Abstract

Recent advances in integrated photoelasticity have opened the possibility of determining tomographically arbitrary three-dimensional stress fields. Since photoelastic tomography is based on experimental measurement of the characteristic parameters, the dependence of these parameters on the stress distribution on a light ray is considered in detail. The possibility of determining certain integrals of the stress components is analyzed, and the linear approximation of integrated photoelasticity has been rigorously treated.

© 2005 Optical Society of America

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References

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  1. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  2. H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).
    [CrossRef]
  3. H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.
  4. H. Aben, A. Errapart, L. Ainola, J. Anton, W. Osten, M. Takeda, eds., Proc. SPIE5457, 1–11 (2004).
  5. R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  6. H. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
    [CrossRef]
  7. H. Aben, “Characteristic directions in optics of twisted birefringent media,“ J. Opt. Soc. Am. A 3, 1414–1421 (1986).
    [CrossRef]
  8. L. Ainola, H. Aben, “Duality in optical theory of twisted birefringent media,” J. Opt. Soc. Am. A 16, 2545–2549 (1999).
    [CrossRef]
  9. L. Ainola, H. Aben, “On the optical theory of photoelastic tomography,” J. Opt. Soc. Am. A 21, 1093–1101 (2004).
    [CrossRef]
  10. A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach Scientific, Amsterdam, 1998).
  11. S. K. Mangal, K. Ramesh, ”Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
    [CrossRef]
  12. R. A. Tomlinson, E. A. Patterson, ”The case of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
    [CrossRef]
  13. H. Poincaré, Théorie mathématique de la lumière, II (Carré et Naud, Paris, 1892).
  14. S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, New York, 1969).
  15. A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech.. 4, 115–126 (1962).
  16. A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Stuttgart) 19, 261–272 (1962).
  17. A. Kuske, “L’analyse des phénomènes optiques en photoélasticité à trios dimensions par la méthode du cercle de ‘J’,” Rev. Fr. Méc. 9, 49–58 (1964).
  18. H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
    [CrossRef]

2004 (1)

2002 (1)

R. A. Tomlinson, E. A. Patterson, ”The case of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

1999 (2)

S. K. Mangal, K. Ramesh, ”Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
[CrossRef]

L. Ainola, H. Aben, “Duality in optical theory of twisted birefringent media,” J. Opt. Soc. Am. A 16, 2545–2549 (1999).
[CrossRef]

1989 (1)

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

1986 (1)

1966 (1)

H. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[CrossRef]

1964 (1)

A. Kuske, “L’analyse des phénomènes optiques en photoélasticité à trios dimensions par la méthode du cercle de ‘J’,” Rev. Fr. Méc. 9, 49–58 (1964).

1962 (2)

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech.. 4, 115–126 (1962).

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Stuttgart) 19, 261–272 (1962).

Aben, H.

L. Ainola, H. Aben, “On the optical theory of photoelastic tomography,” J. Opt. Soc. Am. A 21, 1093–1101 (2004).
[CrossRef]

L. Ainola, H. Aben, “Duality in optical theory of twisted birefringent media,” J. Opt. Soc. Am. A 16, 2545–2549 (1999).
[CrossRef]

H. Aben, “Characteristic directions in optics of twisted birefringent media,“ J. Opt. Soc. Am. A 3, 1414–1421 (1986).
[CrossRef]

H. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[CrossRef]

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).
[CrossRef]

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

H. Aben, A. Errapart, L. Ainola, J. Anton, W. Osten, M. Takeda, eds., Proc. SPIE5457, 1–11 (2004).

Aben, H. K.

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

Ainola, L.

L. Ainola, H. Aben, “On the optical theory of photoelastic tomography,” J. Opt. Soc. Am. A 21, 1093–1101 (2004).
[CrossRef]

L. Ainola, H. Aben, “Duality in optical theory of twisted birefringent media,” J. Opt. Soc. Am. A 16, 2545–2549 (1999).
[CrossRef]

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

H. Aben, A. Errapart, L. Ainola, J. Anton, W. Osten, M. Takeda, eds., Proc. SPIE5457, 1–11 (2004).

Anton, J.

H. Aben, A. Errapart, L. Ainola, J. Anton, W. Osten, M. Takeda, eds., Proc. SPIE5457, 1–11 (2004).

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

Azzam, R. M.A.

R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bashara, N. M.

R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bhagavantam, S.

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, New York, 1969).

Errapart, A.

H. Aben, A. Errapart, L. Ainola, J. Anton, W. Osten, M. Takeda, eds., Proc. SPIE5457, 1–11 (2004).

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

Fuki, A. A.

A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach Scientific, Amsterdam, 1998).

Guillemet, C.

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).
[CrossRef]

Josepson, J. I.

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

Kell, K.-J.

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

Kravtsov, Yu. A.

A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach Scientific, Amsterdam, 1998).

Kuske, A.

A. Kuske, “L’analyse des phénomènes optiques en photoélasticité à trios dimensions par la méthode du cercle de ‘J’,” Rev. Fr. Méc. 9, 49–58 (1964).

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech.. 4, 115–126 (1962).

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Stuttgart) 19, 261–272 (1962).

Mangal, S. K.

S. K. Mangal, K. Ramesh, ”Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
[CrossRef]

Naida, O. N.

A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach Scientific, Amsterdam, 1998).

Osten, W.

H. Aben, A. Errapart, L. Ainola, J. Anton, W. Osten, M. Takeda, eds., Proc. SPIE5457, 1–11 (2004).

Patterson, E. A.

R. A. Tomlinson, E. A. Patterson, ”The case of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

Poincaré, H.

H. Poincaré, Théorie mathématique de la lumière, II (Carré et Naud, Paris, 1892).

Ramesh, K.

S. K. Mangal, K. Ramesh, ”Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
[CrossRef]

Takeda, M.

H. Aben, A. Errapart, L. Ainola, J. Anton, W. Osten, M. Takeda, eds., Proc. SPIE5457, 1–11 (2004).

Tomlinson, R. A.

R. A. Tomlinson, E. A. Patterson, ”The case of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

Venkatarayudu, T.

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, New York, 1969).

Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. (1)

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech.. 4, 115–126 (1962).

Exp. Mech. (2)

H. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[CrossRef]

R. A. Tomlinson, E. A. Patterson, ”The case of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. 42, 43–50 (2002).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lasers Eng. (2)

S. K. Mangal, K. Ramesh, ”Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Lasers Eng. 31, 263–278 (1999).
[CrossRef]

H. K. Aben, J. I. Josepson, K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145–157 (1989).
[CrossRef]

Optik (Stuttgart) (1)

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Stuttgart) 19, 261–272 (1962).

Rev. Fr. Méc. (1)

A. Kuske, “L’analyse des phénomènes optiques en photoélasticité à trios dimensions par la méthode du cercle de ‘J’,” Rev. Fr. Méc. 9, 49–58 (1964).

Other (8)

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).
[CrossRef]

H. Aben, A. Errapart, L. Ainola, J. Anton, “Photoelastic tomography in linear approximation,” in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics ATEM’03 (The Japan Society of Mechanical Engineers, Nagoya, Japan, 2003), CD ROM.

H. Aben, A. Errapart, L. Ainola, J. Anton, W. Osten, M. Takeda, eds., Proc. SPIE5457, 1–11 (2004).

R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

A. A. Fuki, Yu. A. Kravtsov, O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach Scientific, Amsterdam, 1998).

H. Poincaré, Théorie mathématique de la lumière, II (Carré et Naud, Paris, 1892).

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, New York, 1969).

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Figures (2)

Fig. 1
Fig. 1

Illustration of the problem.

Fig. 2
Fig. 2

Illustration of the rotation of the principal stress axes; primary ( x 0 , y 0 ) and secondary ( x * , y * ) characteristic directions.

Equations (73)

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d E d z = V E ,
E = ( E 1 E 2 ) ,
V = i C [ 1 2 ( σ 11 σ 22 ) σ 12 σ 12 1 2 ( σ 11 σ 22 ) ] .
E = U E 0 ,
U = [ u 1 + i u 2 u 3 + i u 4 u 3 + i u 4 u 1 i u 2 ] ,
u 1 2 + u 2 2 + u 3 2 + u 4 2 = 1 .
d U d z = V U .
U = S ( α * ) G ( γ ) S ( α 0 ) .
S ( ϑ ) = [ cos ϑ sin ϑ sin ϑ cos ϑ ] .
G ( γ ) = [ exp ( i γ ) 0 0 exp ( i γ ) ] .
u 1 = cos γ cos ( α * α 0 ) ,
u 2 = sin γ cos ( α * + α 0 ) ,
u 3 = cos γ sin ( α * α 0 ) ,
u 4 = sin γ sin ( α * + α 0 ) .
S ( α * ) E = G ( γ ) S ( α 0 ) E 0 .
E = S ( α * ) E , E 0 = S ( α 0 ) E 0 ,
E = G ( γ ) E 0 .
[ d S ( α * ) d z G ( γ ) S ( α 0 ) + S ( α * ) d G ( γ ) d z S ( α 0 ) + S ( α * ) G ( γ ) d S ( α 0 ) d z ] S ( α 0 ) G ( γ ) S ( α * ) = V .
[ 0 1 1 0 ] d α * d z + i [ cos 2 α * sin 2 α * sin 2 α * cos 2 α * ] d γ d z + [ i sin 2 γ sin 2 α * cos 2 γ + i sin 2 γ cos 2 α * cos 2 γ + i sin 2 γ cos 2 α * i sin 2 γ sin 2 α * ] d α 0 d z = V .
cos 2 α * d γ d z sin 2 γ sin 2 α * d α 0 d z = 1 2 C ( σ 11 σ 22 ) ,
d α * d z cos 2 γ d α 0 d z = 0 ,
sin 2 α * d γ d z + sin 2 γ cos 2 α * d α 0 d z = C σ 12 .
γ ( z 0 ) = 0 , α * ( z 0 ) = α 0 ( z 0 ) .
cos 2 α * ( z 0 ) d γ d z z = z 0 = 1 2 C [ σ 11 ( z 0 ) σ 22 ( z 0 ) ] ,
sin 2 α * ( z 0 ) d γ d z z = z 0 = C σ 12 ( z 0 ) .
tan 2 α * ( z 0 ) = 2 σ 12 ( z 0 ) σ 11 ( z 0 ) σ 22 ( z 0 ) .
cos 2 α * d γ d z tan 2 γ sin 2 α * d α * d z = 1 2 C ( σ 11 σ 22 ) ,
sin 2 α * d γ d z + tan 2 γ cos 2 α * d α * d z = C σ 12 .
α * ( z 1 ) = α * 1 , γ ( z 1 ) = γ 1 .
z 0 z 1 ( σ 11 σ 22 ) d z , z 0 z 1 σ 12 d z ,
C ( σ 11 σ 22 ) = 1 cos 2 γ d d z ( cos 2 α * sin 2 γ ) ,
2 C σ 12 = 1 cos 2 γ d d z ( sin 2 α * sin 2 γ ) .
C z 0 z 1 ( σ 11 σ 22 ) d z = z 0 z 1 1 cos 2 γ d d z ( cos 2 α * sin 2 γ ) d z ,
2 C z 0 z 1 σ 12 d z = z 0 z 1 1 cos 2 γ d d z ( sin 2 α * sin 2 γ ) d z .
C 2 z 0 z 1 ( σ 11 σ 22 ) d z = γ 1 cos 2 ϕ 0 ,
C z 0 z 1 σ 12 d z = γ 1 sin 2 ϕ 0 .
tan 2 ϕ = tan 2 α * ( d γ d z ) + tan 2 γ ( d α * d z ) d γ d z tan 2 α * tan 2 γ ( d α * d z ) .
tan 2 ϕ = tan 2 ϕ 0 ;
C ( σ 11 σ 22 ) cos 2 γ = d d z ( cos 2 α * sin 2 γ ) ,
2 C σ 12 cos 2 γ = d d z ( sin 2 α * sin 2 γ ) .
C z 0 z 1 ( σ 11 σ 22 ) cos 2 γ d z = cos 2 α * 1 sin 2 γ 1 ,
2 C z 0 z 1 σ 12 cos 2 γ d z = sin 2 α * 1 sin 2 γ 1 .
U = I + z 0 z V d z + z 0 z V z 0 z V d z d z + .
cos 2 γ = u 1 2 + u 3 2 ,
sin 2 γ = u 2 2 + u 4 2 ,
cos 2 γ = u 1 2 + u 3 2 u 2 2 u 4 2 .
u 1 = 1 + C 2 4 z 0 z ( σ 11 σ 22 ) z 0 z ( σ 11 σ 22 ) d z d z + ,
u 2 = C 2 z 0 z ( σ 11 σ 22 ) d z + ,
u 3 = C 2 [ 1 4 z 0 z ( σ 11 σ 22 ) z 0 z ( σ 11 σ 22 ) d z d z + z 0 z σ 12 z 0 z σ 12 d z d z ] + ,
u 4 = C z 0 z σ 12 d z + .
cos 2 γ = 1 + C 2 { 1 2 z 0 z ( σ 11 σ 22 ) z 0 z ( σ 11 σ 22 ) d z d z 1 4 [ z 0 z ( σ 11 σ 22 ) d z ] 2 ( z 0 z σ 12 d z ) 2 } + .
C z 0 z 1 ( σ 11 σ 22 ) d z + C 3 z 0 z 1 ( σ 11 σ 22 ) d z + = cos 2 α * 1 sin 2 γ 1 ,
2 C z 0 z 1 σ 12 d z + 2 C 3 z 0 z 1 σ 12 d z + = sin 2 α * 1 sin 2 γ 1 ,
= 1 2 z 0 z ( σ 11 σ 22 ) z 0 z ( σ 11 σ 22 ) d z d z 1 4 [ z 0 z ( σ 11 σ 22 ) d z ] 2 ( z 0 z σ 12 d z ) 2 .
sin 2 γ 2 γ , cos 2 γ 1 ,
C 2 z 0 z 1 ( σ 11 σ 22 ) d z = γ 1 cos 2 α * 1 ,
C z 0 z 1 ( σ 11 σ 22 ) d z = γ 1 sin 2 α * 1 .
C z 0 z 1 ( σ 11 σ 22 ) d z = cos 2 α * 1 tan 2 γ 1 + 2 z 0 z 1 tan 2 2 γ cos 2 α * d γ d z d z ,
2 C z 0 z 1 ( σ 11 σ 22 ) d z = sin 2 α * 1 tan 2 γ 1 + 2 z 0 z 1 tan 2 2 γ sin 2 α * d γ d z d z .
C z 0 z 1 ( σ 11 σ 22 ) d z = cos 2 α * 1 tan 2 γ 1 ,
2 C z 0 z 1 σ 12 d z = sin 2 α * 1 tan 2 γ 1 .
z 0 z 1 tan 2 2 γ cos 2 α * d γ d z d z = tan 2 2 γ 1 d γ d z z = z 1 z 0 z 1 cos 2 α * d z z 0 z 1 z 0 z cos 2 α * d z Π d z ,
z 0 z 1 tan 2 2 γ sin 2 α * d γ d z d z = tan 2 2 γ 1 d γ d z z = z 1 z 0 z 1 sin 2 α * d z z 0 z 1 z 0 z cos 2 α * d z Π d z ,
Π = 4 tan 2 γ cos 2 2 γ ( d γ d z ) 2 + tan 2 2 γ d 2 γ d z 2 .
d γ d z z = z 1 γ 1 z 1 z 0 ,
z 0 z 1 cos 2 α * d z cos 2 α * 1 ( z 1 z 0 ) ,
z 0 z 1 sin 2 α * d z sin 2 α * 1 ( z 1 z 0 ) ,
z 0 z 1 tan 2 2 γ cos 2 α * d γ d z d z = γ 1 cos 2 α * 1 tan 2 2 γ 1 ,
z 0 z 1 tan 2 2 γ sin 2 α * d γ d z d z = γ 1 sin 2 α * 1 tan 2 2 γ 1 .
C z 0 z 1 ( σ 11 σ 22 ) d z = cos 2 α * 1 tan 2 γ 1 ( 1 2 γ 1 tan 2 2 γ 1 ) ,
C z 0 z 1 σ 12 d z = sin 2 α * 1 tan 2 γ 1 ( 1 2 γ 1 tan 2 2 γ 1 ) .
C z 0 z 1 ( σ 11 σ 22 ) d z = cos 2 α * 1 sin 2 γ 1 ,
2 C z 0 z 1 σ 12 d z = sin 2 α * 1 sin 2 γ 1 .

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