E-Journal of Advanced Maintenance Vol. 10 No. 4 (2019) 9-17
Japan Society of Maintenology

Intergranular Strains of Plastically Deformed Austenitic Stainless Steel
Kenji SUZUKI1* and Takahisa Shobu2
1 Niigata University, 8050, Ikarashi-2-no-cho, Nishi-ku, Niigata, 950-2181, Japan
2 Material Sciences Research Center, JAEA, Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan

ABSTRACTA plate specimen of austenitic stainless steel was prepared, and a plastic strain of 2.8% was applied to the specimen under uniaxial tension. The residual stresses of the specimen were measured by the cos2 χ method using
multiple lattice planes. For lattice planes with a lower atomic density, the 2 θ -cos2 χ diagram showed an upward
warp. For lattice planes with a higher atomic density, the 2 θ -cos2 χ diagram showed a downward warp. Thisphenomenon is caused by intergranular strains. The strain distribution for each lattice plane was measured in the
direction from 0◦ to 180◦ with a step of 5◦ . As a result, the strain of the lattice plane for low atomic density
balanced that for high atomic density. The intergranular strain depends on the lattice plane and the azimuth. Theintergranular strain due to plastic deformation affects the X-ray stress measurement.

ARTICLE INFORMATIONKEYWORDS

Articlehistory:intergranular
austenitic
stainlesssteel,
plastic
deformation,
elastic
anisotropy, X-ray
intergranularstrain,strain;micro-residual
micro-residualstress,
stress;austenitic
stainlesssteel;
plasticstressmeasurement
Received 7 May 2018
deformation;
elastic anisotropy; X-ray stress measurement
Accepted 24 January 2019

1. Introduction

Residual stress is classified into Types I, II, and III [1]. Type II stress is caused by differences in crystal
properties. In multi-phase materials, Type II stress arises because of inclusions or binary phases; this iscalled phase stress. Although there is no difference in crystal properties in single-phase materials, TypeII stress arises in these materials as well; this is called intergranular stress. Intergranular stress is caused
by elastic anisotropy.
Micro-residual stress by plastic deformation has been studied for many years. For example, Greenough measured the residual strains in plastically deformed iron wire using the 3 1 0, 2 1 1, and 1 1 0
lattice planes. He found a dependence of the residual strain on the lattice plane, and suggested that theinternal stresses arose mainly from the plastic anisotropy of the metal [2]. He had already used the term
”intergranular stress” in his work [3]. Recently, residual stress was measured using neutron diffraction. Holden et al. measured the residual strain distributions on a U-bent sample [4] and found that thestrains with the 1 1 1 and 0 0 2 lattice planes were quite unequal. To clarify the intergranular strain, thelattice strains were measured by neutron diffraction under tensile testing [5]. A tensile test of austenitic
stainless steel was carried out using neutron diffraction with the time-of-flight method. The strains thatwere measured by multiple h k l lattice planes demonstrated the dependence of the residual stress on thelattice plane; this behaviour was interpreted by a self-consistent model [6, 7]. Many researchers are still
studying intergranular stresses today.
Micro-residual stress does not arise in an elastic regime. However, micro-residual stress does arise
when the polycrystalline aggregate is plastically deformed and unloaded. Micro-residual stress is causedby plastic deformation and elastic anisotropy. Therefore, we must investigate micro-residual stress fromboth aspects, plastic deformation and elastic anisotropy.
The elastic anisotropic parameter, A, is defined by [8].
A=

2 c44
,
c11 − c12(1)

where ci j is the stiffness of a single crystal. Austenitic stainless steel has large elastic anisotropy. For
example, the parameter A for austenitic stainless steel is 3.26, where c11 = 206, c12 = 133, and c44 = 119
GPa [9]. On the other hand, the parameter A for aluminium is 1.23, where c11 = 106.8, c12 = 60.7, andc44 = 28.2 GPa [10]. Figure 1 (a) and (b) show the elastic anisotropy for austenitic stainless steel and* Corresponding author, E-mail: suzuki@ed.niigata-u.ac.jp

ISSN-1883-9894/10 © 2010 – JSM and the authors. All rights
9 reserved.

9

K. Suzuki, et al.
Intergranular Strains of Plastically Deformed Austenitic Stainless Steel
c
c

a
a

b

b

(a) Austenitic stainless steel(b) Aluminium

Fig. 1. Elastic anisotropy of polycrystalline aggregate.

aluminium, respectively. The length of the radius vector indicates the diffraction elastic constant Ehkl ,and the direction of the radius vector indicates the orientation of the h k l lattice plane. The diffraction
elastic constant Ehkl was calculated by the Kröner model [11]. As shown in the figure, aluminium issimilar to a spherical shape, but austenitic stainless steel shows a cubic shape. Large elastic anisotropy
is a feature of austenitic stainless steel that causes intergranular strain.
The authors have investigated the micro-residual stress of austenitic stainless steel due to plastic
deformation. It was found that the micro-residual stresses of plastically deformed austenitic stainlesssteel depended on the lattice plane [12]. The effects of the size of plastic deformation or the strain
rate were investigated by hard synchrotron X-rays using multiple lattice planes [13, 14]. The microresidual stresses were arranged with diffraction elastic constants calculated by the Kröner model. We
clarified that the intergranular stress correlates with the diffraction elastic constants. Previous studies
mainly focused on intergranular strains in the direction of a principal stress. Our understanding of
micro-residual stress is not sufficient. According to the results of our studies, elastic anisotropy of acrystal influences the micro-residual stresses in austenitic stainless steel, which is plastically deformed.
Therefore, it is necessary to know the relation between the elastic anisotropy and the micro-residual
stress. Especially, we should understand a balance of the micro-residual stresses in the omni-azimuth.
In this study, we applied plastic deformation to austenitic stainless steel under uniform tension. The
omni-azimuth strain profiles were measured using multiple lattice planes with a hard synchrotron X-ray.
The relationship of the intergranular strains between lattice planes is discussed.

2.

Analysis and Experiments

2.1. Material and test specimen

The material used in this study was the austenitic stainless steel SUS316L (equivalent to X2CrNiMo1712-2). The dimensions of the test specimen were thickness 3 mm, width 15 mm, and length 55 mm, as
shown in Fig. 2 (a). After machining, the surface of the specimen was polished by buffing and finally
was finished with electrode polishing. In order to release the residual strain, the specimen was annealed
at 913 K for 10 min. The proof stress of the specimen was 202 MPa, and the tensile strength was 523
MPa. The mean grain size of the specimen was 55 µ m. The test specimen was plastically deformed
under uniaxial tension with a tensile testing machine. The stress-strain curve is shown in Fig. 2 (b). The
cross-head speed was 0.2 mm/min (strain rate: 6.06 × 10−5 /s). The plastic strain ε p of the test specimen
measured after unloading was 2.8 %. We wanted to give the specimen uniform plastic deformation inorder to induce the micro-residual stress without preferred orientation. The plastic strain of 2.8% was
decided from such a reason.
The diffraction and mechanical elastic constants of the test specimen were calculated by the Krönermodel [11]. The stiffness ci j values were referred from other experimental data for SUS316 [9]. The
diffraction elastic constants are shown in Table 1. The web system was prepared to calculate the elastic
constants of arbitrary lattice plane (h k l) by the Kröner model [15].

10K. Suzuki, et al.
Intergranular Strains of Plastically Deformed Austenitic Stainless Steel400
SUS316L
L21S200

350
Stress , MPa

55

140

300

15

250
200
150
100

Strain rate 6.67x10-2 /s

50

30

0

t3

0

0.5

1

1.5

2

2.5

3

3.5

Strain , %(a) Test specimen(b) Stress-strain curve under uniaxial tension

Fig. 2. Experimental specimen and its plastic deformation.
Table 1. Elastic constants for austenitic stainless steel 316 calculated by the Kröner model.

E (GPa)
ν

311
182.4
0.306

400
144.1
0.347

331
217.4
0.269

440
209.7
0.277

620
164.9
0.325

Mechanical
194.7
0.294

z
z
Incident X-ray

Diffracted X-ray

Specimen
χ-Cradle

Fig. 3. Experimental setup for the 2 θ -cos2 χ method.
2.2. 2 θ -cos2 χ method

The residual stress in the plastically deformed specimen was measured by the cos2 χ method. Thismethod is suitable for stress measurement using high energy synchrotron X-rays in transmission. The
cos2 χ method has several advantages; for instance, the strain free lattice spacing is unnecessary, andwe can use a wide range of cos2 χ values from 0 to 1.
In the cos2 χ method, the diffraction angle 2 θ is measured by tilting the χ -cradle as shown in Fig. 3.
The stress can be evaluated from the change in the diffraction angle 2 θ with the change in the azimuth
χ . We assumed that the direction of principal stress corresponds to the z-axis. In the plane stress, thestrain εχ in the direction of the χ -angle is expressed as

εχ =

ν
1+ν
(σ1 cos2 χ + σ2 sin2 χ ) − (σ1 + σ2 ) ,
E
E

11

(2)

K. Suzuki, et al.
Intergranular Strains of Plastically Deformed Austenitic Stainless Steel

Fig. 4. Stress measurement using cos2 χ

where σ1 and σ2 are the principal stresses, and E and ν are diffraction elastic constants. The relation
between the strain εχ and the diffraction angle θχ is given byεχ = −(θχ − θ0 ) cot θ0 .

(3)Thus, we can obtain the following equation:2θχ − 2θ0 = −2 (1 + ν )2ν
tan θ0 (σ1 cos2 χ + σ2 sin2 χ ) +tan θ0 (σ1 + σ2 )
E
E

(4)

The stress σ2 is 0 for uniaxial tension, and the partial differentiation of Eq. (4) by cos2 χ becomes∂ 2 θχ
2 (1 + ν )=−tan θ0 σ1 .
2
∂ cos χE

(5)

The stress σ1 can be determined from the gradient of the 2 θ -cos2 χ diagram as follows:

σ1 = K · M ,(6)where K and M are given byK=−∂ 2 θχ
2 (1 + ν )tan θ0 , M =.
E∂ cos2 χ

2.3. X-ray measurements using hard synchrotron X-rays

The residual stresses were measured using hard synchrotron X-rays, because the diffraction curve had tobe measured with transmission for the cos2 χ method as shown in Fig. 4. The experiment was performedat the beam-line BL22XU in SPring-8, owned by the Japan Atomic Energy Agency (JAEA). The X-ray
energy was 66.40 keV, and the dimensions of the incident X-ray beam were height 1.0 mm and width
0.2 mm. The double slit was used as a receiving slit, the dimensions of which were height 1.0 mm
and width 0.2 mm. The diffraction curve was measured with a horizontal diffractometer. The values ofcos2 χ were from 0 to 1 with a step of 0.2. The 2 2 0, 4 0 0, 3 3 1, and 6 2 0 lattice planes of γ -Fe wereused.
An additional experiment was carried out to measure the detailed profile of the lattice strain foromni-azimuth χ . The beam-line was BL02B1 in SPring-8. The X-ray energy was 72.312 keV . The
dimensions of the divergent slit were height 0.5 mm and width 5 mm. The receiving slit was Soller slits.
The χ -cradle was tilted from 0◦ to 180◦ with a step of 5◦ , and the diffraction was measured successively
by a diffractometer in transmission. The lattice planes were 4 0 0, 3 3 1, 4 4 0, and 6 2 0 of γ -Fe.
12

K. Suzuki, et al.
Intergranular Strains of Plastically Deformed Austenitic Stainless Steel

Diffraction angle 2 , deg

152.6
SUS316L21S00-Fe 311 by Mn-K
152.4 K=-302 MPa/deg=-2.5 +/-26.3 MPa
152.2
152.0
151.8
151.6

0

0.1

0.2

0.3
sin

0.4

0.5

0.6
Tensile direction

2(a) 2 θ -sin ψ diagram before plastic deformation
2(b) Specimen surface after plastic deformation

Fig. 5. Initial residual stress before plastic deformation and specimen surface after plastic deformation.

Diffraction angle 2θ, deg

SUS316L, =18.660pm
εp=2.8%

400
220

620
331

18.96011.965620

400

18.95511.96018.95013.0458.460

13.040

8.455
220

331
8.450

13.035
0

0.2

0.4

0.6

0.8

1

0

cos2

0.2

0.4

cos

0.6

0.8

1

2

Fig. 6. 2 θ -cos2 χ diagrams for plastically deformed austenitic stainless steel.

3. Results and Discussion
3.1. cos2 χ diagrams of the plastically deformed specimen

The macro-residual stress in the direction of tensile axis was measured with the sin2 ψ method before
plastic deformation. The result is shown in Fig. 5 (a). The residual stress was measured using the 3 1 1
lattice plane with the Mn-K α characteristic X-ray. The gradient of the 2 θ -sin2 ψ diagram was horizontal, and there was no residual stress in the specimen. However, a small undulation was recognised in the
diagram.
The surface view of the specimen after plastic deformation is shown in Fig. 5 (b). The tensile axis
is the horizontal direction in the photograph. As shown in the photograph, slip bands appeared and thesurface became rough from the plastic deformation.
The micro-residual stresses were measured using the cos2 χ method. Figure 6 shows the 2 θ -cos2 χdiagram for each lattice plane. There should not be a macro-residual stress in the specimen becauseof unloading. Therefore, the relation between the diffraction angle 2 θ and cos2 χ should indicate ahorizontal straight line. However, the relations in the figure clearly show curvature. The cos2 χ diagramsby the 2 2 0 and 3 3 1 lattice planes indicate a downward warp. Oppositely, the cos2 χ diagrams by the
4 0 0 and 6 2 0 lattice planes indicate an upward warp. The feature of intergranular strain appears in the13

K. Suzuki, et al.
Intergranular Strains of Plastically Deformed Austenitic Stainless SteelSUS316L,p=2.8%=0.17183 Å

Strain0.0005

400
331

90

120

60
30

0

-0.00050(a) 4 0 0 and 3 3 1 lattice planes
SUS316L, εp =2.8%=0.17183 Å

Strain0.0005

440
90

120

620
60
30

0

-0.00050(b) 4 4 0 and 6 2 0 lattice planes

Fig. 7. Changes in lattice strains with orientation angle χ . The tensile direction is χ = 0◦ and180◦ .

warp of the diagram. The 2 2 0 and 3 3 1 lattice planes belong to a hard lattice plane. The 4 0 0 and the
6 2 0 lattice planes belong to a soft lattice plane. It is interesting that the lattice strain depends on theχ -angle in the austenitic stainless steel by plastic deformation under uniform tension. The warp of therelation of the 2 θ -cos2 χ diagram is caused by the intergranular strain rightly. The intergranular strainarises from the elastic anisotropy.
3.2.

Balance between micro-residual stresses

The result of Fig. 6 suggests that the micro-residual strain depends on the azimuth angle χ . To investigate the behaviour of the lattice strain with the change in χ -angle, the additional experiment describedin Section 2.3 was carried out. The lattice spacings were measured from χ = 0◦ to 180◦ with a stepof 5◦ , and we obtained the χ angle profiles of the lattice strains. In this study, the mean value of all
measured lattice constants was defined as the strain-free lattice constant a0 .
Figure 7 shows the χ -angle profile of each lattice strain. As shown in Fig. 7, the profile of the
micro-residual strain shows different dependence on the lattice plane. For the 4 0 0 lattice plane, thelattice strain becomes large in the direction of the tensile axis. The 4 0 0 lattice plane has a low atomic
density, and belongs to a soft lattice plane. For the 3 3 1 lattice plane, the lattice strain becomes large
in the direction of the principal shearing stress. The 3 3 1 lattice plane has a high atomic density, andbelongs to a hard lattice plane. The black solid marks in Fig. 7 (b) indicate the strain profile of the 4 4 0
lattice plane. Its profile is similar to that of the 3 3 1 lattice plane, and belongs to the hard plane. The
6 2 0 profile in Fig. 7 (b) contrasts with the 4 4 0 profile.
The elastic anisotropy of the crystal grain causes the micro residual stress due to plastic deformation.
It seems that the intergranular strain due to plastic deformation is determined by the interaction between
the hard and soft lattice planes. It is interesting that the hard lattice planes balance with the soft latticeplanes. The profile of the intergranular strain depends on the lattice plane and the azimuth angle χ .
Larsson et al measured the strain pole figures for the uniaxially compressed stainless steel using
neutron diffraction technique [16]. The measured residual intergranular strains indicated the χ -angle
dependence like us. However, the behaviour of intergranular strains were not obvious, because the pitch
14

K. Suzuki, et al.
Intergranular Strains of Plastically Deformed Austenitic Stainless Steel

BL02B1 in SPring-8

17.36017.355620
17.350

SUS316L, =17.183pm,

p

=2.8%

Diffraction angle 2θ, deg

15.520440
15.51515.51011.950331

11.945
11.94010.96010.955400
10.950
0.0

0.2

0.4

0.6

0.8

1.0

cos2

Fig. 8. Undulation of 2 θ -cos2 χ diagrams due to micro-residual stressesof the χ -angle was 15◦ in their experiment. The relation between the residual strain with the 311 latticeplane and the χ -angle undulated due to plastic deformation, though it seemed that the 311 lattice plane
didn’t influence the intergranular strain [17]. In the X-ray stress measurement of austenitic stainlesssteel, the 311 lattice plane is recommended [18]. Therefore, it is important to know detailes of theintergranular strains due to plastic deformation.
3.3. Influence of micro-residual stresses on X-ray stress measurement

As mentioned previously, micro-residual stress arises from the elastic anisotropy in plastically deformed
austenitic stainless steel. In addition, the lattice strain depends on the azimuth angle χ . This fact isimportant in X-ray stress measurement, because the X-ray stress includes the Type I and II residual
stresses, and we have no technique that distinguishes one from the other. Therefore, we must consider
the influence of the micro-residual stress on X-ray stress measurement.
Figure 8 shows the 2 θ -cos2 χ diagrams obtained from the additional experiment. The result of each2 θ -cos2 χ diagram in Fig. 8 is like that of Fig. 6. We must be able to reproduce the results in Fig. 6.
Because the diffraction angle was measured at a χ -angle from 0◦ to 180◦ with a step of 5◦ , the detailof the profile of the cos2 χ diagram was obtained. The diagrams in Fig. 8 show the true intergranularstrains. As shown in Fig. 8, the gradient of the cos2 χ diagram depends on the range of cos2 χ and thelattice plane. The result in Fig. 8 suggests that the residual stress cannot be determined from the gradientof the cos2 χ diagram for the plastically deformed polycrystalline aggregate. A problem like this also
happens in the sin2 ψ method.
For example, Fig 9 shows the residual stresses in austenitic stainless steels. The X-ray stresses inFig. 9 were determined using the 2 θ -cos2 χ method [12]. The range of the cos2 χ was from 0.5 to 1.0.
The specimens were plastically deformed under uniform tension; plastic strains of the specimens were
1% and 2%. Each micro-residual stress was summarised with the diffraction constant Ehkl , which was
15

K. Suzuki, et al.
Intergranular Strains of Plastically Deformed Austenitic Stainless SteelResidual stress hkl , MPa300
SUS316L, - Fe
= 71.637 keV
cos2 method

200 400
100

511

311

0

331
420

-100
-200
-300
140

=1%
=2%
p

422
220

p

160

Em
180

200

222
220

240

Diffraction elastic constant E hkl , GPa

Fig. 9. Micro-residual stresses of plastically deformed austenitic stainless steel measured by thecos2 χ method. The range of the value of cos2 χ is from 0.5 to 1.

calculated by the Kröner model.
The 4 0 0 lattice plane belongs to the small diffraction elastic constant in the figure. The gradientsof the 2 θ -cos2 χ diagrams in Fig. 8 are negative in the range of cos2 χ = 0.5 ∼ 1.0. As a result, theresidual stress becomes a tension. On the other hand, the 3 3 1 and 4 4 0 lattice planes belong to a large
elastic constant; these gradients in Fig. 8 are positive in the range of cos2 χ = 0.5 ∼ 1.0. Therefore,
the residual stresses by the 3 3 1 and 4 4 0 lattice planes become a compression. The relation of the
2 θ -cos2 χ diagram undulates because of the intergranular strain, and the intergranular strain affects the
X-ray stress measurement.
In plastically deformed austenitic stainless steel, large intergranular stress such as Type II stressarises from the elastic anisotropy. The X-ray diffraction includes the influence of the macro- and microresidual stresses. However, it is impossible to separate the Type I stress from the X-ray stress. We must
elucidate the micro-residual stress due to plastic deformation. In the future, the macro-residual stresswill be obtained from X-ray stress using an understanding of the intergranular strain.

4. Conclusions
In this study, a plastic strain of 2.8% was applied to a plate specimen of austenitic stainless steel
SUS316L under uniaxial tension. The micro-residual stresses were measured using the cos2 χ methodwith multiple lattice planes. The obtained results are as follows:(1) The relation of the 2 θ -cos2 χ diagrams was not a horizontal straight line, but showed warp. The
diagrams of soft lattice planes such as 4 0 0 or 6 2 0 showed upward warp, whereas the diagrams of hardlattice planes such as 3 3 1 or 4 4 0, in contrast, showed downward warp. The warp of the cos2 χ diagram
is caused by the intergranular strains. Therefore, the intergranular strain depends on the azimuth angleχ.
(2) The changes in the lattice strains were measured using the 4 0 0, 3 3 1, 4 4 0, and 6 2 0 lattice planes
while changing the azimuth angle χ . The tensile strain of the soft lattice planes became large in the
direction of the principal axis and became small in the direction of the principal shearing stress. In
contrast, the hard lattice planes showed tension in the direction of the principal shear stress and showed
compression in the direction of the tensile axis. The residual strain of the hard lattice planes balancedwith that of the soft lattice planes.
(3) In a polycrystalline aggregate with elastic anisotropy, micro-residual strain such as intergranularstrain arises from plastic deformation. The intergranular strain depends on the lattice plane and azimuth
angle χ . As a result, the cos2 χ diagram undulates by the intergranular strain. Therefore, the intergranular strain affects the X-ray stress measurement, because the X-ray stress is determined using the change
in lattice spacing with the change in the scattering vector.

16

K. Suzuki, et al.
Intergranular Strains of Plastically Deformed Austenitic Stainless SteelAcknowledgements
The authors would like to acknowledge financial support from the Japan Society for the Promotion of
Science, KAKENHI Grant (No. 17K06046). The synchrotron radiation experiment at the BL22XU
in SPring-8 was performed with the approval of the Japan Atomic Energy Agency (No. 2008AE12),and the experiment at BL02B1 was performed with the approval of the Japan Synchrotron Radiation
Research Institute (Proposal No. 2008A1766).

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