The effect of texture on ridging of ferritic stainless steel

Acta Materialia 51 (2003) 4693–4706 www.actamat-journals.com

The effect of texture on ridging of ferritic stainless steel Hyung-Joon Shin a, Joong-Kyu An b, Soo Ho Park c, Dong Nyung Lee d,∗ b c

a Center for Science in Nanometer Scale, ISRC, Seoul National University, Seoul 151-742, South Korea Metal & Ceramic Research Group, LG Cable Ltd, 555, Hogye-dong, Dongan-gu, Anyang-si, Kyungi 431-080, South Korea Stainless Steel Research Group, Technical Research Laboratories, POSCO, Pohang P.O. Box 36, Kyungbuk 790-785, South Korea d School of Materials Science and Engineering, and RIAM, Seoul National University, Seoul 151-742, South Korea

Received 19 November 2002; accepted 28 February 2003

Abstract Ferritic stainless steel sheets exhibit ridging parallel to the rolling direction when subjected to tension or deep drawing. The origin of ridging behavior has not been clearly explained yet. Many people agree that ridging originates from different plastic anisotropies of grains. In this study, 430 and 409L stainless steels having columnar and equiaxed structures were chosen as initial specimens to elucidate the role of microstructure and composition on ridging. The specimens initially having the columnar structure showed severe ridging and 409L stainless steel showed an inferior surface quality. The existence of band-like colonies of similar orientations was found in the center of the sheets by electron back-scattered diffraction measurement. In addition, the previous models suggested by other researchers were examined quantitatively by the crystal plasticity finite element method. In order to obtain a more realistic ridging simulation, the specimens containing variously oriented colonies in a textured matrix were also considered. The simulated results showed that the lower plastic strain ratio of {001}⬍110⬎ colonies and different shear deformations of {111}⬍110⬎ or {112}⬍110⬎ colonies resulted in ridging.  2003 Published by Elsevier Ltd on behalf of Acta Materialia Inc. Keywords: Ferritic stainless steel; Ridging; Crystal plasticity; Finite element method; Texture

1. Introduction The ferritic stainless steel (FSS) sheet containing 11–17% Cr is known to develop an undesirable surface corrugation known as ridging when pulled or deep drawn. When pulled or deep drawn, FSS shows undulations, with peaks on one side of the sheet coinciding with valleys on the other side

∗

Corresponding author. E-mail address: [email protected] (D.N. Lee).

without change in the thickness. The ridges have a depth in the range of 20–50 µm. Ridging is distinguished from the stretcher strain in that the latter occurs within the range of about 5% elongation, whereas the former becomes noticeable at a stage of about 5% elongation and increases the amplitude on further straining until break of the specimen. This undesirable surface defects obliges the manufacturer to add costs in polishing operation. Many causes of ridging have been suggested over the last three decades. The segregation of alloying elements such as chromium, molyb-

1359-6454/$30.00  2003 Published by Elsevier Ltd on behalf of Acta Materialia Inc. doi:10.1016/S1359-6454(03)00187-3

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denum, or carbon may be one of the possible reasons for ridging [1,2]. But many people agree that the texture is responsible for ridging [3–9]. The columnar grains in FSS slab have ND//⬍100⬎ texture. The columnar grains develop the rotated cube texture after plane strain deformation and are not easily recrystallized [10]. In addition, the slab does not undergo a to g or g to a phase transformation during all processes. Therefore, long grain colonies with similar orientations develop from the columnar grains in the slab during cold rolling and survive even after annealing. These colonies and matrix are likely to show different plastic anisotropies, resulting in ridging. There exist three basic concepts associating ridging phenomena with anisotropy of plastic deformation, which have been proposed by Chao [3], Takechi et al. [4], and Wright [5]. Though these models give good physical pictures of ridging, they do not take interactions between neighboring grains into account. Brochu et al. [8] measured local orientations of FSS by electron backscattered diffraction (EBSD). However, they did not take interactions among grains into account. They calculated surface profiles by considering only strains along the normal and rolling directions, i.e. eND and eRD. Instead of considering the compatibility conditions among grains, they assumed that the ‘center of deformation’ replaced the compatibility conditions. The force equilibria among grains could not be fulfilled either. It has been very difficult so far to consider the interactions among grains. When material deforms, the stress or strain of each grain is different from the macroscopic stress or strain. The operating slip systems vary with the orientation of each grain. The compatibility conditions and force equilibria among grains should be satisfied simultaneously. The crystal plasticity finite element method (CPFEM) is one of the powerful simulation tools. The crystal plasticity provides a micromechanical model for slip-dominated plastic flow and serves as a constitutive theory. The finite element method offers a numerical means to solve partial differential equations, such as the field equations of elasticity and plasticity. These are combined together in the CPFEM. Harren and Asaro [11] conducted detailed polycrystal simulations with models of

idealized two-dimensional crystals. Kalidindi et al. [12] simulated the plane–strain forging process of copper and compared the calculated results with experimental ones. Beaudoin et al. [13] performed FEM simulations of a model crystal deformed in channel die compression. They introduced hybrid formulation, where the stress components are introduced as global field variables by forming a weighted residual on the crystal constitutive relationships. There are numerous CPFEM results for other cases [14–21]. Beaudoin et al. [16] simulated roping phenomenon in aluminum sheet using crystal plasticity, which is similar to ridging of FSS. They measured orientation image mapping (OIM) of the surface layer and simulated surface roughening when materials are under biaxial stretching. They showed the relationship between texture and roping. The purpose of this study is to test the previous ridging models more quantitatively using the CPFEM and to simulate ridging of a FSS sheet using the orientation distribution through the thickness measured by EBSD. The role of various colonies will also be discussed.

2. Experimental procedures The compositions of 409L and 430 FSS used in this study are given in Table 1. The sample were cut from columnar (C) and equiaxed (E) zones of a slab as shown in Fig. 1. The size of the initial sample was 150 mm in length, 200 mm in width, and 25 mm in thickness. This was hot rolled to 3.5 mm in thickness after homogenization treatment. The hot band was annealed and cold rolled in several passes to 0.7 mm in thickness, followed by annealing. The details of process conditions are given in Table 2. Ridging characteristics were determined after 15% tensile elongation. The Table 1 Chemical composition of stainless steel (wt%)

430 409L

C

Si

Mn

Cr

Ti

N

0.048 0.008

0.37 0.56

0.42 0.25

16.4 11.4

– 0.23

0.037 0.009


Fig. 1.

Schematic drawing of casting structure through cross-section of the ingot.

Table 2 Process conditions

3. Numerical procedure STS430

Reheating temp. Finishing temp. Pass schedule HR annealing CR annealing

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STS409L

1200 °C 930 °C 25→17→12→8→5.2→3.5 (mm) 850 °C, 5 h 930 °C, 1 min. 860 °C, 30 s 930 °C, 30 s

We adopted Kalidindi’s method [12] to implement crystal plasticity in FEM. The constitutive equations and time–integration procedures were implemented in the implicit finite element code ABAQUS Standard, by writing the user material subroutine UMAT. We define an elastic deformation gradient by Fe ⫽ FFp⫺1 , det Fe ⬎ 0.

roughness profiles were measured using a surface roughness measuring instrument. The crystallographic textures were measured by X-ray diffraction and electron back-scattered diffraction (EBSD) methods. The (110), (200) and (112) incomplete pole figures were measured in the center and surface layers using Co Ka1 radiation in the back reflection mode. From the pole figures the orientation distribution function (ODF) was computed by the WIMV method [22]. The local textures of the sheet were measured by scanning electron microscope–electron back-scattered diffraction (SEM-EBSD) technique. This technique made it possible to determine the orientations of grains with a precision of 1° and a spatial resolution of 200 nm. The orientation of each grain obtained from EBSD was used for CPFEM.

(1)

The plastic part Fp in this multiplicative decomposition of F represents the cumulative effect of dislocation motion on the active slip systems in the crystal and the elastic part of Fe describes the elastic distortion of the lattice. The evolution of the plastic velocity gradient is given by Lp ⫽ F˙pFp⫺1 ⫽

冘

g˙ aSa0 , Sa0 ⫽ ma0 丢na0 ,

(2)

a

where Sa0 is the Schmid tensor and ma0 and na0 are time-independent orthonormal unit vectors which define the slip direction and slip plane normal of the slip system a in a fixed reference configuration. For a rate-sensitive slip, the usual power law is used to relate the plastic shearing rate on the ath slip system, g˙ a, to the resolved shear stress, ta, as follows [23]:

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ta ta g˙ a ⫽ g˙ 0 a a g g

冉冊

||

1 ⫺1 m

(3)

Elastic constant

where ga and m represent the slip system resistance parameter and the rate sensitivity of slip, respectively. The self and latent hardening can be readily accounted for by suitable evolution of ga values in the constitutive law of Eq. (3) [12]. For ga values, the approach of Kalidindi et al. [12] has been employed. ga ⫽

冘

Table 3 Material parameters

hab|g˙ b|.

Strain rate sensitivity Initial value of slip resistance parameter Saturation value of slip resistance parameter Reference shear strain rate Hardening parameter

(4)

m = 80.69 GPa l = 111.44 GPa m = 0.02 g0 = 110 MPa gsat = 252 MPa g˙ 0 = 0.0001 q = 1.4 a = 2.24 h0 = 1.7 GPa

b

The hab is n × n hardening matrix, where n is the total number of slip systems. It describes the rate of increase of the deformation resistance on slip system a due to shearing on slip system b. The several simple phenomenological forms for the hardening matrix, hab, have been suggested. Peirce et al. [24] used the following simple form for the hardening law. hab ⫽ [q ⫹ (1⫺q)dab]hb

(5)

with hb denoting the self-hardening rate and the parameter q representing the latent hardening parameter. The self hardening rate, hb, can be obtained by

冉冊

hb ⫽ h0 1⫺

gb gsat

a

(6)

where h0, a, and gsat are slip system hardening parameters. In order to investigate the ridging phenomena, we simulated 20% tensile straining of the FSS sheet. First, the previous models suggested by Chao [3], Takechi et al. [4], and Wright [5] were tested. In addition, the roles of various grain colonies, which are believed to play a role in ridging phenomena, were investigated. We compared tensile deformation behavior of textured sheets with and without grain colonies. Finally, we calculated tensile deformation behavior when local orientations obtained from EBSD measurement of a real sample were used for input data of CPFEM. The material parameters used for simulations are given in Table 3. The 24 slip systems (12 {110}⬍111⬎ and 12 {112}⬍111⬎) were

assumed to be active during deformation. A finite element mesh was formed by ABAQUS-C3D8R (continuum, 3-D, eight-noded, reduced integration) element. Each element represents one orientation.

4. Results and discussion 4.1. Experimental results Fig. 2 shows the surface profiles of FSS after tensile deformation. The measured ridging heights of the specimens after tension are given in Table 4. The maximum peak of the surface profile is defined as the ridging height. STS 409L stainless steel sheet shows severer ridging than STS 430 steel sheet. For the same composition, initially columnar structured specimens show higher ridging heights than equiaxed specimens. Fig. 3 shows ODFs of the surface and center layers of FSS after cold rolling and subsequent annealing. The textures are approximated by RD//⬍110⬎ (a-fiber) and ND//⬍111⬎ (g-fiber) orientations with peaks in the {334}⬍483⬎ and {111}⬍112⬎ orientations. The {334}⬍483⬎ and {111}⬍112⬎ orientations appear in the recrystallized IF steel [25], and bcc stainless steel [26] sheets and are closely related to the {558}⬍110⬎ and {001}⬍110⬎ components in deformation texture, respectively [25]. The presence of the {334}⬍483⬎ and {111}⬍112⬎ components indicates that the sheets were recrystallized, which is consistent with the results in Fig. 3. In the center layer of the 409C specimen, the


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Fig. 2. Surface roughness profiles of FSSs after tensile straining by 15%. Table 4 Ridging height after tensile straining by 15% Specimen 430C Ridging height (µm) 22.5

430E 19.8

409C 55.0

409E 42.5

{115}⬍110⬎ orientation is very strong. It is known that the ND//⬍001⬎ orientation, which is the orientation of the columnar grains, rotates to the {001}⬍110⬎ orientation after plane strain compression [10]. The {115}⬍110⬎ component is deviated from the {001}⬍110⬎ orientation by

Fig. 3. ODFs (j 2 = 45°) of FSS. From top left: 430C-S, 430ES, 430C-C, 430E-C. From top right: 409C-S, 409E-S, 409C-C, 409E-C. Bottom: Location of ideal orientations and fibers. Last characters S and C stand for surface and center layers.

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16°. Therefore, the {115}⬍110⬎ orientation is thought to originate from deformation of the columnar grains. The {001}⬍110⬎ component eventually transforms into the {111}⬍112⬎ orientation after recrystallization [25,27]. However, the {001}⬍110⬎ to {111}⬍112⬎ transformation rate is slow, because the {001}⬍110⬎ component has low Taylor factor, and in turn low stored energy [25]. The results in Fig. 4 were measured from RD. Interestingly, the grain size is in the order of the ridging height. The grain size in 409C is the largest of all. The initial grain size also influences the final grain size after processing. The columnar structured specimens have larger grains, because the initial grain size of them is larger than that of the equiaxed specimens. The 409L stainless steel does not experience any transformation during manufacturing, whereas transformation, recovery and recrystallization occur competitively in 430 stain-

less steel. Therefore, the grain refinement in 430 stainless steel could be partially achieved. The difference in the riding heights between the 409L and 430 stainless steels is attributed more to composition than initial grain size. Since the 409C specimen revealed most severe ridging, its broad EBSD mapping was measured from ND and RD (Fig. 5). The major texture of the specimen can be approximated by the ND//{111} and RD//⬍110⬎ orientation. An about 600 µm wide colony of {001}⬍110⬎ is found in the center layer. The position where severe ridging occurs was consistent with that colony. Besides, various colonies are extended along RD. These colonies are not single crystals, but in the form of similar orientations, which is sufficient to bring about macroscopic plastic anisotropy during deformation.

Fig. 4. EBSD mapping of ND measured from RD of stainless steel specimens. (a) 430C, (b) 430E, (c) 409C, (d) 409E, (e) orientation color. Vertical and horizontal directions of each figure indicate ND and TD. White scale bar 80 µm.


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Fig. 5. EBSD mapping of 409C specimen. Upper two figures (ND and RD) measured from RD; lower two figures (ND and RD) measured from ND.

4.2. Examination of the previous models 4.2.1. Chao’s model Chao [3] attributes ridging to different plastic strain ratios between the ND//⬍111⬎ and ND//⬍100⬎ components as shown in Fig. 6a. The most serious problem in his model is that the deformed outline of the surface is far from that of real materials, which shows undulations [5]. It means that the peaks or valleys are symmetric with respect to the center plane. As is well known, ND//⬍111⬎ components of BCC metals show higher r-values than ND//⬍100⬎ components. The calculated result is similar to his prediction (Fig.

6b). The normal strains, eTD and eND, assumed by him results in symmetrically deformed surface. However, the calculated result shows that not only the normal strains, but also the shear strain gTN plays a role in ridging, where the subscripts T and N stand for TD and ND, respectively. The {111}⬍110⬎ components buckle during deformation. When a (111)[011¯ ] crystal is pulled along RD, (211¯ )[11¯ 1] and (21¯ 1)[111¯ ] slip systems are most activated, and are asymmetric with respect to RD. Therefore, the shear strain, gTN, is generated and undulations occur macroscopically in Chao’s model too.

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Fig. 6. (a) Schematic drawing of Chao’s model and (b) result simulated by CPFEM.

Fig. 7. (a) Schematic drawing of Takechi’s model and (b) result simulated by CPFEM.

4.2.2. Takechi’s model Takechi [4] focused on different shear strains between RD//⬍110⬎ fibers. As mentioned above, the {111}⬍110⬎ components undergo shear deformation when pulled along RD. Though his model is very simple, he considered crystal plasticity, so the result calculated in this study is in good agreement with his prediction as shown in Fig. 7. As mentioned above, asymmetric slip brings about the shear strain gTN. The shear direction varies with respect to RD or ND.

4.2.3. Wright’s model According to Wright [5], there occurs the compatibility problem during deformation due to different plastic–strain ratios between the {111}⬍112⬎ matrix and the {001}⬍110⬎ band. The {001}⬍110⬎ orientation gives rise to lower r-vales than the {111}⬍112⬎ orientation. In order to satisfy the compatibility condition, he claimed that the {001}⬍110⬎ band should buckle along ND as shown in Fig. 8a and b. However, CPFEM


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Fig. 8. Deformation (a) before and (b) after tensile straining by 20% along RD in Wright’s model. (c) Upper and lower sides of deformed mesh (displacement: magnification, ×2) and (d) distribution of eTD calculated by CPFEM.

results for Wright’s model show that, instead of buckling of the {001}⬍110⬎ band, it shrinks only along the thickness direction on both sides (Fig. 8c). This discrepancy comes from his strict assumption. The model implies that distortion is limited to the {001}⬍110⬎ band, i.e. AB and CD lines in Fig. 8d must remain straight during deformation. However, the present calculation shows a little different behavior. If the lateral side

of sheet, AB or CD, is free to move, the compatibility is compensated by a little contraction of matrix around the band. 4.3. Effects of different colonies in a textured matrix Though the previous models are instructive, they are too simple in that there are many more grains

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and the colonies do not extend through the thickness in real materials. The EBSD results indicate that the colonies develop in the center layer. Therefore, we will consider more realistic cases where colonies exist in the center layer of the sheet. 4.3.1. The textured matrix We need to obtain a texture similar to the texture of matrix without colonies (Fig. 3). Fig. 9 shows the texture of a bcc steel specimen consisting of the 12600 (70 in TD × 20 in ND × 9 in RD) elements (Fig. 10), calculated using CPFEM, when the specimen, initially randomly oriented, was plane–strain compressed by 50%. The texture in Fig. 9 is similar to the measured ones in Fig. 3 and will be referred to as the matrix texture (MT) hereafter. In the case where grains are randomly distributed so that the specimen can have MT, no ridging occurs and the strain distributions are of regular pattern after 20% tensile strain as shown in Fig. 10. 4.3.2. The {001}⬍110⬎ colonies Fig. 11 shows the results simulated for the {001}⬍110⬎ colonies in the center zone of the MT oriented matrix. The {001}⬍110⬎ colonies shrink more in the thickness direction than the matrix, resulting in ridging. This is an extended

Fig. 10. Calculated shape of and strain distributions in specimen of Fig. 9 after tensile straining by 20%. From top: Initial mesh, deformed mesh (displacement: magnification ×2), distributions of eTD and gTN.

Fig. 9. CPFEM calculated ODF (j 2 = 45°) of bcc steel plane– strain compressed by 50%.

concept of Wright’s model. Through-thickness colonies are considered in Wright’s model, whereas the {001}⬍110⬎ colonies are located in the center zone of the sheet in this case. It follows from this result that Wright’s prediction is different from the behavior of real materials. The compatibility between colonies and matrix is compensated for not by buckling of the {001}⬍110⬎ colonies, but


Fig. 11. Calculated shape of and strain distributions in specimen consisting of matrix of Fig. 9 and {001}⬍110⬎ colonies after tensile straining by 20%. From top: Initial mesh, deformed mesh (displacement: magnification ×2), distributions of eTD and gTN.

by deformation of the colonies and matrix under restriction. Therefore, ridging occurs due to differences in plastic–strain anisotropies between the colonies and the matrix and it is not in the form of undulation, i.e. the upper and lower sides of sheet are symmetric. 4.3.3. The {111}⬍110⬎ colonies The {111}⬍110⬎ colonies in the matrix with MT give rise to undulations as shown in Fig. 12. This is similar to Takechi’s model. However, the degree of ridging is less severe than that in Take-

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Fig. 12. Calculated shape of and strain distributions in specimen consisting of matrix of Fig. 9 and {111}⬍110⬎ colonies after tensile straining by 20%. From top: Initial mesh, deformed mesh (displacement: magnification ×2), distributions of eTD and gTN.

chi’s model, due to interactions between the colonies and matrix grains. The adjacent colonies with different rolling orientations bring about different shear directions after deformation, resulting in distinct undulations. 4.3.4. The {112}⬍110⬎ colonies The {112}⬍110⬎ orientation is one of major components in the a-fiber orientation. Fig. 13 show the simulated deformation of FSS sheet with the {112}⬍110⬎ colonies. The deformed shape of it is similar to the case of the {111}⬍110⬎ colonies. The undulations occur, because the {112}⬍110⬎

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Fig. 14. Calculated deformation of (111)[1¯ 10] and (112)[1¯ 10] oriented crystal after tensile straining of 20%.

Table 5 Strains of (111)[1¯ 10] and (112)[1¯ 10] crystals after tensile straining by 20%

(111)[1¯ 10] (112)[1¯ 10] Fig. 13. Calculated shape of and strain distributions in specimen consisting of matrix of Fig. 9 and {112}⬍110⬎ colonies after tensile straining by 20%. From top: Initial mesh, deformed mesh (displacement: magnification ×2), distributions of eTD and gTN.

colonies bring about shear strain, gTN, during tensile deformation. The difference between effects of the {111}⬍110⬎ and {112}⬍110⬎ colonies is that contraction along TD direction of the {112}⬍110⬎ colonies is smaller than that of the {111}⬍110⬎ colonies. Fig. 14 shows calculated deformations of the (111)[1¯ 10] and (112)[1¯ 10] oriented crystals after a tensile strain of 0.2. The strains of the crystals after a tensile strain of 0.2 are given in Table 5. The transverse strain eTD of the (111)[1¯ 10] crystal is smaller than that of the (112)[1¯ 10] crystal and the strain gTN of the former is slightly smaller than that of the latter. Therefore,

eTD

eND

gTN

⫺0.1342 ⫺0.07578

⫺0.0472 ⫺0.1057

0.1580 0.1779

the ridging heights for both colonies are almost the same, while the widths of the specimens after deformation are different. 4.4. Simulation based on the EBSD data Ridging of 409C sheet is simulated based on EBSD data. It is desirable that the same morphologies and orientations of grains as the measured data are used for CPFEM calculation. However, it is impractical to measure the shapes and orientations of all grains in the specimen from its twodimensional sections. In order to avoid this problem, Bhattacharyya et al. [28] simulated compression of a polycrystalline aluminum specimen with columnar grains using CPFEM. Even if the shapes and orientations of all grains in a specimen


are available, computational time and capacity may make it impossible or too costly to realize deformation behavior of millions of grains in the specimen. The EBSD and X-ray data indicate that 409C specimen has the g- and a-fiber textures as the main components and some colonies. To simplify the problem, the specimen is modeled as MT textured matrix embedded by some colonies measured by EBSD. Fig. 15 shows the initial specimen mesh with colonies of different orientations and strain distributions in the specimen after a tensile strain of 0.2. The simulated result indicates that ridging

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is well simulated by CPFEM. The low plastic– strain ratio of the {001}⬍110⬎ colonies and different shear deformations of the {111}⬍110⬎ or {112}⬍110⬎ colonies give rise to ridging. The height of ridging simulated here is lower than the results simulated in Sections 4.2 and 4.3, because the interactions between differently oriented grains in colonies can weaken the anisotropy of colonies. If grain orientations are randomly distributed, plastic anisotropy of each grain can be weakened and compensated by neighboring grains. But if they are in the form of colonies, their anisotropies cannot be neglected and bring about macroscopic corrugations. Therefore, the colonies should be eliminated for the good surface quality of FSS. Many studies on reducing ridging have focused on eliminating the columnar structure, which is the origin of colonies [26,29–32].

5. Conclusion

Fig. 15. Calculated shape of and strain distributions in specimen consisting of matrix of Fig. 9 and various colonies after tensile straining of 20%. From top: initial mesh, deformed mesh (displacement: magnification ×2), distributions of eTD and gTN.

CPFEM analyses of STS 430 and 409L FSS having different initial microstructures have led to the following conclusions. The initially columnar-structured specimen developed more severe ridging than the equiaxed one and STS409L FSS shows more severe ridging than STS430 FSS. The locations of colonies found in the center zone of specimen are in agreement with those of ridging. Since the previous models, suggested by Chao [3], Takechi et al. [4], and Wright [5], do not take interactions among grains, or compatibilities among grains into account, their predictions differ from the results calculated using CPFEM in which the interactions and compatibilities are taken into account. EBSD and X-ray data indicate that the structure of 409C specimen can be modeled as the MT-textured matrix whose center zone is embedded by the {001}⬍110⬎-, {111}⬍110⬎- and {112}⬍110⬎oriented colonies. The result of CPFEM analysis of this structure shows that the lower plastic–strain ratio of the {001}⬍110⬎ colonies and differences in shear deformation between the {111}⬍110⬎ and {112}⬍110⬎ colonies result in ridging.

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Acknowledgements The authors gratefully acknowledge the financial support by POSCO and Texture Control Laboratory (NRL), Seoul National University. References [1] Chao HC, Defilippi JD. Effect of chromium and molybdenum segregation on the ridging behavior of type 434 stainless steel. Metall Trans 1971;2(11):3209–16. [2] Suzuki K, Asami S, Suzuki K. Formation of ridging related to the banded segregation pattern of chromium and carbon on ferritic stainless steel sheet. Trans ISIJ 1983;23(9):731–7. [3] Chao HC. The mechanism of ridging in ferritic stainless steels. Trans ASM 1967;60(1):37–50. [4] Takechi H, Kato H, Sunami T, Nakayama T. The mechanism of ridging phenomenon in 17% chromium stainless steel sheets. Trans JIM 1967;31(6):717–23. [5] Wright RN. Anisotropic plastic flow in ferritic stainless steels and the roping phenomenon. Metall Trans 1972;3(1):83–91. [6] Bethke K, Ho¨ lcher M, Lu¨ cke K. Local orientation investigation on the ridging phenomenon in iron 17% chromium steel. Mater Sci Forum 1994;157-162(2):1137–44. [7] Kim HM, Szpunar JA. Ridging phenomena in textured ferritic stainless steel sheets. Mater Sci Forum 1994;157162:753–60. [8] Brochu M, Yokota T, Satoh S. Analysis of grain colonies in type 430 ferritic stainless steels by electron back scattering diffraction. ISIJ Int 1997;37(9):872–77. [9] Sztwiertnia K, Pospiech J. Orientation topography and ridging phenomenon in ferritic stainless steel sheets. Arch Metall 1999;44(2):157–65. [10] Tsuji N, Tsuzaki K, Maki T. Effects of rolling reduction and annealing temperature on the recrystallization structure of solidified columnar crystals in a 19% ferritic stainless steel. ISIJ Int 1994;34(12):1008–17. [11] Harren SV, Asaro RJ. Nonuniform deformations in polycrystals and aspects of the validity of the Taylor model. J Mech Phys Solids 1989;37(2):191–232. [12] Kalidindi SR, Bronkhorst CA, Anand L. Crystallographic texture evolution in bulk deformation processing of fcc metals. J Mech Phys Solids 1992;40(3):537–69. [13] Beaudoin AJ, Dawson PR, Mathur KK, Kocks UF. A hybrid finite element formulation for polycrystal plasticity with consideration of macrostructural and microstructural linking. Int J Plasticity 1995;11(5):501–21. [14] Becker R, Panchanadeeswaran S. Effects of grain interactions on deformation and local texture in polycrystals. Acta metal mater 1995;43(7):2701–19. [15] Sarma GB, Dawson PR. Effects of interactions among crystals on the inhomogeneous deformations of polycrystals. Acta mater 1996;44(5):1937–53.

[16] Beaudoin AJ, Bryant JD, Korzekwa DA. Analysis of ridging in aluminium auto body sheet metal. Metall Trans 1998;29A(9):2323–32. [17] Kothary M, Anand L. Elasto-viscoplastic constitutive equations for polycrystalline metals: applications to tantalum. J Mech Phys Solids 1998;46(1):51–84. [18] Bate P. Modelling deformation microstructure with the crystal plasticity finite element method. Phil Trans Roy Soc Lond 1999;357A:1589–601. [19] Harder J. A crystallographic model for the study of local deformation processes in polycrystals. Int J Plasticity 1999;15(6):605–24. [20] Kumar A, Dawson PR. Computational modeling of FCC deformation textures over Rodrigues’ space. Acta mater 2000;48(10):2719–36. [21] Pe´ rocheau F, Driver JH. Texture gradient simulations for extrusion and reversible rolling of FCC metals. Int J Plasticity 2000;16(1):73–89. [22] Matthies S, Vinel GW, Helming K. Standard distributions in texture analysis. Berlin: Akademie-Verlag, 1987. [23] Hutchinson JW. Bounds and self-consistent estimates for creep of polycrystalline materials. Proc Roy Soc Lond 1976;348A:101–27. [24] Pierce D, Asaro RJ, Needleman A. An analysis of nonuniform and localized deformation in ductile single crystals. Acta metall 1982;30(6):1087–119. [25] Hong SH, Lee DN. Recrystallization textures in coldrolled Ti bearing IF steel sheets. ISIJ Int 2002;42(11):1278–87. [26] Huh M-Y, Engler O. Effect of intermediate annealing on texture, formability and ridging of 17%Cr ferritic stainless steel sheet. Mater Sci Eng 2001;A308(1-2):74–87. [27] Lee DN. The evolution of recrystallization textures from deformation textures. Scripta Metall Mater 1995;32(10):1689–94. [28] Bhattacharyya A, El-Danaf E, Kalidindi SR, Doherty RD. Evolution of grain-scale microstructure during large strain simple compression of polycrystalline aluminium with quasi-columnar grains: OIM measurements and numerical simulations. Int J Plasticity 2001;17(6):861–88. [29] Arakawa M, Takemura S, Ooka T. On the ridging properties and the textures of 17%Cr stainless steel strip coldrolled and annealed from continuously cast slab. Trans ISIJ 1971;11(2):890–4. [30] Takeuchi H, Mori H, Ikehara Y, Komano T, Yanai T. The effects of electromagnetic stirring on cast structure of continuously cast SUS430 stainless steel slabs. Tetsu to Hagane 1980;66(6):638–46. [31] Hasegawa H, Maruhashi S, Muranaka Y, Hoshi F, Kunugasa M. Refining the solidification structure of continuously cast 18% chromium stainless steel by electromagnetic stirring. Tetsu to Hagane 1981;67(8):1354–62. [32] Uematsu Y, Yamazaki K. Effects of hot rolling conditions on hot rolled microstructures and ridging properties in 17% chromium ferritic stainless steel. Tetsu to Hagane 1992;78(4):632–9.