Abstract

We present a modified method of shearography, known herein as multiple-image shearography, whereby the curvatures of an object can be measured directly from the resulting fringes. It employs an image-shearing camera that produces three sheared images simultaneously to interfere with each other in the image plane. When film is doubly exposed before and after an object is deformed, three sets of fringes are observed of which one set would depict the second-order derivatives of surface displacement.

The theory of the multiple-image shearography technique and its application to curvature measurements in plate bending are presented.

© 1995 Optical Society of America

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References

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  1. F. P. Chiang, M. Bailangadi, “A method for direct determination of small curvatures,” J. Appl. Mech. 42(1), 29–31 (March1975).
    [CrossRef]
  2. J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (March1986).
    [CrossRef]
  3. F. P. Chiang, T. Y. Kao, “An optical method for generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251–260 (1979).
    [CrossRef]
  4. R. Ritter, “Reflection moire methods for plate bending studies,” Opt. Eng. 21, 663–671 (1982).
  5. T. Y. Kao, F. P. Chiang, “Family of grating techniques of slope and curvature measurements for static and dynamic flexure of plates,” Opt. Eng. 21, 721–742 (1982).
  6. K. V. Sriram, M. P. Kothiyal, R. S. Sirohi, “Talbot interferometry in noncollimated illumination for curvature and focal length measurements,” Appl. Opt. 31, 75–79 (1992).
    [CrossRef] [PubMed]
  7. Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plate,” in Proceedings of the 1982 Joint Conference on Experimental Mechanics, Oahu-Maui, Hawaii (May 1982), pp. 514–519.
  8. A. Assa, J. Politch, A. A. Bester, “Slope and curvature measurement by a double frequency-grating shearing interferometer,” Exp. Mech. 19(4), 129–137 (December1979).
    [CrossRef]
  9. P. M. Boone, “Determination of slope and strain contours by double-exposure shearing interferometry,” Exp. Mech. 15(8), 295–302 (August1975).
    [CrossRef]
  10. P. K. 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]
  11. Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21, 391–395 (1982).
  12. Y. Y. Hung, A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88 (1979).
    [CrossRef]
  13. D. K. Sharma, R. S. Sirohi, M. P. Kothiyal, “Simultaneous measurement of slope and curvature with a three-aperture speckle shearing interferometer,” Appl. Opt. 23, 1542–1546 (1984).
    [CrossRef] [PubMed]
  14. 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]
  15. C. Joenathan, R. K. Mohanty, R. S. Sirohi, “Hololens in speckle and speckle shear interferometry,” Appl. Opt. 24, 1294–1298 (1985).
    [CrossRef] [PubMed]
  16. R. S. Sirohi, ed., Speckle Metrology (Dekker, New York, 1993).
  17. P. A. Klumpp, E. Schnack, “Speckle-interferometric camera for displacement measurements,” Exp. Mech. 30, 411–415 (December1990).
    [CrossRef]
  18. P. A. Klumpp, E. Schnack, “White light reconstruction setup for shearograms,” Appl. Opt. 28, 3270–3271 (1989).
    [CrossRef] [PubMed]
  19. P. A. Klumpp, “Simple spatial filtering for shearograms,” Opt. Laser Technol. 21, 105–111 (1989).
    [CrossRef]
  20. S. L. Toh, C. J. Tay, H. M. Shang, Q. Y. Lin, “Analysis of shearogram reconstruction,” Appl. Opt. 32, 4929–4933 (1993).
    [CrossRef] [PubMed]

1993 (1)

1992 (1)

1991 (1)

P. K. 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]

1990 (1)

P. A. Klumpp, E. Schnack, “Speckle-interferometric camera for displacement measurements,” Exp. Mech. 30, 411–415 (December1990).
[CrossRef]

1989 (2)

P. A. Klumpp, E. Schnack, “White light reconstruction setup for shearograms,” Appl. Opt. 28, 3270–3271 (1989).
[CrossRef] [PubMed]

P. A. Klumpp, “Simple spatial filtering for shearograms,” Opt. Laser Technol. 21, 105–111 (1989).
[CrossRef]

1986 (1)

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (March1986).
[CrossRef]

1985 (2)

1984 (1)

1982 (3)

R. Ritter, “Reflection moire methods for plate bending studies,” Opt. Eng. 21, 663–671 (1982).

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

Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21, 391–395 (1982).

1979 (3)

Y. Y. Hung, A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88 (1979).
[CrossRef]

A. Assa, J. Politch, A. A. Bester, “Slope and curvature measurement by a double frequency-grating shearing interferometer,” Exp. Mech. 19(4), 129–137 (December1979).
[CrossRef]

F. P. Chiang, T. Y. Kao, “An optical method for generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251–260 (1979).
[CrossRef]

1975 (2)

P. M. Boone, “Determination of slope and strain contours by double-exposure shearing interferometry,” Exp. Mech. 15(8), 295–302 (August1975).
[CrossRef]

F. P. Chiang, M. Bailangadi, “A method for direct determination of small curvatures,” J. Appl. Mech. 42(1), 29–31 (March1975).
[CrossRef]

Assa, A.

A. Assa, J. Politch, A. A. Bester, “Slope and curvature measurement by a double frequency-grating shearing interferometer,” Exp. Mech. 19(4), 129–137 (December1979).
[CrossRef]

Bailangadi, M.

F. P. Chiang, M. Bailangadi, “A method for direct determination of small curvatures,” J. Appl. Mech. 42(1), 29–31 (March1975).
[CrossRef]

Bester, A. A.

A. Assa, J. Politch, A. A. Bester, “Slope and curvature measurement by a double frequency-grating shearing interferometer,” Exp. Mech. 19(4), 129–137 (December1979).
[CrossRef]

Boone, P. M.

P. M. Boone, “Determination of slope and strain contours by double-exposure shearing interferometry,” Exp. Mech. 15(8), 295–302 (August1975).
[CrossRef]

Chao, Y. J.

Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plate,” in Proceedings of the 1982 Joint Conference on Experimental Mechanics, Oahu-Maui, Hawaii (May 1982), pp. 514–519.

Chiang, F. P.

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

F. P. Chiang, T. Y. Kao, “An optical method for generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251–260 (1979).
[CrossRef]

F. P. Chiang, M. Bailangadi, “A method for direct determination of small curvatures,” J. Appl. Mech. 42(1), 29–31 (March1975).
[CrossRef]

Durelli, A. J.

Y. Y. Hung, A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88 (1979).
[CrossRef]

Hung, Y. Y.

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (March1986).
[CrossRef]

Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21, 391–395 (1982).

Y. Y. Hung, A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88 (1979).
[CrossRef]

Joenathan, C.

Kao, T. Y.

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

F. P. Chiang, T. Y. Kao, “An optical method for generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251–260 (1979).
[CrossRef]

Klumpp, P. A.

P. A. Klumpp, E. Schnack, “Speckle-interferometric camera for displacement measurements,” Exp. Mech. 30, 411–415 (December1990).
[CrossRef]

P. A. Klumpp, “Simple spatial filtering for shearograms,” Opt. Laser Technol. 21, 105–111 (1989).
[CrossRef]

P. A. Klumpp, E. Schnack, “White light reconstruction setup for shearograms,” Appl. Opt. 28, 3270–3271 (1989).
[CrossRef] [PubMed]

Kothiyal, M. P.

Lin, Q. Y.

Mohanty, R. K.

Politch, J.

A. Assa, J. Politch, A. A. Bester, “Slope and curvature measurement by a double frequency-grating shearing interferometer,” Exp. Mech. 19(4), 129–137 (December1979).
[CrossRef]

Rastogi, P. K.

P. K. 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]

Ritter, R.

R. Ritter, “Reflection moire methods for plate bending studies,” Opt. Eng. 21, 663–671 (1982).

Schnack, E.

P. A. Klumpp, E. Schnack, “Speckle-interferometric camera for displacement measurements,” Exp. Mech. 30, 411–415 (December1990).
[CrossRef]

P. A. Klumpp, E. Schnack, “White light reconstruction setup for shearograms,” Appl. Opt. 28, 3270–3271 (1989).
[CrossRef] [PubMed]

Shang, H. M.

Sharma, D. K.

Sirohi, R. S.

Sriram, K. V.

Sutton, M. A.

Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plate,” in Proceedings of the 1982 Joint Conference on Experimental Mechanics, Oahu-Maui, Hawaii (May 1982), pp. 514–519.

Takezaki, J.

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (March1986).
[CrossRef]

Tay, C. J.

Taylor, C. E.

Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plate,” in Proceedings of the 1982 Joint Conference on Experimental Mechanics, Oahu-Maui, Hawaii (May 1982), pp. 514–519.

Toh, S. L.

Appl. Opt. (6)

Exp. Mech. (3)

A. Assa, J. Politch, A. A. Bester, “Slope and curvature measurement by a double frequency-grating shearing interferometer,” Exp. Mech. 19(4), 129–137 (December1979).
[CrossRef]

P. M. Boone, “Determination of slope and strain contours by double-exposure shearing interferometry,” Exp. Mech. 15(8), 295–302 (August1975).
[CrossRef]

P. A. Klumpp, E. Schnack, “Speckle-interferometric camera for displacement measurements,” Exp. Mech. 30, 411–415 (December1990).
[CrossRef]

Int. J. Solids Struct. (1)

F. P. Chiang, T. Y. Kao, “An optical method for generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251–260 (1979).
[CrossRef]

J. Appl. Mech. (2)

F. P. Chiang, M. Bailangadi, “A method for direct determination of small curvatures,” J. Appl. Mech. 42(1), 29–31 (March1975).
[CrossRef]

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (March1986).
[CrossRef]

J. Mod. Opt. (1)

P. K. 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. Strain Anal. (1)

Y. Y. Hung, A. J. Durelli, “Simultaneous measurement of three displacement derivatives using a multiple image-shearing interferometric camera,” J. Strain Anal. 14, 81–88 (1979).
[CrossRef]

Opt. Eng. (3)

Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21, 391–395 (1982).

R. Ritter, “Reflection moire methods for plate bending studies,” Opt. Eng. 21, 663–671 (1982).

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

Opt. Laser Technol. (1)

P. A. Klumpp, “Simple spatial filtering for shearograms,” Opt. Laser Technol. 21, 105–111 (1989).
[CrossRef]

Other (2)

R. S. Sirohi, ed., Speckle Metrology (Dekker, New York, 1993).

Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plate,” in Proceedings of the 1982 Joint Conference on Experimental Mechanics, Oahu-Maui, Hawaii (May 1982), pp. 514–519.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of multiple-image shearography.

Fig. 2
Fig. 2

Setup for the high-pass Fourier filtering system.

Fig. 3
Fig. 3

Image reconstructed from a multiple-image shearogram showing the second-order derivatives and perturbed fringes.

Fig. 4
Fig. 4

Comparison of theoretical and experimental values of second-order derivatives ∂2 w/∂x 2 along the plate center line.

Fig. 5
Fig. 5

Image reconstructed from a multiple-image shearogram showing the slope fringes.

Equations (19)

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I = a 2 [ 3 + 2 cos ϕ + 2 cos ( ϕ + d ϕ ) + 2 cos ϕ ] ,
ϕ = θ ( x , y ) - θ ( x - δ x , y ) , d ϕ = [ θ ( x + δ x , y ) - θ ( x , y ) ] - [ θ ( x , y ) - θ ( x - δ x , y ) ] , ϕ = θ ( x + δ x , y ) - θ ( x - δ x , y ) ,
I = a 2 [ 3 + 2 cos ( ϕ + Δ ) + 2 cos ( ϕ + d ϕ + Δ + d Δ ) + 2 cos ( ϕ + Δ ) ] ,
I s = I + I = 2 a 2 [ 3 + cos ϕ + cos ( ϕ + d ϕ ) + cos ϕ + cos ( ϕ + Δ ) + cos ( ϕ + d ϕ + Δ + d Δ ) + cos ( ϕ + Δ ) ] = 2 a 2 [ 3 + 2 cos ( ϕ + Δ 2 ) cos Δ 2 + 2 cos ( ϕ + d ϕ + Δ + d Δ 2 ) cos Δ + d Δ 2 + 2 cos ( ϕ + Δ 2 ) cos Δ 2 ] .
I s cos ( ϕ + Δ 2 ) cos Δ 2 + cos ( ϕ + d ϕ + Δ + d Δ 2 ) cos Δ + d Δ 2 ,
I s cos ( ϕ + Δ 2 ) cos Δ 2 .
cos Δ 2 + cos Δ + d Δ 2 = cos ( Δ 2 + d Δ 4 ) cos d Δ 4 = 0.
Δ + d Δ 2 = ( 2 n + 1 ) π ,             n = 0 , ± 1 , ± 2 , .
d Δ = 2 ( 2 n + 1 ) π ,             n = 0 , ± 1 , ± 2 , .
Δ = ( 2 n + 1 ) π ,             n = 0 , ± 1 ,     ± 2 , .
Δ = 2 π λ ( A u x + B v x + C w x ) δ x ,
d Δ = 2 π λ ( A 2 u x 2 + B 2 v x 2 + C 2 w x 2 ) δ x 2 ,
Δ = 4 π λ ( A u x + B v x + C w x ) δ x ,
A = x - x s R s + x - x 0 R 0 , B = y - y s R s + y - y 0 R 0 , C = z - z s R s + z - z 0 R 0 , R s = [ ( x - x s ) 2 + ( y - y s ) 2 + ( z - z s ) 2 ] 1 / 2 , R 0 = [ ( x - x o ) 2 + ( y - y o ) 2 + ( z - z o ) 2 ] 1 / 2 .
2 λ ( A u x + B v x + C w x ) δ x + 1 λ ( A 2 u x 2 + B 2 v x 2 + C 2 w x 2 ) δ x 2 = 2 n + 1 ,
1 λ ( A 2 u x 2 + B 2 v x 2 + C 2 w x 2 ) δ x 2 = 2 n + 1 ,
4 λ ( A u x + B v x + C w x ) δ x = 2 n + 1.
w x δ x + 1 2 2 w x 2 δ x 2 = 2 n + 1 4 λ ,
2 w x 2 δ x 2 = ( n + 1 2 ) λ .

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