Abstract

Aimed at the low accuracy problem of shear strain measurement in Moiré methods, a two-dimensional (2D) Moiré phase analysis method is proposed for full-field deformation measurement with high accuracy. A grid image is first processed by the spatial phase-shifting sampling Moiré technique to get the Moiré phases in two directions, which are then conjointly analyzed for measuring 2D displacement and strain distributions. The strain especially the shear strain measurement accuracy is remarkably improved, and dynamic deformation is measurable from automatic batch processing of single-shot grid images. As an application, the 2D microscale strain distributions of a titanium alloy were measured, and the crack occurrence location was successfully predicted from strain concentration.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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  2. M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
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  13. E. Okunishi, N. Endo, and Y. Kondo, “Atomic column elemental mapping by STEM-Moire Method,” Microsc. Microanal. 20(S3), 586–587 (2014).
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  14. H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
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  15. H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101(10), 103511 (2007).
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    [Crossref]
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    [Crossref]
  27. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999).
    [Crossref]
  28. M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
    [Crossref]
  29. S. Y. Chou, P. R. Krauss, and P. J. Renstrom, “Imprint of sub-25 nm vias and trenches in polymers,” Appl. Phys. Lett. 67(21), 3114–3116 (1995).
    [Crossref]
  30. M. Tang, H. Xie, J. Zhu, X. Li, and Y. Li, “Study of moiré grating fabrication on metal samples using nanoimprint lithography,” Opt. Express 20(3), 2942–2955 (2012).
    [Crossref] [PubMed]

2017 (2)

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

2016 (1)

2015 (2)

Q. Wang, H. Tsuda, and S. Kishimoto, “Moire techniques based on memory function of laser scanning microscope for deformation measurement at micron/submicron scales,” Int. J. Automot. Technol. 9(5), 494–501 (2015).
[Crossref]

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

2014 (2)

2013 (4)

C. Li, Z. Liu, H. Xie, and D. Wu, “Novel 3D SEM Moiré method for micro height measurement,” Opt. Express 21(13), 15734–15746 (2013).
[Crossref] [PubMed]

S. Ri, M. Saka, K. Nanbara, and D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53(9), 1635–1646 (2013).
[Crossref]

Q. Wang, S. Kishimoto, H. Xie, Z. Liu, and X. Lou, “In situ high temperature creep deformation of micro-structure with metal film wire on flexible membrane using geometric phase analysis,” Microelectron. Reliab. 53(4), 652–657 (2013).
[Crossref]

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Guti Lixue Xuebao 26, 570–583 (2013).

2012 (4)

Q. Wang and S. Kishimoto, “Simultaneous analysis of residual stress and stress intensity factor in a resist after UV-nanoimprint lithography based on electron moiré fringes,” J. Micromech. Microeng. 22(10), 105021 (2012).
[Crossref]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52(4), 331–340 (2012).
[Crossref]

M. Tang, H. Xie, J. Zhu, X. Li, and Y. Li, “Study of moiré grating fabrication on metal samples using nanoimprint lithography,” Opt. Express 20(3), 2942–2955 (2012).
[Crossref] [PubMed]

Q. Wang, S. Kishimoto, and Y. Yamauchi, “Three-directional structural characterization of hexagonal packed nanoparticles by hexagonal digital moiré method,” Opt. Lett. 37(4), 548–550 (2012).
[Crossref] [PubMed]

2011 (1)

Z. Liu, X. Huang, H. Xie, X. Lou, and H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[Crossref]

2010 (3)

P. Ifju and B. Han, “Recent applications of moiré interferometry,” Exp. Mech. 50(8), 1129–1147 (2010).
[Crossref]

M. Tang, H. Xie, Q. Wang, and J. Zhu, “Phase-shifting laser scanning confocal microscopy moiré method and its applications,” Meas. Sci. Technol. 21(5), 055110 (2010).
[Crossref]

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

2007 (2)

H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101(10), 103511 (2007).
[Crossref]

S. Kishimoto, Q. Wang, H. Xie, and Y. Zhao, “Study of the surface structure of butterfly wings using the scanning electron microscopic moiré method,” Appl. Opt. 46(28), 7026–7034 (2007).
[Crossref] [PubMed]

2000 (1)

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

1999 (1)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999).
[Crossref]

1995 (1)

S. Y. Chou, P. R. Krauss, and P. J. Renstrom, “Imprint of sub-25 nm vias and trenches in polymers,” Appl. Phys. Lett. 67(21), 3114–3116 (1995).
[Crossref]

1993 (1)

S. Kishimoto, M. Egashira, and N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32(3), 522–526 (1993).
[Crossref]

1991 (1)

1976 (1)

1948 (1)

R. Weller and B. Shepard, “Displacement measurement by mechanical interferometry,” Proc. Soc. Exp. Stress Anal. 6, 35–38 (1948).

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999).
[Crossref]

Asundi, A.

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

A. Asundi and K. H. Yung, “Phase-shifting and logical moiré,” J. Opt. Soc. Am. A 8(10), 1591–1600 (1991).
[Crossref]

Bechtold, M.

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Boay, C. G.

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

Chou, S. Y.

S. Y. Chou, P. R. Krauss, and P. J. Renstrom, “Imprint of sub-25 nm vias and trenches in polymers,” Appl. Phys. Lett. 67(21), 3114–3116 (1995).
[Crossref]

Dai, F.

H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101(10), 103511 (2007).
[Crossref]

Diehl, M.

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Du, H.

Z. Liu, X. Huang, H. Xie, X. Lou, and H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[Crossref]

Egashira, M.

S. Kishimoto, M. Egashira, and N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32(3), 522–526 (1993).
[Crossref]

Endo, N.

E. Okunishi, N. Endo, and Y. Kondo, “Atomic column elemental mapping by STEM-Moire Method,” Microsc. Microanal. 20(S3), 586–587 (2014).
[Crossref] [PubMed]

Fujigaki, M.

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

Hamano, Y.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Han, B.

P. Ifju and B. Han, “Recent applications of moiré interferometry,” Exp. Mech. 50(8), 1129–1147 (2010).
[Crossref]

Hayashi, S.

Huang, X.

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Guti Lixue Xuebao 26, 570–583 (2013).

Z. Liu, X. Huang, H. Xie, X. Lou, and H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[Crossref]

Ifju, P.

P. Ifju and B. Han, “Recent applications of moiré interferometry,” Exp. Mech. 50(8), 1129–1147 (2010).
[Crossref]

Kishimoto, S.

Q. Wang, H. Tsuda, and S. Kishimoto, “Moire techniques based on memory function of laser scanning microscope for deformation measurement at micron/submicron scales,” Int. J. Automot. Technol. 9(5), 494–501 (2015).
[Crossref]

Q. Wang, S. Kishimoto, H. Xie, Z. Liu, and X. Lou, “In situ high temperature creep deformation of micro-structure with metal film wire on flexible membrane using geometric phase analysis,” Microelectron. Reliab. 53(4), 652–657 (2013).
[Crossref]

Q. Wang and S. Kishimoto, “Simultaneous analysis of residual stress and stress intensity factor in a resist after UV-nanoimprint lithography based on electron moiré fringes,” J. Micromech. Microeng. 22(10), 105021 (2012).
[Crossref]

Q. Wang, S. Kishimoto, and Y. Yamauchi, “Three-directional structural characterization of hexagonal packed nanoparticles by hexagonal digital moiré method,” Opt. Lett. 37(4), 548–550 (2012).
[Crossref] [PubMed]

S. Kishimoto, Q. Wang, H. Xie, and Y. Zhao, “Study of the surface structure of butterfly wings using the scanning electron microscopic moiré method,” Appl. Opt. 46(28), 7026–7034 (2007).
[Crossref] [PubMed]

H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101(10), 103511 (2007).
[Crossref]

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

S. Kishimoto, M. Egashira, and N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32(3), 522–526 (1993).
[Crossref]

Kobayashi, D.

S. Ri, M. Saka, K. Nanbara, and D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53(9), 1635–1646 (2013).
[Crossref]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52(4), 331–340 (2012).
[Crossref]

Kondo, Y.

E. Okunishi, N. Endo, and Y. Kondo, “Atomic column elemental mapping by STEM-Moire Method,” Microsc. Microanal. 20(S3), 586–587 (2014).
[Crossref] [PubMed]

Koyama, M.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Krauss, P. R.

S. Y. Chou, P. R. Krauss, and P. J. Renstrom, “Imprint of sub-25 nm vias and trenches in polymers,” Appl. Phys. Lett. 67(21), 3114–3116 (1995).
[Crossref]

Kusaka, Y.

Li, C.

Li, X.

Li, Y.

Liu, Z.

Q. Wang, S. Kishimoto, H. Xie, Z. Liu, and X. Lou, “In situ high temperature creep deformation of micro-structure with metal film wire on flexible membrane using geometric phase analysis,” Microelectron. Reliab. 53(4), 652–657 (2013).
[Crossref]

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Guti Lixue Xuebao 26, 570–583 (2013).

C. Li, Z. Liu, H. Xie, and D. Wu, “Novel 3D SEM Moiré method for micro height measurement,” Opt. Express 21(13), 15734–15746 (2013).
[Crossref] [PubMed]

Z. Liu, X. Huang, H. Xie, X. Lou, and H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[Crossref]

Lou, X.

Q. Wang, S. Kishimoto, H. Xie, Z. Liu, and X. Lou, “In situ high temperature creep deformation of micro-structure with metal film wire on flexible membrane using geometric phase analysis,” Microelectron. Reliab. 53(4), 652–657 (2013).
[Crossref]

Z. Liu, X. Huang, H. Xie, X. Lou, and H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[Crossref]

Mitsuhara, M.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Morimoto, Y.

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

Muramatsu, T.

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52(4), 331–340 (2012).
[Crossref]

Nanbara, K.

S. Ri, M. Saka, K. Nanbara, and D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53(9), 1635–1646 (2013).
[Crossref]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52(4), 331–340 (2012).
[Crossref]

Ngoi, B. K.

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

Noguchi, H.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

Ogihara, S.

Ohkubo, M.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Okunishi, E.

E. Okunishi, N. Endo, and Y. Kondo, “Atomic column elemental mapping by STEM-Moire Method,” Microsc. Microanal. 20(S3), 586–587 (2014).
[Crossref] [PubMed]

Patorski, K.

Peranio, N.

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Ponge, D.

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Raabe, D.

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Renstrom, P. J.

S. Y. Chou, P. R. Krauss, and P. J. Renstrom, “Imprint of sub-25 nm vias and trenches in polymers,” Appl. Phys. Lett. 67(21), 3114–3116 (1995).
[Crossref]

Ri, S.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Q. Wang, S. Ri, and H. Tsuda, “Digital sampling Moiré as a substitute for microscope scanning Moiré for high-sensitivity and full-field deformation measurement at micron/nano scales,” Appl. Opt. 55(25), 6858–6865 (2016).
[Crossref] [PubMed]

S. Ri, S. Hayashi, S. Ogihara, and H. Tsuda, “Accurate full-field optical displacement measurement technique using a digital camera and repeated patterns,” Opt. Express 22(8), 9693–9706 (2014).
[Crossref] [PubMed]

S. Ri, M. Saka, K. Nanbara, and D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53(9), 1635–1646 (2013).
[Crossref]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52(4), 331–340 (2012).
[Crossref]

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

Q. Wang, S. Ri, and H. Tsuda, “Micro/nano-scale strain distribution measurement from sampling Moiré fringes,” J. Vis. Exp.in press.

Roters, F.

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Saka, M.

S. Ri, M. Saka, K. Nanbara, and D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53(9), 1635–1646 (2013).
[Crossref]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52(4), 331–340 (2012).
[Crossref]

Schemmann, L.

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Shepard, B.

R. Weller and B. Shepard, “Displacement measurement by mechanical interferometry,” Proc. Soc. Exp. Stress Anal. 6, 35–38 (1948).

Shinya, N.

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

S. Kishimoto, M. Egashira, and N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32(3), 522–526 (1993).
[Crossref]

Tanaka, Y.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Tang, M.

M. Tang, H. Xie, J. Zhu, X. Li, and Y. Li, “Study of moiré grating fabrication on metal samples using nanoimprint lithography,” Opt. Express 20(3), 2942–2955 (2012).
[Crossref] [PubMed]

M. Tang, H. Xie, Q. Wang, and J. Zhu, “Phase-shifting laser scanning confocal microscopy moiré method and its applications,” Meas. Sci. Technol. 21(5), 055110 (2010).
[Crossref]

Tasan, C. C.

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Tsuda, H.

Q. Wang, S. Ri, and H. Tsuda, “Digital sampling Moiré as a substitute for microscope scanning Moiré for high-sensitivity and full-field deformation measurement at micron/nano scales,” Appl. Opt. 55(25), 6858–6865 (2016).
[Crossref] [PubMed]

Q. Wang, H. Tsuda, and S. Kishimoto, “Moire techniques based on memory function of laser scanning microscope for deformation measurement at micron/submicron scales,” Int. J. Automot. Technol. 9(5), 494–501 (2015).
[Crossref]

S. Ri, S. Hayashi, S. Ogihara, and H. Tsuda, “Accurate full-field optical displacement measurement technique using a digital camera and repeated patterns,” Opt. Express 22(8), 9693–9706 (2014).
[Crossref] [PubMed]

Q. Wang, S. Ri, and H. Tsuda, “Micro/nano-scale strain distribution measurement from sampling Moiré fringes,” J. Vis. Exp.in press.

Tsuzaki, K.

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999).
[Crossref]

Wang, M.

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

Wang, Q.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Q. Wang, S. Ri, and H. Tsuda, “Digital sampling Moiré as a substitute for microscope scanning Moiré for high-sensitivity and full-field deformation measurement at micron/nano scales,” Appl. Opt. 55(25), 6858–6865 (2016).
[Crossref] [PubMed]

Q. Wang, H. Tsuda, and S. Kishimoto, “Moire techniques based on memory function of laser scanning microscope for deformation measurement at micron/submicron scales,” Int. J. Automot. Technol. 9(5), 494–501 (2015).
[Crossref]

Q. Wang, S. Kishimoto, H. Xie, Z. Liu, and X. Lou, “In situ high temperature creep deformation of micro-structure with metal film wire on flexible membrane using geometric phase analysis,” Microelectron. Reliab. 53(4), 652–657 (2013).
[Crossref]

Q. Wang and S. Kishimoto, “Simultaneous analysis of residual stress and stress intensity factor in a resist after UV-nanoimprint lithography based on electron moiré fringes,” J. Micromech. Microeng. 22(10), 105021 (2012).
[Crossref]

Q. Wang, S. Kishimoto, and Y. Yamauchi, “Three-directional structural characterization of hexagonal packed nanoparticles by hexagonal digital moiré method,” Opt. Lett. 37(4), 548–550 (2012).
[Crossref] [PubMed]

M. Tang, H. Xie, Q. Wang, and J. Zhu, “Phase-shifting laser scanning confocal microscopy moiré method and its applications,” Meas. Sci. Technol. 21(5), 055110 (2010).
[Crossref]

H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101(10), 103511 (2007).
[Crossref]

S. Kishimoto, Q. Wang, H. Xie, and Y. Zhao, “Study of the surface structure of butterfly wings using the scanning electron microscopic moiré method,” Appl. Opt. 46(28), 7026–7034 (2007).
[Crossref] [PubMed]

Q. Wang, S. Ri, and H. Tsuda, “Micro/nano-scale strain distribution measurement from sampling Moiré fringes,” J. Vis. Exp.in press.

Weller, R.

R. Weller and B. Shepard, “Displacement measurement by mechanical interferometry,” Proc. Soc. Exp. Stress Anal. 6, 35–38 (1948).

Wu, D.

Xie, H.

C. Li, Z. Liu, H. Xie, and D. Wu, “Novel 3D SEM Moiré method for micro height measurement,” Opt. Express 21(13), 15734–15746 (2013).
[Crossref] [PubMed]

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Guti Lixue Xuebao 26, 570–583 (2013).

Q. Wang, S. Kishimoto, H. Xie, Z. Liu, and X. Lou, “In situ high temperature creep deformation of micro-structure with metal film wire on flexible membrane using geometric phase analysis,” Microelectron. Reliab. 53(4), 652–657 (2013).
[Crossref]

M. Tang, H. Xie, J. Zhu, X. Li, and Y. Li, “Study of moiré grating fabrication on metal samples using nanoimprint lithography,” Opt. Express 20(3), 2942–2955 (2012).
[Crossref] [PubMed]

Z. Liu, X. Huang, H. Xie, X. Lou, and H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[Crossref]

M. Tang, H. Xie, Q. Wang, and J. Zhu, “Phase-shifting laser scanning confocal microscopy moiré method and its applications,” Meas. Sci. Technol. 21(5), 055110 (2010).
[Crossref]

H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101(10), 103511 (2007).
[Crossref]

S. Kishimoto, Q. Wang, H. Xie, and Y. Zhao, “Study of the surface structure of butterfly wings using the scanning electron microscopic moiré method,” Appl. Opt. 46(28), 7026–7034 (2007).
[Crossref] [PubMed]

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

Yamanouchi, K.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Yamasaki, S.

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Yamauchi, Y.

Yan, D.

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Yokozeki, S.

Yu, J.

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

Yung, K. H.

Zhang, Z.

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

Zhao, Y.

Zheng, C.

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Zhu, J.

M. Tang, H. Xie, J. Zhu, X. Li, and Y. Li, “Study of moiré grating fabrication on metal samples using nanoimprint lithography,” Opt. Express 20(3), 2942–2955 (2012).
[Crossref] [PubMed]

M. Tang, H. Xie, Q. Wang, and J. Zhu, “Phase-shifting laser scanning confocal microscopy moiré method and its applications,” Meas. Sci. Technol. 21(5), 055110 (2010).
[Crossref]

Annu. Rev. Mater. Res. (1)

C. C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe, “An overview of dual-phase steels: advances in microstructure-oriented processing and micromechanically guided design,” Annu. Rev. Mater. Res. 45(1), 391–431 (2015).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

S. Y. Chou, P. R. Krauss, and P. J. Renstrom, “Imprint of sub-25 nm vias and trenches in polymers,” Appl. Phys. Lett. 67(21), 3114–3116 (1995).
[Crossref]

Exp. Mech. (4)

P. Ifju and B. Han, “Recent applications of moiré interferometry,” Exp. Mech. 50(8), 1129–1147 (2010).
[Crossref]

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52(4), 331–340 (2012).
[Crossref]

S. Ri, M. Saka, K. Nanbara, and D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53(9), 1635–1646 (2013).
[Crossref]

Guti Lixue Xuebao (1)

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Guti Lixue Xuebao 26, 570–583 (2013).

Int. J. Automot. Technol. (1)

Q. Wang, H. Tsuda, and S. Kishimoto, “Moire techniques based on memory function of laser scanning microscope for deformation measurement at micron/submicron scales,” Int. J. Automot. Technol. 9(5), 494–501 (2015).
[Crossref]

J. Appl. Phys. (1)

H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101(10), 103511 (2007).
[Crossref]

J. Micromech. Microeng. (1)

Q. Wang and S. Kishimoto, “Simultaneous analysis of residual stress and stress intensity factor in a resist after UV-nanoimprint lithography based on electron moiré fringes,” J. Micromech. Microeng. 22(10), 105021 (2012).
[Crossref]

J. Opt. Soc. Am. A (1)

Mater. Charact. (1)

M. Koyama, K. Yamanouchi, Q. Wang, S. Ri, Y. Tanaka, Y. Hamano, S. Yamasaki, M. Mitsuhara, M. Ohkubo, H. Noguchi, and K. Tsuzaki, “Multiscale in situ deformation experiments: A sequential process from strain localization to failure in a laminated Ti-6Al-4V alloy,” Mater. Charact. 128, 217–225 (2017).
[Crossref]

Meas. Sci. Technol. (2)

Z. Liu, X. Huang, H. Xie, X. Lou, and H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[Crossref]

M. Tang, H. Xie, Q. Wang, and J. Zhu, “Phase-shifting laser scanning confocal microscopy moiré method and its applications,” Meas. Sci. Technol. 21(5), 055110 (2010).
[Crossref]

Microelectron. Reliab. (1)

Q. Wang, S. Kishimoto, H. Xie, Z. Liu, and X. Lou, “In situ high temperature creep deformation of micro-structure with metal film wire on flexible membrane using geometric phase analysis,” Microelectron. Reliab. 53(4), 652–657 (2013).
[Crossref]

Microsc. Microanal. (1)

E. Okunishi, N. Endo, and Y. Kondo, “Atomic column elemental mapping by STEM-Moire Method,” Microsc. Microanal. 20(S3), 586–587 (2014).
[Crossref] [PubMed]

Nanotechnology (1)

H. Xie, S. Kishimoto, A. Asundi, C. G. Boay, N. Shinya, J. Yu, and B. K. Ngoi, “In-plane deformation measurement using the atomic force microscope moiré method,” Nanotechnology 11(1), 24–29 (2000).
[Crossref]

Opt. Commun. (1)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999).
[Crossref]

Opt. Eng. (1)

S. Kishimoto, M. Egashira, and N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32(3), 522–526 (1993).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Proc. Soc. Exp. Stress Anal. (1)

R. Weller and B. Shepard, “Displacement measurement by mechanical interferometry,” Proc. Soc. Exp. Stress Anal. 6, 35–38 (1948).

Science (1)

M. Koyama, Z. Zhang, M. Wang, D. Ponge, D. Raabe, K. Tsuzaki, H. Noguchi, and C. C. Tasan, “Bone-like crack resistance in hierarchical metastable nanolaminate steels,” Science 355(6329), 1055–1057 (2017).
[PubMed]

Other (2)

G. Lütjering and J. C. Williams, Titanium, 2nd ed. (Springer-Verlag Berlin Heidelberg, Berlin, 2007).

Q. Wang, S. Ri, and H. Tsuda, “Micro/nano-scale strain distribution measurement from sampling Moiré fringes,” J. Vis. Exp.in press.

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Figures (13)

Fig. 1
Fig. 1

(a) Diagram of a 2D grid including 2 parallel gratings on specimen, (b) Geometric relationship of gratings before and after deformation.

Fig. 2
Fig. 2

Principle of the sampling moiré method for calculating phase distribution from an oblique grating.

Fig. 3
Fig. 3

Flow chart and measurement process of the proposed technique for accurate strain distribution measurement.

Fig. 4
Fig. 4

(a) Diagram of a regular grid under an applied 2D strain status when the oblique angle θ is variable, and (b)-(g) Measurement process of 2D strain distributions.

Fig. 5
Fig. 5

Comparison of the proposed 2D phase analysis technique and the traditional 1D phase analysis method in measurements of the x-direction strain, the y-direction strain and the shear strains.

Fig. 6
Fig. 6

Relative errors of the measured x-direction, y-direction and shear strains from the proposed 2D phase analysis technique and the traditional 1D phase analysis method relative to the theoretical strains along with the grid oblique angle.

Fig. 7
Fig. 7

(a) Diagram of a regular grid with a random noise of σ = 2% under a 2D strain status when the oblique angle is 10° (b)Absolute errors, (c) relative errors and (d) standard deviations of the measured 2D strains from the proposed technique along with the theoretical strains.

Fig. 8
Fig. 8

(a) Specimen geometry of the Ti alloy and (b) the used tensile device under a laser scanning microscope.

Fig. 9
Fig. 9

Region of interest on the Ti alloy and the fabricated 3-μm-pitch grid image.

Fig. 10
Fig. 10

Process of moiré phase analysis on the Ti alloy when the tensile load is 604 MPa. (a) Grid image, (b) and (c) spatial phase-shifting moiré fringes in the x and y direction, respectively, and (d) and (e) moiré phases in the x and y direction, respectively.

Fig. 11
Fig. 11

Distributions of x-direction, y-direction and shear strains of the Ti alloy under tensile loads of (a)-(d) 225 MPa, (e)-(h) 312 MPa, (i)-(l) 478 MPa, (m)-(p) 604 MPa, and (q)-(t) 660 MPa before damage occurrence.

Fig. 12
Fig. 12

(a) Grid image and (b) enlarge image around prior β boundary on the Ti alloy after unloading from 710 MPa.

Fig. 13
Fig. 13

Grid images and distributions of a part of shear strain of the specimen under 655, 682, 669 and 683 MPa along with time.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

I= A X cos[2π( x p Xx + y p Yy )]+ A Y cos[2π( x p Yx + y p Yy )]+B
I X = A X cos[2π( x p Xx + y p Xy )]+ B X = A X cos φ X + B X
I Y = A Y cos[2π( x p Yx + y p Yy )]+ B Y = A Y cos φ Y + B Y
I = A X cos[2π( x p Xx + y p Yy )]+ A Y cos[2π( x p Yx + y p Yy )]+ B
I X = A X cos[2π( x p Xx + y p Xy )]+ B X = A X cos φ X + B X
I Y = A Y cos[2π( x p Yx + y p Yy )]+ B Y = A Y cos φ Y + B Y
Δ φ X = φ X φ X =2π[( x p Xx + y p Xy )( x p Xx + y p Xy )]
Δ φ Y = φ Y φ Y =2π[( x p Yx + y p Yy )( x p Yx + y p Yy )]
I X,mx ( k x )= A X cos[2π( x p Xx + y p Xy x N x + k x N x )]+ B X = A X cos[ φ X,mx +2π k x N x )]+ B X
I X,mx ( k x )= A X cos[2π( x p Xx + y p Xy x N x + k x N x )]+ B X = A X cos[ φ X,mx +2π k x N x )]+ B X ( k x =0, 1,, N x 1 )
I Y,my ( k y )= A Y cos[2π( x p 2x + y p Yy y N y + k y N y )]+ B Y = A Y cos[ φ Y,my +2π k y N y )]+ B Y
I Y,my ( k y )= A Y cos[2π( x p 2x + y p Yy y N y + k y N y )]+ B Y = A Y cos[ φ Y,my +2π k y N y )]+ B Y ( k y =0, 1,..., N y 1 )
φ J,mj =-arctan k j =0 T j 1 I J,mj ( k j )sin(2π k j / N j ) k j =0 T j 1 I J,mj ( k j )cos(2π k j / N j ) φ J,mj =-arctan k j =0 T j 1 I J,mj ( k j )sin(2π k j / N j ) k j =0 T j 1 I J,mj ( k j )cos(2π k j / N j ) { j=x when J=X j=y when J=Y
Δ φ X,mx = φ X,mx φ X,mx =2π( x p Xx + y p Xy x N x )2π( x p Xx + y p Xy x N x ) =2π[( x p Xx + y p Xy )( x p Xx + y p Xy )]=Δ φ X
Δ φ Y,my = φ Y,my φ Y,my =2π( x p Yx + y p Yy y N y )2π( x p Yx + y p Yy y N y ) =2π[( x p Yx + y p Yy )( x p Yx + y p Yy )]=Δ φ Y
φ X =2π( x u x p Xx + y u y p Xy )
φ Y =2π( x u x p Yx + y u y p Yy )
Δ φ X =2π( x u x p Xx + y u y p Xy )2π( x p Xx + y p Xy ) =2π( u x p Xx + u y p Xy )
Δ φ Y =2π( x u x p Yx + y u y p Yy )2π( x p Yx + y p Yy ) =2π( u x p Yx + u y p Yy )
( Δ φ X,mx Δ φ Y,my )=2π( 1/ p Xx 1/ p Xy 1/ p Yx 1/ p Yy )( u x u y )
( u x u y )= 1 2π ( 1/ p Xx 1/ p Xy 1/ p Yx 1/ p Yy ) 1 ( Δ φ X,mx Δ φ Y,my )= M 2π ( Δ φ X,mx Δ φ Y,my )
( ε xx ε xy ε yx ε yy )= M 2π ( Δ φ X,mx x Δ φ X,mx y Δ φ Y,my x Δ φ Y,my y ) γ xy = ε xy + ε yx
M= ( sin θ X / p X cos θ X / p X sin θ Y / p Y cos θ Y / p Y ) 1
M= ( sin(θ+ π 2 )/p cos(θ+ π 2 )/p sinθ/p cosθ/p ) 1 =p ( cosθ sinθ sinθ cosθ ) 1
( u x u y )= p 2π ( cosθ sinθ sinθ cosθ ) 1 ( Δ φ X,mx Δ φ Y,my )
( ε xx ε xy ε yx ε yy )= p 2π ( cosθ sinθ sinθ cosθ ) 1 ( Δ φ X,mx x Δ φ X,mx y Δ φ Y,my x Δ φ Y,my y ) γ xy = ε xy + ε yx

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