theix step drawing process described above. The region of inter-st is indicated by a heavy line in the mesh as shown in Fig. 2.

e element method. The wire center is a symmetric line and only half of the wire

and E

Tewalic

2

mγ

τ

wesntttit

τ

wtitaios{tE

D

ttN

3

fitet

Di

3

tpUeTti

3

aopttw〈cp

{toiap{cciirsttpbr(

3

siaFr

J.-H. Cho et al. / Materials Science

he wire was modeled with four-node, axisymmetric, quadrilat-ral elements. An isotropic elastic–plastic material was assumedith Young’s modulus of 80 GPa, Poisson’s ratio of 0.42, andplastic strain–stress curve approximated by a set of piecewise

inear segments with the maximum strength of 300 MPa. To min-mize the frictional effect between die and wire, a small frictionoefficient of 0.001 in Coulomb frictional law was assigned.

.4. Polycrystal plasticity model (rate sensitive analysis)

The deformation of a rate-sensitive polycrystal is usuallyodeled by a power-law relationship between the shear rate

˙s and the resolved shear stress τs on the slip system s:

s = τ0 sgn(γs)

∣∣∣∣γs

γ0

∣∣∣∣m

= τ0γs

γ0

∣∣∣∣γs

γ0

∣∣∣∣m−1

(1)

here the rate sensitivity parameter, m, is positive, τ0 the ref-rence shear stress, and γ0 is the reference shear rate. The rateensitivity was given by 0.01 in this work. The value of γ0 doesot affect the texture evolution (set equal to 1) and τ0 depends onhe microscopic hardening such as self and latent hardening. Inhis study, hardening is not considered and the value is assumedo be constant during deformation. The resolved shear stress τs

s related to the Cauchy stress tensor, σij of the crystal throughhe relation:

s = msijσij (2)

here the Schmid tensor msij(= bs

insj) is defined by the unit vec-

or nsj that is normal to the slip plane and the unit vector bs

i thats parallel to the slip direction of the slip system s. The signerm in Eq. (1) requires the shear rate to be of the same signs the resolved shear stress, which permits the use of a singlendex s to designate two opposite slip systems with the ns

j andpposite vectors bs

i . In most fcc crystals, there are 12 possiblelip systems (plus their opposites) in each crystal, defined by the1 1 1} planes and 〈1 1 0〉 directions. The strain rate tensor, Dij

hat is associated with a given stress tensor σij, is obtained fromqs. (1) and (2):

ij =∑

s

1

2(ms

ij + msji)γs

= γ0

τ1/m0

∑

s

1

2(ms

ij + msji)m

sklσkl|ms

pqσpq|(1/m)−1 (3)

The strain rate can also be deduced from a stress poten-ial. The stress state that satisfies the above (implicit) equa-ion for a given strain rate is obtained numerically by theewton–Raphson method [34,35].

. Results and discussion

Texture of fcc metals exhibits a mixture of 〈1 1 1〉 and 〈1 0 0〉

ber components during drawing. Gold bonding wire shows a

ypical fcc wire texture after drawing. In what follows, texturevolution measured with both X-ray and EBSD is compared withhe FE predictions as a function of equivalent drawing strain.

mtib

ngineering A 432 (2006) 202–215 205

ouble pass drawing process with alternative directions is alsonvestigated using FE analysis.

.1. X-ray and EBSD analysis

In order to help understand the texture results, Fig. 3 showshe ideal positions of the Goss, Brass and Copper texture com-onents in (1 1 1) pole figures and in ODFs (ϕ2 = 45◦ section).sing a Gaussian distribution centered at the ideal orientation,

ach component is generated with FWHM = 12.5◦ [31,36].riclinic sample symmetry is assumed for shear deformation

extures, so that the range of the first (Bunge) Euler angle (ϕ1)s 0–360◦.

.1.1. X-ray analysis of cast barA micrograph of a cross-section of the as-cast bar and associ-

ted (1 1 1) pole figures are shown in Fig. 4. The grain structuref the as-cast bar is too coarse to measure with EBSD map-ing. Therefore X-ray diffraction was used to characterize theexture. Equiaxed and columnar grains are evident in the cen-er and surface of the casting, respectively. A (1 1 1) pole figureith AD (axial direction)//casting axis shows a well-developed

1 0 0〉 texture (Fig. 4(c)). Clearly the main texture component islose to rotated cube, {1 0 0}〈0 1 1〉. Fig. 4(d) shows the rotatedole figure around TD (transverse direction) of Fig. 4(c).

In this paper, a texture component designated byh k l}〈u v w〉 means that the {h k l} plane normal is parallel tohe radial direction (RD) and 〈u v w〉 is parallel to the drawingr axial direction (AD). This notation is helpful for illustrat-ng the similarity between uniaxial tension and drawing and,lternatively, for making comparisons with plane strain com-ression. According to this notation, the rotated cube component0 0 1}〈1 1 0〉 in Fig. 4(c) is re-written as {1 1 0}〈0 0 1〉, whichorresponds to the Goss component in Fig. 4(d). The strongestomponent in the casting bar is located 15.8◦ away from thedeal Goss component shown in Fig. 3(a). The overall textures still a 〈1 0 0〉 fiber but the local texture is not random, cor-esponding to radial as opposed to cyclic texture. The graintructure and preferred orientation in the as-cast bar are illus-rated in Fig. 4(b). This diagram shows that the rotated cubeype texture, {1 0 0}〈0 1 1〉 will be observed regardless of whicharticular radial direction is chosen. Most grains in the castingar are biaxially aligned with 〈1 0 0〉//AD and 〈1 1 0〉//RD. Theectangles inside the casting bar in Fig. 4(b) represent simplifiedcubic) unit cells.

.1.2. EBSD analysis of wireAn inverse pole figure (IPF) map from EBSD of a cross-

ection of the wire drawn down to 1000 �m diameter is shownn Fig. 5. Orientation color key is also shown in Fig. 5. Grainsre delimited by boundaries with misorientations of 15◦ or more.ig. 5(a) is revealing a mixture of 〈1 1 1〉 and 〈1 0 0〉 fibers withegions of more complex textures, where “complex” simply

eans orientations other than the 〈1 1 1〉 or 〈1 0 0〉 fiber orienta-

ions expected in wire drawing. The latter are those areas remain-ng after excluding the 〈1 1 1〉 and 〈1 0 0〉 oriented regions. Thelue- or red-shaded grains represent 〈1 1 1〉 or 〈1 0 0〉 regions,

206 J.-H. Cho et al. / Materials Science and Engineering A 432 (2006) 202–215

Fig. 3. The 1 1 1 pole figure and ODF (ϕ2 = 45◦ section) for Goss, Brass and Copper orientations generated by Gaussian function (FWHM = 12.5◦). Texture componentsare for cubic and orthorhombic symmetries for crystal and sample, respectively. Euler space is extended just for triclinic sample symmetry, which makes it easy toa ϕ2 ≤3

rwtTar

stcisitFs

dmdcb

cfabftdtaat

otblfC

nalyze the shear deformation of wires, i.e. 0◦ ≤ ϕ1 ≤ 360◦, 0◦ ≤ Φ ≤ 90◦, 0◦ ≤5.26◦, 45◦}.

espectively, and the white grains are complex regions. The smallindow in the left upper side of Fig. 5(a) shows the location of

he measured cross-section of the 1000 �m diameter gold wire.he (1 1 1) pole figures are shown in Fig. 5(b) for all regionsnd in Fig. 5(c) for only the 〈1 1 1〉 + 〈1 0 0〉-oriented regions,espectively. The drawing direction is vertical in the IPF map.

The texture is a mixture of the strong single component per-isting from the cast bar and an expected fiber component (inhe form of rings around the drawing axis). The major textureomponent in Fig. 5(b) is similar to the Goss component ands similar to that in Fig. 4(d). Only the 〈1 1 1〉 + 〈1 0 0〉 regionshow a strong Goss component in Fig. 5(c). Since most of thenitial 〈1 0 0〉 fiber texture in the casting remains in the cen-er in Fig. 5(a), it appears that the Goss component observed inig. 5(c) comes from the remnant cast texture. The 〈1 1 1〉 grainstart to appear throughout the cross-section of the wire.

Fig. 6 shows IPF maps for longitudinal sections at wireiameters of 522, 185, 81 and 25 �m, respectively. Each IPF

ap covers the entire width of the wire. As the wire diameter

ecreases, the equivalent strain increases. The 〈1 1 1〉, 〈1 0 0〉 andomplex grains are shaded with the same orientation color key asefore. Most of the 〈1 0 0〉-oriented grains are located in the wire

〈sbb

90◦. (a) Goss {90◦, 90◦, 45◦}; (b) Brass {54.74◦, 90◦, 45◦}; (c) Copper {90◦,

enter. Some 〈1 0 0〉-oriented grains are found near the wire sur-ace as shown in Fig. 6(d). The presence of the 〈1 0 0〉 componentt the center is likely inherited from the texture of the castingar. The presence of 〈1 0 0〉 at or near the surface is related toriction between the wire and the dies during drawing leading tohe formation of a shear layer. The 〈1 1 1〉 fiber develops duringrawing. Fig. 6(a)–(d) show that 〈1 1 1〉-oriented grains occurhroughout the cross-section. Two bands of complex regions arepparent in Fig. 6(c) and (d) that are located between the surfacend the center of the wire. This will be discussed further withhe aid of the finite element analysis.

Cho et al. have discussed the microstructural characteristicsf cold-drawn gold bonding wires (smaller than 30 �m in diame-er) in detail [37]. Most boundaries are either 〈1 1 1〉 or 〈1 0 0〉 tiltoundaries. �3, 7, 13b, 21a, 31a types and near-coincident siteattice boundaries (CSLs) with 〈1 1 1〉 misorientation axes areound frequently in the 〈1 1 1〉 fiber regions. The most frequentSLs in the 〈1 0 0〉 regions are �5,13a, 17a, 25a and 29a with

1 0 0〉 axes, as expected. Boundaries between grains with theame fiber show smaller misorientation angles than boundariesetween grains with different fibers. Almost all of the boundariesetween the 〈1 1 1〉 and 〈1 0 0〉 components have misorientation

J.-H. Cho et al. / Materials Science and Engineering A 432 (2006) 202–215 207

F axialp rograp

a2

tF〈Tit5ntrstfnpC44a{4ao4

tfipat�peTpfi1TPsFvi

cnww

ig. 4. Micrograph of a 7-mm casting gold bar and its 1 1 1 pole figures. AD isarallel to the AD. Contours: 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 6 and 7. (a) Mic

ngles above 40◦ so those CSL types are mainly �3, 9, 11, 17b,5b, 31b and 33c.

Figs. 7 and 8 show the (1 1 1) pole figures and ODF sec-ions (ϕ2 = 45◦), respectively, calculated from the IPF maps inig. 6. Both figures include three regions: (a) all grains, (b)1 1 1〉 + 〈1 0 0〉 fiber regions only and (c) complex regions only.he 〈1 1 1〉 + 〈1 0 0〉 grains occupy most of the cross-section

n Fig. 6 and, pole figures for these two fibers look similaro those that include all grains. In Fig. 7(a), corresponding to22 �m diameter, the position of the second strongest compo-ent for all regions or the 〈1 1 1〉 + 〈1 0 0〉 regions is similar tohe Copper component. The (1 1 1) pole figure for the complexegion in Fig. 7(a) resembles that of the Brass component ashown in Fig. 3(b). In Fig. 8(a), the corresponding ODFs forhe 522 �m diameter wire show these more clearly. In the ODFor all regions, there are three major peaks, which are locatedear {45◦, 20◦, 45◦}, {90◦, 30◦, 45◦} and {50◦, 90◦, 45◦}. Theeaks at {90◦, 30◦, 45◦} and {270◦, 30◦, 45◦} are close to theopper component, which is ideally located at {90◦, 35.26◦,5◦} and {270◦, 35.26◦, 45◦}, respectively. Likewise {50◦, 90◦,5◦}, {130◦, 90◦, 45◦}, {230◦, 90◦, 45◦} and {310◦, 90◦, 45◦}re close to the Brass, which is located at {54.74◦, 90◦, 45◦},125.26◦, 90◦, 45◦}, {234.74◦, 90◦, 45◦} and {305.26◦, 90◦,

5◦}, respectively. The ODF for the 〈1 1 1〉 + 〈1 0 0〉grains showssimilar texture to that of the whole area. The Goss componentf the 〈1 1 1〉 + 〈1 0 0〉 regions, which is located at {90◦, 90◦,5◦} or {270◦, 90◦, 45◦}, is stronger than that observed for

stta

direction, RD is radial direction and TD is transverse direction. Casting axis ish; (b) grain alignment; (c) experimental 1 1 1 PF; (d) 1 1 1 PF rotated from (c).

he whole area. Although the overall texture of gold wire hasber (cylindrical) symmetry, the local behavior is analogous tolane strain because of the initial alignment with both tensilexis and radial directions aligned with specific crystal direc-ions (1 0 0 and 1 1 0, respectively). �-Fiber (Goss–Brass) and-fiber (Brass–S–Copper) components are found in fcc duringlane strain compression. Similarly, the 〈1 1 1〉 + 〈1 0 0〉 regionsxhibit both �-fiber and �-fiber components during drawing.here are a few Copper and Goss oriented grains in the com-lex region and the Brass component appears to dominate. Polegures show stronger �- and �-fiber texture components in the85 �m diameter wire (Fig. 7(b)) than in the 522 �m (Fig. 7(a)).his is the result expected as drawing deformation increases.ole figures for 81 and 25 �m wires (Fig. 7(c) and (d)) showimilar trends. Fig. 8(b)–(d) show the ODFs that correspond toig. 7(b)–(d). The Brass component in the complex region isery weak in the 25 �m wire. The �- and �-fibers are prevalentn the 〈1 1 1〉 + 〈1 0 0〉 regions.

The variations in volume fraction of the 〈1 1 1〉, 〈1 0 0〉 andomplex regions are shown in Fig. 9. The 〈1 0 0〉 fiber compo-ent decreases monotonically to a small steady-state fraction,hereas the 〈1 1 1〉 fiber component increases monotonically,hile the complex region decreases steadily, especially at high

trains. The volume fraction is based on the ODF in Fig. 8 and aolerance misorientation angle, 15◦ was used to partition orien-ation space. Around the ideal orientation or ideal fiber direction,ll texture components closer than the tolerance angle (15◦) are

208 J.-H. Cho et al. / Materials Science and Engineering A 432 (2006) 202–215

F its 1C 1〉 +o

atFa

Fl

ig. 5. An inverse pole figure (IPF) map from EBSD of a 1-mm gold wire andontours: 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 6 and 7. (a) EBSD map for the 〈1 1nly.

ccumulated for the corresponding texture volume [36]. The ini-ial casting bar has mostly 〈1 0 0〉-oriented grains as shown inig. 4(c) and this component decreases during drawing. Beyonddiameter of 185 �m (ε = 7.3), the 〈1 0 0〉 fiber has a volume

ftic

ig. 6. IPF map from EBSD of gold wires according to their diameters. IPF maps areongitudinal section of the gold wire. (a) 522 �m; (b) 185 �m; (c) 81 �m; (d) 25 �m

1 1 pole figures. EBSD was measured from the cross-section of the gold wire.〈1 0 0〉 region; (b) 1 1 1 PF for all region; (c) 1 1 1 PF for 〈1 1 1〉 + 〈1 0 0〉 region

raction of about 10% and, as the drawing ratio increases beyondhis point, it does not change much. By contrast, the 〈1 1 1〉 fiberncreases monotonically and the volume fraction of the complexomponents decreases.

showing whole diameter of wires, respectively. EBSD was measured from thewires.

J.-H. Cho et al. / Materials Science and Engineering A 432 (2006) 202–215 209

F urs: 0(

n1〈〈

p

ig. 7. The 1 1 1 pole figures for gold wires calculated from EBSD data. Contod) 25 �m.

Fig. 10 shows volume fraction for several orientations,

amely Brass{1 1 0}〈1 1 2〉, S{1 2 3}〈6 3 4〉, Copper{112}〈11〉, Goss{1 1 0}〈0 0 1〉, cube{1 0 0}〈0 0 1〉, rotated cube{1 0 0}

0 1 1〉, rotated Goss{1 1 0}〈1 1 0〉, {1 1 1}〈1 1 2〉, and {1 1 2}1 1 0〉. The Brass, S and Copper components are the major com-

w〈tf

Fig. 8. ODFs (ϕ2 = 45◦ section) for gold wires calculated from EBSD data. Con

.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 6 and 7. (a) 522 �m; (b) 185 �m; (c) 81 �m;

onents for all regions during drawing. Other components are

eaker than these three texture components. In the 〈1 1 1〉 and

1 0 0〉 fiber regions, the Copper component is the strongest, andhe S and Brass are weaker than the Copper. Although the volumeraction in the complex regions decreases with strain, the S and

tours: 1, 2, 5, 10 and 20. (a) 522 �m; (b) 185 �m; (c) 81 �m; (d) 25 �m.

210 J.-H. Cho et al. / Materials Science and E

Fig. 9. Volume fraction for each fiber and complex regions calculated fromEBSD. Strain is given by equivalent strain according to deformation. (a)1000 �m; (b) 522 �m; (c) 185 �m; (d) 81 �m; (e) 25 �m.

Fig. 10. Volume fraction for texture components calculated from EBSD. (a) Allregions; (b) 〈1 1 1〉 + 〈1 0 0〉 regions; (c) complex regions.

BTvcIriustci

3

sswidisc

watTtaamonreiacFbd

faigiidrnCife

ngineering A 432 (2006) 202–215

rass components are prevalent down to 81 �m diameter wire.he Copper component is a part of the 〈1 1 1〉//RD fiber so itsolume belongs to the 〈1 1 1〉 fiber regions. Therefore the Copperomponent in the complex regions is expected to be negligible.n 25 �m wire, the volume fractions of 〈1 1 1〉 and 〈1 0 0〉 fiberegions are about 90% (with 80% 〈1 1 1〉 and 10% 〈1 0 0〉 shownn Fig. 9(e)). About 40% of these 〈1 1 1〉 and 〈1 0 0〉 fiber vol-me consists of the Brass, S, Copper, Goss and {1 1 1}〈1 1 2〉 ashown in Fig. 10(b). The complex regions with 10% volume frac-ion include many other components than the nine deformationomponents described above. It is interesting that {1 1 1}〈1 1 2〉s around 15% and is the strongest of the nine components.

.2. FEM analysis

FE modeling of wire drawing in this study contains all sixteps as mentioned before. The forward and backward drawingteps (Steps 2 and 5) are the major steps that were modeled,hereas the others are preparation and finishing steps for draw-

ng. The finite element modeling with forward and backwardies is a simplified drawing process to study the two-step draw-ng process with the opposite drawing directions. Each drawingtep in the FE modeling covers 10% reduction in area and itorresponds to one die in the actual wire fabrication.

The variation in shear strain, ε12, with radial position in theire is shown in Fig. 11. Radial and axial directions are referred

s 1 and 2 directions, respectively. Shear strains were taken fromhe region of interest or along the heavy line shown in Fig. 2.he radial position is parameterized by the relative radial posi-

ion, or ratio of s/s0. Here, s is the distance from the wire centernd s0 is the wire radius. Thus the center is equivalent to s = 0nd the surface is s = 1, respectively. After forward drawing, theaximum shear strain is near the surface (s = 0.9). The location

f the maximum shear strain in the backward step shifts fromear the surface (s = 0.9) to the inner region (s = 0.55). The mate-ials between these two maximum shear strains are expected toxperience the most severely repeated shear deformation dur-ng drawing. The shape of the meshes changes with the forwardnd backward drawing steps. The deformed mesh shapes areompared with the parallel guide lines in the right side of theig. 11. The wire center shows almost zero shear strain and it cane regarded as undergoing uniaxial tensile deformation duringrawing.

Fig. 12 displays the variations in deformation gradient. Timesor forward and backward drawing steps correspond to 1–3nd 6–8 s, respectively. The plus and minus signs represent thencrease and decrease in the components of the deformationradient, respectively. The deformation gradient is decomposednto rigid body rotation and pure stretch [38]. When consider-ng ideal drawing deformation, the diagonal components of theeformation gradient are mainly the pure stretch. In the centeregion (s = 0) in Fig. 12(a) and (b), the F12 and F21 shear compo-ents are much smaller than the F11 and F22 and are negligible.

onsequently, the deformation pattern is close to the ideal draw-

ng with minimal rigid rotation. The F22 increases during bothorward and backward drawing, whereas F11 decreases duringach step. This means that the material entering the drawing

J.-H. Cho et al. / Materials Science and Engineering A 432 (2006) 202–215 211

mesh

dct(indsft

pnFpib

Fs

Fig. 11. Shear strain variations and

ie experiences tension along 2-direction or axial direction andompression along 1-direction or radial direction, regardless ofhe forward or backward drawing directions. On the surfaces = 1.0), in Fig. 12(c) and (d), shear deformation is expectedn addition to the drawing deformation. The deformations areo longer pure stretches. Instead, the shear components of the

eformation gradient contribute some rotation in addition to thetretch. Unlike the center region, the F11 and F22 on the sur-ace show extra tension and compression at the beginning ofhe drawing steps, respectively. It implies that the material is

lpdg

ig. 12. Variations in deformation gradient. Times for forward and backward drawi= 0; (c) F11 and F22 at s = 1.0; (d) F12 and F21 at s = 1.0.

shape during drawing modeling.

iled up at the entry. Inside the die, the F11 and F22 compo-ents reveal the expected compression and tension, respectively.12 during forward drawing increases and decreases and therofile of it shows a peak. Conversely, that for backward draw-ng decreases and increases and the profile shows a valley. F21ehaves in a complementary fashion to the F12 component, but

eaves more residuals than F12. The main difference between oneass drawing and double pass drawing with the opposite drawingirections results from the shear components of the deformationradient.

ng are 1–3 and 6–8 s, respectively. (a) F11 and F22 at s = 0; (b) F12 and F21 at

212 J.-H. Cho et al. / Materials Science and Engineering A 432 (2006) 202–215

F , 2.1,

wislocIdpfocetisrta

rtituscai(wa

aiitideammacaioiadid

afwvt〈I〈

ig. 13. The 1 1 1 pole figures calculated from FE modeling. Contours: 0.1, 1.1

In metal forming processes, frictional tractions betweenorkpiece and die produce surface textures differing from those

nside the workpiece. In particular, the surface texture of rolledheets has been investigated in experiments [39,40] and simu-ations [41]. For the wire drawing, Mathur and Dawson pointedut that the shear component of the rate of deformation wasomparable to the normal components near the surface [20].n addition, the shear component changed direction during theeformation. This resulted in the smearing of the final textureroduced near the surface. Shear deformation generally comesrom both roll geometry and friction effects. Some combinationf these effects can form distinct surface textures, or almost can-el each other. Lee and Duggan have pointed out that geometricffects of the rolling gap contributed to surface texture forma-ion, in addition to friction effects [42]. The roll geometry effectsnvoke some shear strain with minimal friction. The inclusion ofhear components into the rolling geometry leads to a reducedate of texture sharpening. In our FE modeling, shear deforma-ion was also found near the surface even with minimal frictionnd contributed surface texture.

In order to evaluate the dependence of texture evolution onadial position during FE modeling, deformation gradients wereaken at several positions along region of interest and used asnput to the crystal plasticity model. Four locations at the cen-er of the wire (s = 0), the middle part (s = 0.65), immediatelynder the surface (s = 0.9) and on the surface (s = 1) were cho-en for texture analysis. A random texture with 1000 singlerystals was used as the initial texture. The (1 1 1) pole figuresnd ODFs (ϕ2 = 45◦ section) calculated from the FE model-

ng results are shown in Figs. 13 and 14, respectively. The1 1 1) pole figures of the texture at s = 0 (Fig. 13(a)), show aeak drawing texture after the forward drawing step only andstronger drawing texture after the backward drawing step is

1iai

3.1, 4.1, 5.1, 6.1, 7.1, 8.1, 9.1. (a) s = 0; (b) s = 0.65; (c) s = 0.9 and (d) s = 1.0.

dded. It means that the drawing texture in the center regionncreases as the reduction in area increases, regardless of draw-ng direction. The (1 1 1) pole figures in Fig. 13(a) are similaro that of the experimentally observed 〈1 1 1〉 + 〈1 0 0〉 regionn Fig. 7(d) which shows the typical drawing texture indepen-ent of initial texture of casting bar. The center of the wirexperiences essentially ideal drawing deformation. As the rel-tive position, s, increases, however, the deformation becomesore complicated. Shear deformation increases due to tool andaterial contact with the wire surface. The repeated forward

nd backward drawing steps make the material flow compli-ated between s = 0.55 and 0.9. Fig. 13(b)–(d) show that textureway from the wire center deviates from that of ideal draw-ng. In Fig. 14, ODFs for s = 0 and 0.65 show that the strengthf the drawing texture components (Brass, S and Copper)ncreases after backward drawing. However, ODFs for s = 0.9nd 1.0 shows that the texture strength decreases after backwardrawing. The drawing texture formed during forward draw-ng breaks up after backward drawing due to inhomogeneouseformation.

The volume fraction changes at the four different positionsre shown in Figs. 15 and 16. Fig. 15 shows the volume variationor each of the 〈1 1 1〉, 〈1 0 0〉 and complex texture componentsith position. The volume fraction changes for the nine indi-idual texture components are shown in Fig. 16. The initialextures were random and the initial volume fractions of the1 1 1〉 and 〈1 0 0〉 fibers were about 14% and 10%, respectively.n Fig. 15(a), at the center of the wire after forward drawing, the1 1 1〉 fiber has a volume fraction of about 20% and the 〈1 0 0〉

0%. The 〈1 1 1〉 fiber increases slightly during forward draw-ng whereas the 〈1 0 0〉 fiber shows little change. These trendsre similar for the other positions. Following backward draw-ng, the center, which is assumed to experience ideal drawing

J.-H. Cho et al. / Materials Science and Engineering A 432 (2006) 202–215 213

ontou

dsfd

sctr

FF

uafis

Fig. 14. ODFs (ϕ2 = 45◦ section) calculated from FE modeling. C

eformation, has 40% of 〈1 1 1〉. As the distance from the center,, increases, shear strain increases and the deformation deviatesrom ideal drawing. Both 〈1 1 1〉 and 〈1 0 0〉 volume fractionsecrease.

The volume fraction evolution of the nine texture components

hows that the Brass, S, Copper and {1 1 1}〈1 1 2〉 are majoromponents after forward drawing. After backward drawing,he Brass, S and Copper still increase strongly in the centralegion (s = 0). However, they decrease on the surface (s = 1) and

ig. 15. Volume fraction for each fiber and complex regions calculated fromEM. (a) Forward drawing and (b) forward + backward drawing.

tTm

Fw

rs: 1, 2, 5, 10, 20. (a) s = 0; (b) s = 0.65; (c) s = 0.9 and (d) s = 1.0.

nder the surface (s = 0.9). The {1 1 2}〈1 1 0〉 and {1 1 1}〈1 1 2〉re prevalent in these positions. For 25 �m wire, the volumeraction of the {1 1 1}〈1 1 2〉 component in the complex regionss also largest in Fig. 10(c). This component is related to thehear deformation. IPF maps in Fig. 6(b)–(d) show that most of

he complex regions are located under the surface (s = 0.6–0.9).he change of shear strain is also largest in these areas of FEodeling results as shown in Fig. 11.

ig. 16. Volume fraction for texture components calculated from FEM. (a) For-ard drawing and (b) forward + backward drawing.

214 J.-H. Cho et al. / Materials Science and E

Fig. 17. Elastic moduli calculated from textures of EBSD and FEM results.Erg

3

ciFaatmv

ElRtdEaSwcdi

A(uctgbahm

cFuctrtdaftnvpaudt〈c

4

Etbe

1

2

3

quivalent strain of FEM is 0.21 after backward drawing. Ideal compliance forandom polycrystals are given as 68.4 GPa by Reuss average. Ideal E[1 0 0] isiven as 42.5 GPa. (a) From EBSD; (b) from FEM.

.3. Elastic modulus

For a fine bonding wire such as the 25 �m diameter oneonsidered here, textural or microstructural homogeneity is anmportant factor that affects the bondability during packaging.or example, wire loop swing, which means that wire is twistednd leaned from its original loop position, has been known torise from microstructural inhomogeneity and low strength, ando result in bonding failure. The elastic modulus is an important

echanical property for bonding wire. Generally, high 〈1 1 1〉olume increases the stiffness and improves the strength.

Fig. 17 shows elastic moduli calculated from textures ofBSD and FE results. Ideal elastic modulus or Young’s modu-

us for a random polycrystals of gold is 68.4 GPa by using theeuss estimate (uniform local stress or lower bounds assump-

ion). The elastic modulus of gold is largest along the 〈1 1 1〉irection, E[1 1 1] = 115.2 GPa, and is smallest along 〈1 0 0〉,[1 0 0] = 42.5 GPa. The anisotropic elastic constants (compli-nces) for gold are S11 = 23.55 TPa−1, S12 = −10.81 TPa−1, and44 = 24.10 TPa−1. Fig. 17(a) shows the variation in modulus

ith strain in experiment. The initial elastic modulus was cal-

ulated from the texture of the 7 mm diameter casting bar. It isominated by the 〈1 0 0〉 texture and the value is 44.8 GPa whichs slightly larger than the minimum value, E[1 0 0] = 42.5 GPa.

4

ngineering A 432 (2006) 202–215

s the equivalent strain increases, so does the elastic modulusparallel to the wire axis). The complex regions have lower mod-li than the 〈1 1 1〉 + 〈1 0 0〉 regions. The volume fraction of theomplex grains decreases during drawing and therefore its con-ribution to the elastic modulus of the wire decreases. The 〈1 1 1〉rains increase in volume fraction during drawing so their contri-ution is significant. After ε = 11.3, the volume of 〈1 1 1〉 grainspproaches 80% (Fig. 9(e)) and the elastic modulus reaches itsighest value, 90 GPa. However, this value is still lower than theaximum elastic modulus, E[1 1 1] = 115.2 GPa.Elastic moduli calculated from textures derived from FE

alculations are shown in Fig. 17(b). The initial texture forEM is homogeneous (random) so its predicted elastic mod-lus is 68.4 GPa. This value is higher than that of the initialasting bar (mainly 〈1 0 0〉 fiber), 44.8 GPa. Thus a higher ini-ial elastic modulus computed from FEM texture results in theelatively high elastic moduli during deformation comparedo the corresponding experimental strain level. After forwardrawing, the equivalent strain approaches ε = 0.105 and theverage modulus of the wire from the center (s = 0) to sur-ace (s = 1) shows 76–77 GPa. It is about 10 GPa higher valuehan the initial elastic modulus. Elastic moduli show homoge-eous distributions independent of position. Fig. 15(a) plotsolume fractions after forward drawing and shows that mostarts of the wire have the similar fractions of the 〈1 1 1〉, 〈1 0 0〉nd complex grains regardless of position. The elastic mod-li seem to reflect these trends. When followed by backwardrawing, the elastic moduli increase in most of the wire excepthe surface. The surface region has the smallest fraction of1 1 1〉 grains as shown in Fig. 15(b), and the elastic modulus isorrespondingly low.

. Conclusions

Cold-drawn gold wires have been characterized by XRD andBSD as a function of drawing strain. In order to understand

he deformation process and texture evolution, the forward andackward drawing steps were modeled with a commercial finitelement package, ABAQUS:

. The initial casting bar had a cast microstructure with〈1 0 0〉//AD texture. In addition to this typical alignment, itappeared that most grains also had 〈1 1 0〉 parallel to the radialdirection. For diameters down to 1000 �m, some 〈1 0 0〉 fiberfrom the casting structure remained in the wire center. Dur-ing drawing the volume fraction of the 〈1 1 1〉 fiber increasedwhereas the 〈1 0 0〉 fiber decreased down to 10%.

. The complex orientations were defined as all those except〈1 1 1〉 and 〈1 0 0〉 fiber components. These were foundmainly under the surface and most of them rotated to the〈1 1 1〉 fiber during wire drawing.

. Brass, S and Copper were the major texture components dur-ing drawing, based on the definition of h k l//RD (radial dir.)

and u v w//AD (axial dir.). The Copper texture componentwas dominant in the 〈1 1 1〉 and 〈1 0 0〉 regions.

. A simplified two-step drawing model using FEM showed thatthe greatest amount of shear strain was found from s = 0.55

and

5

6

7

A

iKpbC

R

[

[[

[[[

[[[

[[[

[

[

[[[[[

[[[

[

[[[

[[

[

J.-H. Cho et al. / Materials Science

to 0.9. The complex texture components were found in theseregions in the EBSD IPF maps.

. The main difference between one pass drawing and doublepass drawing with the forward and backward drawing resultsfrom the shear components of the deformation gradient.

. Texture evolution was predicted using the FE-computeddeformation gradients together with a rate-sensitive poly-crystal plasticity model. In the center, an ideal drawing defor-mation was found as expected and drawing textures werecalculated. On the surface (s = 1.0) and under the surface(s = 0.9), {1 1 2}〈1 1 0〉 and {1 1 1}〈1 1 2〉 components areprevalent. These components are related to increased sheardeformation during drawing.

. Elastic moduli based on the uniform local stress assumption(Reuss) were calculated according to the equivalent drawingstrain and relative distance from the wire center for experi-mental EBSD and FEM textures, respectively.

cknowledgements

This research is supported by the BK21 project of the Min-stry of Education & Human Resources Development, Southorea. This work was also supported in part by the MRSECrogram of the National Science Foundation under award num-er DMR-0079996. The authors are grateful to MK Electrono., R&D Center for provision of the material.

eferences

[1] W.F. Hosford, R.M. Caddell, Metal Forming-Mechanics and Metallurgy,Prentice-Hall, Englewood Cliffs, NJ, 1983.

[2] T.H. Ramsey, Solid State Technol. 16 (1973) 43–47.[3] E.M. Fridman, C.V. Kopezky, L.S. Shvindlerman, Z. Metall. 66 (1975)

533–539.[4] B.L. Gehman, Solid State Technol. 23 (1980) 84–91.[5] S. Omiyama, Y. Fukui, Gold Bull. 15 (1980) 43.

[6] K. Busch, H.U. Kunzi, B. Ilschner, Scr. Metall. 22 (1988) 501–505.[7] G. Qi, S. Zhang, J. Mater. Process. Technol. 68 (1997) 288–293.[8] J. Nutting, J.L. Nuttall, Gold Bull. 10 (1977) 2–8.[9] H. Ahlborn, Z. Metall. 56 (1965) 205–215.10] H. Ahlborn, Z. Metall. 56 (1965) 411–420.

[[[[

Engineering A 432 (2006) 202–215 215

11] G. Linßen, H.D. Mengelberg, H.P. Stuwe, Z. Metall. 55 (1964) 600–604.12] U.F. Kocks, C.N. Tome, H.-R. Wenk, Texture and Anisotropy, Cam-

bridge University Press, 1998 (Chapter 5).13] G.I. Taylor, J. Inst. Met. 62 (1938) 307–324.14] A.T. English, G.Y. Chin, Acta Met. 13 (1965) 1013–1016.15] M.G. Stout, J.S. Kallend, U.F. Kocks, M.A. Przystupa, A.D. Rollett,

Eighth International Conference on Textures of Materials (ICOTOM-8),TMS, Warrendale, Pennsylvania, 1988, p. 479.

16] E. Aernouldt, I. Kokubo, H.P. Stuwe, Z. Metall. 57 (1966) 216–220.17] T. Montesin, J.J. Heizmann, J. Appl. Cryst. 25 (1992) 665–673.18] J.J. Heizmann, C. Laruelle, A. Vadon, A. Abdellaoui, Proceedings of the

11th International Conference on Textures of Materials (ICOTOM-11),Xi’an, China, 1996, pp. 266–272.

19] K. Rajan, R. Petkie, Mater. Sci. Eng. A257 (1998) 185–197.20] M. Pietrzyk, L. Sadok, J. Mater. Process. Technol. 22 (1990) 65–73.21] P. Karnezis, D.C.J. Farrugia, J. Mater. Process. Technol. 80–81 (1998)

690–694.22] S.-H. Choi, J.-H. Cho, K.H. Oh, K. Chung, F. Barlat, Int. J. Mech. Sci.

42 (2000) 1571–1592.23] J.-H. Cho, S.-J. Park, S.-H. Choi, K.H. Oh, Proceeding of the Eighth

International Conference on Textures of Materials (ICOTOM-12), Mon-treal, Canada, 1999, pp. 587–592.

24] K.K. Mathur, P.R. Dawson, J. Eng. Mater. Technol. 112 (1990) 292–297.25] W. Voigt, Lehrbuch der Kristallphysik, Teubner, Leipzig, 1928.26] A. Reuss, Z. Angew. Math. Mech. 9 (1929) 49–58.27] R. Hill, Proc. Phys. Soc. (London) A65 (1952) 54–349.28] W.F. Hosford, The Mechanics of Crystals and Textured Polycrystals,

Oxford University Press, 1993.29] Z. Hashin, S. Shtrikman, J. Mech. Phys. Solids 10 (1962) 335.30] Z. Hashin, S. Shtrikman, J. Mech. Phys. Solids 10 (1962) 343.31] S. Matthies, G.W. Vinel, K. Helming, Standard Distributions in Texture

Analysis, Akademie-Verlag, Berlin, 1987.32] REDS, Repressing of EBSD Data in Seoul National University, User

Manual, Texture Control Lab, 2002.33] ABAQUS Standard User Manual, Hibbitt, Karlsson & Sorensen, 1998.34] L.S. Toth, P. Gilormini, J.J. Jonas, Acta Metall. 36 (1988) 3077–3091.35] G.R. Canova, C. Fressengeas, A. Molinari, U.F. Kocks, Acta Metall. 36

(1988) 1961–1970.36] J.-H. Cho, A.D. Rollett, K.H. Oh, Metall. Trans. 35A (2004) 1075–1085.37] J.-H. Cho, J.-S. Cho, J.-T. Moon, J. Lee, Y.H. Cho, Y.W. Kim, A.D.

Rollett, K.H. Oh, Metall. Trans. 34A (2003) 1113–1125.38] A.S. Khan, S. Huang, Continuum Theory of Plasticity, John Wiley and

Sons Inc., NY, 1995.39] J. Hansen, H. Mecking, ICOTOM-4, Cambridge, 1975, pp. 34–47.40] W. Truszkowski, J. Krol, B. Major, Metall. Trans. 11A (1980) 749–758.41] K.K. Mathur, P.R. Dawson, Int. J. Plast. 5 (1989) 67–94.42] C.S. Lee, B.J. Duggan, Metall. Trans. 22A (1991) 2637–2643.