Estimation of Planar Anisotropy of the r-value in Ferritic Stainless Steel Sheets

Transcription

1 Materials Transactions, Vol. 50, No. 4 (2009) pp. 752 to 758 #2009 The Japan Institute of Metals Estimation of Planar Anisotropy of the r-value in Ferritic Stainless Steel Sheets Jun-ichi Hamada 1, Kazuyuki Agata 2 and Hirofumi Inoue 3 1 Research & Development Center, Nippon Steel & Sumikin Stainless Steel Corp., Hikari , Japan 2 Hikari Stainless Steel Technology, Hikari , Japan 3 Department of Materials Science, Graduate School of Engineering, Osaka Prefecture University, Sakai , Japan Planar anisotropies of r-values are calculated from crystallite orientation distribution functions (ODFs) by using the Taylor theory. These values are compared with experimentally obtained values for two kinds of ferritic stainless steel sheets with different planar anisotropies of the r- values and different texture gradients in the thickness direction. Moreover, the most suitable model for predicting the r-value of ferritic stainless steel sheets is examined. For a material with a remarkable texture gradient in the thickness, the ODF in a particular plane perpendicular to the normal direction was unsuitable for predicting the planar anisotropy of the r-value. On the other hand, r-value calculated using the average ODF through sheet thickness measured in a section perpendicular to the transverse direction shows good agreement with the experimentally obtained. The relaxed-constraints model, in which the shear strain components e 13 and e 23, are relaxed with a CRSS ratio of 1.1 in the f211gh111i and f110gh111i slip systems is the most suitable model for predicting the planar anisotropy of the r-value in ferritic stainless steel sheets. [doi: /matertrans.mra ] (Received October 24, 2008; Accepted January 7, 2009; Published March 25, 2009) Keywords: ferritic stainless steel, r-value, planar anisotropy, texture, orientation distribution function, Taylor model 1. Introduction Polycrystalline materials with texture show anisotropy in various properties, depending on their orientation and degree of their density. 1) For example, the plastic strain ratio, also referred to as the r-value (Lankford value), is closely related to the crystal orientation and is the basic index of formability such as deep drawability, earing in drawing, and hole expansion. 1 10) Therefore, in order to obtain the required formability, it is extremely important to understand the quantitative relation between the texture and the anisotropy of the r-value as well as to control the texture in the manufacturing process. Predictions of the anisotropy of the r- value in polycrystalline materials have been made using a crystallite orientation distribution function (ODF) and theoretical treatments based on the Taylor theory ) The pole figures required for calculating the ODF are generally obtained from the X-ray diffraction of a plane parallel to the sheet surface. However, it is known that some ferritic stainless steel sheets and some Fe-3% Si steel sheets have inhomogeneous texture in the thickness direction ) An example of such a sheet is shown in Fig. 1; the figure shows an orientation map obtained by means of electron back scatter diffraction (EBSD) analysis. For materials with inhomogeneous texture, highly precise predictions of their physical and mechanical properties cannot be made by using only the ODF measured in a particular layer parallel to the rolling plane, and the prediction method should consider the inhomogeneity in the thickness direction. Thus, it is necessary to develop prediction techniques that consider the inhomogeneity. Methods for predicting the anisotropy of the r-value and yield strength that use the average (i.e., overall) texture of the sheets in the thickness direction have been suggested for low-carbon steel, aluminum, and stainless steel sheets ,35) Although it is thought that the consideration of the average texture is effective for the prediction of macroscopic properties, there are hardly any studies that have compared the predicted r-value of ferritic stainless steel sheets with the experimentally obtained r-value. In this study, for two kinds of ferritic stainless steel sheets with homogeneous and inhomogeneous textures in the thickness direction and with different anisotropies of the r- value, we predicted the r-value using the Taylor theory by quantitatively determining the average texture in the thickness direction. The full-constraints Taylor model and the RC1323 model, called the pancake version of the relaxedconstraints models, were employed in this simulation. Here, the first two numbers 13 indicate the shear strain component e 13 parallel to the tensile direction in the rolling plane and the next two numbers 23 refer to the shear strain components e 23 perpendicular to the tensile direction in the rolling plane. The most suitable model for predicting the plastic anisotropy of ferritic stainless steel sheets was also determined by considering the slip systems and the critical resolved shear stress ratio of the {211} and {110} glide planes. 2. Experimental Procedures The chemical compositions of commercial ferritic stainless steel sheets used in this study are given in Table 1. The thickness (t) of both samples was mm. The r-values of JIS 13B test pieces cut from these sheets were measured after 14.4% tensile elongation at angles of 0,15,30,45,60, 75, and 90 with respect to the rolling direction. Table 2 shows the r-values in three representative directions, the average r-values (r m ) and the planar anisotropy parameters (r). R m and r were calculated using the following equations. r m ¼ðr 0 þ 2r 45 þ r 90 Þ=4 ð1þ r ¼ðr 0 2r 45 þ r 90 Þ=2 ð2þ Here, r 0, r 45, and r 90 are the r-values in the longitudinal, diagonal, and transverse directions, respectively. Samples A and B had nearly similar compositions and average r-values, but the anisotropy parameters differed significantly. Figure 2 shows the microstructures of these sheets. Both materials are ferrite single-phase recrystallized structures. The average

2 Estimation of Planar Anisotropy of the r-value in Ferritic Stainless Steel Sheets 753 Surface <100>// <211>// <111>// <110>// t/2 300µm RD Fig. 1 Example of orientation map in the longitudinal section of a ferritic stainless steel sheet with a remarkable texture gradient. Black line is high angle (= 15 degrees) boundary. A deviation angle for h211i and other directions are 10 degrees and 15 degrees, respectively. Table 1 Chemical compositions of steels used (mass%). Table 2 Experimental results of r-values. Sample C Si Mn P S Cr Mo Ti Nb N A B Sample r 0 r 45 r 90 r m r A B :4 100µm Fig. 2 Microstructures of sample A and sample B. X-rays TD t/40 ~ t/2 plane RD X-rays grain diameters of samples A and B were 64 mm and 68 mm, respectively, which are similar. The textures were quantitatively examined by using the ODF. Incomplete pole figures of the (110), (200), (211), and (310) diffraction planes for samples of the RD- and TD-sections of the sheets were measured by means of the Schulz reflection method using Mo K radiation. From the four pole figures, the ODF was determined using the iterative series-expansion method. 36,37) For measuring the average texture in the thickness direction of the sheets, 11 sheet pieces were stacked and bonded as shown in Fig. 3, and the pole figures for a cross section perpendicular to the TD (TD-section) were obtained. 35) For comparison with the TD-section, conventional samples (section) parallel to the rolling plane were also prepared from various locations in the thickness direction, as shown in Fig. 3. In both methods, the surface was electrolytically polished for determining the texture. The ODF on the TDbasis was transformed into the normal ODF on the -basis by assuming orthorhombic sample symmetry. 35) Figure 4 shows some important fibers and orientations on the 2 ¼ 45 section in an ODF (Bunge s notation). TD Thickness (t) Fig. 3 Schematic illustrations of pole figure measurements on sectional sample and TD-sectional sample. RD

3 754 J. Hamada, K. Agata and H. Inoue Φ {100}<011> {100}<001> {100}<011> {211}<011> α-fiber {411}<148> {111}<011> {111}<011> {111}<112> γ-fiber ϕ 1 {211}<111> {111}<112> {554}<225> {110}<011> {110}<112> {110}<001> Fig. 4 Some important texture fibers and orientations in Euler space ( 2 ¼ 45 section). X 1 X 1 e 13 X 3 X 3 e 23 Fig. 5 Schematic representation of local strain relaxation in relaxedconstraints models. X 1, X 2 and X 3 are the tensile, width and thickness directions, respectively. X 2 X 2 3. Analytical Procedures In this study, the prediction of the r-values using the Taylor full-constraints (FC) model 11) and the relaxedconstraints (RC) model, 12) in which some shear strain components are relaxed relative to those in the fully constrained state, was carried out by employing ODF data. Software developed by one of the authors was employed in this calculation. 16,17,22 25) Here, the RC1323 model, i.e., the pancake version where e 13 and e 23 (1, 2, and 3 are defined as the tensile, width, and thickness directions, respectively) are left free as shown in Fig. 5, was used in the RC model because it has successfully been used for BCC metals in previous studies. 13,15,22,25) Assuming that the elongation in the tensile direction is d" and that the contractions in the width and thickness directions are qd" and ð1 qþd", respectively, the deformation tensor can be defined as follows. 2 3 d" e 12 e E ¼ 4 0 qd" e 23 5 ð3þ 0 0 ð1 qþd" The r-value is defined by the following equation. r ¼ q=ð1 qþ In the FC model, e 12, e 13, and e 23 are zero, and in the RC1323 model, e 12 is zero. The q-value and r-value in each tensile direction can be calculated from the ODF by minimizing the plastic work of the slip deformation; the minimization can be performed by considering active slip systems. 20,21) Although it has been reported that h111i is the slip direction and {110}, {211}, and {321} are the slip planes in BCC crystals, in the present study, the f110gh111i and f211gh111i slip systems were considered because they take into account the plastic flow in BCC crystals. 38) The values of the critical resolved shear stress (CRSS) ratio of the {211} and {110} glide planes ( f211g = f110g ) were assumed to be 0.9,, and 1.1 in calculations using the RC1323 model. ð4þ 4. Results and Discussions 4.1 Textures Figure 6 shows the 2 ¼ 45 sections for the ODFs determined from the -section pole figures of the surface, quarter, and center layers; the figure also shows the overall ODFs for the direction of the sheet thickness that was determined from the TD-section pole figures for both samples A and B. With regard to the determination of the texture of each layer in the -section, the texture of sample A shows few changes along the sheet thickness, and the recrystallization texture consists mainly of orientations from f111gh011i to f111gh112i, which is termed the -fiber. On the other hand, sample B has a remarkable texture gradient in the thickness direction. In the quarter and center layers of sample B, the -fiber was stronger than that in sample A, and the texture ranging from f111gh112i to f411gh148i, i.e., fh; 1; 1gh1=h; 1; 2i recrystallization texture 39) was formed. Moreover, around the surface layer of sample B, orientations from f100gh011i to f211gh011i, termed the -fiber, were observed. In both samples, the largest orientation density was observed to occur in the center layer. It was presumed that the use of the texture evaluated at a single location in the prediction model was not suitable for a material with an inhomogeneous texture, like sample B. On the other hand, each overall texture calculated from incomplete pole figures of the TD-section qualitatively resembled the ODF for the center layer, and the orientation density was lower than that of the center layer. Sample B with a very large texture gradient was more remarkable than sample A with a small texture gradient because of the influence of the surface texture. 4.2 Predictions of r-value in material with a slight texture gradient Figure 7 shows the planar anisotropies of the r-values measured and calculated using the FC model and RC1323 model from the textures of the surface, quarter, and center

4 Estimation of Planar Anisotropy of the r-value in Ferritic Stainless Steel Sheets 755 Sample section TD t/40 t/4 t/2 Overall A B γ-fiber Fig. 6 Partial textures in the surface, quarter, and center layers and overall textures through the sheet thickness for samples A and B. Contour levels: 110, step 1. +{211} +{211} (c) (d) +{211} +{211} Fig. 7 Planar anisotropy of the r-value measured and calculated using the FC and RC1323 models from partial textures in the surface, quarter, and (c) center layers, and (d) from the overall texture through the sheet thickness for sample A.

5 756 J. Hamada, K. Agata and H. Inoue +{211} +{211} (c) (d) +{211} +{211} Fig. 8 Planar anisotropy of the r-value measured and calculated using the FC and RC1323 models from partial textures in the surface, quarter, and (c) center layers, and (d) from the overall texture through the sheet thickness for sample B. layers, together with the overall texture of sample A, which shows a slight texture gradient. In the case of predictions made using the textures of the quarter and center layers, the FC model overestimates the values, whereas the RC1323 model provides values that are in good agreement with the experimentally obtained values. Although it is observed that calculations in the case of the RC1323 model are less influenced by changes in the slip system and CRSS ratio, a CRSS ratio of 1.1 leads to relatively good agreement with the experimental results when compared with CRSS ratios of 0.9 and. The r-values calculated using the RC1323 model from the texture of the surface layer are underestimated because the density of the -fiber is smaller than that of the quarter and center layers. In contrast, the planar anisotropy of the r-value calculated from the overall texture using the RC1323 model gives a fairly good prediction. This result suggests that the anisotropy of the r-value can be predicted with sufficient accuracy by the RC1323 model using the ODF of the quarter or center layer as well as the average ODF in the thickness direction. 4.3 Predictions of r-value in material with a remarkable texture gradient Figure 8 shows the planar anisotropies of the r-values measured and calculated using the FC model and RC1323 model from textures in the surface, quarter and center layers, together with the overall texture of sample B, which shows a remarkable texture gradient. When ODFs determined from the -section are used for the prediction, the calculated values differ significantly from the experimental values in all models examined in this research. In the case of prediction made using the texture of the quarter or center layer, the obtained r-value curves are qualitatively quite different from the experimentally obtained curve. In the case of prediction made using the texture of the surface layer, although the r- value curves through calculations qualitatively agree with the experimentally observed curve, the r-values are considerably underestimated in both the FC and RC1323 models. Therefore, for a material with texture inhomogeneity, it is difficult to predict the anisotropy of its properties from the texture at a certain location. In contrast, the use of the overall texture in

6 Estimation of Planar Anisotropy of the r-value in Ferritic Stainless Steel Sheets 757 the sheet thickness leads to good prediction of the r-value when the RC1323 model is used with a CRSS ratio of 1.1; however, the FC model provides inaccurate prediction irrespective of the texture used. This suggests that reliable prediction of the anisotropy of r-values can be obtained by using the average ODF in the thickness direction. In previous studies, it has been shown that the RC1323 model is more suitable than the FC model for computing the r-value of IF steel, Al-killed drawing-quality steel, 13,15,22) and ferritic stainless steel 25) with a BCC structure. The results of this study are similar to those of previous studies, regardless of the texture gradient. Because the RC1323 model with relaxed shear strains gives the good agreement compared with the FC model assuming homogeneous deformation in all grains, it is suggested that inhomogeneous deformation occurs for ferritic stainless steel sheets. For the f211gh111i and f110gh111i glide systems, CRSS ratios of ,15) and ) have been reported to be suitable. It is necessary to examine the use of other models and to compare a number of experiments to determine the model and CRSS ratio that provide the best prediction for the r-values. In this study, as a result of examining CRSS ratios of 0.9,, and 1.1, the best results are found to be obtained when the RC1323 model is used with a CRSS ratio of Relation between texture gradient and r-value According to the above observations, even if the texture is inhomogeneous in the thickness direction like in sample B, the r-value is related to the overall recrystallization texture in the thickness direction. In order to identify the part of the sheet thickness that influences the r-value the most, the recrystallization textures at various locations in the thickness direction are determined in sample B. The orientation densities of each preferred orientation in the thickness direction are shown in Fig. 9. While -fiber orientations are observed from the near-surface region to about t/8, f111gh112i and f554gh225i orientations are notably developed from t/4 to t/2. Moreover, the f411gh148i orientation is dominant near the surface as compared to the center layer. Homma et al. have showed that fh; 1; 1gh1=h; 1; 2i recrystallization texture results from the irregular strain at the grain boundaries of the -fiber with f100gh011i to f211gh011i orientations. 39) Figure 10 shows the planar anisotropy of the r-value for some ideal orientations with Gaussian distribution when the RC1323 model is used with a CRSS ratio of 1.1. From this result, it is suggested that the f100gh011i and f411gh148i components have a much lower r-value than the -fiber for all directions. Further, the f211gh011i component shows an extremely low r-value in the longitudinal and transverse directions as compared to that in the diagonal direction. In spite of the larger density of the -fiber with a high r-value in sample B as compared to sample A, the occurrence of the f411gh148i orientation through the thickness and the remarkable texture gradient in the thickness direction in sample B are presumably related to the r value of the sample. Figure 11 shows the planar anisotropies of the r-value calculated at various locations in the thickness direction of sample B by using the RC1323 model with a CRSS ratio of 1.1. From this figure, it is observed that the r-values depend on the location in the sheet thickness Orientation density, f(g) {100}<011> {211}<011> {111}<011> {111}<112> {554}<225> {411}<148> Distance from surface, D / mm Fig. 9 Change of orientation density along the thickness direction for some preferred orientations in sample B. {100}<011> {211}<011> {111}<011> {111}<112> {554}<225> {411}<148> Fig. 10 Calculated r-values using the RC1323 model with a CRSS ratio of 1.1 for representative ideal orientations with a Gaussian spread of 30 as a function of angle to rolling direction. direction. In this study, in order to consider the volume fraction of each layer, the q-values are calculated from the ODFs of five layers; the average q-value (Ave. q-value) and the average r-value (Ave. r-value) are defined by the following equations.

7 758 J. Hamada, K. Agata and H. Inoue t/2 t/4 Ave. thickness were in good agreement with the measured r- values, regardless of the texture gradient. (2) As a result of examining CRSS ratios of 0.9,, and 1.1 in the f211gh111i and f110gh111i slip systems, the RC1323 model with a CRSS ratio of 1.1 was most suitable for predicting the planar anisotropy of the r- value in ferritic stainless steel sheets. Ave. q-value ¼ X5 i¼1 q i V i Ave. r-value ¼ Ave. q-value=ð1 Ave. q-valueþ ð6þ The average r-value calculated by the rule of mixture and by considering the volume fraction is in relatively good agreement with the experimentally obtained value. This result reveals that the r-value as a whole is determined by the average texture through the thickness. Therefore, for the prediction of material properties, the ODFs obtained at various locations in the thickness can be effectively used. However, this is time-consuming, and the degree of change of the texture and the number of measurements are important for obtaining exact prediction of the r-value. It is concluded that the use of the overall texture of a cross section is effective from the viewpoints of precision and efficiency. 5. Conclusions t/8 t/16 t/ Fig. 11 Planar anisotropies of r-values at various layers calculated using the RC1323 model with a CRSS ratio of 1.1 from the textures at each layer for sample B. In this study, theoretical r-values and their planar anisotropies calculated from the overall texture through the thickness by using the Taylor theory have been compared with experimental results using two kinds of ferritic stainless steel sheets with different planar anisotropies of the r-value and different texture gradients in the thickness direction. Moreover, the most suitable model for predicting the planar anisotropy of the r-value in ferritic stainless steel sheets was examined. The main results can be summarized as follows: (1) In the case of a sample with a remarkable texture gradient, it was impossible to precisely predict the planar anisotropy of the r-value from the texture of a particular -section. On the other hand, the r-values calculated from the overall texture through the sheet ð5þ REFERENCES 1) S. Nagashima: J. Soc. Mater. Sci. Jpn. 32 (1983) ) W. T. Lankford, S. C. Snyder and J. A. Bauscher: Trans. ASM 42 (1950) ) D. B. Lewis and F. B. Pickering: Met. Technol. 10 (1983) ) R. L. Whiteley and D. E. Wise: Flat Rolled Products III, ed. By E. W. Earhart, (AIME, New York, 1962) pp ) W. B. Hutchinson: Int. Met. Rev. 29 (1984) ) R. K. Ray, J. J. Jonas and R. E. Hook: Int. Mater. Rev. 39 (1994) ) F. J. Humphreys and M. Hatherly: Recrystallization and Related Annealing Phenomena, (Pergamon, New York, 1995) pp ) P. Juntunen, D. Raabe, P. Karjalainen, T. Kopio and G. Bolle: Metall. Mater. Tran. A 32A (2001) ) Y. Ito and K. Hashiguchi: Kawasaki Steel Giho 3 (1971) ) T. Sawatani, K. Shimizu, T. Nakayama and T. Hirai: Tetsu-to-Hagané 63 (1977) ) G. I. Taylor: J. Inst. Metals 62 (1938) ) H. Honneff and H. Mecking: Proc. 5th Int. Conf. on Textures of Materials, ed. by G. Gottstein and K. Lücke, (Springer-Verlag, Berlin, 1978) Vol. 1, pp ) D. Daniel and J. J. Jonas: Metall. Trans. A 21A (1990) ) H. Inagaki, K. Kurihara and I. Kozasu: Tetsu-to-Hagané 61 (1975) ) R. Schouwenaars, P. Van Houtte, E. Aernoudt, C. Standaert and J. Dilewijns: ISIJ Int. 34 (1994) ) H. Inoue and N. Inakazu: J. Japan Inst. Light Met. 44 (1994) ) H. Inoue: Recrystallization Textures and Their Application to Structure Control, (ISIJ, Tokyo, 1999) pp ) R. K. Ray, J. J. Jonas, M. P. Butrón-Guillén and J. Savoie: ISIJ Int. 34 (1994) ) H. J. Bunge and W. T. Roberts: J. Appl. Cryst. 2 (1969) ) H. J. Bunge: Kristall und Technik 5 (1970) ) H. J. Bunge: Texture Analysis in Materials Science, (Butterworths, London, 1982). 22) H. Inoue, K. Sekine and T. Hasegawa: Proc. 76th JSME Fall Annual Meeting, (The Japan Society of Mechanical Engineers, 1998) pp ) H. Inoue and T. Takasugi: Mater. Trans. 48 (2007) ) H. Inoue: Yield strength and microstructure in steels Basic understanding and review, (ISIJ, Tokyo, 2006) pp ) H. Inoue, J. Hamada and T. Goto: Yield strength and microstructure in steels Results of experiment and simulation, (ISIJ, Tokyo, 2007) pp ) M. Matsuo: Tetsu-to-Hagané 70 (1984) ) T. Sakai, Y. Saito, M. Matsuo and K. Kawasaki: ISIJ Int. 31 (1991) ) D. Raabe: Mater. Sci. Technol. 11 (1995) ) M. Hölscher, D. Raabe and K. Lücke: Steel Res. 62 (1991) ) D. Raabe: Mater. Sci. Technol. 11 (1995) ) D. Raabe and K. Lücke: Mater. Sci. Technol. 9 (1993) ) D. Raabe and K. Lücke: Scr. Metall. Mater. 26 (1992) ) D. Raabe: Steel. Res. 74 (2003) ) J. Hamada, S. Maeda, F. Fudanoki, M. Abe, T. Shindo and S. Hashimoto: Tetsu-to-Hagané 90 (2004) ) H. Inoue and T. Takasugi: Z. Metallkd. 92 (2001) ) H. Inoue and N. Inakazu: J. Japan Inst. Metals. 58 (1994) ) H. Inoue: Materia Japan 40 (2001) ) P. Franciosi: Acta Metall. 31 (1983) ) H. Homma, S. Nakamura and N. Yoshinaga: Tetsu-to-Hagané 90 (2004)