- Email: [email protected]
Measurement of local plastic strain distribution of stainless steel by electron backscatter diffraction
M A TE RI A L S CH A RACT ER IZ A TI O N 60 ( 20 0 9 ) 1 2 5 –1 3 2
Measurement of local plastic strain distribution of stainless steel by electron backscatter diffraction Masayuki Kamaya⁎ Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Mikata-gun, Fukui 919-1205, Japan
AR TIC LE D ATA
ABSTR ACT
Article history:
Electron backscatter diffraction in conjunction with scanning electron microscopy was used
Received 23 June 2008
to assess the plastic strain on a microstructural scale (local plastic strain) induced in
Accepted 24 July 2008
stainless steel deformed up to a nominal strain of 19.7%. Accuracy of the measurement of misorientations was improved by a technique called the Domain Averaging Method (DAM),
Keywords:
in which an average of crystal orientation was calculated for several data measured from the
Electron backscattering
same domain. It was shown that the misorientation evaluated using the crystal orientation
diffraction (EBSD)
of which accuracy was improved by DAM showed localized plastic strain in the vicinity of
Stainless steel
grain boundaries (GB). The distribution of misorientations followed a log-normal
Plastic deformation
distribution and the mean value correlated well with the macroscopic plastic strain
Cold working
induced. By using the correlation between the misorientation and the plastic strain, the
Misorientation
distribution of local plastic strain could be quantified. It was shown that the plastic strain becomes more than 15% locally under a macroscopic strain of 4.9%. A procedure for confirming the accuracy of the measurement is also suggested. © 2008 Elsevier Inc. All rights reserved.
1.
Introduction
Plastic strain accelerates material degradation by stress corrosion cracking [1,2]. The magnitude of the degradation is characterized by the initiation of small cracks and their growth [3]. In order to understand how the plastic strain accelerates the initiation of small cracks, it is important to know the magnitude of plastic strain on a microstructural scale (hereafter, local plastic strain). Even if the macroscopic strain appears uniform and homogeneous, local strain of polycrystalline material is inhomogeneous due to the anisotropy of crystal grains and their random or nearly random orientation distribution [4–6]. On a microstructural scale, plastic deformation causes crystallographic slip and the geometrically necessary dislocations. The crystal orientation is changed due to the dislocations and may show fluctuations of several degrees even in the same grain. Electron backscatter diffraction (EBSD), in conjunction with scanning electron microscopy (SEM), is one of the most
promising techniques for measuring the change in local crystal orientation. By using commercially available equipment for EBSD measurement, we can identify crystal orientations by scanning the surface of samples. It has been shown that scalar parameters obtained from crystal orientations of the scanned area correlate with the magnitude of macroscopic plastic strain induced in materials [7–9]. Therefore, by using this correlation, the macroscopic plastic strain can be estimated from crystal orientations obtained by EBSD measurements. On the other hand, by evaluating the misorientation angle between neighboring points (hereafter, local misorientation), it is possible to know the magnitude of local plastic strain. In a previous study by the present author [10], the spatial distribution of the local misorientation was compared with that of nominal strain measured by the image correlation technique, and it was revealed that the local misorientation was consistent with the density of the geometrically necessary dislocations rather than with the magnitude of nominal strain, which is defined as the deformation per unit length. In order to evaluate the effect of
⁎ Tel.: +81 770 379114; fax: +81 770 372009. E-mail address: [email protected]. 1044-5803/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.matchar.2008.07.010
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MA TE RI A L S CH A R A CT ER IZ A TI O N 60 ( 20 0 9 ) 1 2 5–1 3 2
Table 1 – Chemical content of test material (wt.%) Fe
C
Si
Mn
P
S
Ni
Cr
Mo
Bal.
0.05
0.41
0.83
0.026
0.001
10.08
16.14
2.08
plastic strain on cracking behavior, the density of dislocations is more relevant than the nominal strain. Therefore, in this study, the local misorientation was evaluated for deformed material. Crystal orientation measurement has an error of 0.1–1° depending on various conditions of measurement. For example, the surface condition of samples affects the quality of the EBSD patterns. Deterioration of the diffraction pattern reduces the accuracy of crystal orientation identification. The plastic strain deteriorates the EBSD pattern even with well-prepared surface [11]. The number of pixels of the CCD camera used to acquire EBSD patterns, and parameters for identification of crystal orientation from obtained EBSD patterns, also affect the accuracy of measurements. Therefore, the misorientation calculated from crystal orientations contains substantial error when the measured misorientation is relatively small [12]. Therefore, it is difficult to measure precise misorientations when the spatial resolution of the measurements is fine or induced plastic strain is small. For quantitative measurements, the influence of the error should be excluded and the accuracy of misorientation identification must be improved. In this study, a procedure for measurement of local plastic strain was developed. The material used was Type 316 stainless steel, in which plastic strain was induced by tensile loads up to a nominal plastic strain of 19.7%. The local distribution of misorientations was identified from crystal orientations obtained using EBSD. In order to improve the accuracy of misorientation identification, a technique for data processing was applied in addition to careful measurement of crystal orientation. Then, the local plastic strain was quantified using the correlation between the misorientation and induced plastic strain.
2.
Crystal orientation measurements by EBSD were made with an orientation imaging microscope interfaced to a field emission electron gun SEM. The step size of the measurements was 0.25 μm at minimum.
3.
Local Misorientation Distribution
Fig. 1 shows the local misorientation, ML, calculated by the following equation: ML ðpo Þ ¼
4 1X bðpo ; pi Þ 4 i¼1
ð1Þ
where β(po, pi) denotes the misorientation between a fixed point po and neighboring points pi in the grain as shown in Fig. 2. Misorientations larger than 5° were regarded as grain boundaries. Two maps of the local misorientation were obtained under different conditions of CCD camera for EBSD pattern acquisition under the same step size of 1.0 μm from the specimen of εp = 4.9%.
Experimental Procedure
The material used for these studies was a solution heattreated Type 316 austenitic stainless steel, whose alloying constituents are shown in Table 1. Plate tensile specimens (gauge length = 20 mm and cross section = 2 × 4 mm) were machined and deformed by tensile loading to six nominal global plastic strains, εp, of 0%, 2.8%, 4.9%, 10.3%, 15.2% and 19.7%. The deformation rate was 1.0 mm/min at the crosshead of the tensile test machine and the strain was defined by the change in distance between indentation marks measured by a traveling microscope. After the deformation, mid-plane sections in the region of uniform strain were prepared for EBSD measurement. The surface was polished up to 3 μm diamond paste followed by colloidal silica in order to achieve relatively flat surfaces free from damage. The material had an approximately equiaxed grain structure. EBSD measurements sampling large numbers of grains indicated that the crystallographic texture was very weak.
Fig. 1 – Distribution of local misorientation (plastic strain: εp = 4.9%, step size: d = 1 μm). (a) Coarse pixel condition (b) fine pixel condition.
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127
distribution, for which the probability density function is defined by: " # 1 1 lnML lnMave 2 pffiffiffiffiffiffi exp ð2Þ f ðML Þ ¼ 2 lnS ðlnSÞML 2p where ML and S are the local misorientation and standard deviation, respectively. Mave is the mean value of the distribution and is calculated by the following equation: " # N 1X lnfML ðpi Þg ð3Þ Mave ¼ exp N i¼1
Fig. 2 – Definition of local misorientation.
In Fig. 1(a), the number of pixels of the CCD camera was 128 ×96 (hereafter, “coarse pixel") whereas it was 640× 480 (hereafter, “fine pixel") in Fig. 1(b). The fine pixel CCD camera makes it possible to measure precise crystal orientation. Therefore, a clearer distribution of local misorientation could be obtained in Fig. 1(b) compared with that of Fig. 1(a). Fig. 3 shows the distribution of local misorientation in Fig. 1(a) together with a regression curve optimized using the log-normal
Fig. 3 – Distribution of local misorientations (same data as in Fig. 1 (a)).
where N is the number of data. It should be noted that only grains consisting of more than 10 points were included in the calculation; smaller grains were ignored. The local misorientation distribution seems to be well-represented by a lognormal distribution. This was the same for Fig. 1(b) and other measurements made in this study. The change in averaged local misorientation with step size, d, is shown in Fig. 4. Since the change in crystal orientation depends on the step size, the averaged local misorientation increased with the step size almost linearly. The averaged local misorientation obtained by the coarse pixel condition was larger than that by the fine pixel condition. The difference between the two conditions can be explained by Fig. 5, which schematically shows the influence of the error in crystal orientation measurement on local misorientation. The error in crystal orientation measurement exists regardless of the step size, and brings about an error in local misorientation. If the local misorientation is large enough, the influence of the error in the averaged local misorientation becomes insignificant due to the averaging effect. However, in the case of small local misorientation, the misorientation angle is smaller than the error, and so the averaged local misorientation becomes larger than that of the real misorientation, because the misorientation angle is an absolute value. Furthermore, it was pointed out that the accuracy of misorientation identification is better for larger misorientation angle [12]. As shown in Fig. 3, small local misorientations were included in the calculation of the averaged local misorientation and they raised the averaged local misorientation. This was significant for smaller step size
Fig. 4 – Change in averaged local misorientation with step size (plastic strain: ɛp = 4.9%).
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MA TE RI A L S CH A R A CT ER IZ A TI O N 60 ( 20 0 9 ) 1 2 5–1 3 2
Fig. 5 – Schematic figure representing influence of error in misorientation measurement (N: number of data).
because the ratio of small local misorientation was large. Hence, the increase in the averaged local misorientation by the coarse pixel condition became larger for small step size due to the accumulation of errors in the measurements. The error in measurement affects the local misorientation distribution shown in Fig. 1. Due to the error, the area of the white region (small local misorientation) is relatively small and unclear in coarse pixel data (Fig. 1(a)). Since misorientation becomes small as the spatial resolution of measurement increases, it is important to reduce the error in local misorientation measurement and to exclude the influences of the condition of misorientation identification.
4.
Domain Averaging Method
In general, errors in measurement can be reduced by averaging several data. However, as shown in Fig. 5, the averaging of local
misorientation does not always reduce the error. Therefore, by taking the average of measured crystal orientation, the accuracy of crystal orientation measurement and evaluation of local misorientation was improved. Fig. 6 shows a schematic drawing of the data acquisition and processing procedure referred to as the Domain Averaging Method (DAM). The crystal orientations were obtained as an average of several measurements for each domain, of which interval is d. The number of measurement points included in one domain is represented by RA. In the case of RA= 4, 16 crystal orientations in total are measured to obtain one crystal orientation used for local misorientation calculation. The average of crystal orientations is calculated using a set of quaternion [13]. If grain boundaries existed in the domain, some measured data are discarded as shown in Fig. 7. Misorientations larger than 5° are regarded as grain boundaries. Fig. 8 shows the relationship between the averaged local misorientation and step size for different RA. The averaged local misorientation decreased as RA increased. This was brought about by the reduction in error of crystal orientation
M A TE RI A L S CH A RACT ER IZ A TI O N 60 ( 20 0 9 ) 1 2 5 –1 3 2
129
Fig. 8 – Relationship between averaged local misorientation and step size obtained under different value of RA (plastic strain: ɛp = 4.9%).
Fig. 6 – Schematic drawing of averaging technique for crystal orientation map (Domain Averaging Method: DAM).
measurement; the rate of the reduction seemed to be saturated at RA= 4. The change in averaged local misorientation with RA is shown in Fig. 9 for step size of 1.0 μm. The reduction in averaged local misorientation was larger for coarse pixel measurement than for fine pixel measurement. Although larger RA gives a greater improvement in accuracy, it requires much time for measurement, as measurements have to be repeated RA×RA times in order to apply DAM of RA. RA= 2 seems to be appropriate when crystal orientation measurements are made using the fine pixel condition.
Distributions of the local crystal orientation obtained using DAM of RA= 4 are shown in Fig. 10. By applying DAM, the distribution becomes clearer than that shown in Fig. 1 and almost the same regardless of the number of pixels. This means that DAM enables us to obtain well-converged precise local misorientation irrespective of the accuracy of crystal orientation measurement. In Fig. 9, data obtained from the 0% strained sample is also shown. Even in unstrained material, there is some misorientation; the water quenching during the heat treatment and stress due to the machining process may have caused small plastic strains. The averaged local misorientation of the 0% strained sample decreased by applying DAM, and the converged value was almost 0.1°.
5. Change in Local Misorientation by Deformation Fig. 11 shows the local misorientation along a straight line, which crosses grain boundaries, obtained from Fig. 10(b). The
Fig. 7 – Treatment of grain boundary for DAM (in case of RA= 4).
Fig. 9 – Change in averaged local misorientation with RA (step size: d = 1 μm).
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MA TE RI A L S CH A R A CT ER IZ A TI O N 60 ( 20 0 9 ) 1 2 5–1 3 2
Fig. 12 – Schematic drawing for representing relationship between deformation of material, accumulation of dislocations, and evolution of misorientation.
Fig. 10 – Distribution of local misorientation obtained by DAM of RA = 4 (plastic strain: ɛp = 4.9%, step size: d = 1 μm). (a) Coarse pixel condition (b) fine pixel condition.
distribution of misorientations was inhomogeneous and different grain by grain. It became more than 0.9° at the maximum, although the averaged value was Mave =0.273°. Especially, the misorientation tended to be large near grain boundaries. By plastic deformation, as schematically shown in Fig. 12, dislocations are initiated and move along crystallographic slip planes, then pile up near grain boundaries and form so-called geometrically necessary dislocations. The large misorientation near the grain boundaries was inferred to be caused by such dislocations. Fig. 13 shows the relationship between the averaged local misorientation and step size for each sample obtained by DAM of RA = 2. The magnitude of the local misorientations was dependent on plastic strain as well as the step size of the crystal orientation measurements. The averaged misorientation and step size did not always show a linear correlation. In cases of large strain, such as 19.7% plastic strain, the increase in the misorientation slowed as the step size increased. The threshold angle for grain boundaries was 5° and the local misorientation exceeded the threshold angle locally. The large
Fig. 11 – Misorientation along a line obtained from Fig. 10b.
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131
local misorientation was taken as grain boundaries and was not considered in the calculation of the averaged misorientation. This caused the change in inclination of the relationship between the averaged local misorientation and step size at large plastic strain. In summary, the local misorientation correlated with the step size as well as the magnitude of the macroscopic plastic strain. Therefore, by quantifying the relationship between these three parameters, we can estimate the degree of local plastic strain.
6. Evaluation of Plastic Strain from Local Misorientation Fig. 14 shows the relationship between the nominal plastic strain, εp, and the averaged local misorientation for each step size d. The local misorientation was calculated based on crystal orientation obtained by DAM of RA = 2 using the fine pixel camera condition. The averaged local misorientation for unstrained conditions is set to Mave = 0.1° regardless of step size. Mave shows an excellent correlation with the plastic strain and the relation is almost linear under the plastic strain of 10%. A linear regression of the data under 10% plastic strain leads to the following equation: ep ¼
Mave 0:1 ; 0:0027d2 þ 0:041d
ð4Þ
where the strain is given in percent and distance in μm. This relation is shown by solid lines in Fig. 14, and is expected to be valid up to 15% plastic strain when the step size is less than 3 μm. For the estimation of local plastic strain, Eq. (4) is modified as: epðlocalÞ ¼
ML 0:1 : 0:0027d2 þ 0:041d
ð5Þ
By substituting measured local misorientation for ML in Eq. (5), the local value of plastic strain, εp(local), can be derived. For example, when the step size and the local misorientation are d = 1.0 μm and ML = 0.7°, the local plastic strain is estimated as εp (local) = 15.7%. The relationship between the local misorientation
Fig. 13 – Change in averaged local misorientation with step size obtained by DAM of RA = 2.
Fig. 14 – Relationship between averaged local misorientation and macroscopic plastic strain obtained by DAM of RA = 2 (solid lines correspond to the equation for each step size d).
and plastic strain was almost linear under plastic strain of 15% when the step size was d = 1.0 μm. Therefore, the distribution of the local plastic strain can be obtained by taking ML = 0.7° as εp = 15.7% in Figs. 10 and 11. Although the applied macroscopic plastic strain was 4.9%, it was more than 15% locally. It should be noted that the estimated plastic strain does not correspond to the nominal plastic strain. As mentioned, the local misorientation correlates with the geometrically necessary dislocations rather than the magnitude of deformation. The estimated local plastic strain just shows the typical local misorientation that is observed under the plastic strain. The local plastic strain (misorientation) has a large dispersion, and is determined not only by applied plastic strain but also by geometry of grain structure, crystal orientation and so on. The procedure for measuring the local plastic strain can be summarized as follows: 1. Carefully measure the crystal orientation by EBSD (using fine pixel CCD camera images and well-prepared samples). 2. Apply DAM in order to reduce the unsolved error in crystal orientation measurement.
Fig. 15 – Change in the Modified Crystal Deformation (MCD) with step size (open symbol: fine pixel with DAM of RA = 2, close symbol: coarse pixel without DAM).
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MA TE RI A L S CH A R A CT ER IZ A TI O N 60 ( 20 0 9 ) 1 2 5–1 3 2
8.
Fig. 16 – Change in the Modified Crystal Deformation (MCD) with macroscopic plastic strain.
3. Estimate the plastic strain from the local misorientation using the relationship shown in Fig. 14 or Eq. (4).
7.
Confirmation of Validity of Measurement
It is important to know whether the measurement conditions are adequate for evaluating local misorientation, or whether the error in misorientation measurement has been sufficiently converged by applying DAM. The validity of the measured data can be confirmed by referring to macroscopic strain. Fig. 15 shows the change in MCD, which is a parameter for the macroscopic plastic strain measurement [9], with step size. MCD is the averaged misorientation from a specific orientation referred to as the central orientation assigned to each grain, and only weakly depends on the step size of EBSD measurement as shown in Fig. 15. Since the misorientation from the central orientation is relatively large compared to the local misorientation, accuracy of the EBSD measurement has little influence on MCD as shown in Fig. 5. Therefore, the MCD obtained using the coarse pixel condition was almost identical with that obtained by the fine pixel condition augmented by applying DAM of RA = 2. The relationship between MCD and the macroscopic plastic strain is shown in Fig. 16, and can be approximated by the following equation: MCD ¼ 0:21ep þ 0:2:
ð6Þ
The relationship is almost linear below the plastic strain of 10% regardless of the step size, and is expected to be the same for different EBSD measurement conditions. Therefore, we can confirm the validity of measurement by comparing the macroscopic plastic strains obtained by Eqs. (4) and (6). As shown in Figs. 8 and 9, the parameter Mave decreases as the accuracy of measurement increases. If the accuracy of the measurement is worse than that for Fig. 14, the plastic strain obtained by Eq. (4) becomes larger than that by Eq. (6). By applying DAM with a large value of RA, the plastic strain given by Eq. (4) is expected to converge to that given by Eq. (6) regardless of the measurement conditions.
Conclusions
In order to quantify the local plastic strain induced in Type 316 stainless steel, EBSD in conjunction with SEM was applied. It was shown that DAM reduces the error in misorientation and enables us to obtain a clear distribution of misorientations. The distribution of misorientations followed a log-normal distribution and its mean value correlated well with the macroscopic plastic strain induced in the specimens. By using the correlation between the misorientation and the plastic strain, the distribution of local plastic strain can be estimated. It was shown that the plastic strain is more than 15% locally under a macroscopic strain of 4.9%. The local plastic strain tends to be especially large near grain boundaries. In order to measure the local plastic strain, it is important to measure the crystal orientation carefully. A method of confirming the accuracy of misorientation identification was presented.
REFERENCES [1] Arioka K, Yamada T, Terachi T, Chiba G. Cold work and temperature dependence of stress corrosion crack growth of austenitic stainless steels in hydrogenated and oxygenated high-temperature water. Corrosion 2007;63:1114. [2] Okamura T, Sakashita A, Fukuda T, Yamashita H, Futami T. Latest SCC issues of core shroud and recirculation piping in Japanese BWRs. Trans. of 17th Int. Conf. on Structural Mechanics in Reactor Technology (SMiRT 17); 2003. WG01-1. [3] Kamaya M, Haruna T. Crack initiation model for sensitized 304 stainless steel in high temperature water. Corros Sci 2006;48:2442. [4] Hashimoto K, Margolin H. The role of elastic interaction stresses on the onset of slip in polycrystalline alpha brass (I. Experimental determination of operating slip systems and qualitative analysis). Acta Metall 1983;31:773. [5] Musienko A, Tatschl A, Schmidegg K, Kolednik O, Pippan R, Cailletaud G. Three-dimensional finite element simulation of a polycrystalline copper specimen. Acta Mater 2007;55:4121. [6] Kamaya M, Kitamura T. Three-dimensional local stress analysis on grain boundaries in polycrystalline material. Int J Solids Struct 2007;44:3267. [7] Lehockey EM, Lin Y, Lepik OE. Mapping residual plastic strain in materials using electron backscatter diffraction. In: Schwartz AJ, Kumar M, Adams BL, editors. Electron Backscatter Diffraction in Materials Science. New York: Kluwer Academic /Plenum Publishers; 2000. p. 247. [8] Kamaya M, Wilkinson AJ, Titchmarsh JM. Measurement of plastic strain of polycrystalline material by electron backscatter diffraction. Nucl Eng Des 2005;235:713. [9] Kamaya M, Wilkinson AJ, Titchmarsh JM. Quantification of plastic strain of stainless steel and nickel alloy by electron backscatter diffraction. Acta Mater 2006;54:539. [10] Kamaya M, Quinta Da Fonseca J, Li LM, Preuss M. Local plastic strain measurement by EBSD. Appl Mech Mater 2007;7-8:173. [11] Wilkinson AJ, Dingley DJ. Quantitative deformation studies using electron back scatter patterns. Acta Metal Mater 1991;39:3047. [12] Wilkinson AJ. A new method for determining small misorientations from electron back scatter diffraction patterns. Scripta Mater 2001;44:2379. [13] Sutton AP. Statistical representation and analysis of orientational data. Phil Mag Lett 1996;74:389.
Measurement of local plastic strain distribution of stainless steel by electron backscatter diffraction Masayuki Kamaya⁎ Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Mikata-gun, Fukui 919-1205, Japan
AR TIC LE D ATA
ABSTR ACT
Article history:
Electron backscatter diffraction in conjunction with scanning electron microscopy was used
Received 23 June 2008
to assess the plastic strain on a microstructural scale (local plastic strain) induced in
Accepted 24 July 2008
stainless steel deformed up to a nominal strain of 19.7%. Accuracy of the measurement of misorientations was improved by a technique called the Domain Averaging Method (DAM),
Keywords:
in which an average of crystal orientation was calculated for several data measured from the
Electron backscattering
same domain. It was shown that the misorientation evaluated using the crystal orientation
diffraction (EBSD)
of which accuracy was improved by DAM showed localized plastic strain in the vicinity of
Stainless steel
grain boundaries (GB). The distribution of misorientations followed a log-normal
Plastic deformation
distribution and the mean value correlated well with the macroscopic plastic strain
Cold working
induced. By using the correlation between the misorientation and the plastic strain, the
Misorientation
distribution of local plastic strain could be quantified. It was shown that the plastic strain becomes more than 15% locally under a macroscopic strain of 4.9%. A procedure for confirming the accuracy of the measurement is also suggested. © 2008 Elsevier Inc. All rights reserved.
1.
Introduction
Plastic strain accelerates material degradation by stress corrosion cracking [1,2]. The magnitude of the degradation is characterized by the initiation of small cracks and their growth [3]. In order to understand how the plastic strain accelerates the initiation of small cracks, it is important to know the magnitude of plastic strain on a microstructural scale (hereafter, local plastic strain). Even if the macroscopic strain appears uniform and homogeneous, local strain of polycrystalline material is inhomogeneous due to the anisotropy of crystal grains and their random or nearly random orientation distribution [4–6]. On a microstructural scale, plastic deformation causes crystallographic slip and the geometrically necessary dislocations. The crystal orientation is changed due to the dislocations and may show fluctuations of several degrees even in the same grain. Electron backscatter diffraction (EBSD), in conjunction with scanning electron microscopy (SEM), is one of the most
promising techniques for measuring the change in local crystal orientation. By using commercially available equipment for EBSD measurement, we can identify crystal orientations by scanning the surface of samples. It has been shown that scalar parameters obtained from crystal orientations of the scanned area correlate with the magnitude of macroscopic plastic strain induced in materials [7–9]. Therefore, by using this correlation, the macroscopic plastic strain can be estimated from crystal orientations obtained by EBSD measurements. On the other hand, by evaluating the misorientation angle between neighboring points (hereafter, local misorientation), it is possible to know the magnitude of local plastic strain. In a previous study by the present author [10], the spatial distribution of the local misorientation was compared with that of nominal strain measured by the image correlation technique, and it was revealed that the local misorientation was consistent with the density of the geometrically necessary dislocations rather than with the magnitude of nominal strain, which is defined as the deformation per unit length. In order to evaluate the effect of
⁎ Tel.: +81 770 379114; fax: +81 770 372009. E-mail address: [email protected]. 1044-5803/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.matchar.2008.07.010
126
MA TE RI A L S CH A R A CT ER IZ A TI O N 60 ( 20 0 9 ) 1 2 5–1 3 2
Table 1 – Chemical content of test material (wt.%) Fe
C
Si
Mn
P
S
Ni
Cr
Mo
Bal.
0.05
0.41
0.83
0.026
0.001
10.08
16.14
2.08
plastic strain on cracking behavior, the density of dislocations is more relevant than the nominal strain. Therefore, in this study, the local misorientation was evaluated for deformed material. Crystal orientation measurement has an error of 0.1–1° depending on various conditions of measurement. For example, the surface condition of samples affects the quality of the EBSD patterns. Deterioration of the diffraction pattern reduces the accuracy of crystal orientation identification. The plastic strain deteriorates the EBSD pattern even with well-prepared surface [11]. The number of pixels of the CCD camera used to acquire EBSD patterns, and parameters for identification of crystal orientation from obtained EBSD patterns, also affect the accuracy of measurements. Therefore, the misorientation calculated from crystal orientations contains substantial error when the measured misorientation is relatively small [12]. Therefore, it is difficult to measure precise misorientations when the spatial resolution of the measurements is fine or induced plastic strain is small. For quantitative measurements, the influence of the error should be excluded and the accuracy of misorientation identification must be improved. In this study, a procedure for measurement of local plastic strain was developed. The material used was Type 316 stainless steel, in which plastic strain was induced by tensile loads up to a nominal plastic strain of 19.7%. The local distribution of misorientations was identified from crystal orientations obtained using EBSD. In order to improve the accuracy of misorientation identification, a technique for data processing was applied in addition to careful measurement of crystal orientation. Then, the local plastic strain was quantified using the correlation between the misorientation and induced plastic strain.
2.
Crystal orientation measurements by EBSD were made with an orientation imaging microscope interfaced to a field emission electron gun SEM. The step size of the measurements was 0.25 μm at minimum.
3.
Local Misorientation Distribution
Fig. 1 shows the local misorientation, ML, calculated by the following equation: ML ðpo Þ ¼
4 1X bðpo ; pi Þ 4 i¼1
ð1Þ
where β(po, pi) denotes the misorientation between a fixed point po and neighboring points pi in the grain as shown in Fig. 2. Misorientations larger than 5° were regarded as grain boundaries. Two maps of the local misorientation were obtained under different conditions of CCD camera for EBSD pattern acquisition under the same step size of 1.0 μm from the specimen of εp = 4.9%.
Experimental Procedure
The material used for these studies was a solution heattreated Type 316 austenitic stainless steel, whose alloying constituents are shown in Table 1. Plate tensile specimens (gauge length = 20 mm and cross section = 2 × 4 mm) were machined and deformed by tensile loading to six nominal global plastic strains, εp, of 0%, 2.8%, 4.9%, 10.3%, 15.2% and 19.7%. The deformation rate was 1.0 mm/min at the crosshead of the tensile test machine and the strain was defined by the change in distance between indentation marks measured by a traveling microscope. After the deformation, mid-plane sections in the region of uniform strain were prepared for EBSD measurement. The surface was polished up to 3 μm diamond paste followed by colloidal silica in order to achieve relatively flat surfaces free from damage. The material had an approximately equiaxed grain structure. EBSD measurements sampling large numbers of grains indicated that the crystallographic texture was very weak.
Fig. 1 – Distribution of local misorientation (plastic strain: εp = 4.9%, step size: d = 1 μm). (a) Coarse pixel condition (b) fine pixel condition.
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distribution, for which the probability density function is defined by: " # 1 1 lnML lnMave 2 pffiffiffiffiffiffi exp ð2Þ f ðML Þ ¼ 2 lnS ðlnSÞML 2p where ML and S are the local misorientation and standard deviation, respectively. Mave is the mean value of the distribution and is calculated by the following equation: " # N 1X lnfML ðpi Þg ð3Þ Mave ¼ exp N i¼1
Fig. 2 – Definition of local misorientation.
In Fig. 1(a), the number of pixels of the CCD camera was 128 ×96 (hereafter, “coarse pixel") whereas it was 640× 480 (hereafter, “fine pixel") in Fig. 1(b). The fine pixel CCD camera makes it possible to measure precise crystal orientation. Therefore, a clearer distribution of local misorientation could be obtained in Fig. 1(b) compared with that of Fig. 1(a). Fig. 3 shows the distribution of local misorientation in Fig. 1(a) together with a regression curve optimized using the log-normal
Fig. 3 – Distribution of local misorientations (same data as in Fig. 1 (a)).
where N is the number of data. It should be noted that only grains consisting of more than 10 points were included in the calculation; smaller grains were ignored. The local misorientation distribution seems to be well-represented by a lognormal distribution. This was the same for Fig. 1(b) and other measurements made in this study. The change in averaged local misorientation with step size, d, is shown in Fig. 4. Since the change in crystal orientation depends on the step size, the averaged local misorientation increased with the step size almost linearly. The averaged local misorientation obtained by the coarse pixel condition was larger than that by the fine pixel condition. The difference between the two conditions can be explained by Fig. 5, which schematically shows the influence of the error in crystal orientation measurement on local misorientation. The error in crystal orientation measurement exists regardless of the step size, and brings about an error in local misorientation. If the local misorientation is large enough, the influence of the error in the averaged local misorientation becomes insignificant due to the averaging effect. However, in the case of small local misorientation, the misorientation angle is smaller than the error, and so the averaged local misorientation becomes larger than that of the real misorientation, because the misorientation angle is an absolute value. Furthermore, it was pointed out that the accuracy of misorientation identification is better for larger misorientation angle [12]. As shown in Fig. 3, small local misorientations were included in the calculation of the averaged local misorientation and they raised the averaged local misorientation. This was significant for smaller step size
Fig. 4 – Change in averaged local misorientation with step size (plastic strain: ɛp = 4.9%).
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Fig. 5 – Schematic figure representing influence of error in misorientation measurement (N: number of data).
because the ratio of small local misorientation was large. Hence, the increase in the averaged local misorientation by the coarse pixel condition became larger for small step size due to the accumulation of errors in the measurements. The error in measurement affects the local misorientation distribution shown in Fig. 1. Due to the error, the area of the white region (small local misorientation) is relatively small and unclear in coarse pixel data (Fig. 1(a)). Since misorientation becomes small as the spatial resolution of measurement increases, it is important to reduce the error in local misorientation measurement and to exclude the influences of the condition of misorientation identification.
4.
Domain Averaging Method
In general, errors in measurement can be reduced by averaging several data. However, as shown in Fig. 5, the averaging of local
misorientation does not always reduce the error. Therefore, by taking the average of measured crystal orientation, the accuracy of crystal orientation measurement and evaluation of local misorientation was improved. Fig. 6 shows a schematic drawing of the data acquisition and processing procedure referred to as the Domain Averaging Method (DAM). The crystal orientations were obtained as an average of several measurements for each domain, of which interval is d. The number of measurement points included in one domain is represented by RA. In the case of RA= 4, 16 crystal orientations in total are measured to obtain one crystal orientation used for local misorientation calculation. The average of crystal orientations is calculated using a set of quaternion [13]. If grain boundaries existed in the domain, some measured data are discarded as shown in Fig. 7. Misorientations larger than 5° are regarded as grain boundaries. Fig. 8 shows the relationship between the averaged local misorientation and step size for different RA. The averaged local misorientation decreased as RA increased. This was brought about by the reduction in error of crystal orientation
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Fig. 8 – Relationship between averaged local misorientation and step size obtained under different value of RA (plastic strain: ɛp = 4.9%).
Fig. 6 – Schematic drawing of averaging technique for crystal orientation map (Domain Averaging Method: DAM).
measurement; the rate of the reduction seemed to be saturated at RA= 4. The change in averaged local misorientation with RA is shown in Fig. 9 for step size of 1.0 μm. The reduction in averaged local misorientation was larger for coarse pixel measurement than for fine pixel measurement. Although larger RA gives a greater improvement in accuracy, it requires much time for measurement, as measurements have to be repeated RA×RA times in order to apply DAM of RA. RA= 2 seems to be appropriate when crystal orientation measurements are made using the fine pixel condition.
Distributions of the local crystal orientation obtained using DAM of RA= 4 are shown in Fig. 10. By applying DAM, the distribution becomes clearer than that shown in Fig. 1 and almost the same regardless of the number of pixels. This means that DAM enables us to obtain well-converged precise local misorientation irrespective of the accuracy of crystal orientation measurement. In Fig. 9, data obtained from the 0% strained sample is also shown. Even in unstrained material, there is some misorientation; the water quenching during the heat treatment and stress due to the machining process may have caused small plastic strains. The averaged local misorientation of the 0% strained sample decreased by applying DAM, and the converged value was almost 0.1°.
5. Change in Local Misorientation by Deformation Fig. 11 shows the local misorientation along a straight line, which crosses grain boundaries, obtained from Fig. 10(b). The
Fig. 7 – Treatment of grain boundary for DAM (in case of RA= 4).
Fig. 9 – Change in averaged local misorientation with RA (step size: d = 1 μm).
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Fig. 12 – Schematic drawing for representing relationship between deformation of material, accumulation of dislocations, and evolution of misorientation.
Fig. 10 – Distribution of local misorientation obtained by DAM of RA = 4 (plastic strain: ɛp = 4.9%, step size: d = 1 μm). (a) Coarse pixel condition (b) fine pixel condition.
distribution of misorientations was inhomogeneous and different grain by grain. It became more than 0.9° at the maximum, although the averaged value was Mave =0.273°. Especially, the misorientation tended to be large near grain boundaries. By plastic deformation, as schematically shown in Fig. 12, dislocations are initiated and move along crystallographic slip planes, then pile up near grain boundaries and form so-called geometrically necessary dislocations. The large misorientation near the grain boundaries was inferred to be caused by such dislocations. Fig. 13 shows the relationship between the averaged local misorientation and step size for each sample obtained by DAM of RA = 2. The magnitude of the local misorientations was dependent on plastic strain as well as the step size of the crystal orientation measurements. The averaged misorientation and step size did not always show a linear correlation. In cases of large strain, such as 19.7% plastic strain, the increase in the misorientation slowed as the step size increased. The threshold angle for grain boundaries was 5° and the local misorientation exceeded the threshold angle locally. The large
Fig. 11 – Misorientation along a line obtained from Fig. 10b.
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local misorientation was taken as grain boundaries and was not considered in the calculation of the averaged misorientation. This caused the change in inclination of the relationship between the averaged local misorientation and step size at large plastic strain. In summary, the local misorientation correlated with the step size as well as the magnitude of the macroscopic plastic strain. Therefore, by quantifying the relationship between these three parameters, we can estimate the degree of local plastic strain.
6. Evaluation of Plastic Strain from Local Misorientation Fig. 14 shows the relationship between the nominal plastic strain, εp, and the averaged local misorientation for each step size d. The local misorientation was calculated based on crystal orientation obtained by DAM of RA = 2 using the fine pixel camera condition. The averaged local misorientation for unstrained conditions is set to Mave = 0.1° regardless of step size. Mave shows an excellent correlation with the plastic strain and the relation is almost linear under the plastic strain of 10%. A linear regression of the data under 10% plastic strain leads to the following equation: ep ¼
Mave 0:1 ; 0:0027d2 þ 0:041d
ð4Þ
where the strain is given in percent and distance in μm. This relation is shown by solid lines in Fig. 14, and is expected to be valid up to 15% plastic strain when the step size is less than 3 μm. For the estimation of local plastic strain, Eq. (4) is modified as: epðlocalÞ ¼
ML 0:1 : 0:0027d2 þ 0:041d
ð5Þ
By substituting measured local misorientation for ML in Eq. (5), the local value of plastic strain, εp(local), can be derived. For example, when the step size and the local misorientation are d = 1.0 μm and ML = 0.7°, the local plastic strain is estimated as εp (local) = 15.7%. The relationship between the local misorientation
Fig. 13 – Change in averaged local misorientation with step size obtained by DAM of RA = 2.
Fig. 14 – Relationship between averaged local misorientation and macroscopic plastic strain obtained by DAM of RA = 2 (solid lines correspond to the equation for each step size d).
and plastic strain was almost linear under plastic strain of 15% when the step size was d = 1.0 μm. Therefore, the distribution of the local plastic strain can be obtained by taking ML = 0.7° as εp = 15.7% in Figs. 10 and 11. Although the applied macroscopic plastic strain was 4.9%, it was more than 15% locally. It should be noted that the estimated plastic strain does not correspond to the nominal plastic strain. As mentioned, the local misorientation correlates with the geometrically necessary dislocations rather than the magnitude of deformation. The estimated local plastic strain just shows the typical local misorientation that is observed under the plastic strain. The local plastic strain (misorientation) has a large dispersion, and is determined not only by applied plastic strain but also by geometry of grain structure, crystal orientation and so on. The procedure for measuring the local plastic strain can be summarized as follows: 1. Carefully measure the crystal orientation by EBSD (using fine pixel CCD camera images and well-prepared samples). 2. Apply DAM in order to reduce the unsolved error in crystal orientation measurement.
Fig. 15 – Change in the Modified Crystal Deformation (MCD) with step size (open symbol: fine pixel with DAM of RA = 2, close symbol: coarse pixel without DAM).
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8.
Fig. 16 – Change in the Modified Crystal Deformation (MCD) with macroscopic plastic strain.
3. Estimate the plastic strain from the local misorientation using the relationship shown in Fig. 14 or Eq. (4).
7.
Confirmation of Validity of Measurement
It is important to know whether the measurement conditions are adequate for evaluating local misorientation, or whether the error in misorientation measurement has been sufficiently converged by applying DAM. The validity of the measured data can be confirmed by referring to macroscopic strain. Fig. 15 shows the change in MCD, which is a parameter for the macroscopic plastic strain measurement [9], with step size. MCD is the averaged misorientation from a specific orientation referred to as the central orientation assigned to each grain, and only weakly depends on the step size of EBSD measurement as shown in Fig. 15. Since the misorientation from the central orientation is relatively large compared to the local misorientation, accuracy of the EBSD measurement has little influence on MCD as shown in Fig. 5. Therefore, the MCD obtained using the coarse pixel condition was almost identical with that obtained by the fine pixel condition augmented by applying DAM of RA = 2. The relationship between MCD and the macroscopic plastic strain is shown in Fig. 16, and can be approximated by the following equation: MCD ¼ 0:21ep þ 0:2:
ð6Þ
The relationship is almost linear below the plastic strain of 10% regardless of the step size, and is expected to be the same for different EBSD measurement conditions. Therefore, we can confirm the validity of measurement by comparing the macroscopic plastic strains obtained by Eqs. (4) and (6). As shown in Figs. 8 and 9, the parameter Mave decreases as the accuracy of measurement increases. If the accuracy of the measurement is worse than that for Fig. 14, the plastic strain obtained by Eq. (4) becomes larger than that by Eq. (6). By applying DAM with a large value of RA, the plastic strain given by Eq. (4) is expected to converge to that given by Eq. (6) regardless of the measurement conditions.
Conclusions
In order to quantify the local plastic strain induced in Type 316 stainless steel, EBSD in conjunction with SEM was applied. It was shown that DAM reduces the error in misorientation and enables us to obtain a clear distribution of misorientations. The distribution of misorientations followed a log-normal distribution and its mean value correlated well with the macroscopic plastic strain induced in the specimens. By using the correlation between the misorientation and the plastic strain, the distribution of local plastic strain can be estimated. It was shown that the plastic strain is more than 15% locally under a macroscopic strain of 4.9%. The local plastic strain tends to be especially large near grain boundaries. In order to measure the local plastic strain, it is important to measure the crystal orientation carefully. A method of confirming the accuracy of misorientation identification was presented.
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