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Measurements of thermal residual elastic strains in ferrite–austenite Fe–Cr–Ni alloys by neutron and X-ray diffractions
PII:
Acta mater. Vol. 47, No. 1, pp. 353±362, 1999 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00300-0 1359-6454/99 $19.00 + 0.00
MEASUREMENTS OF THERMAL RESIDUAL ELASTIC STRAINS IN FERRITE±AUSTENITE Fe±Cr±Ni ALLOYS BY NEUTRON AND X-RAY DIFFRACTIONS S. HARJO1, Y. TOMOTA1{ and M. ONO2 Department of Materials Science, Faculty of Engineering, Ibaraki University, 4-12-1, Nakanarusawa-cho, Hitachi, Ibaraki 316-8511, Japan and 2Kyoto University Research Reactor Institute, Kumatori-cho, Sennan-gun, Osaka-fu 590-0494, Japan 1
(Received 31 March 1998; accepted 3 August 1998) AbstractÐThe thermal residual elastic strains in ferrite (a) and austenite (g) phases in three kinds of a±g Fe±Cr±Ni alloys generated by quenching specimens from 1273 K into water (273 K), have been measured by means of a neutron diraction method. The phase-stresses are successfully determined by employing carefully prepared alloys with volume fractions of a in a range between 0% and 100%, whose chemical compositions are located on an equilibrium tie line of the Fe±Cr±Ni ternary phase diagram. The phasestresses obtained are compressive for a phase and tensile for g phase, showing good agreement with those predicted by Eshelby and Mori±Tanaka theories. The stress measurements for these alloys were also carried out by X-ray diraction method. It is found that the conventional X-ray sin2 c method under the assumption of plane stress condition is not applicable. The phase-stresses obtained by a triaxial X-ray stress measurement method are in good agreement with those obtained by neutron diraction method. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
When a multi-phase alloy is heat-treated, internal stresses usually remain due to dierence in thermal expansion coecient between constituent phases. The distribution of such internal stresses in the cross section of a dual phase alloy specimen after quenched into water from an elevated temperature, are schematically drawn in Fig. 1. As can be seen, macroscopic residual stress (sM ij ) varies smoothly from the surface to internal region of the specimen. On heating or cooling, temperature changes more quickly at the surface than in the internal region of a specimen to result in yielding sM ij . If the elastic moduli are dierent between the two constituent phases, partitioning of sM ij would occur [1, 3]. In a dual phase alloy, microscopic residual stresses arise due to mis®t strains among grains of the constituent phases with dierent thermal expansion coecients. These stresses may vary from grain to grain and their averaged value over the each constituent phase is called thermal phase-stress (sph ij ). It is very useful to measure sph ij in multi-phase alloys, composites or functionally graded materials because sph ij are closely related with the strength of the materials. X-ray diraction has widely been used for such stress measurement so far. However, information only near surfaces can be obtained because its penetration depth is limited within a {To whom all correspondence should be addressed. 353
shallow region near surface of the specimen. Thus, the stresses measured by X-ray diraction method ph become complicated consisting of sM ij and sij that are partially relaxed due to the free surface [3]. On the other hand, application of neutron diraction to sph ij measurement is attractive because of its high penetration power into a specimen. Residual stress measurements by neutron diraction method have been made not only for composites [1, 3±5] but also for commercially available materials [6±10]. Residual elastic strains related to sph in metal ij matrix composites (MMCs) have successfully been measured by several workers [3±5, 11]. On the other hand, the stress measurements for dual phase alloys have not presented reasonable results yet [2, 12]. This is due to diculty of preparation of a stressfree reference material having identical chemical composition with that of a constituent phase in a dual phase alloy. In the case of MMCs, the reasonable results have been obtained because the reference materials are easily prepared. In the present study, therefore, ®ve Fe±Cr±Ni alloys were prepared for sph ij measurement, i.e. they are located along an equilibrium tie line of the ternary phase diagram [13], as shown in Fig. 2. This means that the volume fraction of ferrite (a) can be varied without changing its chemical compositions. The a and austenite (g) single phase alloys are then provided for the reference materials to evaluate thermal residual stresses in a and g phases in the a± g dual phase alloys.
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Fig. 1. Schematic illustration to explain residual stress distribution near surface in a dual phase alloy. (a) Microstructure of a cross section, (b) s11 or s22 and (c) s33, where subscripts ``ph'', ``M'', ``a'' and ``g'' refer to phase-stress, macrostress, a phase (ferrite) and g phase (austenite), respectively.
In order to improve angular/energy resolution with a simultaneous increase of the detector signal with respect to the conventional scattering devices with pyrolitic graphite monochromator [4, 12], we have introduced a Bragg diraction optics employing bent perfect crystals [14]. X-ray diraction methods including a conventional sin2 c method under the assumption of plane stress condition have been applied to compare the results with those by neutron diraction method. Finally, Eshelby inclusion theory coupled with Mori±Tanaka mean ®eld theory is applied to predict thermal phase-stresses and the predictions are compared with experimental results. The results obtained by neutron, X-ray diraction and the prediction are found in a good agreement.
S1 and S5 are g and a single phase alloys, respectively, while alloys S2, S3 and S4 are a±g dual phase alloys with various volume fractions of a, as will be described later. The ingots of these alloys were hotrolled at 1523 K to plates with 15 mm in thickness. Rod specimens with 10 mm in diameter and 50 mm in length were prepared for stress measurements using neutron diraction method in such a way that the vertical direction of the rod becomes parallel to the rolling direction. Plate specimens with 15 mm 15 mm 2 mm were prepared for stress measurement using an X-ray diraction method in such a way that a plane with 15 mm 15 mm is parallel to the rolling plane of hot-rolled plates. The specimens were mechanically polished with emery papers up to ]600 and annealed at 1273 K for 3.6 ks in vacuum followed by quenching into water. After the heat treatment, the rod specimens for neutron diraction testing were mechanically polished with emery papers (]1000) to eliminate their oxide layer. On the other hand, the plate specimens for X-ray diraction testing were slightly polished with emery papers followed by electropolishing. Microstructures of the specimens were observed by using an optical microscope (OM). Typical microstructures for the alloys are shown in Fig. 3, in which the observed plane is indicated in (f). Observations were also performed on the other two mutually orthogonal planes shown in Fig. 3(f) and it was found that grains were a little elongated along the rolling direction. The volume fractions of a (fa) of these alloys after the heat treatment were measured by a point counting method on optical micrographs and the results obtained are shown in Table 2. The fa can also be determined from ratios of relevant integrated intensities of peaks corresponding to the constituent phases in a dual phase alloy. The results are also listed in Table 2, showing good agreements with those obtained by the optical metallography method. The optical metallography method data were installed for the evaluation of sph ij
2. EXPERIMENTAL PROCEDURES
2.1. Specimens Five alloys were prepared by using an induction furnace aiming at having chemical compositions which lie on a tie line in the Fe±Cr±Ni ternary phase diagram [13], as shown in Fig. 2. The results of chemical analysis for ingots obtained are listed in Table 1, showing satisfactory compositions. Alloys
Fig. 2. Fe±Cr±Ni phase diagram [13] and ®ve alloys used in this investigation.
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355
Table 1. Chemical compositions of Fe±Cr±Ni alloys used in this investigation (mass %) Alloy S1 S2 S3 S4 S5
C
Si
Mn
Cr
Ni
Mo
Cu
P
N
Fe
0.012 0.013 0.009 0.018 0.016
0.39 0.38 0.36 0.33 0.32
0.33 0.32 0.31 0.34 0.31
22.43 27.82 29.63 31.41 35.96
18.38 13.16 10.88 9.25 4.10
0.02 0.02 0.01 0.01 0.01
0.02 0.02 0.01 0.01 0.01
0.009 0.010 0.007 0.008 0.006
0.027 0.035 0.039 0.051 0.049
bal. bal. bal. bal. bal.
value as described later. The grain sizes are approximately 10±30 mm for these alloys. 2.2. Dilatometry test
e
Dilatometry tests were carried out to obtain thermal expansion coecients of these alloys. Rod specimens with 5 mm in diameter and 15 mm in length were prepared from hot-rolled plates, as shown in Fig. 3(f). Using a dilatometer with a heating rate of 0.167 K sÿ1 (10 K minÿ1) in a temperature range between 296 K and 1273 K, thermal expansion coecients were measured in an argon gas atmosphere. A rod of Al2O3 was used for a reference material. 2.3. Stress measurement using neutron diraction method The principle of stress measurement is based on precise determinations of relative values of interplanar lattice spacings. Hence, the following Bragg's law [15] is essential. 2dhkl sin y l;
strain along the normal direction to (hkl) plane is then given as follows,
1
where dhkl, y and l refer to the spacing of (hkl) plane, the diraction angle (Bragg angle), and the incident neutron wavelength, respectively. The
dhkl ÿ d0 ÿ cot y Dy d0
2
where d0 represents the spacing of (hkl) plane for a stress-free material. Then a change of Bragg angle (Dy) provides us with the elastic strain of a sample [7, 16]. Neutron diraction measurements were performed using the residual stress diractometer installed at super mirror guide hall of Kyoto University Research Reactor Institute (light water moderated, 5 MW). Thermal neutron ¯ux at the exit of the guider is 5 107 neutrons cmÿ2 sÿ1 and their wavelengths of approximately 0.12±1.0 nm. A Si single crystal (30 mm in height, 200 mm in length and 4 mm in thickness) was elastically bent in order to focus the neutron beam to a sample [14, 17]. The wavelength of the monochromatized neutron beam is 0.124 nm. A one-dimensional position-sensitive proportional counter with a length of 500 mm was set horizontally to detect neutrons diracted in a range of diraction angles 2y 1 67 2 148. Signals from the counter were processed via logic units (2048 ch, 24 bits/ch) into positional neutron counts,
Fig. 3. Microstructures observed by an optical microscope (O.M.): (a) alloy S1, (b) S2, (c) S3, (d) S4 and (e) S5. Observed plane of a plate is shown in (f). The specimens for dilatometry and neutron and X-ray diractions are also shown in (f).
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Table 2. Volume fractions of a and thermal expansion coecients of alloys used in this investigation Volume fraction of a (%) Alloy S1 S2 S3 S4 S5
1
2
3
Thermal expansion coecient (10ÿ6 Kÿ1)
0.0 32.6 52.6 67.3 100
0.0 32.2 50.1 72.4 100
0.0 34.3 54.1 71.0 100
18.0 17.2 15.2 14.9 12.4
1. Measured by a point counting method using optical microscopy. 2. Calculated from peak intensities ratio obtained by X-ray diraction. 3. Calculated from peak intensities ratio obtained by neutron diraction.
which allowed us to get a pro®le of angular intensity distribution of a diracted beam. Figure 4(a) shows the measuring system used in this investigation. For conversion of the Bragg peak positions (channel numbers) to lattice spacings or Bragg angles y under the ®xed geometrical condition, a calibration curve was prepared from the several well-de®ned Bragg peaks from powder samples of pure iron and copper. The peak positions were determined by using a Gaussian curve ®tting technique. In order to acquire enough diraction intensity, the measurement was run more than 32.4 ks (9 h) for each sample. Two slits (2.5 mm in width and 30 mm in height) were placed before and after a sample, screening the beam to keep a sampling volume constant.
2.4. Stress measurement using X-ray diraction method A local coordinate (Li) and a global one (Si) for a specimen used in this study are shown in Fig. 4(b). Measuring an interplanar lattice spacing along L3(f, c) direction (dyc,hkl) enables us to determine a strain eyc by equation (2) that is rewritten here as, efc
dfc;hkl ÿ d0 ; d0
3
which can be written by using strain components eij in a global coordinate in Fig. 4(b) as follows. efc e11 cos2 f sin2 c e22 sin2 f sin2 c e33 cos2 c e12 sin 2f sin2 c e13 cos f sin 2c e23 sin f sin 2c:
4
Equation (4) can be rewritten by using stress component sij in the global coordinate through Hooke's law for isotropic materials: efc
1 f s11 cos2 f s2 sin2 f ÿ s33 sin2 c E s33 s31 cos f s23 sin f sin 2cg ÿ
s11 s22 s33 : E
5
A conventional sin2 c method [18] assumes plane stress condition (s33=s23=s31=0), and thus s11 can be determined from a slope of a efc vs sin2 c plotting when f = 0. Therefore, the in¯uence of preciseness of d0 on measuring s11 is relatively small. This is why this method has widely been used for stress measuring in engineering materials. First, this conventional method was used in this study. It is however revealed that this method cannot be applied to the present sph ij measurement, as will be described later in Section 3.2. Then, secondly, X-ray triaxial stress measurement was performed. A stress component s33 which changes from zero at free surface to some value inside a sample with increasing a distance from the surface is assumed to present [see Fig. 1(c)]. Here, an averaged value of such a gradient s33 in X-ray penetrating region is expressed by hs33i. According to the procedure proposed by Noyan [19], efc was measured for conditions of various +c and ÿc under f = 08, 458 and 908. Averaging the stresses obtained for +c and those for ÿc gives the following relation; fef c ef ÿcg 1 E 2 s11 cos2 f s12 sin 2f s22 sin2 f ÿ s33 sin2 c
Fig. 4. Geometrical arrangements for stress measurements; (a) for neutron diraction method and (b) for X-ray diffraction method.
1 s33 ÿ s11 s22 s33 : E E
6
A slope obtained from {ef(+c) + ef(ÿc)}/2 vs sin2
HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS Table 3. Conditions for X-ray diraction stress measurement Characteristic X-ray Tube voltage Tube current Diraction planes
Cr±Ka 40 kV 30 mA (211) for a phase (220) for g phase f = 08, 458, 908 c = ÿ 408 to +408 5 mm by 5 mm 300 2 1 K
Angles f and c Irradiation area Temperature
c plottings give, s11 ÿ hs33 i s 0
for f 0
7
3.1. Residual elastic strains measured by neutron diffraction method
for f 90
8
Figures 5(a)±(e) show neutron diraction pro®les of the S1, S2, S3, S4 and S5 alloys obtained. These pro®les exhibit much higher S/N ratios in comparison of the previous results with use of pyrolitic graphite monochromator [4, 12, 20]. Remarkable improvement has been achieved by using the Si bent crystal monochromator in this study. As can be seen in Figs 5(b), (c) and (d), diraction peaks corresponding to a and g phases are clearly identi®ed in alloys S2, S3 and S4. As long as g phase is concerned, peak heights of (311) show the largest followed by (220) and (222) commonly in alloys S1, S2, S3 and S4. This means that there exists the similar texture in the alloys. Concerning a phase, on the other hand, (200) shows the largest in alloy S5 while (211) show the largest in alloys S2, S3 and S4. This means that the texture of the a single phase alloy is dierent from that of the dual phase alloys. It is found that the intensities for
Because normal directional strain with respect to surface can be directly known from the change of an interplanar lattice spacing, i.e. he33 i dc0 ÿ d0 =d0
9
hs33i can be determined by the following equation derived from equations (7)±(9); hs33 i
Conditions for the stress measurements by X-ray diraction method are summarized in Table 3. After subtracting background from diraction pro®les obtained, Pseudo Voigt function was employed for curve ®tting and then a center of half-value width method was used to determine the relevant peak positions.
3. EXPERIMENTAL RESULTS
or, s22 ÿ hs33 i s 90
357
E he33 i fs 0 s 90g; 1 ÿ 2 1 ÿ 2
10
where Hooke's law of he33i = (1 + n)/Eh s33i ÿ n/ E(s11+s22+hs33i) is used. Thus s11(=s(0) + hs33i) and s22(=s(90) + hs33i) can be obtained. Similarly to this procedure, shear components, s23, s31 and s12 can be obtained by using {ef(+c) ÿ ef(ÿc)}/2 for f = 08 and 908, and {e45(+c) + e45(ÿc)}/2 for f = 458, respectively.
Fig. 5. Examples of diraction patterns obtained by neutron diraction for alloy S1 (a), S2 (b), S3 (c), S4 (d) and S5 (e).
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HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS
(220) and (311) of g decrease with increasing fa while the ones for (211) of a increase. The peak positions in the line counter corresponding to the diraction angles of (220), (311) and (222) for g and those of (200), (211) and (220) for a were used to evaluate residual elastic strains. The diraction angles obtained in alloys S1 and S5 were used as references to determine the residual elastic strains for g and a phases in the dual phase alloys, respectively. The peak (220) for a phase is detected in the three dual phase alloys but not in the a single phase alloy S5. Then, the interplanar lattice spacing for (220) in alloy S5 is calculated from those for well-de®ned (200) and (211) peaks, i.e. using the lattice parameters derived from (200) and (211) peaks obtained, the interplanar lattice spacing for (220) is estimated. The strain measured here is called a total residual elastic strain (eij) which includes macro-strain (eM ij ) associated with ph sM and phase strain (eph ij ij ) associated with sij . Because elastic moduli for a are estimated nearly identical with those for g, the strain associated with the partitioning of sM ij to a and g grains can be neglected. Therefore, the measured eij can be written as: eij
eM ij
eijph
11
The eij evaluated with equation (2) for the alloys S2, S3 and S4 are shown in Fig. 6. It is found that tensile (positive) strains exist in g phase while compressive (negative) ones in a phase. The strains for dierent (hkl) planes are fairly in good agreement with each other except those for g phase in alloy S4. The measured strains depend on the volume fraction of a (fa). The results which were independently obtained for the present alloys by our collaborated work which was performed at NPI, Rez, Czech Rep. [21] are also plotted in Fig. 6, showing excellent agreements.
Fig. 6. Total residual elastic strains, aeij and geij for alloys S2, S3 and S4 obtained by stress measurement using neutron diraction method.
If the shape of a specimen is spherical, sM ij in its internal region is considered to be isotropic. When the shapes of grains of constituent phases are spherical, sph ij in the internal region of a specimen is also considered to be isotropic. That is, e11=e22=e33 and shear strains are zero. For simplicity, we assume here the isotropy both for sM ij and sph although the actual shapes of specimen and ij grains are not exactly spherical. Thus, isotropic total residual stresses (s11=s22=s33) either in a phase or in g phase could roughly be evaluated by, s11 fE= 1 ÿ 2ge11 :
12
Then, by using Young's modulus (E) = 200 GPa and Poisson's ratio (n) = 0.30 equally for a and g phases in equation (12), we obtained total thermal residual stresses, as11 and gs11 as shown in Fig. 7, where the measured strains from three dierent (hkl) planes in Fig. 6 are averaged for inputting into e11. Similarly to equation (11), sij could be expressed as; ph sij sM ij sij ;
13
where sph ij refers to the thermal phase-stress which is caused by thermal mismatching between the constituent phases during the quenching. The equilibrium condition for thermal phase-stresses within all over the sampling volume in a specimen of a dual phase alloy can be written as: a ph sij fa
gsijph 1 ÿ fa 0;
14
g ph where asph ij and sij stand for thermal phase-stress in a phase and that in g phase, respectively. Then, using equation (13) sM ij can be calculated by the following equation, a sij ÿ sM 15 fa g sij ÿM ij ij 1 ÿ fa 0:
Fig. 7. Total residual stresses, asij and gsij for alloys S2, S3 and S4 obtained by stress measurement using neutron diffraction method.
HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS
Using the procedure described above and the measured sij in Fig. 7, the sM ij for the three alloys calculated from equation (15) is shown in Fig. 8 by a dashed line, together with the sph ij obtained after subtracting sM from s . It is found in Fig. 8 that ij ij the sM ij averaged over the sampling volume shown in Fig. 4(a) is positive commonly for the three alloys, revealing that the cooling rate in an internal region of a sample was slower than at the surface resulting in tensile residual stress in the sampling volume. That is, sM ij in the central region is tensile while compressive at the surface, which will be discussed again in the following section of stress measurement by X-ray. Experimental results obtained from dilatometry tests show that the thermal expansion coecient of alloy S1(g) is higher than that of alloy S5 (a), as listed in Table 2. This means that g grains shrink more largely than a grains in a dual phase alloy during quenching, yielding residual tensile strain in g grains while compressive one in a grains. The sph ij in the present experiment is quantitatively consistent with this prediction. The sph obtained is ij obviously dependent on fa as shown in Fig. 8; with increasing fa, the absolute value of sph ij increases in g phase while it decreases in a phase. 3.2. Residual stresses measured by X-ray diraction method
359
ph Fig. 8. Macro- and phase-stresses, sM ij and sij obtained after separating them from sij by using equations (13) and (15).
When sM ij obtained by using equation (15) is subg ph tracted from asij or gsij, the stresses of asph ij or sij can be obtained. The results are shown in Fig. 9(b). Such stresses have been reported in many previous papers [24, 26]; although the stresses shown in Fig. 9(b) look excellent with respect to their equilibrium, these values are incorrect because hsph 33 i as ph sketched in Fig. 1 is neglected. The sph 11 or s22
3.2.1. Stresses obtained by the conventional sin2 c method under the assumption of plane stress condition. In many previous reports [22, 23] including Tomota et al.'s paper [24], phase-stresses formed by heat treatment and/or plastic deformation in dual phase alloys have been measured by X-ray diraction under the assumption of plane stress condition. However, it has recently been made clear, particularly in MMCs, that the plane stress condition is not applicable to some multi-phase materials [19, 25]. That is, gradient phase-stress of M sph ij (=s33ÿs33) exists and its average within an Xray penetrating region, hsph 33 i, cannot be ignored [see Fig. 1(c)]. In this section, some results by the conventional method are presented only to emphasize how they are dierent from the results obtained under consideration of triaxial stress condition. Keeping f = 08, diraction angle (y) was measured by changing c [see Fig. 4(b)]. By adopting the conventional sin2 c method, y was plotted against sin2 c, so that the stress s11 (f = 08) can be obtained from a slope by a least-square linear ®tting. When we take f = 908, s22 can be obtained. Such stresses (asij and gsij) shown in Fig. 9(a) cong ph M sist of asph ij or sij and sij . Equation (13) can be rewritten as: a
M sij a sph ij sij
for a phase
16a
g
M sij g sph ij sij
for g phase:
16b
Fig. 9. Residual stresses obtained by X-ray stress measurements under an assumption of plane stress condition for alloys S2, S3 and S4. (a) sij including macroscopic residual ph stresses, sM ij and (b) phase-stresses sij calculated by subtracting sM ij from the total residual stresses shown in (a).
360
HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS
obtained by this method is smaller than that obtained by the following triaxial stress measuring method. 3.2.2. Stresses obtained by triaxial stress method. When the number of grains within an X-ray penetrating region is taken into consideration, s33 along normal direction with respect to surface cannot be ignored [25] [Fig. 1(c)]. In the case of a commercially available dual phase stainless steel, it is not easy to perform triaxial stress measurement because no appropriate d0 value is available [2, 12]. However, in this study, we can use the data for the alloys S1 or S5 for appropriate determination of d0. Similarly to Section 3.2.1, the measured sij is sepph arated into sM ij and sij by using equation (15). As a result, sM is compressive being dierent from the ij result obtained in the previous section (Section 3.1). M For instance, sM 11= ÿ 218.1 MPa, s22= ÿ 191.3 MPa and hsM i = ÿ 15.3 MPa were obtained for 33 near the surface alloy S3. The stresses show that sM ij is approximately in the plane stress condition; i.e. M M hsM 33i is negligible compared with s11 and s22. The phase-stresses obtained from the triaxial stress analysis are shown in Fig. 10. As can be seen, hsph 33 i is not zero so that the plane stress condition assumed in Section 3.2.1 is revealed to be incorrect. 4. DISCUSSION
4.1. Comparison of the results obtained by neutron and X-ray diraction methods
Fig. 10. Thermal phase-stresses, sph ij obtained from 3-axial stress measurement by X-ray diraction for alloys S2, S3 and S4.
4.2. Predictions of phase-stresses by Eshelby and Mori-Tanaka theories It has well been established that phase-stresses, i.e. mean internal stresses, in an alloy including many second phase ellipsoidal grains with uniform mis®t strains, so called eigen strains e*ij, can be calculated by using Eshelby's inclusion theory [27] coupled with Mori-Tanaka mean ®eld theory [28]. In the present case, e*ij is introduced from the dierence in thermal expansion coecients between a and g phases given in Table 2. Assuming that elastic moduli for both phases are identical, phase-stresses in a and g phases produced by quenching are calculated as follows. ÿ eij K g ÿ K a DT e dij ; 17
ph The sph 11 and s22 obtained by X-ray diraction triaxial stress measurement method are averaged as ph ph (sph 11 +s22 )/2 and plotted in Fig. 11. The s11 by neutron method are also shown in this ®gure. The g ph sij fa Ca S ÿ Ie dij 18a agreements between the two results are surprisingly excellent, but the following problems should be a ph sij ÿ 1 ÿ fa Ca S ÿ Ie dij ; 18b taken into consideration. It is speculated that sph ij measured by X-ray diraction method should be smaller in their absolute values than those obtained by neutron diraction method because stress relaxation occurs near surface; such a dierence has actually been observed in a SiCw/A2014 MMC [3]. On the other hand, Fitzpatrick et al. who measured sph ij in a SiCp/A2124 MMC using neutron diraction method, have claimed that the eective temperature drop during quenching which generated the thermal phase-stresses was slightly larger at the surface than that in the central region of a sample leading to slightly larger phase-stresses near the surface [1]. This manifests that sph ij measured by Xray shows almost the same or higher magnitude with that measured by neutron since the neutron measurement was performed at the central region of sampling volume of a specimen. Although the present results from neutron and X-ray are comparable, it seems that the neutron results are more re- Fig. 11. Comparisons between thermal phase-stresses sph ij liable than the X-ray results due to the dierence of predicted by using Eshelby and Mori±Tanaka theories and their penetration powers. those measured by neutron or X-ray diraction.
HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS
where Kg, Ka, DT, dij, Ca, S and I represent the thermal expansion coecient of a, that of g, the temperature drop, Kronecker's delta, the elastic stiness tensor of a, Eshelby's tensor and the identity tensor, respectively. For simplicity, the shape of grains is assumed to be spherical, then S1111=S2222=S3333=(7 ÿ 5n)/15(1 ÿ n), S1122= S2233=S3311= ÿ (1 ÿ 5n)/15(1 ÿ n), S1212=S2323= S3131=(4 ÿ 5n)/15(1 ÿ n), and others are zero. Inputting Ka=12.4 10ÿ6 Kÿ1, Kg=18.0 10ÿ6 Kÿ1 and DT = 1000 K in equation (17), e*ij=5.6 10ÿ3 dij is obtained. Then, by using Young's modulus: E = 200 GPa and Poisson's ratio; n = 0.30 equally for a and g phases in equation (18a) and (18b), we g ph have obtained asph ij and sij presented in Fig. 11. It is found that the agreements between the predicted stresses and the measured stresses are fairly good. During the quenching from 1273 K to room temperature, thermal internal stress may be partially relaxed by diusion and/or dislocation punchingout. According to the results of our collaborated work performed at NPI, Rez, Czech Rep. [21], the dislocation density predicted from pro®le analysis for neutron diraction experiments is higher in the a±g dual phase alloys (S2, S3, and S4) than in the single phase alloys (S1 or S5). Although their results suggest that plastic relaxation by dislocation punching-out takes place, Fig. 11 shows that the in¯uence of such a relaxation mechanism on sph ij is small.
5. CONCLUSIONS
Five Fe±Cr±Ni alloys, i.e. three ferrite(a)±austenite(g) alloys, a single phase alloy and g single phase alloy, were carefully prepared, whose chemical compositions lie on an equilibrium tie line in the ternary phase diagram. The thermal phase-stresses of specimens generated by quenching from 1273 K to 273 K were measured by neutron and X-ray diffractions. The main results obtained are summarized as follows; (i) Compressive residual elastic strain in a and tensile one in g are measured in the present a±g dual phase alloys by neutron diraction method. The results are considered to be reasonable from the consideration based on thermal expansion coecients of 18.0 10ÿ6 Kÿ1 for g and 12.4 10ÿ6 Kÿ1 for a; during quenching g grains shrink more largely than a grains, yielding residual tensile strain in g grains while compressive one in a grains. The preparation of appropriate reference materials was a point for obtaining such good results. (ii) The employment of a Si-bent crystal for focusing neutron beam to a sample is found to be eective to increase the precision in stress measurement.
361
(iii) The conventional X-ray sin2 c method under the assumption of plane stress condition cannot be applied to the measurement of phase-stresses. (iv) X-ray triaxial stress analysis provides similar results to those by neutron diraction method but seems to be insucient because of the stress gradient near surface. (v) Phase-stresses predicted by Eshelby and Mori± Tanaka theories are found to show good agreement with those measured by neutron diraction method, suggesting that plastic relaxation is not so large. AcknowledgementsÐWe would like to thank NIDAK Co. for melting the alloys used in this study and Dr P. Lukas of Nuclear Physics Institute, Czech Republic, for valuable discussions. The neutron stress measurements were performed as a coworking project at KURRI (Kyoto University, Research Reactor Institute). This work was partially supported by the special coordinate fund of the Science and Technology Agency Japan.
REFERENCES 1. Fitzpatrick, M.E., Hutching, M.T. and Whiters, P.J., Acta metall. mater., 1997, 45, 4867. 2. Tomota, Y., Rudiono, Harjo, S., Tokumura, K., Sato, H. and Ono, M., J. Japan Soc. Heat Treatment, 1998, 38, 174. 3. Ohnuki, T., Fujita, M., Tomota, Y. and Ono, M., J. Japan Inst. Metals, 1998, 62, 261. 4. Ohnuki, T., Tomota, Y. and Ono, M., J. Japan Inst. Metals, 1996, 60, 56. 5. Wang, K.J., Wakashima, K. and Ono, M., J. Japan Inst. Metals, 1995, 59, 525. 6. Webster, G.A. and Ezeilo, A.N., Phisica, 1997, B234(236), 949. 7. Stacey, A., Macgillivray, H.J., Webster, G.A., Webster, P.J. and Ziebeck, K.R.A., J. Strain Analysis, 1985, 20, 93. 8. Ezeilo, A. N., Webster, G. A., Webster, P. J. and Webster, P. S., in Proc. 5th Int. Conf. on Shot Peening, Oxford University, 1993, p. 274. 9. Webster, P. J., Low, K. S., Mills, G. and Webster, G. A., in Proc. Mat. Res. Soc. Symp., Vol. 166, 1990, p. 311. 10. Winholtz, R.A. and Krawitz, A.D., Metall. Trans., 1995, 26A, 1287. 11. Maeda, K., Wakashima, K. and Ono, M., Scripta mater., 1997, 36, 335. 12. For, instance, Rudiono, Ohnuki, T., Tomota, Y. and Ono, M., CAMP ISIJ, 1995, 8, 1530. 13. Pugh, J.D. and Nisbet, J.W., Trans. AIME, 1950, 188, 269. 14. Ono, M., Mikula, P., Harjo, S., Saroun, J., Yoneda, K. and Sawano, J., in Proc. 5th Neutron Symp., JAERI, Tokai, 1997, p. 111. 15. Bacon, G. E., in Neutron Diraction, 3rd edn. Clarendon Press, Oxford, 1975. 16. Allen, A., Andreani, C., Hutchings, M.T. and Windsor, C.G., NDT Int., 1981, 14, 249. 17. Ono, M., in Proc. of ICRS-5, Linkoping, Sweden, 1997, p. 715. 18. Winholtz, R.A. and Cohen, J.B., Adv. J. Phys., 1988, 41, 189. 19. Noyan, I.C., Metall. Trans., 1983, 14A, 249.
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20. Harjo, S., Rudiono, Tomota, Y. and Ono, M., in Proc. 32nd Symp. X-Ray Studies on Mechanical Behaviour of Materials, 1996, p. 83. 21. Lukas, P., Ono, M., Tomota, Y., Harjo, S., Kouril, Z., Vrana, M. and Mikula, P. in press. Proc. of ICRS5. Sweden. 22. Cox, B.N., James, M.R., Marshall, D.B. and Addison, R.C. Jr, Metall. Trans., 1990, 21A, 2701. 23. Bobet, J-L., Naslain, R., Guette, A., Ji, N. and Lebrun, J-L., Acta metall., 1995, 43, 2255.
24. Tomoto, Y., Nakamura, S., Kuroki, K. and Tamura, I., Mater. Sci. Eng., 1980, 46, 69. 25. Hanabusa, T., Nishioka, K. and Fujiwara, H., Z. Metallkde., 1983, 74, 307. 26. Harjo, S., Ono, M. and Tomota, Y., in Proc. of ICRS-5, Linkoping, Sweden, 1997, p. 46. 27. Eshelby, J. D., in Proc. R. Soc. Lond., Vol. A241, 1957, p. 376. 28. Mori, T. and Tanaka, K., Acta metall., 1973, 21, 571.
Acta mater. Vol. 47, No. 1, pp. 353±362, 1999 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00300-0 1359-6454/99 $19.00 + 0.00
MEASUREMENTS OF THERMAL RESIDUAL ELASTIC STRAINS IN FERRITE±AUSTENITE Fe±Cr±Ni ALLOYS BY NEUTRON AND X-RAY DIFFRACTIONS S. HARJO1, Y. TOMOTA1{ and M. ONO2 Department of Materials Science, Faculty of Engineering, Ibaraki University, 4-12-1, Nakanarusawa-cho, Hitachi, Ibaraki 316-8511, Japan and 2Kyoto University Research Reactor Institute, Kumatori-cho, Sennan-gun, Osaka-fu 590-0494, Japan 1
(Received 31 March 1998; accepted 3 August 1998) AbstractÐThe thermal residual elastic strains in ferrite (a) and austenite (g) phases in three kinds of a±g Fe±Cr±Ni alloys generated by quenching specimens from 1273 K into water (273 K), have been measured by means of a neutron diraction method. The phase-stresses are successfully determined by employing carefully prepared alloys with volume fractions of a in a range between 0% and 100%, whose chemical compositions are located on an equilibrium tie line of the Fe±Cr±Ni ternary phase diagram. The phasestresses obtained are compressive for a phase and tensile for g phase, showing good agreement with those predicted by Eshelby and Mori±Tanaka theories. The stress measurements for these alloys were also carried out by X-ray diraction method. It is found that the conventional X-ray sin2 c method under the assumption of plane stress condition is not applicable. The phase-stresses obtained by a triaxial X-ray stress measurement method are in good agreement with those obtained by neutron diraction method. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
When a multi-phase alloy is heat-treated, internal stresses usually remain due to dierence in thermal expansion coecient between constituent phases. The distribution of such internal stresses in the cross section of a dual phase alloy specimen after quenched into water from an elevated temperature, are schematically drawn in Fig. 1. As can be seen, macroscopic residual stress (sM ij ) varies smoothly from the surface to internal region of the specimen. On heating or cooling, temperature changes more quickly at the surface than in the internal region of a specimen to result in yielding sM ij . If the elastic moduli are dierent between the two constituent phases, partitioning of sM ij would occur [1, 3]. In a dual phase alloy, microscopic residual stresses arise due to mis®t strains among grains of the constituent phases with dierent thermal expansion coecients. These stresses may vary from grain to grain and their averaged value over the each constituent phase is called thermal phase-stress (sph ij ). It is very useful to measure sph ij in multi-phase alloys, composites or functionally graded materials because sph ij are closely related with the strength of the materials. X-ray diraction has widely been used for such stress measurement so far. However, information only near surfaces can be obtained because its penetration depth is limited within a {To whom all correspondence should be addressed. 353
shallow region near surface of the specimen. Thus, the stresses measured by X-ray diraction method ph become complicated consisting of sM ij and sij that are partially relaxed due to the free surface [3]. On the other hand, application of neutron diraction to sph ij measurement is attractive because of its high penetration power into a specimen. Residual stress measurements by neutron diraction method have been made not only for composites [1, 3±5] but also for commercially available materials [6±10]. Residual elastic strains related to sph in metal ij matrix composites (MMCs) have successfully been measured by several workers [3±5, 11]. On the other hand, the stress measurements for dual phase alloys have not presented reasonable results yet [2, 12]. This is due to diculty of preparation of a stressfree reference material having identical chemical composition with that of a constituent phase in a dual phase alloy. In the case of MMCs, the reasonable results have been obtained because the reference materials are easily prepared. In the present study, therefore, ®ve Fe±Cr±Ni alloys were prepared for sph ij measurement, i.e. they are located along an equilibrium tie line of the ternary phase diagram [13], as shown in Fig. 2. This means that the volume fraction of ferrite (a) can be varied without changing its chemical compositions. The a and austenite (g) single phase alloys are then provided for the reference materials to evaluate thermal residual stresses in a and g phases in the a± g dual phase alloys.
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Fig. 1. Schematic illustration to explain residual stress distribution near surface in a dual phase alloy. (a) Microstructure of a cross section, (b) s11 or s22 and (c) s33, where subscripts ``ph'', ``M'', ``a'' and ``g'' refer to phase-stress, macrostress, a phase (ferrite) and g phase (austenite), respectively.
In order to improve angular/energy resolution with a simultaneous increase of the detector signal with respect to the conventional scattering devices with pyrolitic graphite monochromator [4, 12], we have introduced a Bragg diraction optics employing bent perfect crystals [14]. X-ray diraction methods including a conventional sin2 c method under the assumption of plane stress condition have been applied to compare the results with those by neutron diraction method. Finally, Eshelby inclusion theory coupled with Mori±Tanaka mean ®eld theory is applied to predict thermal phase-stresses and the predictions are compared with experimental results. The results obtained by neutron, X-ray diraction and the prediction are found in a good agreement.
S1 and S5 are g and a single phase alloys, respectively, while alloys S2, S3 and S4 are a±g dual phase alloys with various volume fractions of a, as will be described later. The ingots of these alloys were hotrolled at 1523 K to plates with 15 mm in thickness. Rod specimens with 10 mm in diameter and 50 mm in length were prepared for stress measurements using neutron diraction method in such a way that the vertical direction of the rod becomes parallel to the rolling direction. Plate specimens with 15 mm 15 mm 2 mm were prepared for stress measurement using an X-ray diraction method in such a way that a plane with 15 mm 15 mm is parallel to the rolling plane of hot-rolled plates. The specimens were mechanically polished with emery papers up to ]600 and annealed at 1273 K for 3.6 ks in vacuum followed by quenching into water. After the heat treatment, the rod specimens for neutron diraction testing were mechanically polished with emery papers (]1000) to eliminate their oxide layer. On the other hand, the plate specimens for X-ray diraction testing were slightly polished with emery papers followed by electropolishing. Microstructures of the specimens were observed by using an optical microscope (OM). Typical microstructures for the alloys are shown in Fig. 3, in which the observed plane is indicated in (f). Observations were also performed on the other two mutually orthogonal planes shown in Fig. 3(f) and it was found that grains were a little elongated along the rolling direction. The volume fractions of a (fa) of these alloys after the heat treatment were measured by a point counting method on optical micrographs and the results obtained are shown in Table 2. The fa can also be determined from ratios of relevant integrated intensities of peaks corresponding to the constituent phases in a dual phase alloy. The results are also listed in Table 2, showing good agreements with those obtained by the optical metallography method. The optical metallography method data were installed for the evaluation of sph ij
2. EXPERIMENTAL PROCEDURES
2.1. Specimens Five alloys were prepared by using an induction furnace aiming at having chemical compositions which lie on a tie line in the Fe±Cr±Ni ternary phase diagram [13], as shown in Fig. 2. The results of chemical analysis for ingots obtained are listed in Table 1, showing satisfactory compositions. Alloys
Fig. 2. Fe±Cr±Ni phase diagram [13] and ®ve alloys used in this investigation.
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355
Table 1. Chemical compositions of Fe±Cr±Ni alloys used in this investigation (mass %) Alloy S1 S2 S3 S4 S5
C
Si
Mn
Cr
Ni
Mo
Cu
P
N
Fe
0.012 0.013 0.009 0.018 0.016
0.39 0.38 0.36 0.33 0.32
0.33 0.32 0.31 0.34 0.31
22.43 27.82 29.63 31.41 35.96
18.38 13.16 10.88 9.25 4.10
0.02 0.02 0.01 0.01 0.01
0.02 0.02 0.01 0.01 0.01
0.009 0.010 0.007 0.008 0.006
0.027 0.035 0.039 0.051 0.049
bal. bal. bal. bal. bal.
value as described later. The grain sizes are approximately 10±30 mm for these alloys. 2.2. Dilatometry test
e
Dilatometry tests were carried out to obtain thermal expansion coecients of these alloys. Rod specimens with 5 mm in diameter and 15 mm in length were prepared from hot-rolled plates, as shown in Fig. 3(f). Using a dilatometer with a heating rate of 0.167 K sÿ1 (10 K minÿ1) in a temperature range between 296 K and 1273 K, thermal expansion coecients were measured in an argon gas atmosphere. A rod of Al2O3 was used for a reference material. 2.3. Stress measurement using neutron diraction method The principle of stress measurement is based on precise determinations of relative values of interplanar lattice spacings. Hence, the following Bragg's law [15] is essential. 2dhkl sin y l;
strain along the normal direction to (hkl) plane is then given as follows,
1
where dhkl, y and l refer to the spacing of (hkl) plane, the diraction angle (Bragg angle), and the incident neutron wavelength, respectively. The
dhkl ÿ d0 ÿ cot y Dy d0
2
where d0 represents the spacing of (hkl) plane for a stress-free material. Then a change of Bragg angle (Dy) provides us with the elastic strain of a sample [7, 16]. Neutron diraction measurements were performed using the residual stress diractometer installed at super mirror guide hall of Kyoto University Research Reactor Institute (light water moderated, 5 MW). Thermal neutron ¯ux at the exit of the guider is 5 107 neutrons cmÿ2 sÿ1 and their wavelengths of approximately 0.12±1.0 nm. A Si single crystal (30 mm in height, 200 mm in length and 4 mm in thickness) was elastically bent in order to focus the neutron beam to a sample [14, 17]. The wavelength of the monochromatized neutron beam is 0.124 nm. A one-dimensional position-sensitive proportional counter with a length of 500 mm was set horizontally to detect neutrons diracted in a range of diraction angles 2y 1 67 2 148. Signals from the counter were processed via logic units (2048 ch, 24 bits/ch) into positional neutron counts,
Fig. 3. Microstructures observed by an optical microscope (O.M.): (a) alloy S1, (b) S2, (c) S3, (d) S4 and (e) S5. Observed plane of a plate is shown in (f). The specimens for dilatometry and neutron and X-ray diractions are also shown in (f).
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Table 2. Volume fractions of a and thermal expansion coecients of alloys used in this investigation Volume fraction of a (%) Alloy S1 S2 S3 S4 S5
1
2
3
Thermal expansion coecient (10ÿ6 Kÿ1)
0.0 32.6 52.6 67.3 100
0.0 32.2 50.1 72.4 100
0.0 34.3 54.1 71.0 100
18.0 17.2 15.2 14.9 12.4
1. Measured by a point counting method using optical microscopy. 2. Calculated from peak intensities ratio obtained by X-ray diraction. 3. Calculated from peak intensities ratio obtained by neutron diraction.
which allowed us to get a pro®le of angular intensity distribution of a diracted beam. Figure 4(a) shows the measuring system used in this investigation. For conversion of the Bragg peak positions (channel numbers) to lattice spacings or Bragg angles y under the ®xed geometrical condition, a calibration curve was prepared from the several well-de®ned Bragg peaks from powder samples of pure iron and copper. The peak positions were determined by using a Gaussian curve ®tting technique. In order to acquire enough diraction intensity, the measurement was run more than 32.4 ks (9 h) for each sample. Two slits (2.5 mm in width and 30 mm in height) were placed before and after a sample, screening the beam to keep a sampling volume constant.
2.4. Stress measurement using X-ray diraction method A local coordinate (Li) and a global one (Si) for a specimen used in this study are shown in Fig. 4(b). Measuring an interplanar lattice spacing along L3(f, c) direction (dyc,hkl) enables us to determine a strain eyc by equation (2) that is rewritten here as, efc
dfc;hkl ÿ d0 ; d0
3
which can be written by using strain components eij in a global coordinate in Fig. 4(b) as follows. efc e11 cos2 f sin2 c e22 sin2 f sin2 c e33 cos2 c e12 sin 2f sin2 c e13 cos f sin 2c e23 sin f sin 2c:
4
Equation (4) can be rewritten by using stress component sij in the global coordinate through Hooke's law for isotropic materials: efc
1 f s11 cos2 f s2 sin2 f ÿ s33 sin2 c E s33 s31 cos f s23 sin f sin 2cg ÿ
s11 s22 s33 : E
5
A conventional sin2 c method [18] assumes plane stress condition (s33=s23=s31=0), and thus s11 can be determined from a slope of a efc vs sin2 c plotting when f = 0. Therefore, the in¯uence of preciseness of d0 on measuring s11 is relatively small. This is why this method has widely been used for stress measuring in engineering materials. First, this conventional method was used in this study. It is however revealed that this method cannot be applied to the present sph ij measurement, as will be described later in Section 3.2. Then, secondly, X-ray triaxial stress measurement was performed. A stress component s33 which changes from zero at free surface to some value inside a sample with increasing a distance from the surface is assumed to present [see Fig. 1(c)]. Here, an averaged value of such a gradient s33 in X-ray penetrating region is expressed by hs33i. According to the procedure proposed by Noyan [19], efc was measured for conditions of various +c and ÿc under f = 08, 458 and 908. Averaging the stresses obtained for +c and those for ÿc gives the following relation; fef c ef ÿcg 1 E 2 s11 cos2 f s12 sin 2f s22 sin2 f ÿ s33 sin2 c
Fig. 4. Geometrical arrangements for stress measurements; (a) for neutron diraction method and (b) for X-ray diffraction method.
1 s33 ÿ s11 s22 s33 : E E
6
A slope obtained from {ef(+c) + ef(ÿc)}/2 vs sin2
HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS Table 3. Conditions for X-ray diraction stress measurement Characteristic X-ray Tube voltage Tube current Diraction planes
Cr±Ka 40 kV 30 mA (211) for a phase (220) for g phase f = 08, 458, 908 c = ÿ 408 to +408 5 mm by 5 mm 300 2 1 K
Angles f and c Irradiation area Temperature
c plottings give, s11 ÿ hs33 i s 0
for f 0
7
3.1. Residual elastic strains measured by neutron diffraction method
for f 90
8
Figures 5(a)±(e) show neutron diraction pro®les of the S1, S2, S3, S4 and S5 alloys obtained. These pro®les exhibit much higher S/N ratios in comparison of the previous results with use of pyrolitic graphite monochromator [4, 12, 20]. Remarkable improvement has been achieved by using the Si bent crystal monochromator in this study. As can be seen in Figs 5(b), (c) and (d), diraction peaks corresponding to a and g phases are clearly identi®ed in alloys S2, S3 and S4. As long as g phase is concerned, peak heights of (311) show the largest followed by (220) and (222) commonly in alloys S1, S2, S3 and S4. This means that there exists the similar texture in the alloys. Concerning a phase, on the other hand, (200) shows the largest in alloy S5 while (211) show the largest in alloys S2, S3 and S4. This means that the texture of the a single phase alloy is dierent from that of the dual phase alloys. It is found that the intensities for
Because normal directional strain with respect to surface can be directly known from the change of an interplanar lattice spacing, i.e. he33 i dc0 ÿ d0 =d0
9
hs33i can be determined by the following equation derived from equations (7)±(9); hs33 i
Conditions for the stress measurements by X-ray diraction method are summarized in Table 3. After subtracting background from diraction pro®les obtained, Pseudo Voigt function was employed for curve ®tting and then a center of half-value width method was used to determine the relevant peak positions.
3. EXPERIMENTAL RESULTS
or, s22 ÿ hs33 i s 90
357
E he33 i fs 0 s 90g; 1 ÿ 2 1 ÿ 2
10
where Hooke's law of he33i = (1 + n)/Eh s33i ÿ n/ E(s11+s22+hs33i) is used. Thus s11(=s(0) + hs33i) and s22(=s(90) + hs33i) can be obtained. Similarly to this procedure, shear components, s23, s31 and s12 can be obtained by using {ef(+c) ÿ ef(ÿc)}/2 for f = 08 and 908, and {e45(+c) + e45(ÿc)}/2 for f = 458, respectively.
Fig. 5. Examples of diraction patterns obtained by neutron diraction for alloy S1 (a), S2 (b), S3 (c), S4 (d) and S5 (e).
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HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS
(220) and (311) of g decrease with increasing fa while the ones for (211) of a increase. The peak positions in the line counter corresponding to the diraction angles of (220), (311) and (222) for g and those of (200), (211) and (220) for a were used to evaluate residual elastic strains. The diraction angles obtained in alloys S1 and S5 were used as references to determine the residual elastic strains for g and a phases in the dual phase alloys, respectively. The peak (220) for a phase is detected in the three dual phase alloys but not in the a single phase alloy S5. Then, the interplanar lattice spacing for (220) in alloy S5 is calculated from those for well-de®ned (200) and (211) peaks, i.e. using the lattice parameters derived from (200) and (211) peaks obtained, the interplanar lattice spacing for (220) is estimated. The strain measured here is called a total residual elastic strain (eij) which includes macro-strain (eM ij ) associated with ph sM and phase strain (eph ij ij ) associated with sij . Because elastic moduli for a are estimated nearly identical with those for g, the strain associated with the partitioning of sM ij to a and g grains can be neglected. Therefore, the measured eij can be written as: eij
eM ij
eijph
11
The eij evaluated with equation (2) for the alloys S2, S3 and S4 are shown in Fig. 6. It is found that tensile (positive) strains exist in g phase while compressive (negative) ones in a phase. The strains for dierent (hkl) planes are fairly in good agreement with each other except those for g phase in alloy S4. The measured strains depend on the volume fraction of a (fa). The results which were independently obtained for the present alloys by our collaborated work which was performed at NPI, Rez, Czech Rep. [21] are also plotted in Fig. 6, showing excellent agreements.
Fig. 6. Total residual elastic strains, aeij and geij for alloys S2, S3 and S4 obtained by stress measurement using neutron diraction method.
If the shape of a specimen is spherical, sM ij in its internal region is considered to be isotropic. When the shapes of grains of constituent phases are spherical, sph ij in the internal region of a specimen is also considered to be isotropic. That is, e11=e22=e33 and shear strains are zero. For simplicity, we assume here the isotropy both for sM ij and sph although the actual shapes of specimen and ij grains are not exactly spherical. Thus, isotropic total residual stresses (s11=s22=s33) either in a phase or in g phase could roughly be evaluated by, s11 fE= 1 ÿ 2ge11 :
12
Then, by using Young's modulus (E) = 200 GPa and Poisson's ratio (n) = 0.30 equally for a and g phases in equation (12), we obtained total thermal residual stresses, as11 and gs11 as shown in Fig. 7, where the measured strains from three dierent (hkl) planes in Fig. 6 are averaged for inputting into e11. Similarly to equation (11), sij could be expressed as; ph sij sM ij sij ;
13
where sph ij refers to the thermal phase-stress which is caused by thermal mismatching between the constituent phases during the quenching. The equilibrium condition for thermal phase-stresses within all over the sampling volume in a specimen of a dual phase alloy can be written as: a ph sij fa
gsijph 1 ÿ fa 0;
14
g ph where asph ij and sij stand for thermal phase-stress in a phase and that in g phase, respectively. Then, using equation (13) sM ij can be calculated by the following equation, a sij ÿ sM 15 fa g sij ÿM ij ij 1 ÿ fa 0:
Fig. 7. Total residual stresses, asij and gsij for alloys S2, S3 and S4 obtained by stress measurement using neutron diffraction method.
HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS
Using the procedure described above and the measured sij in Fig. 7, the sM ij for the three alloys calculated from equation (15) is shown in Fig. 8 by a dashed line, together with the sph ij obtained after subtracting sM from s . It is found in Fig. 8 that ij ij the sM ij averaged over the sampling volume shown in Fig. 4(a) is positive commonly for the three alloys, revealing that the cooling rate in an internal region of a sample was slower than at the surface resulting in tensile residual stress in the sampling volume. That is, sM ij in the central region is tensile while compressive at the surface, which will be discussed again in the following section of stress measurement by X-ray. Experimental results obtained from dilatometry tests show that the thermal expansion coecient of alloy S1(g) is higher than that of alloy S5 (a), as listed in Table 2. This means that g grains shrink more largely than a grains in a dual phase alloy during quenching, yielding residual tensile strain in g grains while compressive one in a grains. The sph ij in the present experiment is quantitatively consistent with this prediction. The sph obtained is ij obviously dependent on fa as shown in Fig. 8; with increasing fa, the absolute value of sph ij increases in g phase while it decreases in a phase. 3.2. Residual stresses measured by X-ray diraction method
359
ph Fig. 8. Macro- and phase-stresses, sM ij and sij obtained after separating them from sij by using equations (13) and (15).
When sM ij obtained by using equation (15) is subg ph tracted from asij or gsij, the stresses of asph ij or sij can be obtained. The results are shown in Fig. 9(b). Such stresses have been reported in many previous papers [24, 26]; although the stresses shown in Fig. 9(b) look excellent with respect to their equilibrium, these values are incorrect because hsph 33 i as ph sketched in Fig. 1 is neglected. The sph 11 or s22
3.2.1. Stresses obtained by the conventional sin2 c method under the assumption of plane stress condition. In many previous reports [22, 23] including Tomota et al.'s paper [24], phase-stresses formed by heat treatment and/or plastic deformation in dual phase alloys have been measured by X-ray diraction under the assumption of plane stress condition. However, it has recently been made clear, particularly in MMCs, that the plane stress condition is not applicable to some multi-phase materials [19, 25]. That is, gradient phase-stress of M sph ij (=s33ÿs33) exists and its average within an Xray penetrating region, hsph 33 i, cannot be ignored [see Fig. 1(c)]. In this section, some results by the conventional method are presented only to emphasize how they are dierent from the results obtained under consideration of triaxial stress condition. Keeping f = 08, diraction angle (y) was measured by changing c [see Fig. 4(b)]. By adopting the conventional sin2 c method, y was plotted against sin2 c, so that the stress s11 (f = 08) can be obtained from a slope by a least-square linear ®tting. When we take f = 908, s22 can be obtained. Such stresses (asij and gsij) shown in Fig. 9(a) cong ph M sist of asph ij or sij and sij . Equation (13) can be rewritten as: a
M sij a sph ij sij
for a phase
16a
g
M sij g sph ij sij
for g phase:
16b
Fig. 9. Residual stresses obtained by X-ray stress measurements under an assumption of plane stress condition for alloys S2, S3 and S4. (a) sij including macroscopic residual ph stresses, sM ij and (b) phase-stresses sij calculated by subtracting sM ij from the total residual stresses shown in (a).
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obtained by this method is smaller than that obtained by the following triaxial stress measuring method. 3.2.2. Stresses obtained by triaxial stress method. When the number of grains within an X-ray penetrating region is taken into consideration, s33 along normal direction with respect to surface cannot be ignored [25] [Fig. 1(c)]. In the case of a commercially available dual phase stainless steel, it is not easy to perform triaxial stress measurement because no appropriate d0 value is available [2, 12]. However, in this study, we can use the data for the alloys S1 or S5 for appropriate determination of d0. Similarly to Section 3.2.1, the measured sij is sepph arated into sM ij and sij by using equation (15). As a result, sM is compressive being dierent from the ij result obtained in the previous section (Section 3.1). M For instance, sM 11= ÿ 218.1 MPa, s22= ÿ 191.3 MPa and hsM i = ÿ 15.3 MPa were obtained for 33 near the surface alloy S3. The stresses show that sM ij is approximately in the plane stress condition; i.e. M M hsM 33i is negligible compared with s11 and s22. The phase-stresses obtained from the triaxial stress analysis are shown in Fig. 10. As can be seen, hsph 33 i is not zero so that the plane stress condition assumed in Section 3.2.1 is revealed to be incorrect. 4. DISCUSSION
4.1. Comparison of the results obtained by neutron and X-ray diraction methods
Fig. 10. Thermal phase-stresses, sph ij obtained from 3-axial stress measurement by X-ray diraction for alloys S2, S3 and S4.
4.2. Predictions of phase-stresses by Eshelby and Mori-Tanaka theories It has well been established that phase-stresses, i.e. mean internal stresses, in an alloy including many second phase ellipsoidal grains with uniform mis®t strains, so called eigen strains e*ij, can be calculated by using Eshelby's inclusion theory [27] coupled with Mori-Tanaka mean ®eld theory [28]. In the present case, e*ij is introduced from the dierence in thermal expansion coecients between a and g phases given in Table 2. Assuming that elastic moduli for both phases are identical, phase-stresses in a and g phases produced by quenching are calculated as follows. ÿ eij K g ÿ K a DT e dij ; 17
ph The sph 11 and s22 obtained by X-ray diraction triaxial stress measurement method are averaged as ph ph (sph 11 +s22 )/2 and plotted in Fig. 11. The s11 by neutron method are also shown in this ®gure. The g ph sij fa Ca S ÿ Ie dij 18a agreements between the two results are surprisingly excellent, but the following problems should be a ph sij ÿ 1 ÿ fa Ca S ÿ Ie dij ; 18b taken into consideration. It is speculated that sph ij measured by X-ray diraction method should be smaller in their absolute values than those obtained by neutron diraction method because stress relaxation occurs near surface; such a dierence has actually been observed in a SiCw/A2014 MMC [3]. On the other hand, Fitzpatrick et al. who measured sph ij in a SiCp/A2124 MMC using neutron diraction method, have claimed that the eective temperature drop during quenching which generated the thermal phase-stresses was slightly larger at the surface than that in the central region of a sample leading to slightly larger phase-stresses near the surface [1]. This manifests that sph ij measured by Xray shows almost the same or higher magnitude with that measured by neutron since the neutron measurement was performed at the central region of sampling volume of a specimen. Although the present results from neutron and X-ray are comparable, it seems that the neutron results are more re- Fig. 11. Comparisons between thermal phase-stresses sph ij liable than the X-ray results due to the dierence of predicted by using Eshelby and Mori±Tanaka theories and their penetration powers. those measured by neutron or X-ray diraction.
HARJO et al.: THERMAL RESIDUAL ELASTIC STRAINS
where Kg, Ka, DT, dij, Ca, S and I represent the thermal expansion coecient of a, that of g, the temperature drop, Kronecker's delta, the elastic stiness tensor of a, Eshelby's tensor and the identity tensor, respectively. For simplicity, the shape of grains is assumed to be spherical, then S1111=S2222=S3333=(7 ÿ 5n)/15(1 ÿ n), S1122= S2233=S3311= ÿ (1 ÿ 5n)/15(1 ÿ n), S1212=S2323= S3131=(4 ÿ 5n)/15(1 ÿ n), and others are zero. Inputting Ka=12.4 10ÿ6 Kÿ1, Kg=18.0 10ÿ6 Kÿ1 and DT = 1000 K in equation (17), e*ij=5.6 10ÿ3 dij is obtained. Then, by using Young's modulus: E = 200 GPa and Poisson's ratio; n = 0.30 equally for a and g phases in equation (18a) and (18b), we g ph have obtained asph ij and sij presented in Fig. 11. It is found that the agreements between the predicted stresses and the measured stresses are fairly good. During the quenching from 1273 K to room temperature, thermal internal stress may be partially relaxed by diusion and/or dislocation punchingout. According to the results of our collaborated work performed at NPI, Rez, Czech Rep. [21], the dislocation density predicted from pro®le analysis for neutron diraction experiments is higher in the a±g dual phase alloys (S2, S3, and S4) than in the single phase alloys (S1 or S5). Although their results suggest that plastic relaxation by dislocation punching-out takes place, Fig. 11 shows that the in¯uence of such a relaxation mechanism on sph ij is small.
5. CONCLUSIONS
Five Fe±Cr±Ni alloys, i.e. three ferrite(a)±austenite(g) alloys, a single phase alloy and g single phase alloy, were carefully prepared, whose chemical compositions lie on an equilibrium tie line in the ternary phase diagram. The thermal phase-stresses of specimens generated by quenching from 1273 K to 273 K were measured by neutron and X-ray diffractions. The main results obtained are summarized as follows; (i) Compressive residual elastic strain in a and tensile one in g are measured in the present a±g dual phase alloys by neutron diraction method. The results are considered to be reasonable from the consideration based on thermal expansion coecients of 18.0 10ÿ6 Kÿ1 for g and 12.4 10ÿ6 Kÿ1 for a; during quenching g grains shrink more largely than a grains, yielding residual tensile strain in g grains while compressive one in a grains. The preparation of appropriate reference materials was a point for obtaining such good results. (ii) The employment of a Si-bent crystal for focusing neutron beam to a sample is found to be eective to increase the precision in stress measurement.
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(iii) The conventional X-ray sin2 c method under the assumption of plane stress condition cannot be applied to the measurement of phase-stresses. (iv) X-ray triaxial stress analysis provides similar results to those by neutron diraction method but seems to be insucient because of the stress gradient near surface. (v) Phase-stresses predicted by Eshelby and Mori± Tanaka theories are found to show good agreement with those measured by neutron diraction method, suggesting that plastic relaxation is not so large. AcknowledgementsÐWe would like to thank NIDAK Co. for melting the alloys used in this study and Dr P. Lukas of Nuclear Physics Institute, Czech Republic, for valuable discussions. The neutron stress measurements were performed as a coworking project at KURRI (Kyoto University, Research Reactor Institute). This work was partially supported by the special coordinate fund of the Science and Technology Agency Japan.
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