Journal of Neutron Research, January–September 2004 Vol. 12 (1–3), pp. 201–205
Residual Stress Variation in Polycrystalline Copper during Recrystallization ´ SKIa, R. WAWSZCZAKa, B. BACROIXb K. WIERZBANOWSKIa,*, A. BACZMAN c and A. LODINI a
WFiTJ, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krako´w, Poland; bLPMTM-CNRS, Universite´ Paris XIII, 99, av. J.B. Clement, 93 430 Villetaneuse, France; cLACM, Universite´ de Reims Champagne Ardenne, 9, bd. de la Paix, 51100 Reims, France
A general multi-reflection method was applied to determine the stress level in deformed and recrystallized polycrystalline copper samples. Different reflections hkl were simultaneously used in the fitting procedure. The anisotropic diffraction elastic constants were calculated using the self-consistent model and crystallographic texture. Important decrease of the first order residual stresses was observed during recovery and recrystallization. Independently, diffraction peak widths and intensities were examined for a few characteristic texture components during the recrystallization process. The importance of cubic orientation grains in the recrystallization process was confirmed. Keywords: Residual stress; Multireflection method; Recrystallization; Crystallographic texture; Peak width; Stored energy
INTRODUCTION Recrystallization of previously deformed polycrystalline material is a very complex process. Microstructure, residual stresses, stored energy and crystallographic texture are very strongly modified during annealing. Diffraction technique offers a deep insight in many aspects of this process. In the present work a general multireflection method for determination of the stress field was used. The advantage of the new method is that various reflections hkl are simultaneously used in fitting procedure, consequently the results are statistically more representative in comparison with the single reflection method. The residual stress was measured in rolled and recrystallized copper samples. Residual stress is in general accompanied by dislocation field. This latter is often associated with the stored energy, remaining after deformation process. To have a full description of recrystallization process also the dislocation density and texture transformation have to be examined. This information was obtained by the study of peak width and intensity corresponding to main texture maxima.
*Corresponding author. Fax: þ 48-126340010. E-mail:
[email protected] ISSN 1023-8166 print/ISSN 1477-2655 online q 2004 Taylor & Francis Ltd DOI: 10.1080/10238160410001734685
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MULTI-REFLECTION METHOD FOR STRESS DETERMINATION The standard X ray sin2 c diffraction method of stress analysis is based on the measurement of peak positions for a given hkl reflection and for various directions of the scattering vector (defined by F and c angles). Interplanar spacings are calculated using the Bragg’s law and the macro-stress (first order one) is determined applying a linear or elliptical regression procedure [1 – 3]. The multi-reflection methods, using various hkl reflections and sin2 c geometry was discussed by Kamminga et al. [4]. This method, generalized for anisotropic materials, was used in the present work. Many reflections hkl were simultaneously used in the fitting procedure, hence the results were statistically more reliable. Cobalt radiation was used and sample surfaces were polished by etching before measurement. Anisotropic material was considered, hence the influence of crystallographic texture f(g) and of hkl anisotropy on diffraction elastic constants were taken into account [5]. The above experimental data treatment can be applied for sin2 c method [6] and for grazing incidence geometry as well [7]. The strain determined by diffraction method (i.e. along the scattering vector) is: k1ðc; fÞl{hkl} ¼ F ij ðhkl; c; f; f ðgÞÞs Iij
ð1Þ
where s Iij is the first order residual stress tensor and Fij are diffraction elastic constants. These latter depend in general on hkl reflection, orientation of the scattering vector—c, f and on texture function f(g). The repeated index summation convention is applied in this work. Taking into account that: k1ðc; fÞl{hkl} ¼
0 kdðc; fÞl{hkl} 2 d{hkl} 0 d{hkl}
¼
kaðc; fÞl{hkl} 2 a 0 ; one obtains : a0
kaðc; fÞlhkl ¼ F ij ðhkl; c; f; f ðgÞÞs Iij a 0 þ a 0
ð2Þ
where ka(c,f)l{hkl} is the equivalent inter-planar distance and for the cubic structure: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 h 2 þ k 2 þ l2 : kaðc; fÞl{hkl} ¼ kdðc; fÞl{hkl} h 2 þ k 2 þ l2 and a 0 ¼ d {hkl} Measured and theoretical ka (c,f)l{hkl} parameters, determined for various hkl, c and f can be fitted applying the least square procedure. Using previously calculated Fij (hkl,c,f, f(g)) diffraction elastic constants, the values of a0 and the macro-stress s Iij can be found. Nonlinear Fij(hkl,c,f, f(g)) diffraction elastic constants [5,8] were calculated using the selfconsistent model [9]. These constants were calculated for all considered intermediary textures, corresponding to different annealing temperatures. No strong anisotropy of Fij (hkl,c,f, f(g)) was observed; this is visible on nearly linear ka(f,c)l{311} vs. sin2 c graphs obtained from X ray measurements and the least squares procedure (Fig. 1 a– c).
RESULTS AND DISCUSSION Rolled polycrystalline copper samples were studied in order to evaluate residual stresses during the whole annealing process. The rolling, transverse and normal directions define x, y, z reference system for stress determination. The material was examined after cold rolling (70% reduction) and then after annealing at different temperatures in the range of 100 –2508C. In each case the annealing time was 15 min. The equivalent lattice parameters
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FIGURE 1 Measured kal311 vs. sin2 c for cold rolled polycrystalline copper samples for f ¼ 08 (a), f ¼ 458 (b) and f ¼ 908 (c); (d) Evaluated a0 parameter vs. annealing temperature.
kal200, kal220 and kal311 vs. sin2 c were determined for f ¼ 08, 458 and 908. Exemplary results for kal311 are shown in Fig. 1 a– c. I I I Based on the complete data set, the residual stress components s11 ; s12 and s22 were evaluated. Their variation in function of annealing temperature is shown in Fig. 2. A I I considerable decrease of main residual stress components (i.e. s11 and s22 ) takes place in the temperature range between 150 and 2308C; at the latter temperature the whole sample I is already completely recrystallized. Contrary to that, the s12 shear component has negligible values. It should be underlined that the decrease of main component values starts
I I I FIGURE 2 First order residual stress components s11 ; s12 and s22 vs. annealing temperature for rolled polycrystalline copper (vertical line shows the beginning of recrystallization).
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I already before the beginning of recrystallization, especially for s11 component. The final reduction of residual stress takes place during the recrystallization process (for temperature above 2108C). It is important to note, that approximately the same value of a 0 has been found using the multi-reflection procedure for all the studied samples (Fig. 1d), which confirms that our method evaluates accurately the lattice constant of a stress free material even in the presence of different residual stresses. One can appreciate better the near constant value of a 0, if it is compared with the variation range of kal311 lattice parameter vs. sin2 c (Fig. 1 a– c). It is well known that a strong cubic texture forms during recrystallization of the rolled polycrystalline copper (the cubic texture component has the following Euler angles: ðw1 ; f; w2 Þ ¼ ð08; 08; 08Þ—see e.g. [10]). In general a small amount of cubic orientation is already present after deformation and intensity of this orientation strongly increases during annealing. This tendency is confirmed by the present X-ray measurements. Intensities of 200 diffraction peaks for the cubic, copper and brass orientations are plotted vs. annealing temperature in Fig. 3b (copper and brass texture components mentioned above are defined by (908, 358, 458) and (358, 458, 08) Euler angles, respectively). The intensity of the cubic orientation is low till the temperature of about 2008C and then it grows rapidly. Contrary to that, intensities of the copper and brass orientations decrease in temperatures above 2208C. At the end of recrystallization process the cubic component dominates the final texture. The peak width was also studied (Fig. 3a). The examined material had the mean grain size of about 50 mm after rolling and it was not much modified during the primary recrystallization (average grain shape changed, however). It is known that peak width depends on the square root of the dislocation density. According to Fig. 3a, the peak widths (and consequently dislocation densities) of main texture components do not change much during recovery. This suggests that the observed residual stress decrease at this stage (Fig. 2) is caused by some relaxation processes in grain boundary regions. It can be noted, however, that peak width of the cubic component slightly decreases already during recovery (starting from temperature of 1508C—see Fig. 3a) and then is much reduced during recrystallization. This delicate effect in the recovery stage (i.e. before recrystallization) will be the subject of further investigation. It suggests that dislocation density is slightly reduced in grains of cubic orientation already during recovery process. It is interesting to note that independent synchrotron measurements give the lowest dislocation density for the cubic orientation grains [11]. The important reduction of the peak width during primary recrystallization (above temperature of 2108) is observed for three examined peaks but is the most marked for
FIGURE 3 Peak width—FWHM (a) and peak intensity (b) for three main texture components (cubic, copper and brass) vs. annealing temperature.
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the cubic component. The observed strong decrease of peak widths is mostly caused by reduction of dislocation density. Further study of this effect is planned.
CONCLUSIONS Multi-reflection X-ray method was successfully used for the residual stress measurement in deformed and annealed polycrystalline copper. The main residual stress components are strongly reduced in the annealing temperature range 150 –2308C. The stresses start to decrease already before the recrystallization process and it is probably due to relaxation phenomena in grain boundary regions. Our study confirms exceptional role of the cubic texture component. The peak width for this orientation starts to decrease already during recovery and is strongly reduced during primary recrystallization. Cubic orientation also dominates the final recrystallization copper texture. Acknowledgements The present work was supported by the Polish Commitee of Scientific Research (KBN) and by French-Polish joint research project POLONIUM (No. 3804.III/ 03270RD).
REFERENCES [1] Noyan, I.C. and Cohen, J.B. (1987) Residual Streess (SpringerVerlag, New York). [2] Do¨lle, H. (1979) “The influence of multiaxial stress states, stress gradients and elastic anisotropy on the evaluation of (residual) stresses by X-rays”, J. Appl. Cryst. 12, 489. [3] Brakman, C.M. (1985) “A general treatment of X-ray (residual) macro-stress determination in textured cubic materials: general expressions, cubic invariancy and application to X-ray strain pole figures”, Cryst. Res. Technol. 20, 593. [4] Kamminga, J.-D., de Keijser, Th.H., Mittemeijer, E.J. and Delhez, R. (2000) “New methods for diffraction stress measurement: a critical evaluation of new and existing methods”, J. Appl. Cryst. 33, 1059. [5] Baczman´ski, A., Wierzbanowski, K., Haije, W.G., Helmholdt, R.B., Ekambaranathan, G. and Pathiraj, B. (1993) “Diffraction elastic constants for textured materials different methods of calculation”, Cryst. Res. Technol. 28, 229. [6] Baczman´ski, A., Braham, C. and Seiler, W. (2003) “Microstresses in textured polycrystals studied by multireflection diffraction method and self consistent model”, accepted for publication in, Philos. Mag. A. [7] Skrzypek, S.J., Baczman´ski, A., Ratuszek, W. and Kusior, E. (2001) “New approach to stress analysis based on grazing-incidence X-ray diffraction”, J. Appl. Cryst. 34, 427. [8] Baczman´ski, A., Wierzbanowski, K., Lipinski, P., Helmholdt, R.B., Ekambaranathan, G., Pathiraj, B., et al. (1994) “Examination of the residual stress field in plastically deformed polycrystalline material”, Philos. Mag. A 69, 437. [9] Lipinski, P. and Berveiller, M. (1989) “Elastoplasticity of micro-inhomogeneous metals at large strains”, Int. J. Plast. 5, 149. [10] Bunge, H.J. (1982) Texture analysis in materials science (Butterworths, London). [11] Gerber, Ph., Tarasiuk, J., Chauveau, Th. and Bacroix, B. (2003) “A quantitative analysis of the evolution of texture and stored energy during annealing of cold rolled copper”, Acta Mater. 51, 6359–6371.
