Abstract

The influence of displacement and its first- and second-order derivative components on curvature fringe formations in speckle shearography is discussed. The results show that (a) all the displacement components have no direct influence on curvature fringe formations; (b) only the first-order derivative component along the centerline of three apertures has an influence on curvature fringe formations, whereas all the other first-order components have no influence; and (c) all the second-order derivative components have no influence on curvature fringe formations. Results from theory and experiments are in good agreement.

© 2002 Optical Society of America

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References

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  1. R. S. Sirohi, “Speckle methods in experimental mechanics,” in Speckle Metrology, R. S. Sirohi, ed. (Marcel Dekker, New York, 1993), pp. 99–155.
  2. 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]
  3. C. J. Tay, S. L. Toh, H. M. Shang, Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91–98 (1994).
    [CrossRef]
  4. K. F. Wang, A. K. Tieu, E. B. Li, “Measurement of pure curvature fringe distribution by using a double whole-field filtering technique,” Opt. Laser Technol. 31, 289–294 (1999).
    [CrossRef]
  5. K. F. Wang, A. K. Tieu, E. B. Li, “Simultaneous measurement of pure curvature and twist fringe distribution fields by a five-aperture shearing and two-Fourier filtering technique,” Appl. Opt. 39, 2577–2583 (2000).
    [CrossRef]
  6. Y. Iwahashi, K. Iwata, R. Nagata, “Influence of in-plane displacement in single-aperture and double-aperture speckle shearing interferometry,” Appl. Opt. 24, 2189–2192 (1985).
    [CrossRef] [PubMed]
  7. N. K. Mohan, P. J. Masalkar, V. M. Murukeshan, R. S. Sirohi, “Separation of the influence of in-plane displacement in multiaperture speckle shear interferometry,” Opt. Eng. 33, 1973–1982 (1994).
    [CrossRef]
  8. N. K. Mohan, R. S. Sirohi, “Fringe formation in symmetric three-aperture speckle shear interferometry: an analysis,” Opt. Lasers Eng. 26, 437–447 (1997).
    [CrossRef]
  9. N. K. Mohan, R. S. Sirohi, “Fringe formation in multiaperture speckle shear interferometry,” Appl. Opt. 35, 1617–1622 (1996).
    [CrossRef] [PubMed]
  10. K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and strain components on slope fringe distributions in double-aperture speckle wedge-shearing interferometry,” Opt. Laser Technol. 31, 549–554 (1999).
    [CrossRef]
  11. K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and double-aperture orientation on slope fringe formation in double-shearing-aperture speckle interferometry,” Opt. Eng. 39, 2124–2128 (2000).
    [CrossRef]
  12. K. F. Wang, A. K. Tieu, E. B. Li, “Influence of shearing direction on slope fringe distributions in the presence of in-plane displacement and strain components in double-aperture speckle shearing interferometry,” Opt. Commun. 174, 69–74 (2000).
    [CrossRef]
  13. H. Reismann, Elastic Plates: Theory and Application (Wiley, New York, 1988), pp. 121–123.

2000

K. F. Wang, A. K. Tieu, E. B. Li, “Simultaneous measurement of pure curvature and twist fringe distribution fields by a five-aperture shearing and two-Fourier filtering technique,” Appl. Opt. 39, 2577–2583 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and double-aperture orientation on slope fringe formation in double-shearing-aperture speckle interferometry,” Opt. Eng. 39, 2124–2128 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of shearing direction on slope fringe distributions in the presence of in-plane displacement and strain components in double-aperture speckle shearing interferometry,” Opt. Commun. 174, 69–74 (2000).
[CrossRef]

1999

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and strain components on slope fringe distributions in double-aperture speckle wedge-shearing interferometry,” Opt. Laser Technol. 31, 549–554 (1999).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Measurement of pure curvature fringe distribution by using a double whole-field filtering technique,” Opt. Laser Technol. 31, 289–294 (1999).
[CrossRef]

1997

N. K. Mohan, R. S. Sirohi, “Fringe formation in symmetric three-aperture speckle shear interferometry: an analysis,” Opt. Lasers Eng. 26, 437–447 (1997).
[CrossRef]

1996

1994

N. K. Mohan, P. J. Masalkar, V. M. Murukeshan, R. S. Sirohi, “Separation of the influence of in-plane displacement in multiaperture speckle shear interferometry,” Opt. Eng. 33, 1973–1982 (1994).
[CrossRef]

C. J. Tay, S. L. Toh, H. M. Shang, Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91–98 (1994).
[CrossRef]

1985

1984

Iwahashi, Y.

Iwata, K.

Kothiyal, M. P.

Li, E. B.

K. F. Wang, A. K. Tieu, E. B. Li, “Simultaneous measurement of pure curvature and twist fringe distribution fields by a five-aperture shearing and two-Fourier filtering technique,” Appl. Opt. 39, 2577–2583 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and double-aperture orientation on slope fringe formation in double-shearing-aperture speckle interferometry,” Opt. Eng. 39, 2124–2128 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of shearing direction on slope fringe distributions in the presence of in-plane displacement and strain components in double-aperture speckle shearing interferometry,” Opt. Commun. 174, 69–74 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and strain components on slope fringe distributions in double-aperture speckle wedge-shearing interferometry,” Opt. Laser Technol. 31, 549–554 (1999).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Measurement of pure curvature fringe distribution by using a double whole-field filtering technique,” Opt. Laser Technol. 31, 289–294 (1999).
[CrossRef]

Lin, Q. Y.

C. J. Tay, S. L. Toh, H. M. Shang, Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91–98 (1994).
[CrossRef]

Masalkar, P. J.

N. K. Mohan, P. J. Masalkar, V. M. Murukeshan, R. S. Sirohi, “Separation of the influence of in-plane displacement in multiaperture speckle shear interferometry,” Opt. Eng. 33, 1973–1982 (1994).
[CrossRef]

Mohan, N. K.

N. K. Mohan, R. S. Sirohi, “Fringe formation in symmetric three-aperture speckle shear interferometry: an analysis,” Opt. Lasers Eng. 26, 437–447 (1997).
[CrossRef]

N. K. Mohan, R. S. Sirohi, “Fringe formation in multiaperture speckle shear interferometry,” Appl. Opt. 35, 1617–1622 (1996).
[CrossRef] [PubMed]

N. K. Mohan, P. J. Masalkar, V. M. Murukeshan, R. S. Sirohi, “Separation of the influence of in-plane displacement in multiaperture speckle shear interferometry,” Opt. Eng. 33, 1973–1982 (1994).
[CrossRef]

Murukeshan, V. M.

N. K. Mohan, P. J. Masalkar, V. M. Murukeshan, R. S. Sirohi, “Separation of the influence of in-plane displacement in multiaperture speckle shear interferometry,” Opt. Eng. 33, 1973–1982 (1994).
[CrossRef]

Nagata, R.

Reismann, H.

H. Reismann, Elastic Plates: Theory and Application (Wiley, New York, 1988), pp. 121–123.

Shang, H. M.

C. J. Tay, S. L. Toh, H. M. Shang, Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91–98 (1994).
[CrossRef]

Sharma, D. K.

Sirohi, R. S.

N. K. Mohan, R. S. Sirohi, “Fringe formation in symmetric three-aperture speckle shear interferometry: an analysis,” Opt. Lasers Eng. 26, 437–447 (1997).
[CrossRef]

N. K. Mohan, R. S. Sirohi, “Fringe formation in multiaperture speckle shear interferometry,” Appl. Opt. 35, 1617–1622 (1996).
[CrossRef] [PubMed]

N. K. Mohan, P. J. Masalkar, V. M. Murukeshan, R. S. Sirohi, “Separation of the influence of in-plane displacement in multiaperture speckle shear interferometry,” Opt. Eng. 33, 1973–1982 (1994).
[CrossRef]

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]

R. S. Sirohi, “Speckle methods in experimental mechanics,” in Speckle Metrology, R. S. Sirohi, ed. (Marcel Dekker, New York, 1993), pp. 99–155.

Tay, C. J.

C. J. Tay, S. L. Toh, H. M. Shang, Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91–98 (1994).
[CrossRef]

Tieu, A. K.

K. F. Wang, A. K. Tieu, E. B. Li, “Simultaneous measurement of pure curvature and twist fringe distribution fields by a five-aperture shearing and two-Fourier filtering technique,” Appl. Opt. 39, 2577–2583 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of shearing direction on slope fringe distributions in the presence of in-plane displacement and strain components in double-aperture speckle shearing interferometry,” Opt. Commun. 174, 69–74 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and double-aperture orientation on slope fringe formation in double-shearing-aperture speckle interferometry,” Opt. Eng. 39, 2124–2128 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and strain components on slope fringe distributions in double-aperture speckle wedge-shearing interferometry,” Opt. Laser Technol. 31, 549–554 (1999).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Measurement of pure curvature fringe distribution by using a double whole-field filtering technique,” Opt. Laser Technol. 31, 289–294 (1999).
[CrossRef]

Toh, S. L.

C. J. Tay, S. L. Toh, H. M. Shang, Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91–98 (1994).
[CrossRef]

Wang, K. F.

K. F. Wang, A. K. Tieu, E. B. Li, “Simultaneous measurement of pure curvature and twist fringe distribution fields by a five-aperture shearing and two-Fourier filtering technique,” Appl. Opt. 39, 2577–2583 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of shearing direction on slope fringe distributions in the presence of in-plane displacement and strain components in double-aperture speckle shearing interferometry,” Opt. Commun. 174, 69–74 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and double-aperture orientation on slope fringe formation in double-shearing-aperture speckle interferometry,” Opt. Eng. 39, 2124–2128 (2000).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and strain components on slope fringe distributions in double-aperture speckle wedge-shearing interferometry,” Opt. Laser Technol. 31, 549–554 (1999).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Measurement of pure curvature fringe distribution by using a double whole-field filtering technique,” Opt. Laser Technol. 31, 289–294 (1999).
[CrossRef]

Appl. Opt.

Opt. Commun.

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of shearing direction on slope fringe distributions in the presence of in-plane displacement and strain components in double-aperture speckle shearing interferometry,” Opt. Commun. 174, 69–74 (2000).
[CrossRef]

Opt. Eng.

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and double-aperture orientation on slope fringe formation in double-shearing-aperture speckle interferometry,” Opt. Eng. 39, 2124–2128 (2000).
[CrossRef]

N. K. Mohan, P. J. Masalkar, V. M. Murukeshan, R. S. Sirohi, “Separation of the influence of in-plane displacement in multiaperture speckle shear interferometry,” Opt. Eng. 33, 1973–1982 (1994).
[CrossRef]

Opt. Laser Technol.

K. F. Wang, A. K. Tieu, E. B. Li, “Influence of in-plane displacement and strain components on slope fringe distributions in double-aperture speckle wedge-shearing interferometry,” Opt. Laser Technol. 31, 549–554 (1999).
[CrossRef]

C. J. Tay, S. L. Toh, H. M. Shang, Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91–98 (1994).
[CrossRef]

K. F. Wang, A. K. Tieu, E. B. Li, “Measurement of pure curvature fringe distribution by using a double whole-field filtering technique,” Opt. Laser Technol. 31, 289–294 (1999).
[CrossRef]

Opt. Lasers Eng.

N. K. Mohan, R. S. Sirohi, “Fringe formation in symmetric three-aperture speckle shear interferometry: an analysis,” Opt. Lasers Eng. 26, 437–447 (1997).
[CrossRef]

Other

R. S. Sirohi, “Speckle methods in experimental mechanics,” in Speckle Metrology, R. S. Sirohi, ed. (Marcel Dekker, New York, 1993), pp. 99–155.

H. Reismann, Elastic Plates: Theory and Application (Wiley, New York, 1988), pp. 121–123.

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Figures (5)

Fig. 1
Fig. 1

Imaging system.

Fig. 2
Fig. 2

Filtering system.

Fig. 3
Fig. 3

Theoretical curvature fringe contours at ∂u(x, y)/∂x = (a) 0, (b) 755 μ∊, (c) 1510 μ∊, and (d) 2265 μ∊.

Fig. 4
Fig. 4

Pure curvature fringe contours at ∂u(x, y)/∂x = 0: (a) theoretical and (b) experimental.

Fig. 5
Fig. 5

Compound curvature fringe contours at ∂u(x, y)/∂x = 755 μ∊: (a) theoretical and (b) experimental.

Equations (27)

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U1=a1 expiϕ1+a2 expiϕ2+β+a3 expiϕ3-β,
I1=i=13 ai2+2a1a2 cosϕ21+β+2a1a3 cosϕ13+β+2a2a3 cosϕ23+2β,
U2=a1 expiϕ1+δ1+a2 expiϕ2+δ2+β+a3 expiϕ3+δ3-β,
δ1=K1-K0 · Lx, y, δ2=K2-K0 · Lx+Δ, y, δ3=K3-K0 · Lx-Δ, y,
I2=i=13 ai2+2a1a2 cosϕ21+δ21+β+2a1a3×cosϕ13+δ13+β+2a2a3 cosϕ23+δ23+2β,
δ21=K2-K0 · Lx+Δ/2, yx Δ+K2-K1 · Lx, y, δ13=K3-K0 · Lx-Δ/2, yx Δ+K1-K3 · Lx, y.
U1f=a1a2 expiϕ21+β+a1a3 expiϕ13+β,
I1f=a12a22+a32+2a12a2a3 cosϕ21-ϕ13.
U2f=a1a2 expiϕ21+δ21+β+a1a3 expiϕ13+δ13+β,
I2f=a12a22+a32+2a12a2a3 cosϕ21-ϕ13+δ21-δ13.
It=I1f+I2f=2a12a22+a32+2a12a2a3×cosϕ21-ϕ13+cosϕ21-ϕ13+δ21-δ13.
Uf=a12a2a3expiϕ21-ϕ13+expiϕ21-ϕ13+δ21-δ13,
If=2a14a22a321+cosδ21-δ13.
δ21-δ13=K2+K3-2K02 · 2Lx, yx2 Δ2+K2-K3· Lx, yx Δ+K2+K3-2K1 · Lx, y.
Ki, k  1 i=1, 2, 3,
K2+K3-2K10.
δ21-δ13=K1-K0 · 2Lx, yx2 Δ2+K2-K3· Lx, yx Δ,
K0=-2πλk, K1=2πλk, K2=2πλDdoi+k, K3=2πλ-Ddoi+k,
δ21-δ13=4πΔ2λ2wx, yx2+DdoΔux, yx.
δ21-δ13=2Nπ N=0, ±1, ±2, ,
2wx, yx2+DdoΔux, yx=Nλ2Δ2N=0, ±1, ±2, .
wx, y=wmax1-x2+y2r2+x2+y2r2lnx2+y2r2x2+y2r, 
2wx, yx2=2wmaxr22x2x2+y2+lnx2+y2r2x2+y2r. 
ux, y=x x2+y2r,
ux, yx= x2+y2r.
2wmaxr22x2x2+y2+lnx2+y2r2+DdoΔ=Nλ2Δ2x2+y2r, N=0, ±1, ±2, .
2x2x2+y2+lnx2+y2r2+563=1.71Nx2+y2r, N=0, ±1, ±2, .

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