M A T E R I A L S C H A R A C T E R I Z A T I O N 6 0 ( 2 0 0 9 ) 1 2 5 – 1 3 2

Measurement of local plastic strain distribution of stainlesssteel by electron backscatter diffraction

Masayuki Kamaya⁎

Institute of Nuclear Safety System, Inc., 64 Sata, Mihama-cho, Mikata-gun, Fukui 919-1205, Japan

A R T I C L E D A T A

⁎ Tel.: +81 770 379114; fax: +81 770 372009.E-mail address: kamaya@inss.co.jp.

1044-5803/$ – see front matter © 2008 Elsevidoi:10.1016/j.matchar.2008.07.010

A B S T R A C T

Article history:Received 23 June 2008Accepted 24 July 2008

Electron backscatter diffraction in conjunction with scanning electronmicroscopy was usedto assess the plastic strain on a microstructural scale (local plastic strain) induced instainless steel deformed up to a nominal strain of 19.7%. Accuracy of the measurement ofmisorientations was improved by a technique called the Domain Averaging Method (DAM),inwhich an average of crystal orientationwas calculated for several datameasured from thesame domain. It was shown that the misorientation evaluated using the crystal orientationof which accuracy was improved by DAM showed localized plastic strain in the vicinity ofgrain boundaries (GB). The distribution of misorientations followed a log-normaldistribution and the mean value correlated well with the macroscopic plastic straininduced. By using the correlation between the misorientation and the plastic strain, thedistribution of local plastic strain could be quantified. It was shown that the plastic strainbecomes more than 15% locally under a macroscopic strain of 4.9%. A procedure forconfirming the accuracy of the measurement is also suggested.

© 2008 Elsevier Inc. All rights reserved.

Keywords:Electron backscatteringdiffraction (EBSD)Stainless steelPlastic deformationCold workingMisorientation

1. Introduction

Plastic strain accelerates material degradation by stresscorrosion cracking [1,2]. The magnitude of the degradation ischaracterized by the initiation of small cracks and theirgrowth [3]. In order to understand how the plastic strainaccelerates the initiation of small cracks, it is important toknow the magnitude of plastic strain on a microstructuralscale (hereafter, local plastic strain). Even if the macroscopicstrain appears uniform and homogeneous, local strain ofpolycrystalline material is inhomogeneous due to the aniso-tropy of crystal grains and their random or nearly randomorientation distribution [4–6]. On a microstructural scale,plastic deformation causes crystallographic slip and thegeometrically necessary dislocations. The crystal orientationis changed due to the dislocations and may show fluctuationsof several degrees even in the same grain.

Electron backscatter diffraction (EBSD), in conjunction withscanning electron microscopy (SEM), is one of the most

er Inc. All rights reserved

promising techniques for measuring the change in local crystalorientation. By using commercially available equipment forEBSD measurement, we can identify crystal orientations byscanning the surface of samples. It has been shown that scalarparameters obtained from crystal orientations of the scannedarea correlate with themagnitude ofmacroscopic plastic straininduced in materials [7–9]. Therefore, by using this correlation,the macroscopic plastic strain can be estimated from crystalorientations obtained by EBSD measurements. On the otherhand, by evaluating the misorientation angle between neigh-boring points (hereafter, local misorientation), it is possible toknow the magnitude of local plastic strain. In a previous studyby the present author [10], the spatial distribution of the localmisorientation was compared with that of nominal strainmeasured by the image correlation technique, and it wasrevealed that the local misorientation was consistent with thedensity of the geometrically necessary dislocations rather thanwith the magnitude of nominal strain, which is defined as thedeformation per unit length. In order to evaluate the effect of

.

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

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plastic strain on cracking behavior, the density of dislocations ismore 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 exam-ple, the surface condition of samples affects the quality of theEBSD patterns. Deterioration of the diffraction pattern reducesthe accuracy of crystal orientation identification. The plasticstrain deteriorates the EBSD pattern even with well-preparedsurface [11]. The number of pixels of the CCD camera used toacquire EBSD patterns, and parameters for identification ofcrystal orientation from obtained EBSD patterns, also affectthe accuracy of measurements. Therefore, the misorientationcalculated from crystal orientations contains substantial errorwhen the measured misorientation is relatively small [12].Therefore, it is difficult to measure precise misorientationswhen the spatial resolution of the measurements is fine orinduced plastic strain is small. For quantitative measure-ments, the influence of the error should be excluded and theaccuracy of misorientation identification must be improved.

In this study, a procedure for measurement of local plasticstrain was developed. The material used was Type 316stainless steel, in which plastic strain was induced by tensileloads up to a nominal plastic strain of 19.7%. The localdistribution of misorientations was identified from crystalorientations obtained using EBSD. In order to improve theaccuracy of misorientation identification, a technique for dataprocessing was applied in addition to careful measurement ofcrystal orientation. Then, the local plastic strain was quanti-fied using the correlation between the misorientation andinduced plastic strain.

2. Experimental Procedure

Fig. 1 –Distribution of local misorientation (plastic strain:εp=4.9%, step size: d=1μm). (a) Coarse pixel condition (b) finepixel condition.

The material used for these studies was a solution heat-treated Type 316 austenitic stainless steel, whose alloyingconstituents are shown in Table 1. Plate tensile specimens(gauge length=20 mm and cross section=2×4 mm) weremachined and deformed by tensile loading to six nominalglobal plastic strains, εp, of 0%, 2.8%, 4.9%, 10.3%, 15.2% and19.7%. The deformation rate was 1.0 mm/min at the cross-head of the tensile test machine and the strain was defined bythe change in distance between indentation marks measuredby a traveling microscope. After the deformation, mid-planesections in the region of uniform strain were prepared forEBSD measurement. The surface was polished up to 3 μmdiamond paste followed by colloidal silica in order to achieverelatively flat surfaces free from damage. The material had anapproximately equiaxed grain structure. EBSD measurementssampling large numbers of grains indicated that the crystal-lographic texture was very weak.

Crystal orientation measurements by EBSD were madewith an orientation imaging microscope interfaced to a fieldemission electron gun SEM. The step size of the measure-ments was 0.25 μm at minimum.

3. Local Misorientation Distribution

Fig. 1 shows the local misorientation, ML, calculated by thefollowing equation:

ML poð Þ ¼ 14

X4i¼1

b po;pið Þ ð1Þ

where β(po, pi) denotes the misorientation between a fixed pointpo and neighboring points pi in the grain as shown in Fig. 2.Misorientations larger than5°were regardedasgrainboundaries.Two maps of the local misorientation were obtained underdifferent conditions of CCD camera for EBSD pattern acquisitionunder the samestepsizeof 1.0μmfromthe specimenof εp=4.9%.

Fig. 2 –Definition of local misorientation.

127M A T E R I A L S C H A R A C T E R I Z A T I O N 6 0 ( 2 0 0 9 ) 1 2 5 – 1 3 2

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 itpossible to measure precise crystal orientation. Therefore, aclearer distribution of local misorientation could be obtained inFig. 1(b) compared with that of Fig. 1(a).

Fig. 3 shows thedistributionof localmisorientation inFig. 1(a)togetherwith a regression curve optimized using the log-normal

Fig. 3 –Distribution of local misorientations (same data as inFig. 1 (a)).

distribution, forwhich theprobabilitydensity function isdefinedby:

f MLð Þ ¼ 1lnSð ÞML

ffiffiffiffiffiffi2p

p exp � 12

lnML � lnMave

lnS

� �2" #

ð2Þ

where ML and S are the local misorientation and standarddeviation, respectively. Mave is the mean value of thedistribution and is calculated by the following equation:

Mave ¼ exp1N

XNi¼1

ln ML pið Þf g" #

ð3Þ

where N is the number of data. It should be noted that onlygrains consisting of more than 10 points were included in thecalculation; smaller grains were ignored. The local misorien-tation distribution seems to be well-represented by a log-normal distribution. This was the same for Fig. 1(b) and othermeasurements made in this study.

The change in averaged local misorientation with step size,d, is shown in Fig. 4. Since the change in crystal orientationdepends on the step size, the averaged local misorientationincreased with the step size almost linearly. The averaged localmisorientationobtainedby thecoarsepixel conditionwas largerthan that by the finepixel condition.Thedifferencebetween thetwo conditions can be explained by Fig. 5, which schematicallyshows the influence of the error in crystal orientationmeasure-ment on local misorientation. The error in crystal orientationmeasurementexists regardlessof the stepsize, andbringsaboutan error in local misorientation. If the local misorientation islarge enough, the influence of the error in the averaged localmisorientation becomes insignificant due to the averagingeffect. However, in the case of small local misorientation, themisorientation angle is smaller than the error, and so theaveraged local misorientation becomes larger than that of thereal misorientation, because the misorientation angle is anabsolute value. Furthermore, it was pointed out that theaccuracy of misorientation identification is better for largermisorientation angle [12]. As shown in Fig. 3, small localmisorientations were included in the calculation of the aver-aged local misorientation and they raised the averaged localmisorientation. This was significant for smaller step size

Fig. 4 –Change in averaged local misorientation with stepsize (plastic strain: ɛp=4.9%).

Fig. 5 –Schematic figure representing influence of error in misorientation measurement (N: number of data).

128 M A T E R I A L S C H A R A C T E R I Z A T I O N 6 0 ( 2 0 0 9 ) 1 2 5 – 1 3 2

because the ratio of small localmisorientationwas large. Hence,the increase in the averaged local misorientation by the coarsepixel condition became larger for small step size due to theaccumulation of errors in the measurements.

The error in measurement affects the local misorientationdistribution shown in Fig. 1. Due to the error, the area of thewhite region (small local misorientation) is relatively smalland unclear in coarse pixel data (Fig. 1(a)). Since misorienta-tion becomes small as the spatial resolution of measurementincreases, it is important to reduce the error in localmisorientation measurement and to exclude the influencesof the condition of misorientation identification.

4. Domain Averaging Method

In general, errors in measurement can be reduced by averagingseveral data. However, as shown in Fig. 5, the averaging of local

misorientation does not always reduce the error. Therefore, bytaking the averageofmeasuredcrystal orientation, the accuracyof crystal orientation measurement and evaluation of localmisorientationwas improved. Fig. 6 shows a schematic drawingof the data acquisition and processing procedure referred to asthe Domain Averaging Method (DAM). The crystal orientationswere obtained as an average of several measurements for eachdomain, of which interval is d. The number of measurementpoints included in one domain is represented by RA. In the caseof RA=4, 16 crystal orientations in total are measured to obtainonecrystal orientationused for localmisorientation calculation.The average of crystal orientations is calculated using a set ofquaternion [13]. If grain boundaries existed in thedomain, somemeasured data are discarded as shown in Fig. 7. Misorientationslarger than 5° are regarded as grain boundaries.

Fig. 8 shows the relationship between the averaged localmisorientation and step size for differentRA. The averaged localmisorientation decreased as RA increased. This was broughtabout by the reduction in error of crystal orientation

Fig. 6 –Schematic drawing of averaging technique for crystalorientation map (Domain Averaging Method: DAM).

Fig. 8 –Relationship between averaged local misorientationand step size obtained under different value of RA (plasticstrain: ɛp=4.9%).

129M A T E R I A L S C H A R A C T E R I Z A T I O N 6 0 ( 2 0 0 9 ) 1 2 5 – 1 3 2

measurement; the rate of the reduction seemed to be saturatedatRA=4. The change in averaged localmisorientationwithRA isshown in Fig. 9 for step size of 1.0 μm.The reduction in averagedlocal misorientation was larger for coarse pixel measurementthan for fine pixel measurement. Although larger RA gives agreater improvement in accuracy, it requires much time formeasurement, as measurements have to be repeated RA×RAtimes in order to apply DAM of RA. RA=2 seems to be appro-priate when crystal orientation measurements are made usingthe fine pixel condition.

Fig. 7 –Treatment of grain boundary for DAM (in case of RA=4).

Distributions of the local crystal orientation obtained usingDAM of RA=4 are shown in Fig. 10. By applying DAM, thedistribution becomes clearer than that shown in Fig. 1 andalmost the same regardless of the number of pixels. Thismeansthat DAM enables us to obtain well-converged precise localmisorientation irrespectiveof theaccuracyofcrystal orientationmeasurement. In Fig. 9, data obtained from the 0% strainedsample is also shown. Even in unstrained material, there issome misorientation; the water quenching during the heattreatment and stress due to the machining process may havecaused small plastic strains. The averaged local misorientationof the 0% strained sample decreased by applying DAM, and theconverged value was almost 0.1°.

5. Change in Local Misorientationby Deformation

Fig. 11 shows the localmisorientationalonga straight line,whichcrosses grain boundaries, obtained from Fig. 10(b). The

Fig. 9 –Change in averaged local misorientation with RA (stepsize: d=1 μm).

Fig. 10 –Distribution of local misorientation obtained by DAMof RA=4 (plastic strain: ɛp=4.9%, step size: d=1 μm).(a) Coarse pixel condition (b) fine pixel condition.

Fig. 11 –Misorientation along a

Fig. 12 –Schematic drawing for representing relationshipbetween deformation of material, accumulationof dislocations, and evolution of misorientation.

130 M A T E R I A L S C H A R A C T E R I Z A T I O N 6 0 ( 2 0 0 9 ) 1 2 5 – 1 3 2

distribution of misorientations was inhomogeneous and differ-ent grain by grain. It became more than 0.9° at the maximum,although the averaged value was Mave=0.273°. Especially, themisorientation tended to be large near grain boundaries. Byplastic deformation, as schematically shown in Fig. 12, disloca-tions are initiated and move along crystallographic slip planes,then pile up near grain boundaries and form so-called geome-trically necessary dislocations. The largemisorientation near thegrain boundaries was inferred to be caused by such dislocations.

Fig. 13 shows the relationship between the averaged localmisorientation and step size for each sample obtained by DAMof RA=2. The magnitude of the local misorientations wasdependent on plastic strain as well as the step size of thecrystal orientation measurements. The averaged misorienta-tion and step size did not always show a linear correlation. Incases of large strain, such as 19.7% plastic strain, the increasein the misorientation slowed as the step size increased. Thethreshold angle for grain boundaries was 5° and the localmisorientation exceeded the threshold angle locally. The large

line obtained from Fig. 10b.

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local misorientation was taken as grain boundaries and wasnot considered in the calculation of the averaged misorienta-tion. This caused the change in inclination of the relationshipbetween the averaged local misorientation and step size atlarge plastic strain. In summary, the local misorientationcorrelated with the step size as well as the magnitude of themacroscopic plastic strain. Therefore, by quantifying therelationship between these three parameters, we can estimatethe degree of local plastic strain.

Fig. 14 –Relationship between averaged local misorientationand macroscopic plastic strain obtained by DAM of RA=2(solid lines correspond to the equation for each step size d).

6. Evaluation of Plastic Strain fromLocal Misorientation

Fig. 14 shows the relationship between the nominal plasticstrain, εp, and the averaged local misorientation for each stepsize d. The local misorientation was calculated based oncrystal orientation obtained by DAM of RA=2 using the finepixel camera condition. The averaged local misorientation forunstrained conditions is set to Mave=0.1° regardless of stepsize. Mave shows an excellent correlation with the plasticstrain and the relation is almost linear under the plastic strainof 10%. A linear regression of the data under 10% plastic strainleads 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. Thisrelation is shown by solid lines in Fig. 14, and is expected to bevalid up to 15% plastic strain when the step size is less than3 μm. For the estimation of local plastic strain, Eq. (4) ismodified as:

epðlocalÞ ¼ML � 0:1

�0:0027d2 þ 0:041d: ð5Þ

By substituting measured local misorientation for ML inEq. (5), the local valueofplastic strain, εp(local), canbederived. Forexample, when the step size and the local misorientation ared=1.0 μmandML=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 stepsize obtained by DAM of RA=2.

and plastic strain was almost linear under plastic strain of 15%when the step size was d=1.0 μm. Therefore, the distribution ofthe local plastic strain can be obtained by taking ML=0.7° asεp=15.7% in Figs. 10 and 11. Although the applied macroscopicplastic strain was 4.9%, it was more than 15% locally.

It should be noted that the estimated plastic strain does notcorrespond to the nominal plastic strain. As mentioned, thelocal misorientation correlates with the geometrically neces-sary dislocations rather than the magnitude of deformation.The estimated local plastic strain just shows the typical localmisorientation that is observed under the plastic strain. Thelocal plastic strain (misorientation) has a large dispersion, andis determined not only by applied plastic strain but also bygeometry of grain structure, crystal orientation and so on.

The procedure for measuring the local plastic strain can besummarized as follows:

1. Carefully measure the crystal orientation by EBSD (usingfine pixel CCD camera images and well-prepared samples).

2. Apply DAM in order to reduce the unsolved error in crystalorientation 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).

Fig. 16 –Change in the Modified Crystal Deformation (MCD)with macroscopic plastic strain.

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3. Estimate the plastic strain from the local misorientationusing the relationship shown in Fig. 14 or Eq. (4).

7. Confirmation of Validity of Measurement

It is important to know whether the measurement conditionsare adequate for evaluating local misorientation, or whetherthe error in misorientation measurement has been suffi-ciently converged by applying DAM. The validity of themeasured data can be confirmed by referring to macroscopicstrain. Fig. 15 shows the change in MCD, which is a parameterfor the macroscopic plastic strain measurement [9], with stepsize. MCD is the averaged misorientation from a specificorientation referred to as the central orientation assigned toeach grain, and only weakly depends on the step size of EBSDmeasurement as shown in Fig. 15. Since the misorientationfrom the central orientation is relatively large compared to thelocal misorientation, accuracy of the EBSD measurement haslittle influence on MCD as shown in Fig. 5. Therefore, the MCDobtained using the coarse pixel condition was almost identicalwith that obtained by the fine pixel condition augmented byapplying DAM of RA=2. The relationship between MCD andthe macroscopic plastic strain is shown in Fig. 16, and can beapproximated by the following equation:

MCD ¼ 0:21ep þ 0:2: ð6ÞThe relationship is almost linear below the plastic strain of

10% regardless of the step size, and is expected to be the samefor different EBSDmeasurement conditions. Therefore, we canconfirm the validity of measurement by comparing themacroscopic plastic strains obtained by Eqs. (4) and (6). Asshown in Figs. 8 and 9, the parameter Mave decreases as theaccuracy of measurement increases. If the accuracy of themeasurement is worse than that for Fig. 14, the plastic strainobtained by Eq. (4) becomes larger than that by Eq. (6). Byapplying DAMwith a large value of RA, the plastic strain givenby Eq. (4) is expected to converge to that given by Eq. (6)regardless of the measurement conditions.

8. Conclusions

In order to quantify the local plastic strain induced in Type 316stainless steel, EBSD in conjunction with SEM was applied. Itwas shown that DAM reduces the error in misorientation andenables us to obtain a clear distribution of misorientations.The distribution of misorientations followed a log-normaldistribution and its mean value correlated well with themacroscopic plastic strain induced in the specimens. By usingthe correlation between the misorientation and the plasticstrain, the distribution of local plastic strain can be estimated.It was shown that the plastic strain is more than 15% locallyunder a macroscopic strain of 4.9%. The local plastic straintends to be especially large near grain boundaries.

In order to measure the local plastic strain, it is importantto measure the crystal orientation carefully. A method ofconfirming the accuracy of misorientation identification waspresented.

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