†

∗ • ∗∗ • ∗∗∗

Stress Measurement of Coarse Grains Using Double Exposure Method

by

Kenji SUZUKI∗, Takahisa SHOBU∗∗ and Ayumi SHIRO∗∗∗

The double exposure method (DEM) is proposed herein as a new X-ray stress measurement method for coarsegrain materials. A diffraction angle can be obtained from an incident and a spotty diffracted beam. Each X-ray beamis measured by an area detector on a linear motion stage on the 2 θ-arm in the DEM. To examine the validity of theDEM, the residual stress of the plastically bent specimen was measured. In addition, the residual stress distribution ofthe indentation specimen was measured. The result by the DEM was similar to the result simulated by the finite elementmethod. As a result, the DEM is useful for the X-ray stress measurement method for coarse grain material.

Key words:X-Ray stress measurement, Coarse grain, Double exposure method, Hard syn-chrotron X-ray, Residual stress

1

1)

2)

2

2 X

cosα 3) 2D 4) 5)

2

(DSTM)6) 2

DSTM

1)7) 2)

1)

2)

2

22 ·1

Fig. 1

O1,

O2 ℓ1 P1, P2 ℓ2

ℓ1, ℓ2 2 θ PC

O1, O2

2 P1 P2

Fig. 1 The principle of DEM (double exposure method).

† 30 8 24 Received Aug. 24, 2018. c⃝2019 The Society of Materials Science, Japan∗ 950-2181∗ Faculty of Education, Niigata University, Nishi-ku, Niigata, 950-2181.∗∗ 679-5148∗∗ Material Sciences Research Center, Japan Atomic Energy Agency, Sayo-gun, Hyogo, 679-5148.∗∗∗ 679-5148∗∗∗ Synchrotron Radiation Research Center, National Institutes for Quantum and Radiological Science and Technology, Sayo-gun, Hyogo, 679-5148.

2θ PC

Fig. 1

2 θ

2 2

O1 O2

2 θ P1, P2

4

2 θ

PC (DEM:

double exposure method)

2 ·2y

3 Fig. 2

P 1, P 2

P 1 = (x1, y1, z1), P 2 = (x2, y2, z2) . (1)

P1, P2 r1, r2 Fig. 2

r1 =√

x21 + z21 , r2 =

√x22 + z22 . (2)

2 θ

2θ = arctan

(r2 − r1y2 − y1

)(3)

ℓy O y

ℓy t

ℓy = (0, t, 0) (4)

ℓ P 1, P 2

ℓ = P 1 + t (P 2 − P 1) (5)

Fig. 3 ℓy ℓ

e =(x2 − x1, y2 − y1, z2 − z1)

∥LD∥ , eX = (0, 1, 0) (6)

PC

OX

0

Fig. 2 Geometry for DEM with hard synchrotron X-ray.

Fig. 3 Intersection of X-ray beam ℓx and diffracted beam ℓ.

e · −−−−→OXPC = e · (PC −OX) = 0 (7)

eX · −−−−→OXPC = eX · (PC −OX) = 0 (8)

Fig. 3 PC , OX

PC = P 1 − ℓ1 e , OX = ℓ0 eX (9)

(7), (8) ℓ0, ℓ1PC, OX

ℓ0 =eX · P 1 −

(e · P 1

)(e · eX)

1− (e · eX)2(10)

ℓ1 =e · P 1 −

(eX · P 1

)(e · eX)

1− (e · eX)2(11)

ℓ1 =cos 2θ

L sin2 2θ

[(x2 − x1)x1 + (z2 − z1) z1

](12)

ℓ0 = y1 −cot2 2θ

L

[(x2 − x1)x1 + (z2 − z1) z1

](13)

PC = (xC , yC , zC) (9), (12)

xC = x1 −cot2 2θ

L2

[(x2 − x1)x1

+(z2 − z1) z1](x2 − x1) (14)

yC = y1 −cot2 2θ

L

[(x2 − x1)x1

+(z2 − z1) z1]

(15)

zC = z1 −cot2 2θ

L2

[(x2 − x1)x1

+(z2 − z1) z1](z2 − z1) (16)

3DXRD8),9)

2

DEM

DEM

2 ·3DEM

SPring-8

BL22XU

2 θ 800 mm

｢材料｣ (Journal of the Society of Materials Science, Japan), Vol. 68, No. 4, pp. 312-317, Apr. 2019

論　　文

01-2018-0107-(p.312-317).indd 312 2019/03/27 10:56:50

†

∗ • ∗∗ • ∗∗∗

Stress Measurement of Coarse Grains Using Double Exposure Method

by

Kenji SUZUKI∗, Takahisa SHOBU∗∗ and Ayumi SHIRO∗∗∗

The double exposure method (DEM) is proposed herein as a new X-ray stress measurement method for coarsegrain materials. A diffraction angle can be obtained from an incident and a spotty diffracted beam. Each X-ray beamis measured by an area detector on a linear motion stage on the 2 θ-arm in the DEM. To examine the validity of theDEM, the residual stress of the plastically bent specimen was measured. In addition, the residual stress distribution ofthe indentation specimen was measured. The result by the DEM was similar to the result simulated by the finite elementmethod. As a result, the DEM is useful for the X-ray stress measurement method for coarse grain material.

Key words:X-Ray stress measurement, Coarse grain, Double exposure method, Hard syn-chrotron X-ray, Residual stress

1

1)

2)

2

2 X

cosα 3) 2D 4) 5)

2

(DSTM)6) 2

DSTM

1)7) 2)

1)

2)

2

22 ·1

Fig. 1

O1,

O2 ℓ1 P1, P2 ℓ2

ℓ1, ℓ2 2 θ PC

O1, O2

2 P1 P2

Fig. 1 The principle of DEM (double exposure method).

† 30 8 24 Received Aug. 24, 2018. c⃝2019 The Society of Materials Science, Japan∗ 950-2181∗ Faculty of Education, Niigata University, Nishi-ku, Niigata, 950-2181.∗∗ 679-5148∗∗ Material Sciences Research Center, Japan Atomic Energy Agency, Sayo-gun, Hyogo, 679-5148.∗∗∗ 679-5148∗∗∗ Synchrotron Radiation Research Center, National Institutes for Quantum and Radiological Science and Technology, Sayo-gun, Hyogo, 679-5148.

2θ PC

Fig. 1

2 θ

2 2

O1 O2

2 θ P1, P2

4

2 θ

PC (DEM:

double exposure method)

2 ·2y

3 Fig. 2

P 1, P 2

P 1 = (x1, y1, z1), P 2 = (x2, y2, z2) . (1)

P1, P2 r1, r2 Fig. 2

r1 =√

x21 + z21 , r2 =

√x22 + z22 . (2)

2 θ

2θ = arctan

(r2 − r1y2 − y1

)(3)

ℓy O y

ℓy t

ℓy = (0, t, 0) (4)

ℓ P 1, P 2

ℓ = P 1 + t (P 2 − P 1) (5)

Fig. 3 ℓy ℓ

e =(x2 − x1, y2 − y1, z2 − z1)

∥LD∥ , eX = (0, 1, 0) (6)

PC

OX

0

Fig. 2 Geometry for DEM with hard synchrotron X-ray.

Fig. 3 Intersection of X-ray beam ℓx and diffracted beam ℓ.

e · −−−−→OXPC = e · (PC −OX) = 0 (7)

eX · −−−−→OXPC = eX · (PC −OX) = 0 (8)

Fig. 3 PC , OX

PC = P 1 − ℓ1 e , OX = ℓ0 eX (9)

(7), (8) ℓ0, ℓ1PC, OX

ℓ0 =eX · P 1 −

(e · P 1

)(e · eX)

1− (e · eX)2(10)

ℓ1 =e · P 1 −

(eX · P 1

)(e · eX)

1− (e · eX)2(11)

ℓ1 =cos 2θ

L sin2 2θ

[(x2 − x1)x1 + (z2 − z1) z1

](12)

ℓ0 = y1 −cot2 2θ

L

[(x2 − x1)x1 + (z2 − z1) z1

](13)

PC = (xC , yC , zC) (9), (12)

xC = x1 −cot2 2θ

L2

[(x2 − x1)x1

+(z2 − z1) z1](x2 − x1) (14)

yC = y1 −cot2 2θ

L

[(x2 − x1)x1

+(z2 − z1) z1]

(15)

zC = z1 −cot2 2θ

L2

[(x2 − x1)x1

+(z2 − z1) z1](z2 − z1) (16)

3DXRD8),9)

2

DEM

DEM

2 ·3DEM

SPring-8

BL22XU

2 θ 800 mm

Ｐ1,Ｐ2

313二重露光法による粗大粒材の応力測定

01-2018-0107-(p.312-317).indd 313 2019/03/27 10:56:50

Fig. 4 Experiment for DEM.

2 PILATUS-300K

(Fig. 4) 2 θ

0◦ O1 = (0, y1, 0),

O2 = (0, y2, 0) Aℓ 331

2 θ 26◦

LD (= 791.889 mm) P1 P2

L0 = 707.989

mm L = 711.746 mm

30.034 keV (0.4127988A)

0.2×0.2mm2 PILATUS-300K

83.8 × 106.5 mm2 487 × 619

pixel 172 µm/pixel

4.75 mm 3 mm

(A5052)

623 K 1

27 µm Aℓ

331

331

2 θ0 = 25.62767◦ (= 93.06326 pm)

cij10) c11 =

106.78 GPa, c12 = 60.74 GPa, c44 = 28.21 GPa

Kroner 331

E = 71.39 GPa,

ν = 0.344

EM = 70.07 GPa νM = 0.347

33 ·1 DEM

Fig 5 DEM P1, P2

(a) P1

(b) P2

PLIATUS-300K 3

Fig 5

DEM P1 P2

(x1, z1) (x2, z2)

(a) P1 (b) P2

Fig. 5 Diffraction images measured by DEM. The box inthe figure (a) indicates the detected area at P2.

(a) P1 (b) P2

Fig. 6 Diffraction spot centers determined by the Demsys,which is matching program. Each diffraction centeris indicated with the cross mark.

Fig. 5

P1 P2

P1 P2

2 θ

PC(xC , yC , zC)

(Demsys) Fig. 6 P1

P2

2 θ

PC(xC , yC , zC)

Demsys

3 ·2DEM

Fig. 7

4.7 mm

3 mm

Fig. 7 Plastically bent specimen.

25.60

25.61

25.62

25.63

25.64

25.65

25.66

25.67

0 1 2 3 4 5

=0

=90

Diffr

action a

ngle

2, deg

Distance from tesile side x, mm

Bent specimen (#AM1, A5052)30.034keV, DEM, Al (331)

20=25.62767deg

Fig. 8 Results of plastically bent specimen measured byDEM.

0.1 mm 0.5 mm 9

1 mm 3

27 χ = 0◦

( ) χ = 90◦ ( ) 2

Fig. 8 2θ

2 θ χ = 0◦

χ = 90◦ εχ=0, εχ=90

εχ=0, εχ=90 ε1

ε2

ε1 ε2

Fig. 9 σ1

DEM

DEM

-80

-60

-40

-20

0

20

40

60

80

0 1 2 3 4 5

1

2

Re

sid

ul str

ess

R,

MP

a

Distance from tensile edge x, mm

Bent specimen (#AM1, A5052)30.034keV, DEM, Al (331)

Fig. 9 Distributions of residual stresses of plastically bentspecimen measured by DEM.

-3

-2

-1

0

1

2

3

0 1 2 3 4 5

Thic

kness p

ositio

n y

, m

m

Distance from tensile side x, mm

Bent specimen (#AM1, A5052)30.034keV, DEM, Al (331)

Plastic zone Plastic zoneElastic zone

Fig. 10 Diffraction positions of plastically bent specimen deter-mined by DEM. The rectangle drawn with a broken lineindicates the cross section of the plastically bent speci-men. The vertical broken lines indicate the X-ray beams.

DEM

DEM

PC

(z ) (xC , yC) Fig. 10

PC

Fig. 10

PC

xC , yCzC

(h = 0.2 mm)

zC

Fig.5

Fig.5

314 鈴木　賢治，菖蒲　敬久，城　　鮎美

01-2018-0107-(p.312-317).indd 314 2019/03/27 10:56:51

Fig. 4 Experiment for DEM.

2 PILATUS-300K

(Fig. 4) 2 θ

0◦ O1 = (0, y1, 0),

O2 = (0, y2, 0) Aℓ 331

2 θ 26◦

LD (= 791.889 mm) P1 P2

L0 = 707.989

mm L = 711.746 mm

30.034 keV (0.4127988A)

0.2×0.2mm2 PILATUS-300K

83.8 × 106.5 mm2 487 × 619

pixel 172 µm/pixel

4.75 mm 3 mm

(A5052)

623 K 1

27 µm Aℓ

331

331

2 θ0 = 25.62767◦ (= 93.06326 pm)

cij10) c11 =

106.78 GPa, c12 = 60.74 GPa, c44 = 28.21 GPa

Kroner 331

E = 71.39 GPa,

ν = 0.344

EM = 70.07 GPa νM = 0.347

33 ·1 DEM

Fig 5 DEM P1, P2

(a) P1

(b) P2

PLIATUS-300K 3

Fig 5

DEM P1 P2

(x1, z1) (x2, z2)

(a) P1 (b) P2

Fig. 5 Diffraction images measured by DEM. The box inthe figure (a) indicates the detected area at P2.

(a) P1 (b) P2

Fig. 6 Diffraction spot centers determined by the Demsys,which is matching program. Each diffraction centeris indicated with the cross mark.

Fig. 5

P1 P2

P1 P2

2 θ

PC(xC , yC , zC)

(Demsys) Fig. 6 P1

P2

2 θ

PC(xC , yC , zC)

Demsys

3 ·2DEM

Fig. 7

4.7 mm

3 mm

Fig. 7 Plastically bent specimen.

25.60

25.61

25.62

25.63

25.64

25.65

25.66

25.67

0 1 2 3 4 5

=0

=90

Diffr

action a

ngle

2, deg

Distance from tesile side x, mm

Bent specimen (#AM1, A5052)30.034keV, DEM, Al (331)

20=25.62767deg

Fig. 8 Results of plastically bent specimen measured byDEM.

0.1 mm 0.5 mm 9

1 mm 3

27 χ = 0◦

( ) χ = 90◦ ( ) 2

Fig. 8 2θ

2 θ χ = 0◦

χ = 90◦ εχ=0, εχ=90

εχ=0, εχ=90 ε1

ε2

ε1 ε2

Fig. 9 σ1

DEM

DEM

-80

-60

-40

-20

0

20

40

60

80

0 1 2 3 4 5

1

2

Re

sid

ul str

ess

R,

MP

a

Distance from tensile edge x, mm

Bent specimen (#AM1, A5052)30.034keV, DEM, Al (331)

Fig. 9 Distributions of residual stresses of plastically bentspecimen measured by DEM.

-3

-2

-1

0

1

2

3

0 1 2 3 4 5

Thic

kness p

ositio

n y

, m

m

Distance from tensile side x, mm

Bent specimen (#AM1, A5052)30.034keV, DEM, Al (331)

Plastic zone Plastic zoneElastic zone

Fig. 10 Diffraction positions of plastically bent specimen deter-mined by DEM. The rectangle drawn with a broken lineindicates the cross section of the plastically bent speci-men. The vertical broken lines indicate the X-ray beams.

DEM

DEM

PC

(z ) (xC , yC) Fig. 10

PC

Fig. 10

PC

xC , yCzC

(h = 0.2 mm)

zC

につい

315二重露光法による粗大粒材の応力測定

01-2018-0107-(p.312-317).indd 315 2019/03/27 10:56:51

3 ·3DEM

623 K 1

A5052 4.75×3 mm2

Fig. 11 (a) 2

1 2

Fig.

Spcimen

Indenter

Amvil

1st impact

2nd impact

Fig. 11 Indentation specimen.

Indentation

Indentation 600

400

200

0

-200

-400

-600

ε1

ε2

Strain, με

Fig. 12 Residual strain maps of indentation specimen using DEM.

σ1

σ2

Stress, MPa

100

50

0

-50

-100

Indentation

Indentation

Fig. 13 Residual stress maps of indentation specimen using DEM.

11 (b)

DEM 3 mm

0.2

mm χ

0◦ 90◦ 2 Aℓ 331

ε1 ε2

Fig. 12 ε1

ε2

45

ε1 ε2

σ1 σ2

Fig. 13 σ1

σ2

DEM

1 0.25 mm

4 1701

1600

σ1

σ2

-5 0 5

-5 0 5

Stress, MPa

100

50

0

-50

-100

Fig. 14 Residual stress map of indented specimen by FEM simu-lation.

Impact 11) Impact

E = 70.07 GPa, ρ = 2.68 g/cm3,

ν = 0.3466, σy = 100 MPa, p = 0.350

2

Fig. 14 σ1

σ1

2 DEM

σ2

DEM

45◦

σ2

DEM

Fig. 13

DEM

42

(DEM) DEM

2 2

Aℓ (A5052)

DEM

DEM

DEM

DEM

DEM

29

(C) 17K06046

(2017A-E10)

1) Standard method for X-ray stress measurement ,JSMS-SD-10-05 (2005), The Society of Materials Sci-ence, Japan.http://x-ray.jsms.jp/standard/sample English.pdf(Feb. 24, 2018)

2) P.J. Withers, Use of synchrotron X-ray radiation forstress measurement , in: Analysis of Residual Stressby Diffraction using Neutron and Synchrotron Radi-ation, ed. by M.E. Fitzpatrick and A. Lodini, pp. 170-189 (2003), Taylor & Francis.

3) S. Taira and K. Tanaka, Local residual stress nearfatigue crack tip , Transactions of the Iron and SteelInstitute of Japan, Vol. 19, pp. 411-418 (1979).

4) B.B. He and K.L. Smith, A new method for resid-ual stress measurement using an area detector ,in: Proceedings of The 5th International Conferenceon Residual Stresses (ICRS-5), ed. by T. Ericsson,M. Oden, A. Andersson, Linkoping, Sweden, pp. 634-639 (1997).

5) K. Suzuki, Proposal for a direct-method for stressmeasurement using an X-ray area detector , NDTand E International, Vol. 92, pp. 104-110 (2017).

6) K. Suzuki, T. Shobu, A. Shiro and S. Zhang, Inter-nal stress measurement of weld part using diffractionspot trace method , Material Science Forum, Vol. 777,pp. 155-160 (2014).

7) K. Suzuki, X-ray study on strain measurement ofcoarse-grain material using area detector , Proc. the50th Symposium on X-Ray Studies on Mechanical Be-haviour of Materials, pp. 105-108 (2016), The Societyof Materials Science, Japan.

8) D. Naragani, M.D. Sangid, P.A. Shade, J.C. Schuren,H. Sharma, J-S. Park, P. Kenesei, J.V. Bernier,T.J. Turner and I. Parr, Investigation of fatigue crackinitiation from a non-metallic inclusion via high en-ergy x-ray diffraction microscopy , Acta Materialia137, pp. 71-84 (2017).

9) Y. Hayashi, Y. Yoshiharu, D. Setoyama and Y. Seno,Orientation and stress mapping in polycrystalline

materials by the scanning 3DXRD method , Journalof the Japanese Society for Synchrotron Radiation Re-search, Vol. 31 No. 4, pp. 257-265 (2018).

10) G.N. Kamm and G.A. Alers, Low-temperature elas-tic moduli of aluminum , Journal of Applied Physics,Vol. 35, pp. 327-330 (1964).

11) https://sourceforge.net/projects/impact/

316 鈴木　賢治，菖蒲　敬久，城　　鮎美

01-2018-0107-(p.312-317).indd 316 2019/03/27 10:57:00

3 ·3DEM

623 K 1

A5052 4.75×3 mm2

Fig. 11 (a) 2

1 2

Fig.

Spcimen

Indenter

Amvil

1st impact

2nd impact

Fig. 11 Indentation specimen.

Indentation

Indentation 600

400

200

0

-200

-400

-600

ε1

ε2

Strain, με

Fig. 12 Residual strain maps of indentation specimen using DEM.

σ1

σ2

Stress, MPa

100

50

0

-50

-100

Indentation

Indentation

Fig. 13 Residual stress maps of indentation specimen using DEM.

11 (b)

DEM 3 mm

0.2

mm χ

0◦ 90◦ 2 Aℓ 331

ε1 ε2

Fig. 12 ε1

ε2

45

ε1 ε2

σ1 σ2

Fig. 13 σ1

σ2

DEM

1 0.25 mm

4 1701

1600

σ1

σ2

-5 0 5

-5 0 5

Stress, MPa

100

50

0

-50

-100

Fig. 14 Residual stress map of indented specimen by FEM simu-lation.

Impact 11) Impact

E = 70.07 GPa, ρ = 2.68 g/cm3,

ν = 0.3466, σy = 100 MPa, p = 0.350

2

Fig. 14 σ1

σ1

2 DEM

σ2

DEM

45◦

σ2

DEM

Fig. 13

DEM

42

(DEM) DEM

2 2

Aℓ (A5052)

DEM

DEM

DEM

DEM

DEM

29

(C) 17K06046

(2017A-E10)

1) Standard method for X-ray stress measurement ,JSMS-SD-10-05 (2005), The Society of Materials Sci-ence, Japan.http://x-ray.jsms.jp/standard/sample English.pdf(Feb. 24, 2018)

2) P.J. Withers, Use of synchrotron X-ray radiation forstress measurement , in: Analysis of Residual Stressby Diffraction using Neutron and Synchrotron Radi-ation, ed. by M.E. Fitzpatrick and A. Lodini, pp. 170-189 (2003), Taylor & Francis.

3) S. Taira and K. Tanaka, Local residual stress nearfatigue crack tip , Transactions of the Iron and SteelInstitute of Japan, Vol. 19, pp. 411-418 (1979).

4) B.B. He and K.L. Smith, A new method for resid-ual stress measurement using an area detector ,in: Proceedings of The 5th International Conferenceon Residual Stresses (ICRS-5), ed. by T. Ericsson,M. Oden, A. Andersson, Linkoping, Sweden, pp. 634-639 (1997).

5) K. Suzuki, Proposal for a direct-method for stressmeasurement using an X-ray area detector , NDTand E International, Vol. 92, pp. 104-110 (2017).

6) K. Suzuki, T. Shobu, A. Shiro and S. Zhang, Inter-nal stress measurement of weld part using diffractionspot trace method , Material Science Forum, Vol. 777,pp. 155-160 (2014).

7) K. Suzuki, X-ray study on strain measurement ofcoarse-grain material using area detector , Proc. the50th Symposium on X-Ray Studies on Mechanical Be-haviour of Materials, pp. 105-108 (2016), The Societyof Materials Science, Japan.

8) D. Naragani, M.D. Sangid, P.A. Shade, J.C. Schuren,H. Sharma, J-S. Park, P. Kenesei, J.V. Bernier,T.J. Turner and I. Parr, Investigation of fatigue crackinitiation from a non-metallic inclusion via high en-ergy x-ray diffraction microscopy , Acta Materialia137, pp. 71-84 (2017).

9) Y. Hayashi, Y. Yoshiharu, D. Setoyama and Y. Seno,Orientation and stress mapping in polycrystalline

materials by the scanning 3DXRD method , Journalof the Japanese Society for Synchrotron Radiation Re-search, Vol. 31 No. 4, pp. 257-265 (2018).

10) G.N. Kamm and G.A. Alers, Low-temperature elas-tic moduli of aluminum , Journal of Applied Physics,Vol. 35, pp. 327-330 (1964).

11) https://sourceforge.net/projects/impact/

による

317二重露光法による粗大粒材の応力測定

01-2018-0107-(p.312-317).indd 317 2019/03/27 10:57:06