E jiri, Toshihiko Sasaki2, Hirofumi Inoue3, Yoshio Shirasuna and Yukio Hirose

Transcription

1 RN NOC GN MRE O NERO NO NOUBR O LLO NOCN LEE EE Copyright(c)JCPDS-nternational Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol X-RAY S ESS R US DE NA N OF LD-R C ED A DS FU S SH Shouichi E jiri, oshihiko Sasaki2, Hirofumi noue3, Yoshio Shirasuna and Yukio Hirose Seisen Jogakuin College, Nagano, Japan 2Department of Materials Science, Kanazawa University, Kanazawa, Japan 3Department of Metallurgy and Materials Science, Graduate School of Engineering, Osaka Prefecture University, Osaka, Japan. ABS RAC Difficulties often attended determining the residual stress of cold-rolled steel sheets with crystallographic textures from the conventional X-ray stress measurement, i.e., the sin* d, technique, because the technique is constructed on assumption that specimens are isotropic elastic polycrystalline materials. herefore, the aim of this investigation is to develop an effective X-ray stress measurement for textured materials. At first, the relation between the stress and lattice strain measured by the X-ray diffraction is formulated by an average method. n the method, the strain is averaged as the expected value with a weight function from the crystallite orientation distribution function (ODF). hen, in order to lower the effect on the unstressed lattice spacing, an X-ray stress determination is introduced by using the derivative of the X-ray stress-strain relation. Moreover, the X-ray stress measurement is actually applied to cold-rolled steel sheets, and the stresses are determined. ODU 9 _ he conventional X-ray stress measurement, so-called sin* technique requires the specimen to be macroscopically an isotropic polycrystalline material, which is composed of anisotropic single crystallites. However, actual polycrystalline materials are macroscopically anisotropic materials having the bias of the crystallite orientation distribution in greater or lesser degree. For instance, there are coating films by physical vapor deposition (P VD) and chemical vapor deposition (C VD) which have preferred orientations along the normal to the film surfaces, casting parts which have the anisotropy due to one-way solidification and so on.2-4 he applicable X-ray stress measurement should be necessary for such anisotropic materials. n the present study, therefore, an effective X-ray stress measurement is formulated for textured materials on assumption that the crystalline anisotropy is represented by the crystallite orientation distribution function (ODF). At first, the relation between the stress in the specimen and the lattice strain measured by the X-ray diffraction method, which is the fundamental equation of the X- ray stress measurement, is derived from Hook s law in the single crystal using ODF. hen, in order to lower the effect on the measurement error with respect to the unstressed lattice spacing, the differential form of the X-ray stress-strain relation is introduced as an X-ray stress determination. Moreover, the X-ray stress measurement is actually applied to the cold-rolled steel sheets, and discussed.

2 his document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the nternational Centre for Diffraction Data (CDD). his document is provided by CDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by CDD. Usage is restricted for the purposes of education and scientific research. DXC Website CDD Website -

3 M k NEMER Copyright(c)JCPDS-nternational Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol X-RAY S ESS R EASU A. X-ray stress analysis for cubic polycrystalline materials hree orthogonal coordinate systems are introduced in the beginning. As shown in Figure 1, the sample coordinate system (S system) depends on the form or the rolled direction of the specimen. n order that the new & axis is parallel to the normal direction of the diffraction plane (h kl), the transformed system by the rotation with the Eulerian angles ( 4, (1, 0) is the laboratory coordinate system (L system). hree axes of the crystal coordinate system (C system) correspond to the crystallographic axes of a cubic crystal. he relation among these systems is shown in Figure 2 with three matrices of orthogonal transformations. hese matrices are represented by the rotation matrix Rij(a, b, c) with the Eulerian angles (a, b, c), i.e. qj =Q W,0) n- y =R,(Y,O,@) (1) yii = R, h P, > Y where Y 3i is also represented by Miller indices (h ko 3i Y = & J ch A7 1. of diffraction plane as follows: (2) Fig. 1 Relation between the sample and laboratry coordinate systems. Fig.2 Definitions of orthogonal transformation tensors among three coordinate systems.

4 E E 3LE 3LE EL33 _ E _ Ej 3 YMY Copyright(c)JCPDS-nternational Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol n the next place, the relation between the stress (3 ~ in the S system and the strain EL33 along the 3L axis is formulated for polycrystalline materials. he generalized Hooke s law with relation to the stress ocb and strain ~~~ in the C system is assumed. Cy = SCij kdkl (3) where AcV k, denotes the elastic compliance of single crystal. Under the transformations: C 3 = y3iy3 Cij, -1-1 s (r kl =?.t mn k /id mn, (4) the X-ray stress-strain relation for single crystals is obtained from q. E 1. 3 = SX 3ijcsS~ ) (5) SXvk, = Y,Y bscabe X- ekn- di j. (6) Not all crystallites contribute the diffraction phenomenon on the Bragg s law, only crystallites that have crystal planes with normal to the JL axis, namely the measured lattice strain has the rotation sym metry about the axis. hus the measured lattice strain is obtain from the average with a weight function6: - ~"fwld EL33 = o 2n (7) o f(y >dy where the weight functiona Y ) is given by ODF F( pr, 0, Q). f(y) =~(W P(Y)) (8) Q q he Eulerian angles (V, 0, ) have a parameter y by the transformation rule. n -1 =ykicok j (9) ntroducing q.7 into q.5, E therefore, the relation between the X-ray stress and strain for polycrystalline -- materials is obtained by = Sx3 3ijdij (10) for the Reuss model which assumes constant stress. B. X-ray stress determination Measuring the variation f from the unstressed diffraction angle f3 0, the lattice strain ~~3 3 is determined by the difference of the Bragg diffraction condition where d and do are lattice spacings in stressed and unstressed state, respectively. On the other hand, the X-ray elastic compliance,sx3 3, is determined by q.6. E herefore, the X-ray stress is derived from q. 10. However, in order to lower the measurement error on the unstressed lattice spacing, the differential of q. E 11 for 4 or sin* ( : - w3 _ 1 ad ---.-= -a8cot(j aw do av aw is introduced. hus -_ the X-ray stress is determined by the differential of q. E e3 asx3 3ij au, ayj (5 v s (11) (12) (13)

5 E P Copyright(c)JCPDS-nternational Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol Especially, the X-ray elastic compliances for diffraction planes { 11 l} and { lo O} is derived the independent equation on the rotation angle Y from q.6 E as follows: SC0 =P,, -s c l SC (16) 2 where 6 ij and (fir,, $r2, Fs3) denote the Kronecker s delta and the elastic constants of cubic crystal, respectively. hen, the ~~3 3 -sin2 d, diagram shows linearity without the influence of the elastic anisotropy. he result is equivalent to the equation by Honda and Dolle et al 9 Furthermore, since the averaged X-ray elastic compliance for isotropic polycrystalline materials with f( y ) =l is given by sx3 3ij = ( SC12 +scor >s ( (14) (15) ii + SC,1 -SC,2-3sc0r )w j 3 ) (17) -Y 3,2Y Y Y Y Y 3,2 9 (18) q. 13 with q. E 17 in plane stress state is equal to the X-ray stress determination of the sin2 (1, technique in the Reuss model: (3 X = -cot 8, A0 (19) SC,, -SC12-3S J Asin, \ if/ C# 2= q % go Fig. 3 Crystallite orientation distribution function of the ferrite phase for cold-rolled steel sheet

6 NEMREPXE & % _ Copyright(c)JCPDS-nternational Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol AL c? --(3S,,cos2~+(S,2sin2~+oS22sin2~. (20) A very low carbon steel sheet with 10 X 50 X 0.90 mm was employed as the specimen after a cold-rolled treatment up to 80%. he ODF results of the ferrite phase are shown in Figure 3. he diffraction angles 2 0 of a Fe21 were measured in 4 =0 degrees by an X-ray diffractometer with the iso-inclination method. hen, the St axis was fixed along the rolled direction. able 1 listed the X-ray stress measurement condition. ($11,,?t~, F4 4)=(7.67, , 8.57) l/ Pa was used as the elastic constants of Q Fe single crystal. o verify the present X-ray stress measurement, the X-ray stress values were compared with the applied stress values by loading experiments with a small four-point jig. he applied stress was determined by Hooke s law: d,, =?,, E (21) where the applied strain E~ was measured by a strain gage on the specimen surface and the mechanical elastic constant E* =192.9 Pa G was calculated by the Reuss model for macroscopic elastic constants. & able 1 X-ray str X-ray tube Filter Diffraction plane Diffraction angle ube voltage current Step width Peak determination Fixed time Scanning method V & 1s measurement condition. Cr-K (y. Vanadium afe deg 30k 1O ma Peak 0.3, B.G. 0.5 deg Gaussian fitting 3s Fixed (1, method m-e zl -n A-- Applied straiq E A11 j? l 25&i? a 0 5oox106 cu n 750x c ;-... ;... -a % ' ' ' ' ' ' * '...' ) 0' = 0.6 Sin*y Fig.4 Non linearity of diffraction anlgles for each applied strain.

7 L O Copyright(c)JCPDS-nternational Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol ESU R S AND DSSC USS NS A. Non-linearity in the diagram By means of the 2 8 -sin2 4 diagrams for applied stress values, the measurement results of cy Fe21 1 diffraction is shown in Figure 4. hese 2 6 values for each sin2 (, values are calculated by the Gaussian fitting for maximum diffraction intensities in the 2 8 -intensity scanning. he diagrams do not show linearity for strong texture. Consequently, it is impossible to apply the conventional sin2 (, technique, which demands the linearity in the diagram, to the textured material as the X-ray stress determination. 1 1 # Sin2W Fig.5 Gradient of lattice strain on 4 for cz Fe21 1. afe211 ($, v)=(o,lo)deg Rotation angle y,deg. Fig.6 Relative intensity distribution about the normal direction of the diffraction plane cr. Fe2 11 by the ODF analysis in ( $,G )=(O,lO) degrees.

8 LCNOC O Copyright(c)JCPDS-nternational Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol able 2 X-ray stress values for applied strains. Applied strain, X-ray stress by present method X-ray stress by the sin* techx1o-6 o 11,M Pa nique o 1,M Pa Applied stress ant i,m Pa Fig.7 Variations of X-ray stress values for applied stresses by ODF analysis and linear Reuss model. B. Compression with applied stress he X-ray stress values were determined from q. E 13 by the least-squares method. n the calculation, the differential values of the measured lattice strain were calculated from q.12 E by the polynomial approximation, as shown in Figure 5. n the same way, the differential values of the X-ray elastic compliance were calculate after averaging q.6. E n illustration of the weight function, Figure 6 shows the weight function from ODF of a Fe21 1 in ( 4, (1,)=(O, O) deg. he X-ray stress values by the present method and the sin2 4 technique in the Reuss model are shown in able 2. hen, the difference between the applied and X-ray stress is shown in Figure 7. he relations between the applied and X-ray stress have linearity in each method, and these gradients are 1.21 and 0.58 for the present method and the sin2 d, technique, respectively. n result, it is evident that the calculated values by the present method is nearer to the measured values than by the sin2 9 technique. US NS he X-ray stress measurement for cubic polycrystalline materials, which consists of the averaged lattice strain with ODF and the differential form of the stress-strain relation, was devel-

9 R 3[ 4[ 8[ 9[ CNER 9 9 Copyright(c)JCPDS-nternational Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol oped in the Reuss model. Furthermore, he X-ray stress measurement was applied a cold-rolled steel sheet, and the X-ray stress was determined. (1) he relation between the X-ray stress and the measured lattice strain was derived from ODF. n general, the relation had non-linearity in the sin2 d, diagram, but the relations for { 111 } and { lo O} had linearity. hen, the relation for isotropic materials was coincident with the conventional sin2 (, technique. (2) n order to control the measurement error of the Bragg angle in unstressed state, the differential form of the relation between the X-ray stress and lattice strain was introduced. n result, the X-ray stress was possible to determine for textured materials with the nonlinear sin2 diagrams even if the sin2 (1, technique was impossible to apply. (3) Actually, it was hard to apply the sin2 d, technique to the cold-rolled steel sheet employed because of the nonlinearity in the sin2 (1, diagram. (4) As a result of loading tests, the X-ray stress by the present method was nearer the applied stress than the sin2 (1, technique. EFE ES [l] Noyan,.C. and Cohen, J.B., Residual Stress, Springer-Verlag, New York, [2] Hanabusa,., ominaga, K. and Fujiwara, H., J. Sot. Muter. Sci., Japan, 19 3, 42, ] jiri, E S., Lin, Z., Sasaki,. and Hirose, Y., J. Sot. Muter. Sci., Japan, 19 97,46, ] jiri, E S., Sasaki,., Miyano, M., and Hirose, Y., J. Sot. Muter. Sci., Japan, 19 9,4 8, [5] Bunge, H.J., exture Analysis in Materials Science, Butterworths, [6] Yashiro,., J. JSND, 19 81, 30, [7] Reuss, A., 2. Angew. Math. Mech., 1929,9,4 9. ] Honda, K., Hosokawa,. and Sarai,., J. JSND, 1977,26, ] Diille, H. and Hauk, V., 2. Metallkde, 1978, 69,41 O [lo] Hellwge, K.H., Landolt-Bornstein vol. 11, Springer-Verlag, New York, 1979.