Abstract

We demonstrate a simple method for obtaining slope contours of bent plates using Talbot interferometry. The technique has been used to map slope contours of polymethyl methacrylate specimens of different shapes. The Talbot image of a coarse grating is projected onto a specimen such that the self-image is backreflected onto the same grating again. As a Talbot interferometer is basically a grating shearing inter ferometer, it results in the generation of characteristic slope maps of the specimen under test. Results of the investigation match well with other slope-mapping techniques. Validation of experimental results with theoretical predictions in the case of a cantilever beam specimen has been undertaken. Accuracy of about 4.7% with respect to theoretical predictions is obtained.

© 2010 Optical Society of America

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  1. F. K. Ligtenberg, “The moiré method: a new experimental method for the determination of moments in small slab models,” Proc. SESA 12, 83–98 (1954).
  2. P. S. Theocaris, A. Koutsabessis “Slope measurement by means of moiré fringes,” J. Sci. Instrum. 42, 607–610 (1965).
    [CrossRef]
  3. R. Ritter, R. Schettler-Koehler, “Curvature measurement by moiré effect,” Exp. Mech. 23, 165–170 (1983).
    [CrossRef]
  4. A. Assa, A. A. Bester, J. Politch, “Recording slope and curvature contours of fixed plates using a grating shearing interferometer,” Appl. Opt. 16, 2504–2513 (1977).
    [CrossRef] [PubMed]
  5. A. Assa, T. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
    [CrossRef]
  6. F. P. Chiang, T. Y. Kao, “Family of grating techniques of slope and curvature measurements for static and dynamic flexure of plates,” Opt. Eng. 21, 721–742 (1982).
  7. K. P. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251–1263 (1991).
    [CrossRef]
  8. P. R. Sreedhar, N. K. Mohan, R. S. Sirohi, “Displacement and slope measurements using image plane holography with a BaTiO3 crystal,” Opt. Laser Technol. 29, 111–115 (1997).
    [CrossRef]
  9. F. P. Chiang, R. M. Juang, “Laser speckle interferometry for plate bending problems,” Appl. Opt. 15, 2199–2204 (1976).
    [CrossRef] [PubMed]
  10. R. K. Mohanty, C. Joenathan, R. S. Sirohi, “Speckle and speckle-shearing interferometers combined for the simultaneous determination of out-of-plane displacement and slope,” Appl. Opt. 24, 3106–3109 (1985).
    [CrossRef] [PubMed]
  11. C. J. Tay, H. M. Shang, A. N. Poo, M. Luo “Measurement of surface coordinates and slopes by shearography,” Opt. Laser Technol. 24, 209–213 (1992).
    [CrossRef]
  12. K. Szmodits, “Zur theorie des moiré verfahrens,” Proceedings of Symposium on Shell Research, A. M. Hass and A. L. Bowma, eds. (North Holland, 1961), pp. 208–214.
  13. G. Reider, R. Ritter, “Krummungsmessung an belasteter platten nach dem ligentenbergschen moiré verfahren,” Forsch Ing. Wes. 31, 33–44 (1965).
    [CrossRef]
  14. T. Y. Kao, F. P. Chiang, “Family of grating techniques of slope and curvature measurements for static and dynamic flexures of plates,” Opt. Eng. 21, 721–742 (1982).
  15. Vasco Ronchi, “Forty years of history of a grating interfer ometer,” Appl. Opt. 3, 437–451 (1964).
    [CrossRef]
  16. G. Subramanian, S. Krishnakumar, “An image shearing moiré method to record slopes in flexed plates,” Strain 20, 69–73 (1984).
    [CrossRef]
  17. G. Subramanian, P. N. Akella, “A selective diffraction order based lens-plane grating shearing interferometer for the study of bent plates,” Strain 23, 55–59 (1987).
    [CrossRef]
  18. G. Subramanian, K. Jancy Rose, A. Subramanian, “A multiplexed grating shearing interferometer for reflection moiré analysis of partial slopes and curvatures of bent plates,” Exp. Tech. 24, 27–30 (2000).
    [CrossRef]
  19. L. Wang, D. Yun, “Measurement of the deformation of thin plates by moiré interferometry,” Opt. Laser Technol. 23, 151–154 (1991).
    [CrossRef]
  20. L. Wang, D. Z. Yun, “Application of moiré shearing interferometry to slope measurement of shallow shells,” J. Strain Anal. Eng. Des. 27, 45–48 (1992).
    [CrossRef]
  21. R. Ritter, W. Wilke, “Slope and contour measurement by reflection grating method and the photogrammetric principle,” Opt. Lasers Eng. 15, 103–113 (1991).
    [CrossRef]
  22. G. Subramanian, P. Ramana Kumar, “Intensity integration technique for contouring two dimensional reflective surfaces,” Exp. Tech. 30, 56–60 (2006).
    [CrossRef]
  23. H. V. Tippur, “Simultaneous and real-time measurement of slope and curvature fringes in thin structures using shearing interferometry,” Opt. Eng. 43, 3014–3020 (2004).
    [CrossRef]
  24. H. F. Talbot, “Facts relating to optical Science,” Philos. Mag. 9, 403–405 (1836).
  25. K. Patorski, “Self imaging and its applications,” Progress in Optics Vol. 27, E. Wolf, ed. (Elsevier, 1989).
    [CrossRef]
  26. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells2nd ed. (McGraw-Hill, 1959).
  27. K. Kalestynski, S. Smolinska, “Self restoration of auto idolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).
  28. C. Menzel, E. Menzel, “Beugungsercheinungen optischer Gitter nach intensitat and phase,” Optik (Jena) 3, 247–59 (1948).
  29. J. Cowley, A. Moodie, “Fourier images II—The out of focus patterns,” Proc. Phys. Soc. London Sect. B 70, 497–504 (1957).
    [CrossRef]
  30. Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111–4116 (1999).
    [CrossRef]
  31. Q. Liu, R. Ohba, S. Kakuma, “Effects of unparallel grating planes in Talbot interferometry. II,” Appl. Opt. 39, 2084–2090 (2000).
    [CrossRef]
  32. Q. Liu, R. Ohba, “Effects of a small inclination misalignment in Talbot interferometry by use of gratings with arbitrary line orientation. II. Experimental verification,” Appl. Opt. 40, 4534–4539 (2001).
    [CrossRef]

2006 (1)

G. Subramanian, P. Ramana Kumar, “Intensity integration technique for contouring two dimensional reflective surfaces,” Exp. Tech. 30, 56–60 (2006).
[CrossRef]

2004 (1)

H. V. Tippur, “Simultaneous and real-time measurement of slope and curvature fringes in thin structures using shearing interferometry,” Opt. Eng. 43, 3014–3020 (2004).
[CrossRef]

2001 (1)

2000 (2)

Q. Liu, R. Ohba, S. Kakuma, “Effects of unparallel grating planes in Talbot interferometry. II,” Appl. Opt. 39, 2084–2090 (2000).
[CrossRef]

G. Subramanian, K. Jancy Rose, A. Subramanian, “A multiplexed grating shearing interferometer for reflection moiré analysis of partial slopes and curvatures of bent plates,” Exp. Tech. 24, 27–30 (2000).
[CrossRef]

1999 (1)

1997 (1)

P. R. Sreedhar, N. K. Mohan, R. S. Sirohi, “Displacement and slope measurements using image plane holography with a BaTiO3 crystal,” Opt. Laser Technol. 29, 111–115 (1997).
[CrossRef]

1992 (2)

C. J. Tay, H. M. Shang, A. N. Poo, M. Luo “Measurement of surface coordinates and slopes by shearography,” Opt. Laser Technol. 24, 209–213 (1992).
[CrossRef]

L. Wang, D. Z. Yun, “Application of moiré shearing interferometry to slope measurement of shallow shells,” J. Strain Anal. Eng. Des. 27, 45–48 (1992).
[CrossRef]

1991 (3)

R. Ritter, W. Wilke, “Slope and contour measurement by reflection grating method and the photogrammetric principle,” Opt. Lasers Eng. 15, 103–113 (1991).
[CrossRef]

L. Wang, D. Yun, “Measurement of the deformation of thin plates by moiré interferometry,” Opt. Laser Technol. 23, 151–154 (1991).
[CrossRef]

K. P. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251–1263 (1991).
[CrossRef]

1987 (1)

G. Subramanian, P. N. Akella, “A selective diffraction order based lens-plane grating shearing interferometer for the study of bent plates,” Strain 23, 55–59 (1987).
[CrossRef]

1985 (1)

1984 (1)

G. Subramanian, S. Krishnakumar, “An image shearing moiré method to record slopes in flexed plates,” Strain 20, 69–73 (1984).
[CrossRef]

1983 (1)

R. Ritter, R. Schettler-Koehler, “Curvature measurement by moiré effect,” Exp. Mech. 23, 165–170 (1983).
[CrossRef]

1982 (2)

F. P. Chiang, T. Y. Kao, “Family of grating techniques of slope and curvature measurements for static and dynamic flexure of plates,” Opt. Eng. 21, 721–742 (1982).

T. Y. Kao, F. P. Chiang, “Family of grating techniques of slope and curvature measurements for static and dynamic flexures of plates,” Opt. Eng. 21, 721–742 (1982).

1979 (1)

A. Assa, T. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[CrossRef]

1978 (1)

K. Kalestynski, S. Smolinska, “Self restoration of auto idolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).

1977 (1)

1976 (1)

1965 (2)

P. S. Theocaris, A. Koutsabessis “Slope measurement by means of moiré fringes,” J. Sci. Instrum. 42, 607–610 (1965).
[CrossRef]

G. Reider, R. Ritter, “Krummungsmessung an belasteter platten nach dem ligentenbergschen moiré verfahren,” Forsch Ing. Wes. 31, 33–44 (1965).
[CrossRef]

1964 (1)

1957 (1)

J. Cowley, A. Moodie, “Fourier images II—The out of focus patterns,” Proc. Phys. Soc. London Sect. B 70, 497–504 (1957).
[CrossRef]

1954 (1)

F. K. Ligtenberg, “The moiré method: a new experimental method for the determination of moments in small slab models,” Proc. SESA 12, 83–98 (1954).

1948 (1)

C. Menzel, E. Menzel, “Beugungsercheinungen optischer Gitter nach intensitat and phase,” Optik (Jena) 3, 247–59 (1948).

1836 (1)

H. F. Talbot, “Facts relating to optical Science,” Philos. Mag. 9, 403–405 (1836).

Akella, P. N.

G. Subramanian, P. N. Akella, “A selective diffraction order based lens-plane grating shearing interferometer for the study of bent plates,” Strain 23, 55–59 (1987).
[CrossRef]

Assa, A.

A. Assa, T. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[CrossRef]

A. Assa, A. A. Bester, J. Politch, “Recording slope and curvature contours of fixed plates using a grating shearing interferometer,” Appl. Opt. 16, 2504–2513 (1977).
[CrossRef] [PubMed]

Bester, A. A.

Betser, A. A.

A. Assa, T. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[CrossRef]

Chiang, F. P.

F. P. Chiang, T. Y. Kao, “Family of grating techniques of slope and curvature measurements for static and dynamic flexure of plates,” Opt. Eng. 21, 721–742 (1982).

T. Y. Kao, F. P. Chiang, “Family of grating techniques of slope and curvature measurements for static and dynamic flexures of plates,” Opt. Eng. 21, 721–742 (1982).

F. P. Chiang, R. M. Juang, “Laser speckle interferometry for plate bending problems,” Appl. Opt. 15, 2199–2204 (1976).
[CrossRef] [PubMed]

Cowley, J.

J. Cowley, A. Moodie, “Fourier images II—The out of focus patterns,” Proc. Phys. Soc. London Sect. B 70, 497–504 (1957).
[CrossRef]

Joenathan, C.

Juang, R. M.

Kakuma, S.

Kalestynski, K.

K. Kalestynski, S. Smolinska, “Self restoration of auto idolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).

Kao, T. Y.

T. Y. Kao, F. P. Chiang, “Family of grating techniques of slope and curvature measurements for static and dynamic flexures of plates,” Opt. Eng. 21, 721–742 (1982).

F. P. Chiang, T. Y. Kao, “Family of grating techniques of slope and curvature measurements for static and dynamic flexure of plates,” Opt. Eng. 21, 721–742 (1982).

Koutsabessis, A.

P. S. Theocaris, A. Koutsabessis “Slope measurement by means of moiré fringes,” J. Sci. Instrum. 42, 607–610 (1965).
[CrossRef]

Krishnakumar, S.

G. Subramanian, S. Krishnakumar, “An image shearing moiré method to record slopes in flexed plates,” Strain 20, 69–73 (1984).
[CrossRef]

Kumar, P. Ramana

G. Subramanian, P. Ramana Kumar, “Intensity integration technique for contouring two dimensional reflective surfaces,” Exp. Tech. 30, 56–60 (2006).
[CrossRef]

Ligtenberg, F. K.

F. K. Ligtenberg, “The moiré method: a new experimental method for the determination of moments in small slab models,” Proc. SESA 12, 83–98 (1954).

Liu, Q.

Luo, M.

C. J. Tay, H. M. Shang, A. N. Poo, M. Luo “Measurement of surface coordinates and slopes by shearography,” Opt. Laser Technol. 24, 209–213 (1992).
[CrossRef]

Menzel, C.

C. Menzel, E. Menzel, “Beugungsercheinungen optischer Gitter nach intensitat and phase,” Optik (Jena) 3, 247–59 (1948).

Menzel, E.

C. Menzel, E. Menzel, “Beugungsercheinungen optischer Gitter nach intensitat and phase,” Optik (Jena) 3, 247–59 (1948).

Mohan, N. K.

P. R. Sreedhar, N. K. Mohan, R. S. Sirohi, “Displacement and slope measurements using image plane holography with a BaTiO3 crystal,” Opt. Laser Technol. 29, 111–115 (1997).
[CrossRef]

Mohanty, R. K.

Moodie, A.

J. Cowley, A. Moodie, “Fourier images II—The out of focus patterns,” Proc. Phys. Soc. London Sect. B 70, 497–504 (1957).
[CrossRef]

Ohba, R.

Patorski, K.

K. Patorski, “Self imaging and its applications,” Progress in Optics Vol. 27, E. Wolf, ed. (Elsevier, 1989).
[CrossRef]

Politch, J.

Politch, T.

A. Assa, T. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[CrossRef]

Poo, A. N.

C. J. Tay, H. M. Shang, A. N. Poo, M. Luo “Measurement of surface coordinates and slopes by shearography,” Opt. Laser Technol. 24, 209–213 (1992).
[CrossRef]

Rastogi, K. P.

K. P. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251–1263 (1991).
[CrossRef]

Reider, G.

G. Reider, R. Ritter, “Krummungsmessung an belasteter platten nach dem ligentenbergschen moiré verfahren,” Forsch Ing. Wes. 31, 33–44 (1965).
[CrossRef]

Ritter, R.

R. Ritter, W. Wilke, “Slope and contour measurement by reflection grating method and the photogrammetric principle,” Opt. Lasers Eng. 15, 103–113 (1991).
[CrossRef]

R. Ritter, R. Schettler-Koehler, “Curvature measurement by moiré effect,” Exp. Mech. 23, 165–170 (1983).
[CrossRef]

G. Reider, R. Ritter, “Krummungsmessung an belasteter platten nach dem ligentenbergschen moiré verfahren,” Forsch Ing. Wes. 31, 33–44 (1965).
[CrossRef]

Ronchi, Vasco

Rose, K. Jancy

G. Subramanian, K. Jancy Rose, A. Subramanian, “A multiplexed grating shearing interferometer for reflection moiré analysis of partial slopes and curvatures of bent plates,” Exp. Tech. 24, 27–30 (2000).
[CrossRef]

Schettler-Koehler, R.

R. Ritter, R. Schettler-Koehler, “Curvature measurement by moiré effect,” Exp. Mech. 23, 165–170 (1983).
[CrossRef]

Shang, H. M.

C. J. Tay, H. M. Shang, A. N. Poo, M. Luo “Measurement of surface coordinates and slopes by shearography,” Opt. Laser Technol. 24, 209–213 (1992).
[CrossRef]

Sirohi, R. S.

P. R. Sreedhar, N. K. Mohan, R. S. Sirohi, “Displacement and slope measurements using image plane holography with a BaTiO3 crystal,” Opt. Laser Technol. 29, 111–115 (1997).
[CrossRef]

R. K. Mohanty, C. Joenathan, R. S. Sirohi, “Speckle and speckle-shearing interferometers combined for the simultaneous determination of out-of-plane displacement and slope,” Appl. Opt. 24, 3106–3109 (1985).
[CrossRef] [PubMed]

Smolinska, S.

K. Kalestynski, S. Smolinska, “Self restoration of auto idolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).

Sreedhar, P. R.

P. R. Sreedhar, N. K. Mohan, R. S. Sirohi, “Displacement and slope measurements using image plane holography with a BaTiO3 crystal,” Opt. Laser Technol. 29, 111–115 (1997).
[CrossRef]

Subramanian, A.

G. Subramanian, K. Jancy Rose, A. Subramanian, “A multiplexed grating shearing interferometer for reflection moiré analysis of partial slopes and curvatures of bent plates,” Exp. Tech. 24, 27–30 (2000).
[CrossRef]

Subramanian, G.

G. Subramanian, P. Ramana Kumar, “Intensity integration technique for contouring two dimensional reflective surfaces,” Exp. Tech. 30, 56–60 (2006).
[CrossRef]

G. Subramanian, K. Jancy Rose, A. Subramanian, “A multiplexed grating shearing interferometer for reflection moiré analysis of partial slopes and curvatures of bent plates,” Exp. Tech. 24, 27–30 (2000).
[CrossRef]

G. Subramanian, P. N. Akella, “A selective diffraction order based lens-plane grating shearing interferometer for the study of bent plates,” Strain 23, 55–59 (1987).
[CrossRef]

G. Subramanian, S. Krishnakumar, “An image shearing moiré method to record slopes in flexed plates,” Strain 20, 69–73 (1984).
[CrossRef]

Szmodits, K.

K. Szmodits, “Zur theorie des moiré verfahrens,” Proceedings of Symposium on Shell Research, A. M. Hass and A. L. Bowma, eds. (North Holland, 1961), pp. 208–214.

Talbot, H. F.

H. F. Talbot, “Facts relating to optical Science,” Philos. Mag. 9, 403–405 (1836).

Tay, C. J.

C. J. Tay, H. M. Shang, A. N. Poo, M. Luo “Measurement of surface coordinates and slopes by shearography,” Opt. Laser Technol. 24, 209–213 (1992).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, A. Koutsabessis “Slope measurement by means of moiré fringes,” J. Sci. Instrum. 42, 607–610 (1965).
[CrossRef]

Timoshenko, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells2nd ed. (McGraw-Hill, 1959).

Tippur, H. V.

H. V. Tippur, “Simultaneous and real-time measurement of slope and curvature fringes in thin structures using shearing interferometry,” Opt. Eng. 43, 3014–3020 (2004).
[CrossRef]

Wang, L.

L. Wang, D. Z. Yun, “Application of moiré shearing interferometry to slope measurement of shallow shells,” J. Strain Anal. Eng. Des. 27, 45–48 (1992).
[CrossRef]

L. Wang, D. Yun, “Measurement of the deformation of thin plates by moiré interferometry,” Opt. Laser Technol. 23, 151–154 (1991).
[CrossRef]

Wilke, W.

R. Ritter, W. Wilke, “Slope and contour measurement by reflection grating method and the photogrammetric principle,” Opt. Lasers Eng. 15, 103–113 (1991).
[CrossRef]

Woinowsky-Krieger, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells2nd ed. (McGraw-Hill, 1959).

Yun, D.

L. Wang, D. Yun, “Measurement of the deformation of thin plates by moiré interferometry,” Opt. Laser Technol. 23, 151–154 (1991).
[CrossRef]

Yun, D. Z.

L. Wang, D. Z. Yun, “Application of moiré shearing interferometry to slope measurement of shallow shells,” J. Strain Anal. Eng. Des. 27, 45–48 (1992).
[CrossRef]

Appl. Opt. (7)

Exp. Mech. (2)

R. Ritter, R. Schettler-Koehler, “Curvature measurement by moiré effect,” Exp. Mech. 23, 165–170 (1983).
[CrossRef]

A. Assa, T. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[CrossRef]

Exp. Tech. (2)

G. Subramanian, K. Jancy Rose, A. Subramanian, “A multiplexed grating shearing interferometer for reflection moiré analysis of partial slopes and curvatures of bent plates,” Exp. Tech. 24, 27–30 (2000).
[CrossRef]

G. Subramanian, P. Ramana Kumar, “Intensity integration technique for contouring two dimensional reflective surfaces,” Exp. Tech. 30, 56–60 (2006).
[CrossRef]

Forsch Ing. Wes. (1)

G. Reider, R. Ritter, “Krummungsmessung an belasteter platten nach dem ligentenbergschen moiré verfahren,” Forsch Ing. Wes. 31, 33–44 (1965).
[CrossRef]

J. Mod. Opt. (1)

K. P. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251–1263 (1991).
[CrossRef]

J. Sci. Instrum. (1)

P. S. Theocaris, A. Koutsabessis “Slope measurement by means of moiré fringes,” J. Sci. Instrum. 42, 607–610 (1965).
[CrossRef]

J. Strain Anal. Eng. Des. (1)

L. Wang, D. Z. Yun, “Application of moiré shearing interferometry to slope measurement of shallow shells,” J. Strain Anal. Eng. Des. 27, 45–48 (1992).
[CrossRef]

Opt. Acta (1)

K. Kalestynski, S. Smolinska, “Self restoration of auto idolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978).

Opt. Eng. (3)

H. V. Tippur, “Simultaneous and real-time measurement of slope and curvature fringes in thin structures using shearing interferometry,” Opt. Eng. 43, 3014–3020 (2004).
[CrossRef]

F. P. Chiang, T. Y. Kao, “Family of grating techniques of slope and curvature measurements for static and dynamic flexure of plates,” Opt. Eng. 21, 721–742 (1982).

T. Y. Kao, F. P. Chiang, “Family of grating techniques of slope and curvature measurements for static and dynamic flexures of plates,” Opt. Eng. 21, 721–742 (1982).

Opt. Laser Technol. (3)

P. R. Sreedhar, N. K. Mohan, R. S. Sirohi, “Displacement and slope measurements using image plane holography with a BaTiO3 crystal,” Opt. Laser Technol. 29, 111–115 (1997).
[CrossRef]

L. Wang, D. Yun, “Measurement of the deformation of thin plates by moiré interferometry,” Opt. Laser Technol. 23, 151–154 (1991).
[CrossRef]

C. J. Tay, H. M. Shang, A. N. Poo, M. Luo “Measurement of surface coordinates and slopes by shearography,” Opt. Laser Technol. 24, 209–213 (1992).
[CrossRef]

Opt. Lasers Eng. (1)

R. Ritter, W. Wilke, “Slope and contour measurement by reflection grating method and the photogrammetric principle,” Opt. Lasers Eng. 15, 103–113 (1991).
[CrossRef]

Optik (Jena) (1)

C. Menzel, E. Menzel, “Beugungsercheinungen optischer Gitter nach intensitat and phase,” Optik (Jena) 3, 247–59 (1948).

Philos. Mag. (1)

H. F. Talbot, “Facts relating to optical Science,” Philos. Mag. 9, 403–405 (1836).

Proc. Phys. Soc. London Sect. B (1)

J. Cowley, A. Moodie, “Fourier images II—The out of focus patterns,” Proc. Phys. Soc. London Sect. B 70, 497–504 (1957).
[CrossRef]

Proc. SESA (1)

F. K. Ligtenberg, “The moiré method: a new experimental method for the determination of moments in small slab models,” Proc. SESA 12, 83–98 (1954).

Strain (2)

G. Subramanian, S. Krishnakumar, “An image shearing moiré method to record slopes in flexed plates,” Strain 20, 69–73 (1984).
[CrossRef]

G. Subramanian, P. N. Akella, “A selective diffraction order based lens-plane grating shearing interferometer for the study of bent plates,” Strain 23, 55–59 (1987).
[CrossRef]

Other (3)

K. Szmodits, “Zur theorie des moiré verfahrens,” Proceedings of Symposium on Shell Research, A. M. Hass and A. L. Bowma, eds. (North Holland, 1961), pp. 208–214.

K. Patorski, “Self imaging and its applications,” Progress in Optics Vol. 27, E. Wolf, ed. (Elsevier, 1989).
[CrossRef]

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells2nd ed. (McGraw-Hill, 1959).

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Figures (7)

Fig. 1
Fig. 1

Schematic showing orientation of grating G in the y z plane.

Fig. 2
Fig. 2

Schematic of the experimental arrangement.

Fig. 3
Fig. 3

(a) Slope pattern for a cantilever beam of length 45 mm , width 15 mm , and thickness 2 mm of PMMA material bound along one side. The specimen was normally loaded on the opposite side, as shown by arrow, using a precision translating device. (b) Schematic of the loading configuration.

Fig. 4
Fig. 4

Plot showing comparison of experimental and theoretical investigations for a cantilever beam specimen.

Fig. 5
Fig. 5

(a) Slope pattern for front-reflecting, simply supported, centrally loaded circular plate. (b) Schematic of the loading configuration.

Fig. 6
Fig. 6

Slope pattern for a simply supported circular plate, centrally loaded with out-of-plane displacement of 0.18 mm for (a) a flawless plate, (b) a plate with few defects 9 mm from the center and approximately half the depth of circular plate, (c) a plate with more defects, randomly scattered, of depth 0.5 1.0 mm , and (d) a plate with a crack.

Fig. 7
Fig. 7

Slope pattern for an equilateral-triangular-based plate of side 38 mm and thickness 2 mm of PMMA material bound along the corners. The slope curve (a) along the horizontal direction and (b) along the vertical direction. The specimen was normally loaded on the opposite side, at the center.

Equations (6)

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y cos θ 2 = z sin θ 2 + k p ; k = 0 , ± 1 , ± 2 , ± 3 .
( y + z m φ 2 ) cos θ 2 = z sin θ 2 + m p .
( z m φ 2 ) cos θ 2 = ( m k ) p
φ = 2 w y = 2 l p z m .
w y = l p z m .
w y = P ( 2 L d ) d 2 E I ,

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