Abstract

An overdeterministic least-squares phase-stepping method for automated photoelasticity is described. Problems associated with isochromatic–isoclinic interaction are solved by use of a three-wavelength method to calculate the value of the isochromatic parameter and the isoclinic angle. The ramped isoclinic phase map can now be unwrapped to give the orientation of the principal stresses with respect to a reference axis of the polariscope unambiguously. A three-wavelength approach to determination of the absolute value of the isochromatic parameter is shown to give reliable results also.

© 1997 Optical Society of America

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References

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  1. A. S. Voloshin, C. P. Burger, “Half-fringe photoelasticity: a new approach to whole-field stress analysis,” Expt. Mech. 23, 304–313 (1983).
    [CrossRef]
  2. A. S. Redner, “Photoelastic measurements by means of computer-assisted spectral contents analysis,” Expt. Mech. 25, 148–153 (1985).
    [CrossRef]
  3. D. W. Robinson, D. C. Williams, M. Kujawinska, “A phase measurement method for use in photo elastic stress analysis,” (National Physical Laboratory, Teddington, U.K., 1988).
  4. E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–56 (1991).
    [CrossRef]
  5. S. J. Haake, Z. F. Wang, E. A. Patterson, “Evaluation of full field automated photoelastic analysis based on phase stepping,” Expt. Technol. 17, 19–25 (1993).
    [CrossRef]
  6. S. J. Haake, E. A. Patterson, “Photoelastic analysis using a full-field spectral contents analyser,” in Proceedings of the International Conference on Photoelasticity: New Instrumentation, Materials and Data Processing Techniques, London, 1993 (SIRA Communications, Chislehurst, U.K., 1993), Vol. 1, pp. 1.1–1.2.
  7. C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
    [CrossRef]
  8. J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
    [CrossRef]
  9. Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
    [CrossRef]
  10. A. Ajovalasit, S. Barone, G. Petrucci, “Automated photoelasticity in white light: influence of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
    [CrossRef]
  11. A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
    [CrossRef]
  12. C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
    [CrossRef]
  13. Z. F. Wang, E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
    [CrossRef]
  14. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]

1996

C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
[CrossRef]

1995

Z. F. Wang, E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
[CrossRef]

A. Ajovalasit, S. Barone, G. Petrucci, “Automated photoelasticity in white light: influence of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

1994

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

1993

S. J. Haake, Z. F. Wang, E. A. Patterson, “Evaluation of full field automated photoelastic analysis based on phase stepping,” Expt. Technol. 17, 19–25 (1993).
[CrossRef]

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

1991

E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–56 (1991).
[CrossRef]

1989

1985

A. S. Redner, “Photoelastic measurements by means of computer-assisted spectral contents analysis,” Expt. Mech. 25, 148–153 (1985).
[CrossRef]

1983

A. S. Voloshin, C. P. Burger, “Half-fringe photoelasticity: a new approach to whole-field stress analysis,” Expt. Mech. 23, 304–313 (1983).
[CrossRef]

Ajovalasit, A.

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

A. Ajovalasit, S. Barone, G. Petrucci, “Automated photoelasticity in white light: influence of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

Barone, S.

A. Ajovalasit, S. Barone, G. Petrucci, “Automated photoelasticity in white light: influence of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

Bryanston-Cross, P. J.

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Buckberry, C.

C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
[CrossRef]

Burger, C. P.

A. S. Voloshin, C. P. Burger, “Half-fringe photoelasticity: a new approach to whole-field stress analysis,” Expt. Mech. 23, 304–313 (1983).
[CrossRef]

Carazo-Alvarez, J.

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

Fujisawa, M.

Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Haake, S. J.

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

S. J. Haake, Z. F. Wang, E. A. Patterson, “Evaluation of full field automated photoelastic analysis based on phase stepping,” Expt. Technol. 17, 19–25 (1993).
[CrossRef]

S. J. Haake, E. A. Patterson, “Photoelastic analysis using a full-field spectral contents analyser,” in Proceedings of the International Conference on Photoelasticity: New Instrumentation, Materials and Data Processing Techniques, London, 1993 (SIRA Communications, Chislehurst, U.K., 1993), Vol. 1, pp. 1.1–1.2.

Huntley, J. M.

Judge, T. R.

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Kujawinska, M.

D. W. Robinson, D. C. Williams, M. Kujawinska, “A phase measurement method for use in photo elastic stress analysis,” (National Physical Laboratory, Teddington, U.K., 1988).

Morimoto, Y.

Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Patterson, E. A.

Z. F. Wang, E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
[CrossRef]

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

S. J. Haake, Z. F. Wang, E. A. Patterson, “Evaluation of full field automated photoelastic analysis based on phase stepping,” Expt. Technol. 17, 19–25 (1993).
[CrossRef]

E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–56 (1991).
[CrossRef]

S. J. Haake, E. A. Patterson, “Photoelastic analysis using a full-field spectral contents analyser,” in Proceedings of the International Conference on Photoelasticity: New Instrumentation, Materials and Data Processing Techniques, London, 1993 (SIRA Communications, Chislehurst, U.K., 1993), Vol. 1, pp. 1.1–1.2.

Petrucci, G.

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

A. Ajovalasit, S. Barone, G. Petrucci, “Automated photoelasticity in white light: influence of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

Quan, C.

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Redner, A. S.

A. S. Redner, “Photoelastic measurements by means of computer-assisted spectral contents analysis,” Expt. Mech. 25, 148–153 (1985).
[CrossRef]

Robinson, D. W.

D. W. Robinson, D. C. Williams, M. Kujawinska, “A phase measurement method for use in photo elastic stress analysis,” (National Physical Laboratory, Teddington, U.K., 1988).

Towers, D.

C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
[CrossRef]

Voloshin, A. S.

A. S. Voloshin, C. P. Burger, “Half-fringe photoelasticity: a new approach to whole-field stress analysis,” Expt. Mech. 23, 304–313 (1983).
[CrossRef]

Wang, Z. F.

Z. F. Wang, E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
[CrossRef]

S. J. Haake, Z. F. Wang, E. A. Patterson, “Evaluation of full field automated photoelastic analysis based on phase stepping,” Expt. Technol. 17, 19–25 (1993).
[CrossRef]

E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–56 (1991).
[CrossRef]

Williams, D. C.

D. W. Robinson, D. C. Williams, M. Kujawinska, “A phase measurement method for use in photo elastic stress analysis,” (National Physical Laboratory, Teddington, U.K., 1988).

Appl. Opt.

Exp. Mech.

A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
[CrossRef]

Expt. Mech.

A. S. Voloshin, C. P. Burger, “Half-fringe photoelasticity: a new approach to whole-field stress analysis,” Expt. Mech. 23, 304–313 (1983).
[CrossRef]

A. S. Redner, “Photoelastic measurements by means of computer-assisted spectral contents analysis,” Expt. Mech. 25, 148–153 (1985).
[CrossRef]

Expt. Technol.

S. J. Haake, Z. F. Wang, E. A. Patterson, “Evaluation of full field automated photoelastic analysis based on phase stepping,” Expt. Technol. 17, 19–25 (1993).
[CrossRef]

J. Strain Anal.

A. Ajovalasit, S. Barone, G. Petrucci, “Automated photoelasticity in white light: influence of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

Opt. Eng.

Y. Morimoto, M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 3709–3714 (1994).
[CrossRef]

Opt. Lasers Eng.

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

C. Buckberry, D. Towers, “New approaches to the full-field analysis of photoelastic stress patterns,” Opt. Lasers Eng. 24, 415–428 (1996).
[CrossRef]

Z. F. Wang, E. A. Patterson, “Use of phase-stepping with demodulation and fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
[CrossRef]

Strain

E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–56 (1991).
[CrossRef]

Other

S. J. Haake, E. A. Patterson, “Photoelastic analysis using a full-field spectral contents analyser,” in Proceedings of the International Conference on Photoelasticity: New Instrumentation, Materials and Data Processing Techniques, London, 1993 (SIRA Communications, Chislehurst, U.K., 1993), Vol. 1, pp. 1.1–1.2.

D. W. Robinson, D. C. Williams, M. Kujawinska, “A phase measurement method for use in photo elastic stress analysis,” (National Physical Laboratory, Teddington, U.K., 1988).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the plane polariscope and data-collection apparatus.

Fig. 2
Fig. 2

Sample results for the longest (R) wavelength: (a) Arccosine phase map for the isochromatic parameter. (b) Arctangent phase map for the isoclinic angle.

Fig. 3
Fig. 3

Results for the isochromatic-parameter phase map: (a) Conversion to a ramped phase map with the three-wavelength method of Buckberry and Towers.12 (b) Unwrapped phase distribution for a horizontal line.

Fig. 4
Fig. 4

Results for the isoclinic-angle phase map obtained by use of (a) the method of Buckberry and Towers with three wavelengths, (b) the proposed method with three wavelengths, (c) the unwrapped phase map, and (d) a graph of the isoclinic results for a vertical line.

Equations (26)

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

I = I0cos2 β +0.5sin 2θ - ϕcos 2θ - ϕ ×cos 2β -sin2 2θ - ϕsin 2βcos α -1),
Iia =I0cos2 βi -0.5sin 2θ cos 2θ cos 2βi -sin2 2θ sin 2i1 - cos α,
Iib =I0cos2 βi + 0.5sin 2θ cos 2θ cos 2βi +sin2 2θ sin 2i1 -cos α,
V1 = I0 sin 2θ cos 2θ 1 -cos α/2,
V2 =I0 sin2 2θ1-cos α/2,
V3=I0 cos2 2θ1-cos α/2,
V4= I0,
θ= 12arctanV2/V1,
θ = 12arctanV1/V3,
θ= ±12arctanV2/V31/2,
θ=± 14arcsin2V1/V2 +V31/2,
θ = ± 12arcsinV2/V2+ V31/2,
θ = ± 12arccosV3/V2 + V31/2.
α = arccos1 - 2V2 + V3/V4.
S=i=13Iia-V1 sin 2βi+ V2 cos 2βi-V4 sin2 βi2+Iib + V1 sin 2βi+ V3 cos 2βi-V4 sin2 βi2.
SV1 = 2 i=13sin 2βi- Iia-V1 sin 2βi+V2 cos 2βi-V4 sin2 βi+Iib+V1 sin 2βi+ V3 cos 2βi- V4 sin2 βi =0,
SV2=2 i=13cos 2βiIia-V1 sin 2βi+V2 cos 2βi-V4 sin2 βi=0,
SV3=2 i=13cos 2βiIib-V1 sin 2βi+V2 cos 2βi-V4 sin2 βi=0,
SV4=2 i=13sin 2βi-Iia -V1 sin 2βi+V2 cos 2βi-V4 sin2 βi-Iib +V1 sin 2βi+ V3 cos 2βi- V4 sin2 βi =0,
DI-DTDV=0,
D =I1aV1I3aV1I1bV1I3bV1I1aV4I3aV4I1bV4I3bV4
I=Ia1I3aI1bI3b,
V=V1V4
V=DTD-1DI.
θ= 12arctanV2/V12arctanV2/V1arcsinV2V2+V3,
θ= 12arctanV2V1

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