Abstract

Two-dimensional photoelasticity is based on the classical Wertheim law. The integral Wertheim law can be used in integrated photoelasticity only in the case when the directions of the secondary principal stresses are constant on the light beam. We generalize the integral Wertheim law for the case when a slight rotation of the secondary principal directions takes place.

© 2008 Optical Society of America

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References

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  1. M. G. Wertheim, “Mémoire sur la double réfraction temporairement produite dans les corps isotropes, et sur la relation entre l'élasticité mécanique et entre l'élasticité optique,” Ann. Chim. Phys. 40, 156-221 (1854).
  2. A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).
  3. H. Wolf, Spannungsoptik (Springer-Verlag, 1961).
  4. H. Aben, Integrated Photoelasticity (McGraw-Hill, 1979).
  5. H. Aben and C. Guillemet, Photoelasticity of Glass (Springer-Verlag, 1993).
  6. H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. (Bellingham) 44, 093601 (2005).
    [CrossRef]
  7. H. Aben, A. Errapart, and L. Ainola, “On real and imaginary algorithms of optical tensor field tomography,” Proc. Est. Acad. Sci., Phys., Math. 55, 112-127 (2006).
  8. H. K. Aben, J. I. Josepson, and K.-J. Kell, “The case of weak birefringence in integrated photoelasticity,” Opt. Lasers Eng. 11, 145-157 (1989).
    [CrossRef]
  9. A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach Scientific, 1998).
  10. H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414-1421 (1986).
    [CrossRef]
  11. S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, 1969).
  12. P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).
  13. L. Ainola and H. Aben, “Principal formulas of integrated photoelasticity in terms of characteristic paramerters,” J. Opt. Soc. Am. A 22, 1181-1186 (2005).
    [CrossRef]
  14. A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math., Phys., Techn. 4, 115-126 (1962).
  15. A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik (Jena) 19, 261-272 (1962).
  16. A. Kuske, “L'analyse des phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de 'J',” Rev. Franç. Méc. 9, 49-58 (1964).

2006

H. Aben, A. Errapart, and L. Ainola, “On real and imaginary algorithms of optical tensor field tomography,” Proc. Est. Acad. Sci., Phys., Math. 55, 112-127 (2006).

2005

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. (Bellingham) 44, 093601 (2005).
[CrossRef]

L. Ainola and H. Aben, “Principal formulas of integrated photoelasticity in terms of characteristic paramerters,” J. Opt. Soc. Am. A 22, 1181-1186 (2005).
[CrossRef]

1998

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

1993

H. Aben and C. Guillemet, Photoelasticity of Glass (Springer-Verlag, 1993).

1989

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

1986

1979

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

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

1974

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).

1969

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

1964

A. Kuske, “L'analyse des phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de 'J',” Rev. Franç. Méc. 9, 49-58 (1964).

1962

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

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

1961

H. Wolf, Spannungsoptik (Springer-Verlag, 1961).

1854

M. G. Wertheim, “Mémoire sur la double réfraction temporairement produite dans les corps isotropes, et sur la relation entre l'élasticité mécanique et entre l'élasticité optique,” Ann. Chim. Phys. 40, 156-221 (1854).

Aben, H.

H. Aben, A. Errapart, and L. Ainola, “On real and imaginary algorithms of optical tensor field tomography,” Proc. Est. Acad. Sci., Phys., Math. 55, 112-127 (2006).

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. (Bellingham) 44, 093601 (2005).
[CrossRef]

L. Ainola and H. Aben, “Principal formulas of integrated photoelasticity in terms of characteristic paramerters,” J. Opt. Soc. Am. A 22, 1181-1186 (2005).
[CrossRef]

H. Aben and C. Guillemet, Photoelasticity of Glass (Springer-Verlag, 1993).

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

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

Aben, H. K.

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

Ainola, L.

H. Aben, A. Errapart, and L. Ainola, “On real and imaginary algorithms of optical tensor field tomography,” Proc. Est. Acad. Sci., Phys., Math. 55, 112-127 (2006).

L. Ainola and H. Aben, “Principal formulas of integrated photoelasticity in terms of characteristic paramerters,” J. Opt. Soc. Am. A 22, 1181-1186 (2005).
[CrossRef]

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. (Bellingham) 44, 093601 (2005).
[CrossRef]

Anton, J.

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. (Bellingham) 44, 093601 (2005).
[CrossRef]

Bhagavantam, S.

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

Errapart, A.

H. Aben, A. Errapart, and L. Ainola, “On real and imaginary algorithms of optical tensor field tomography,” Proc. Est. Acad. Sci., Phys., Math. 55, 112-127 (2006).

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. (Bellingham) 44, 093601 (2005).
[CrossRef]

Fuki, A. A.

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

Gdoutos, E. E.

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

Guillemet, C.

H. Aben and C. Guillemet, Photoelasticity of Glass (Springer-Verlag, 1993).

Josepson, J. I.

H. K. Aben, J. I. Josepson, and 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, and 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, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach Scientific, 1998).

Kuske, A.

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).

A. Kuske, “L'analyse des phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de 'J',” Rev. Franç. Méc. 9, 49-58 (1964).

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

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

Naida, O. N.

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

Robertson, G.

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).

Theocaris, P. S.

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

Venkatarayudu, T.

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

Wertheim, M. G.

M. G. Wertheim, “Mémoire sur la double réfraction temporairement produite dans les corps isotropes, et sur la relation entre l'élasticité mécanique et entre l'élasticité optique,” Ann. Chim. Phys. 40, 156-221 (1854).

Wolf, H.

H. Wolf, Spannungsoptik (Springer-Verlag, 1961).

Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math., Phys., Techn.

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

Ann. Chim. Phys.

M. G. Wertheim, “Mémoire sur la double réfraction temporairement produite dans les corps isotropes, et sur la relation entre l'élasticité mécanique et entre l'élasticité optique,” Ann. Chim. Phys. 40, 156-221 (1854).

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. (Bellingham) 44, 093601 (2005).
[CrossRef]

Opt. Lasers Eng.

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

Optik (Jena)

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

Proc. Est. Acad. Sci., Phys., Math.

H. Aben, A. Errapart, and L. Ainola, “On real and imaginary algorithms of optical tensor field tomography,” Proc. Est. Acad. Sci., Phys., Math. 55, 112-127 (2006).

Rev. Franç. Méc.

A. Kuske, “L'analyse des phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de 'J',” Rev. Franç. Méc. 9, 49-58 (1964).

Other

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

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

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

A. Kuske and G. Robertson, Photoelastic Stress Analysis (Wiley, 1974).

H. Wolf, Spannungsoptik (Springer-Verlag, 1961).

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

H. Aben and C. Guillemet, Photoelasticity of Glass (Springer-Verlag, 1993).

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Equations (123)

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δ = C ( σ 1 σ 2 ) t ,
δ cos 2 φ = C ( σ 11 σ 22 ) t ,
δ sin 2 φ = 2 C σ 12 t .
δ cos 2 φ = C ( σ 11 σ 22 ) d z ,
δ sin 2 φ = 2 C σ 12 d z ,
δ = C ( σ 1 σ 2 ) d z .
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 .
d U d z U 1 = V ,
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 ) .
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 .
U ( z 0 ) = ( 1 0 0 1 ) .
γ ( z 0 ) = 0 , α * ( z 0 ) = α 0 ( 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 = η ,
sin 2 α * d γ d z + tan 2 γ cos 2 α * d α * d z = ϑ ,
η = 1 2 C ( σ 11 σ 22 ) , ϑ = C σ 12 .
γ ( z 0 ) = 0 , α * ( z 0 ) = 1 2 arctan ϑ ( z 0 ) η ( z 0 ) .
d d z ( cos 2 α * sin 2 γ ) = 2 η cos 2 γ ,
d d z ( sin 2 α * sin 2 γ ) = 2 ϑ cos 2 γ .
cos 2 α * sin 2 γ = 2 z 0 z η cos 2 γ d z ,
sin 2 α * sin 2 γ = 2 z 0 z ϑ cos 2 γ d z .
sin 2 2 γ = 4 ( z 0 z η cos 2 γ d z ) 2 + 4 ( z 0 z ϑ cos 2 γ d z ) 2 .
κ = cos 2 γ ,
1 κ 2 = 4 ( z 0 z η κ d z ) 2 + 4 ( z 0 z ϑ κ d z ) 2 .
d κ d z = 4 ( η z 0 z η κ d z + ϑ z 0 z ϑ κ d z ) .
d 2 κ d z 2 = 4 [ ( η 2 + ϑ 2 ) κ + η z 0 z η κ d z + ϑ z 0 z ϑ κ d z ] ,
d 3 κ d z 3 = 4 [ ( η 2 + ϑ 2 ) d κ d z + 3 ( η η + ϑ ϑ ) κ + η z 0 z η κ d z + ϑ z 0 z ϑ κ d z ] .
η ϑ η ϑ 0 ,
z 0 z η κ d z = ϑ d 2 κ d z 2 ϑ d κ d z + 4 ϑ ( η 2 + ϑ 2 ) κ 4 ( η ϑ η ϑ ) ,
z 0 z ϑ κ d z = η d 2 κ d z 2 + η d κ d z 4 η ( η 2 + ϑ 2 ) κ 4 ( η ϑ η ϑ ) .
a ( η , ϑ ) d 3 κ d z 3 + b ( η , ϑ ) d 2 κ d z 2 + c ( η , ϑ ) d κ d z + d ( η , ϑ ) κ = 0 .
a ( η , ϑ ) = η ϑ η ϑ ,
b ( η , ϑ ) = η ϑ η ϑ ,
c ( η , ϑ ) = 4 ( η 2 + ϑ 2 ) ( η ϑ η ϑ ) + η ϑ η ϑ ,
d ( η , ϑ ) = 12 ( η η + ϑ ϑ ) ( η ϑ η ϑ ) 4 ( η 2 + ϑ 2 ) ( η ϑ η ϑ ) .
κ ( z 0 ) = 1 , d κ d z z = z 0 = 0 ,
d 2 κ d z 2 z = z 0 = 4 { [ η ( z 0 ) ] 2 + [ ϑ ( z 0 ) ] 2 } .
2 γ = arccos κ .
η ϑ η ϑ = 0 .
ϑ = k η ,
z 0 z ϑ k d z = k z 0 z η κ d z .
d κ d z = 4 ( 1 + k 2 ) η z 0 z η κ d z ,
d 2 κ d z 2 = 4 ( 1 + k 2 ) ( η 2 κ + η z 0 z η κ d z ) .
d 2 κ d z 2 η η d κ d z + 4 ( 1 + k 2 ) η 2 κ = 0 .
κ ( z 0 ) = 1 , d κ d z z = z 0 = 0 .
κ = A 1 sin 2 1 + k 2 z 0 z η d z + A 2 cos 2 1 + k 2 z 0 z η d z .
κ = cos 2 1 + k 2 z 0 z η d z .
2 γ = 2 1 + k 2 z 0 z η d z ,
2 γ = 2 z 0 z η 2 + ϑ 2 d z .
2 γ = C z 0 z ( σ 11 σ 22 ) 2 + 4 σ 12 2 d z .
( σ 11 σ 22 ) 2 + 4 σ 12 2 = σ 1 σ 2 ,
2 γ = C z 0 z ( σ 1 σ 2 ) d z .
2 σ 1 2 σ 11 σ 22 = k = const .
2 σ 12 σ 11 σ 22 = tan 2 φ ,
d 2 κ d z 2 η η d κ d z + 4 ( η 2 + ϑ 2 ) κ + 4 η ( ϑ η ) z 0 z ϑ κ d z = 0 .
ϑ = k η + ϑ 1 ,
min k z 0 z * ( ϑ k η ) 2 d z ,
k = z 0 z * ϑ η d z z 0 z * η 2 d z .
L ( κ ) = F ( η , ϑ 1 , κ ) ,
L ( κ ) = d 2 κ d z 2 η η d κ d z + 4 ( 1 + k 2 ) η 2 κ ,
F ( η , ϑ 1 , κ ) = 4 { k η [ 2 ϑ 1 κ + ( ϑ 1 η ) z 0 z η κ d z ] + [ ϑ 1 2 κ + η ( ϑ 1 η ) z 0 z ϑ 1 κ d z ] } .
F ( η , ϑ 1 , κ ) = ε F 1 ( η , ϑ 1 , κ ) ,
L ( κ ) = ε F 1 ( η , ϑ 1 , κ ) ,
κ = κ 0 + ε κ 1 + ε 2 κ 2 + .
L ( κ 0 ) = 0 ,
κ 0 ( z 0 ) = 1 , d κ 0 d z z = z 0 = 0 ,
L ( κ 1 ) = F 1 ( η , ϑ 1 , κ 0 ) ,
κ 1 ( z 0 ) = 0 , d κ 1 d z z = z 0 = 0 ,
L ( κ 2 ) = F 1 ( η , ϑ 1 , κ 1 ) ,
κ 2 ( z 0 ) = 0 , d κ 2 d z z = z 0 = 0 .
κ 0 = cos ξ ,
ξ = 2 1 + k 2 z 0 z η d z .
F 1 ( η , ϑ 1 , κ 0 ) = 4 { κ η [ 2 ϑ 1 cos ξ + 1 2 1 + k 2 ( ϑ 1 η ) sin ξ ] + ϑ 1 2 cos ξ + η ( ϑ 1 η ) z 0 z ϑ 1 cos ξ d z } .
κ 1 = 1 2 1 + k 2 [ z 0 z 1 η cos ξ F 1 ( η , ϑ 1 , κ 0 ) d z sin ξ z 0 z 1 η sin ξ F 1 ( η , ϑ 1 , κ 0 ) d z cos ξ ] .
2 γ = arccos ( cos ξ + ε κ 1 ) .
η = a , ϑ = b + c z ,
z 0 = 0 , z * = h .
k = b a + 1 2 c a h ,
ϑ 1 = d + c z ,
d = b k a .
ξ = g z ,
g = 2 1 + k 2 a .
F ( η , ϑ 1 , κ 0 ) = 4 p 0 + ( q 0 + q 1 z ) sin g z + ( r 0 + r 1 z + r 2 z 2 ) cos g z ,
p 0 = c 2 g , q 0 = k a + d g c , q 1 = c 2 g ,
r 0 = ( 2 k a + d ) d , r 1 = 2 ( 2 k a + d ) c ,
r 2 = c 2 .
κ 1 = 2 a 1 + k 2 ( I 1 sin g h I 2 cos g h ) ,
I 1 = A 1 + A 2 sin g h + A 3 sin 2 g h ,
I 2 = B 1 + B 2 cos g h + B 3 cos 2 g h .
A 1 = 1 2 r 0 h + 1 4 r 1 h 2 + 1 6 r 2 h 3 + 1 4 g q 0 1 8 r 1 g 2 ,
A 2 = p 0 g ,
A 3 = 1 8 q 1 g 2 + 1 4 r 0 g + 1 4 r 1 h g + 1 4 r 2 ( h 2 g 1 g 2 ) ,
B 1 = p 0 g + 1 2 q 0 h + 1 4 q 1 h 2 + 1 8 q 1 g 2 + r 0 4 g 1 4 r 2 g 2 ,
B 2 = p 0 g ,
B 3 = 1 8 q 1 g 2 r 2 4 g 1 4 r 1 h g 1 4 r 2 ( h 2 g 1 g 2 ) .
η = 1 , ϑ = 0.5 + 0.2 z , h = 1 .
k = 0.6 , ϑ 1 = 0.1 + 0.2 z .
g = 2.3324 , κ 0 = 0.6901 , κ 1 = 0.0134 .
2 γ = arccos ( 0.6766 ) .
2 γ = 132.6 ° .
2 γ = 2 0 h c 2 z 2 + 2 b c z + ( a 2 + b 2 ) d z ,
2 γ = a 2 c ( Arsh b + c h a Arsh b a ) + b + c h c c 2 z 2 + 2 b c z + ( a 2 + b 2 ) b a a 2 + b 2 .
2 γ = 2.3257 , 2 γ = 133.2 ° .
η = 1 , ϑ = 0.5 + 0.2 z , h = 4 .
k = 0.9 , ϑ 1 = 0.4 + 0.2 z ,
g = 2.6907 , κ 0 = 0.2307 , κ 1 = 0.2543 .
2 γ = 4 π arccos ( 0.4846 ) .
2 γ = 601.2 ° .
2 γ = 9.8108 , 2 γ = 562.1 .

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