Abstract

In any automated algorithm for interpreting photoelastic fringe patterns it is necessary to understand and quantify sources of error in the measurement system. We have been considering how the various components of the coating affect the photoelastic measurement, because this source of error has received fairly little attention in the literature. Because the reflective backing is not a perfect retroreflector, it does not preserve the polarization of light and thereby introduces noise into the measurement that depends on the angle of obliqueness and roughness of the reflective surface. This is of particular concern in resolving the stress tensor through the combination of thermoelasticity and photoelasticity where the components are sensitive to errors in the principal angle and difference of the principal stresses. We have developed a physical model that accounts for this and other sources of measurement error to be introduced in a systematic way so that the individual effects on the fringe patterns can be quantified. Simulations show altered photoelastic fringes when backing roughness and oblique incident angles are incorporated into the model.

© 2000 Optical Society of America

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References

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  1. M. M. Frocht, Photoelasticity (Wiley, New York, 1941), Vol. 1.
  2. R. K. Müller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 19, 245–251 (1979).
    [CrossRef]
  3. E. A. Patterson, “Automated photoelastic analysis,” Strain 24, 15–20 (1988).
    [CrossRef]
  4. A. C. Gillies, “Image processing approach to fringe patterns,” Opt. Eng. 27, 861–866 (1988).
    [CrossRef]
  5. T. Y. Chen, C. E. Taylor, “Computerized fringe analysis,” Exp. Mech. 29, 323–329 (1989).
    [CrossRef]
  6. A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the techniques,” Exp. Mech. 29, 252–257 (1989).
    [CrossRef]
  7. T. Kihara, “Automatic whole-field measurement of photoelasticity using linear polarized light,” in Proceedings of the 9th International Conference on Experimental Mechanics, (Society for Experimental Mechanics, Bethel, Conn., 1990), Vol. 12, pp. 821–827.
  8. S. J. Haake, E. A. Patterson, “The determination of principal stresses for photoelastic data,” Strain 28, 153–158 (1992).
    [CrossRef]
  9. A. Asundi, “Phase shifting in photoelasticity,” Exp. Tech. 17, 19–23 (1993).
    [CrossRef]
  10. S. J. Haake, Z. F. Wang, E. A. Patterson, “Evaluation of full field automated photoelastic analysis based on phase stepping,” Exp. Tech. 17, 19–25 (1993).
    [CrossRef]
  11. J. Carazo-Alverez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
    [CrossRef]
  12. A. Ajovalasit, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech. 35, 193–200 (1995).
    [CrossRef]
  13. K. Ramesh, S. S. Deshmukh, “Three fringe photoelasticity—use of colour image processing hardware to automate ordering of isochromatics,” Strain 32, 79–86 (1996).
    [CrossRef]
  14. S. A. Sparling, C. F. Small, “Photoelastic analysis using chromatic interpretation of digitized video,” in 1995 IEEE Engineering in Medicine and Biology: 17th Annual Conference (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 417–418.
  15. A. D. Nurse, “Full-field automated photoelasticity by use of a three-wavelength approach to phase stepping,” Appl. Opt. 36, 5781–5786 (1997).
    [CrossRef] [PubMed]
  16. G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
    [CrossRef]
  17. T. Y. Chen, “Digital determination of photoelastic birefringence using two wavelengths,” Exp. Mech. 37, 232–236 (1997).
    [CrossRef]
  18. T. W. Ng, “Derivation of retardation phase in computer-aided photoelasticity by using carrier fringe phase shifting,” Appl. Opt. 36, 8259–8263 (1997).
    [CrossRef]
  19. E. A. Patterson, W. Ji, Z. F. Wang, “On image analysis for birefringence measurements in photoelasticity,” Opt. Lasers Eng. 28, 17–36 (1997).
    [CrossRef]
  20. A. Ajovalasit, S. Barone, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
    [CrossRef]
  21. W. Ji, E. A. Patterson, “Simulation of errors in automated photoelasticity,” Exp. Mech. 38, 132–139 (1998).
    [CrossRef]
  22. D. F. Woolard, M. K. Hinders, “Coating for combined thermoelastic and photoelastic stress measurement,” in Nondestructive Evaluation of Bridges and Highways III, S. B. Chase, ed., Proc. SPIE3587, 88–96 (1999).
    [CrossRef]
  23. D. Woolard, M. Hinders, C. Welch, “Combined thermoelastic and photoelastic full-field stress measurement,” Rev. Prog. Quant. Nondestr. Eval. 18, 1431–1438 (1999).
    [CrossRef]
  24. A. S. Redner, “Photoelastic coatings,” Exp. Mech. 20, 403–408 (1980).
    [CrossRef]
  25. M. Wolna, “Polymer materials in practical uses of photoelasticity,” Opt. Eng. 34, 3427–3432 (1995).
    [CrossRef]
  26. J. Cernosek, “Inexpensive photoelastic coating with long contouring window and short polymerization time,” Exp. Tech. 15, 32–35 (1991).
    [CrossRef]
  27. F. Zandman, S. Redner, J. W. Dally, Photoelastic Coatings (Iowa State Univ. Press, Ames, Iowa, 1977).
  28. E. Liasi, W. North, P. I. Makrygiannis, T. Rocheleau, G. Womack, “Photoelasticity using retroreflection,” Exp. Tech. 21, 17–19 (1997).
    [CrossRef]
  29. Measurements Group Inc., “Introduction to stress analysis by the photostress method,” (Measurements Group, Inc., Raleigh, N.C., 1989).
  30. G. D. Lewis, D. L. Jordan, E. Jakeman, “Backscatter linear and circular polarization analysis of roughened aluminum,” Appl. Opt. 37, 5985–5992 (1998).
    [CrossRef]
  31. J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, New York, 1990).
  32. R. D. Guenther, Modern Optics (Wiley, New York, 1990).
  33. Y. C. Fung, A First Course in Continuum Mechanics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1994).
  34. COSMOS/M 2.0, Structural Research & Analysis Corporation, Los Angeles, Calif. (1998).
  35. P. Beckmann, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968).
  36. D. Woolard, “Thermoelastic and photoelastic full-field stress measurement,” Ph.D. dissertation (College of William and Mary, Williamsburg, Va., 1999).
  37. D. Woolard, M. Hinders, “Stress separation errors resulting from imperfect backings,” in Proceedings of SEM Annual Conference on Theoretical, Experimental, and Computational Mechanics (Society for Experimental Mechanics, Bethel, Conn., 1999), pp. 605–608.

1999 (1)

D. Woolard, M. Hinders, C. Welch, “Combined thermoelastic and photoelastic full-field stress measurement,” Rev. Prog. Quant. Nondestr. Eval. 18, 1431–1438 (1999).
[CrossRef]

1998 (3)

G. D. Lewis, D. L. Jordan, E. Jakeman, “Backscatter linear and circular polarization analysis of roughened aluminum,” Appl. Opt. 37, 5985–5992 (1998).
[CrossRef]

A. Ajovalasit, S. Barone, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

W. Ji, E. A. Patterson, “Simulation of errors in automated photoelasticity,” Exp. Mech. 38, 132–139 (1998).
[CrossRef]

1997 (6)

A. D. Nurse, “Full-field automated photoelasticity by use of a three-wavelength approach to phase stepping,” Appl. Opt. 36, 5781–5786 (1997).
[CrossRef] [PubMed]

G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
[CrossRef]

T. Y. Chen, “Digital determination of photoelastic birefringence using two wavelengths,” Exp. Mech. 37, 232–236 (1997).
[CrossRef]

T. W. Ng, “Derivation of retardation phase in computer-aided photoelasticity by using carrier fringe phase shifting,” Appl. Opt. 36, 8259–8263 (1997).
[CrossRef]

E. A. Patterson, W. Ji, Z. F. Wang, “On image analysis for birefringence measurements in photoelasticity,” Opt. Lasers Eng. 28, 17–36 (1997).
[CrossRef]

E. Liasi, W. North, P. I. Makrygiannis, T. Rocheleau, G. Womack, “Photoelasticity using retroreflection,” Exp. Tech. 21, 17–19 (1997).
[CrossRef]

1996 (1)

K. Ramesh, S. S. Deshmukh, “Three fringe photoelasticity—use of colour image processing hardware to automate ordering of isochromatics,” Strain 32, 79–86 (1996).
[CrossRef]

1995 (2)

M. Wolna, “Polymer materials in practical uses of photoelasticity,” Opt. Eng. 34, 3427–3432 (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 (1)

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

1993 (2)

A. Asundi, “Phase shifting in photoelasticity,” Exp. Tech. 17, 19–23 (1993).
[CrossRef]

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

1992 (1)

S. J. Haake, E. A. Patterson, “The determination of principal stresses for photoelastic data,” Strain 28, 153–158 (1992).
[CrossRef]

1991 (1)

J. Cernosek, “Inexpensive photoelastic coating with long contouring window and short polymerization time,” Exp. Tech. 15, 32–35 (1991).
[CrossRef]

1989 (2)

T. Y. Chen, C. E. Taylor, “Computerized fringe analysis,” Exp. Mech. 29, 323–329 (1989).
[CrossRef]

A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the techniques,” Exp. Mech. 29, 252–257 (1989).
[CrossRef]

1988 (2)

E. A. Patterson, “Automated photoelastic analysis,” Strain 24, 15–20 (1988).
[CrossRef]

A. C. Gillies, “Image processing approach to fringe patterns,” Opt. Eng. 27, 861–866 (1988).
[CrossRef]

1980 (1)

A. S. Redner, “Photoelastic coatings,” Exp. Mech. 20, 403–408 (1980).
[CrossRef]

1979 (1)

R. K. Müller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 19, 245–251 (1979).
[CrossRef]

Ajovalasit, A.

A. Ajovalasit, S. Barone, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

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

Asundi, A.

A. Asundi, “Phase shifting in photoelasticity,” Exp. Tech. 17, 19–23 (1993).
[CrossRef]

Barone, S.

A. Ajovalasit, S. Barone, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

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

Beckmann, P.

P. Beckmann, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968).

Carazo-Alverez, J.

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

Cernosek, J.

J. Cernosek, “Inexpensive photoelastic coating with long contouring window and short polymerization time,” Exp. Tech. 15, 32–35 (1991).
[CrossRef]

Chen, T. Y.

T. Y. Chen, “Digital determination of photoelastic birefringence using two wavelengths,” Exp. Mech. 37, 232–236 (1997).
[CrossRef]

T. Y. Chen, C. E. Taylor, “Computerized fringe analysis,” Exp. Mech. 29, 323–329 (1989).
[CrossRef]

Dally, J. W.

F. Zandman, S. Redner, J. W. Dally, Photoelastic Coatings (Iowa State Univ. Press, Ames, Iowa, 1977).

Deshmukh, S. S.

K. Ramesh, S. S. Deshmukh, “Three fringe photoelasticity—use of colour image processing hardware to automate ordering of isochromatics,” Strain 32, 79–86 (1996).
[CrossRef]

Frocht, M. M.

M. M. Frocht, Photoelasticity (Wiley, New York, 1941), Vol. 1.

Fung, Y. C.

Y. C. Fung, A First Course in Continuum Mechanics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1994).

Gillies, A. C.

A. C. Gillies, “Image processing approach to fringe patterns,” Opt. Eng. 27, 861–866 (1988).
[CrossRef]

Guenther, R. D.

R. D. Guenther, Modern Optics (Wiley, New York, 1990).

Haake, S. J.

J. Carazo-Alverez, 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,” Exp. Tech. 17, 19–25 (1993).
[CrossRef]

S. J. Haake, E. A. Patterson, “The determination of principal stresses for photoelastic data,” Strain 28, 153–158 (1992).
[CrossRef]

Hinders, M.

D. Woolard, M. Hinders, C. Welch, “Combined thermoelastic and photoelastic full-field stress measurement,” Rev. Prog. Quant. Nondestr. Eval. 18, 1431–1438 (1999).
[CrossRef]

D. Woolard, M. Hinders, “Stress separation errors resulting from imperfect backings,” in Proceedings of SEM Annual Conference on Theoretical, Experimental, and Computational Mechanics (Society for Experimental Mechanics, Bethel, Conn., 1999), pp. 605–608.

Hinders, M. K.

D. F. Woolard, M. K. Hinders, “Coating for combined thermoelastic and photoelastic stress measurement,” in Nondestructive Evaluation of Bridges and Highways III, S. B. Chase, ed., Proc. SPIE3587, 88–96 (1999).
[CrossRef]

Jakeman, E.

Ji, W.

W. Ji, E. A. Patterson, “Simulation of errors in automated photoelasticity,” Exp. Mech. 38, 132–139 (1998).
[CrossRef]

E. A. Patterson, W. Ji, Z. F. Wang, “On image analysis for birefringence measurements in photoelasticity,” Opt. Lasers Eng. 28, 17–36 (1997).
[CrossRef]

Jordan, D. L.

Kihara, T.

T. Kihara, “Automatic whole-field measurement of photoelasticity using linear polarized light,” in Proceedings of the 9th International Conference on Experimental Mechanics, (Society for Experimental Mechanics, Bethel, Conn., 1990), Vol. 12, pp. 821–827.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, New York, 1990).

Lewis, G. D.

Liasi, E.

E. Liasi, W. North, P. I. Makrygiannis, T. Rocheleau, G. Womack, “Photoelasticity using retroreflection,” Exp. Tech. 21, 17–19 (1997).
[CrossRef]

Makrygiannis, P. I.

E. Liasi, W. North, P. I. Makrygiannis, T. Rocheleau, G. Womack, “Photoelasticity using retroreflection,” Exp. Tech. 21, 17–19 (1997).
[CrossRef]

Müller, R. K.

R. K. Müller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 19, 245–251 (1979).
[CrossRef]

Ng, T. W.

North, W.

E. Liasi, W. North, P. I. Makrygiannis, T. Rocheleau, G. Womack, “Photoelasticity using retroreflection,” Exp. Tech. 21, 17–19 (1997).
[CrossRef]

Nurse, A. D.

Patterson, E. A.

W. Ji, E. A. Patterson, “Simulation of errors in automated photoelasticity,” Exp. Mech. 38, 132–139 (1998).
[CrossRef]

E. A. Patterson, W. Ji, Z. F. Wang, “On image analysis for birefringence measurements in photoelasticity,” Opt. Lasers Eng. 28, 17–36 (1997).
[CrossRef]

J. Carazo-Alverez, 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,” Exp. Tech. 17, 19–25 (1993).
[CrossRef]

S. J. Haake, E. A. Patterson, “The determination of principal stresses for photoelastic data,” Strain 28, 153–158 (1992).
[CrossRef]

E. A. Patterson, “Automated photoelastic analysis,” Strain 24, 15–20 (1988).
[CrossRef]

Petrucci, G.

A. Ajovalasit, S. Barone, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
[CrossRef]

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

Ramesh, K.

K. Ramesh, S. S. Deshmukh, “Three fringe photoelasticity—use of colour image processing hardware to automate ordering of isochromatics,” Strain 32, 79–86 (1996).
[CrossRef]

Redner, A. S.

A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the techniques,” Exp. Mech. 29, 252–257 (1989).
[CrossRef]

A. S. Redner, “Photoelastic coatings,” Exp. Mech. 20, 403–408 (1980).
[CrossRef]

Redner, S.

F. Zandman, S. Redner, J. W. Dally, Photoelastic Coatings (Iowa State Univ. Press, Ames, Iowa, 1977).

Rocheleau, T.

E. Liasi, W. North, P. I. Makrygiannis, T. Rocheleau, G. Womack, “Photoelasticity using retroreflection,” Exp. Tech. 21, 17–19 (1997).
[CrossRef]

Saackel, L. R.

R. K. Müller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 19, 245–251 (1979).
[CrossRef]

Small, C. F.

S. A. Sparling, C. F. Small, “Photoelastic analysis using chromatic interpretation of digitized video,” in 1995 IEEE Engineering in Medicine and Biology: 17th Annual Conference (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 417–418.

Sparling, S. A.

S. A. Sparling, C. F. Small, “Photoelastic analysis using chromatic interpretation of digitized video,” in 1995 IEEE Engineering in Medicine and Biology: 17th Annual Conference (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 417–418.

Taylor, C. E.

T. Y. Chen, C. E. Taylor, “Computerized fringe analysis,” Exp. Mech. 29, 323–329 (1989).
[CrossRef]

Voloshin, A. S.

A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the techniques,” Exp. Mech. 29, 252–257 (1989).
[CrossRef]

Wang, Z. F.

E. A. Patterson, W. Ji, Z. F. Wang, “On image analysis for birefringence measurements in photoelasticity,” Opt. Lasers Eng. 28, 17–36 (1997).
[CrossRef]

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

Welch, C.

D. Woolard, M. Hinders, C. Welch, “Combined thermoelastic and photoelastic full-field stress measurement,” Rev. Prog. Quant. Nondestr. Eval. 18, 1431–1438 (1999).
[CrossRef]

Wolna, M.

M. Wolna, “Polymer materials in practical uses of photoelasticity,” Opt. Eng. 34, 3427–3432 (1995).
[CrossRef]

Womack, G.

E. Liasi, W. North, P. I. Makrygiannis, T. Rocheleau, G. Womack, “Photoelasticity using retroreflection,” Exp. Tech. 21, 17–19 (1997).
[CrossRef]

Woolard, D.

D. Woolard, M. Hinders, C. Welch, “Combined thermoelastic and photoelastic full-field stress measurement,” Rev. Prog. Quant. Nondestr. Eval. 18, 1431–1438 (1999).
[CrossRef]

D. Woolard, “Thermoelastic and photoelastic full-field stress measurement,” Ph.D. dissertation (College of William and Mary, Williamsburg, Va., 1999).

D. Woolard, M. Hinders, “Stress separation errors resulting from imperfect backings,” in Proceedings of SEM Annual Conference on Theoretical, Experimental, and Computational Mechanics (Society for Experimental Mechanics, Bethel, Conn., 1999), pp. 605–608.

Woolard, D. F.

D. F. Woolard, M. K. Hinders, “Coating for combined thermoelastic and photoelastic stress measurement,” in Nondestructive Evaluation of Bridges and Highways III, S. B. Chase, ed., Proc. SPIE3587, 88–96 (1999).
[CrossRef]

Zandman, F.

F. Zandman, S. Redner, J. W. Dally, Photoelastic Coatings (Iowa State Univ. Press, Ames, Iowa, 1977).

Appl. Opt. (3)

Exp. Mech. (8)

W. Ji, E. A. Patterson, “Simulation of errors in automated photoelasticity,” Exp. Mech. 38, 132–139 (1998).
[CrossRef]

A. S. Redner, “Photoelastic coatings,” Exp. Mech. 20, 403–408 (1980).
[CrossRef]

G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
[CrossRef]

T. Y. Chen, “Digital determination of photoelastic birefringence using two wavelengths,” Exp. Mech. 37, 232–236 (1997).
[CrossRef]

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

R. K. Müller, L. R. Saackel, “Complete automatic analysis of photoelastic fringes,” Exp. Mech. 19, 245–251 (1979).
[CrossRef]

T. Y. Chen, C. E. Taylor, “Computerized fringe analysis,” Exp. Mech. 29, 323–329 (1989).
[CrossRef]

A. S. Voloshin, A. S. Redner, “Automated measurement of birefringence: development and experimental evaluation of the techniques,” Exp. Mech. 29, 252–257 (1989).
[CrossRef]

Exp. Tech. (4)

A. Asundi, “Phase shifting in photoelasticity,” Exp. Tech. 17, 19–23 (1993).
[CrossRef]

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

J. Cernosek, “Inexpensive photoelastic coating with long contouring window and short polymerization time,” Exp. Tech. 15, 32–35 (1991).
[CrossRef]

E. Liasi, W. North, P. I. Makrygiannis, T. Rocheleau, G. Womack, “Photoelasticity using retroreflection,” Exp. Tech. 21, 17–19 (1997).
[CrossRef]

J. Strain Anal. (1)

A. Ajovalasit, S. Barone, G. Petrucci, “A method for reducing the influence of quarter-wave plate errors in phase stepping photoelasticity,” J. Strain Anal. 33, 207–216 (1998).
[CrossRef]

Opt. Eng. (2)

A. C. Gillies, “Image processing approach to fringe patterns,” Opt. Eng. 27, 861–866 (1988).
[CrossRef]

M. Wolna, “Polymer materials in practical uses of photoelasticity,” Opt. Eng. 34, 3427–3432 (1995).
[CrossRef]

Opt. Lasers Eng. (2)

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

E. A. Patterson, W. Ji, Z. F. Wang, “On image analysis for birefringence measurements in photoelasticity,” Opt. Lasers Eng. 28, 17–36 (1997).
[CrossRef]

Rev. Prog. Quant. Nondestr. Eval. (1)

D. Woolard, M. Hinders, C. Welch, “Combined thermoelastic and photoelastic full-field stress measurement,” Rev. Prog. Quant. Nondestr. Eval. 18, 1431–1438 (1999).
[CrossRef]

Strain (3)

S. J. Haake, E. A. Patterson, “The determination of principal stresses for photoelastic data,” Strain 28, 153–158 (1992).
[CrossRef]

K. Ramesh, S. S. Deshmukh, “Three fringe photoelasticity—use of colour image processing hardware to automate ordering of isochromatics,” Strain 32, 79–86 (1996).
[CrossRef]

E. A. Patterson, “Automated photoelastic analysis,” Strain 24, 15–20 (1988).
[CrossRef]

Other (13)

M. M. Frocht, Photoelasticity (Wiley, New York, 1941), Vol. 1.

T. Kihara, “Automatic whole-field measurement of photoelasticity using linear polarized light,” in Proceedings of the 9th International Conference on Experimental Mechanics, (Society for Experimental Mechanics, Bethel, Conn., 1990), Vol. 12, pp. 821–827.

S. A. Sparling, C. F. Small, “Photoelastic analysis using chromatic interpretation of digitized video,” in 1995 IEEE Engineering in Medicine and Biology: 17th Annual Conference (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 417–418.

D. F. Woolard, M. K. Hinders, “Coating for combined thermoelastic and photoelastic stress measurement,” in Nondestructive Evaluation of Bridges and Highways III, S. B. Chase, ed., Proc. SPIE3587, 88–96 (1999).
[CrossRef]

Measurements Group Inc., “Introduction to stress analysis by the photostress method,” (Measurements Group, Inc., Raleigh, N.C., 1989).

F. Zandman, S. Redner, J. W. Dally, Photoelastic Coatings (Iowa State Univ. Press, Ames, Iowa, 1977).

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, New York, 1990).

R. D. Guenther, Modern Optics (Wiley, New York, 1990).

Y. C. Fung, A First Course in Continuum Mechanics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1994).

COSMOS/M 2.0, Structural Research & Analysis Corporation, Los Angeles, Calif. (1998).

P. Beckmann, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968).

D. Woolard, “Thermoelastic and photoelastic full-field stress measurement,” Ph.D. dissertation (College of William and Mary, Williamsburg, Va., 1999).

D. Woolard, M. Hinders, “Stress separation errors resulting from imperfect backings,” in Proceedings of SEM Annual Conference on Theoretical, Experimental, and Computational Mechanics (Society for Experimental Mechanics, Bethel, Conn., 1999), pp. 605–608.

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Figures (12)

Fig. 1
Fig. 1

Plane wave traveling along the negative z direction is incident upon a birefringent coating with the x and y axes in the plane of the material. The polarizer and analyzer have a perpendicular orientation relative to each other. Rotation to the material coordinates is done through the angle θ where the x′ axis and y′ axis dictate the propagation of the extraordinary and ordinary waves, respectively.

Fig. 2
Fig. 2

Propagating waves associated with the theoretical model.

Fig. 3
Fig. 3

Comparison of (a) experimental versus (b) theoretical photoelastic fringe pattern for a hole in a poly(methyl methacrylate) (PMMA) plate under vertical tension. The experimental image was obtained with the Measurements Group Model 030-Polariscope, and the theoretical image was calculated from the Kirsch problem.

Fig. 4
Fig. 4

Comparison of (a) experimental versus (b) theoretical photoelastic fringe pattern for a hole in a PMMA plate under vertical tension. The theoretical image was calculated with finite-element analysis.

Fig. 5
Fig. 5

Comparison of (a) experimental versus (b) theoretical photoelastic fringe pattern for a hole with a notch in a PMMA plate under vertical tension. The theoretical image was calculated with finite-element analysis.

Fig. 6
Fig. 6

Cross-polarization ratio versus incidence angle Θ. As the rms slope increases, so does the depolarization. In this simulation, the relative permittivity was 4. □, rms slope, 0.08 (4.6°); ◇, rms slope, 0.14 (8°); ○, rms slope, 0.2 (11.5°); and △, rms slope, 0.25 (14.3°).

Fig. 7
Fig. 7

Change in path length that is due to off-normal incidence. The new path length is -d sec ψ.

Fig. 8
Fig. 8

Degradation of the fringes from the Kirsch problem for (a) zero depolarization and normal incidence, (b) zero depolarization and 5° from normal incidence, (c) 25% depolarization and normal incidence, (d) 25% depolarization and 5°, (e) 50% depolarization and normal incidence, and (f) 50% depolarization and 5°.

Fig. 9
Fig. 9

Isoclinic lines for a hole in a plate under vertical tension. The angles correspond to the rotation of the polarizer and analyzer about the z axis.1

Fig. 10
Fig. 10

Best-fit isoclinic lines for various depolarizations and angles of off-normal incidence for the Kirsch problem.

Fig. 11
Fig. 11

Degradation of the finite-element analysis model for the notched hole at (a) zero depolarization and normal incidence and (b) 25% depolarization and normal incidence. □, 0%–0 off; ◇, 0%–5 off; ○, 0%–10 off; △, 25%–0 off; ⊞, 25%–5 off; ⋄+, 50%–0 off; and ⊕, 50%–5 off.

Fig. 12
Fig. 12

Best-fit lines for various depolarizations and off-normal incidence for the notched hole. □, 0%–0 off; ◇, 0%–5 off; ○, 25%–0 off; △, 25%–5 off; ⊞, 50%–0 off; and ao-39-13-2043-i001, 50%–5 off.

Equations (62)

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Φ=dne-no,
ne-no=Kεx-εy,
εx-εy=Φ2dK=Nλ2dK,
Ei=κijDj,
ε0κik=ε0κik0+qijkmTjm for stress,
ε0κik=ε0κik0+pijkmSjm for strain.
pijkl=qijmncmnkl.
EI=xˆ cos ϕ+yˆ sin ϕA exp-ik0z.
EI=xˆ cosθ-ϕ+yˆ sinθ-ϕA exp-ik0z,
ER=xˆβx+yˆβyexpik0z,
EM=xˆκxxGo exp-iwzνκxx+yˆκyyGe exp-iwzνκyy+xˆκxxGo expiwzνκxx+yˆκyyGe expiωzνκyy,
ET=xˆFx+yˆFyexp-iktz.
nˆ·B2-B1=0, nˆ×E2-E1=0,nˆ·D2-D1=0, nˆ×H2-H1=0,
EIxz=0+ERxz=0=EMxz=0,
EIyz=0+ERyz=0=EMyz=0,
HIxz=0+HRxz=0=HMxz=0,
HIyz=0+HRyz=0=HMyz=0,
EMxz=-d=ETxz=-d,
EMyz=-d=ETyz=-d,
HMxz=-d=HTxz=-d,
HMyz=-d=HTyz=-d.
βx=A cosθ-ϕcoswdνκxx+wkoνκxxi sinwdνκxxcoswdνκxx-wkoνκxxi sinwdνκxx,
βy=-A sinθ-ϕcoswdνκyy+wkoνκyyi sinwdνκyycoswdνκyy-wkoνκyyi sinwdνκyy.
ER=xˆ cosθ-ϕexpiΓe+yˆ sinθ-ϕ×expiΓoA expik0z,
Γe=arctan-ne sin 2Kedsin2 Ked-ne2 cos2 Ked,
Γo=arctan-no sin 2Kodsin2 Kod-no2 cos2 Kod,
Ke=wνκxx,
Ko=wνκyy,
ne=Kek0,
no=Kek0.
Eanalyzer=xˆERx cosϕ-90°+yˆERy sinϕ-90°,
I=Ex2+Ey2+2ExEycosδ,
Ilinear=Ae2cos2 θ sin2 ϕ+sin2 θ cos2ϕ+Ao2sin2 θ sin2 ϕ+cos2 θ cos2 ϕ+AeAo sin 2θ cos 2ϕ cosΓe-Γo,
Ae=A cosϕ-θ,Ao=A sinϕ-θ.
Ilinear=Ae2cos2 θ sin2 ϕ+sin2 θ cos2 ϕ+Ao2sin2 θ sin2 ϕ+cos2 θ cos2 ϕ+AeAo sin 2θ cos 2ϕ-2AeAo sin 2θ cos 2ϕ sin2Γe-Γo2.
Ilinear=A2 sin2θ-ϕsin 2θ cos 2ϕ sin2Γe-Γo2.
Tx=12Tx+Ty+12Tx-Ty×cos 2θ+Txy sin 2θ,
Ty=12Tx+Ty-12Tx-Ty×cos 2θ-Txy sin 2θ,
tan 2θ=2TxyTx-Ty,
κxx=1εon2+1εoqxxTx+qxyTy,
κyy=1εon2+1εoqxyTx+qyyTy.
Ep=A2  cos δR-R-R+Rcos 2Ψexp-2ik·rdS,Ec=A2  cos δR+Rsin 2Ψ exp-2ik·rdS,
R|=˜ cos Θ-˜-sin2 Θ1/2˜ cos Θ+˜-sin2 Θ1/2,
R=cos Θ-ε˜-sin2 Θ1/2cos Θ+˜-sin2 Θ1/2,
˜=rp,
P2=EcEc*EpEp*,
EcEc*=A22|cos δR+Rsin 2Ψ|2×  exp2ik·r2-r1dS1dS2,
EpEp*=A22|cos δR-R-R+Rcos 2Ψ|2×  exp2ik·r2-r1dS1dS2,
P2|cos δR+Rsin 2Ψ|2|cos δR-R-R+Rcos 2Ψ|2,
cos δ=Zx sin Θ+cos Θ1+Zz2+Zy21/2,
ER=xˆ cosθ-ϕexpiΓe+yˆ sinθ-ϕexpiΓoA expikoz.
ER=xˆCe cosθ-ϕexpiΩe+yˆCo sinθ-ϕexpiΩoexpikoz,
Ce=A N12 cos4 a+N22 sin4 a-sin22a12N1N2-16ne21+P21/2cos2 aP+ne2+P2+sin2 a2+P1+ne2,
Co=A M12 cos4 b+M22 sin4 b+sin22b12M1M2-16no2P-121/2cos2 bP+noP-22+sin2 bP1-no-22,
Ωe=arctan-4ne1+Psin2aN2 sin2 a-N1 cos2 a,
Ωo=arctan4noP-1sin2bM2 sin2 b+M1 cos2 b,
N1=-P2+ne22+P2,
N2=41+P-P2ne2-1,
a=Ked sec ψ,
M1=P2-no2P-22,
M2=41-P-P2no2-1,
b=Kod sec ψ.

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