Abstract

In integrated photoelasticity the measurement data may be strongly influenced by the rotation of the principal stress axes. By pointwise measurements, usually the characteristic directions and optical retardation are determined and on their basis conclusions about the stress distribution along the light beam are made. However, often the integrated fringe pattern, obtained in a circular polariscope, can be recorded. This fringe pattern is determined by the integrals of the principal stresses as well as by the rotation of the principal stress directions. We consider the effect of rotation of the principal stress axes in the general form. It is shown that the rotation diminishes the distance between interference fringes and the contrast of the fringe pattern is also diminished.

© 2009 Optical Society of America

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References

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  1. F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystellinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3-254 (1843).
  2. H. J. Menges, “Die experimentelle Ermittlung räumlicher Spannungszustände an durchsichtigen Modellen mit Hilfe des Tyndallefektes,” Z. Angew. Math. Mech. 20, 210-217 (1940).
    [CrossRef]
  3. A. Kammerer, Recherches sur la Photoélasticimétrie (Herman et Cie, 1944).
  4. R. Plechata, “Elliptical polarization at a three-dimensionally loaded continuum,” Acta Tech. CSAV 9, 43-50 (1964).
  5. D. C. Drucker and R. D. Mindlin, “Stress analysis by three-dimensional photoelastic methods,” J. Appl. Phys. 11, 724-732 (1940).
    [CrossRef]
  6. V. L. Ginzburg, “Investigation of stress by the optical method,” Zh. Tekh. Fiz. 14, 181-192 (1944) (in Russian).
  7. R. C. Jones, “A new calculus for the treatment of optical systems, VII,” J. Opt. Soc. Am. 38, 671-685 (1948).
    [CrossRef]
  8. R. D. Mindlin and L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89-95 (1949).
    [CrossRef]
  9. H. K. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13-22 (1966).
    [CrossRef]
  10. H. Aben, Integrated Photoelasticity (McGraw-Hill, 1979).
  11. H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414-1421 (1986).
    [CrossRef]
  12. H. Lawrence and H. N. Lee, “Effects of rotation of principal stresses on photoelastic retardation,” Proc. Soc. Exp. Stress Anal. 21, 306-312 (1964).
  13. H. Kubo and R. Nagata, “Considerations of propagation of light waves in photoelastic materials and crystals,” Optik 52, 37-47 (1978).
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    [CrossRef]
  15. L. Ainola and H. Aben, “On the generalized Wertheim law in integrated photoelasticity,” J. Opt. Soc. Am. A 25, 1843-1849 (2008).
  16. L. Ainola and H. Aben, “Principal formulas of integrated photoelasticity in terms of characteristic parameters,” J. Opt. Soc. Am. A 22, 1181-1186 (2005).
    [CrossRef]
  17. S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Its Application to Physical Problems (Andhra University, 1951).
  18. P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).
  19. H. Aben and L. Ainola, “Isochromatic fringes in photoelasticity,” J. Opt. Soc. Am. A 17, 750-755 (2000).
    [CrossRef]
  20. H. Aben and J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172-7179 (1997).
    [CrossRef]
  21. H. Aben and L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404-2411 (1998).
    [CrossRef]
  22. P. Hartman, Ordinary Differential Equations (Wiley, 1964).

2008

2005

2001

2000

1998

1997

1986

1978

H. Kubo and R. Nagata, “Considerations of propagation of light waves in photoelastic materials and crystals,” Optik 52, 37-47 (1978).

1966

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

1964

R. Plechata, “Elliptical polarization at a three-dimensionally loaded continuum,” Acta Tech. CSAV 9, 43-50 (1964).

H. Lawrence and H. N. Lee, “Effects of rotation of principal stresses on photoelastic retardation,” Proc. Soc. Exp. Stress Anal. 21, 306-312 (1964).

1949

R. D. Mindlin and L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89-95 (1949).
[CrossRef]

1948

1944

V. L. Ginzburg, “Investigation of stress by the optical method,” Zh. Tekh. Fiz. 14, 181-192 (1944) (in Russian).

1940

D. C. Drucker and R. D. Mindlin, “Stress analysis by three-dimensional photoelastic methods,” J. Appl. Phys. 11, 724-732 (1940).
[CrossRef]

H. J. Menges, “Die experimentelle Ermittlung räumlicher Spannungszustände an durchsichtigen Modellen mit Hilfe des Tyndallefektes,” Z. Angew. Math. Mech. 20, 210-217 (1940).
[CrossRef]

1843

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystellinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3-254 (1843).

Aben, H.

Aben, H. K.

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

Ainola, L.

Bhagavantam, S.

S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Its Application to Physical Problems (Andhra University, 1951).

Drucker, D. C.

D. C. Drucker and R. D. Mindlin, “Stress analysis by three-dimensional photoelastic methods,” J. Appl. Phys. 11, 724-732 (1940).
[CrossRef]

Gdoutos, E. E.

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

Ginzburg, V. L.

V. L. Ginzburg, “Investigation of stress by the optical method,” Zh. Tekh. Fiz. 14, 181-192 (1944) (in Russian).

Goodman, L. E.

R. D. Mindlin and L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89-95 (1949).
[CrossRef]

Hartman, P.

P. Hartman, Ordinary Differential Equations (Wiley, 1964).

Jones, R. C.

Josepson, J.

Kammerer, A.

A. Kammerer, Recherches sur la Photoélasticimétrie (Herman et Cie, 1944).

Kubo, H.

H. Kubo and R. Nagata, “Considerations of propagation of light waves in photoelastic materials and crystals,” Optik 52, 37-47 (1978).

Lawrence, H.

H. Lawrence and H. N. Lee, “Effects of rotation of principal stresses on photoelastic retardation,” Proc. Soc. Exp. Stress Anal. 21, 306-312 (1964).

Lee, H. N.

H. Lawrence and H. N. Lee, “Effects of rotation of principal stresses on photoelastic retardation,” Proc. Soc. Exp. Stress Anal. 21, 306-312 (1964).

Menges, H. J.

H. J. Menges, “Die experimentelle Ermittlung räumlicher Spannungszustände an durchsichtigen Modellen mit Hilfe des Tyndallefektes,” Z. Angew. Math. Mech. 20, 210-217 (1940).
[CrossRef]

Mindlin, R. D.

R. D. Mindlin and L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89-95 (1949).
[CrossRef]

D. C. Drucker and R. D. Mindlin, “Stress analysis by three-dimensional photoelastic methods,” J. Appl. Phys. 11, 724-732 (1940).
[CrossRef]

Nagata, R.

H. Kubo and R. Nagata, “Considerations of propagation of light waves in photoelastic materials and crystals,” Optik 52, 37-47 (1978).

Neumann, F. E.

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystellinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3-254 (1843).

Plechata, R.

R. Plechata, “Elliptical polarization at a three-dimensionally loaded continuum,” Acta Tech. CSAV 9, 43-50 (1964).

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 Its Application to Physical Problems (Andhra University, 1951).

Abh. Kön. Akad. Wiss. Berlin

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystellinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3-254 (1843).

Acta Tech. CSAV

R. Plechata, “Elliptical polarization at a three-dimensionally loaded continuum,” Acta Tech. CSAV 9, 43-50 (1964).

Appl. Opt.

Exp. Mech.

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

J. Appl. Phys.

R. D. Mindlin and L. E. Goodman, “The optical equations of three-dimensional photoelasticity,” J. Appl. Phys. 20, 89-95 (1949).
[CrossRef]

D. C. Drucker and R. D. Mindlin, “Stress analysis by three-dimensional photoelastic methods,” J. Appl. Phys. 11, 724-732 (1940).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Optik

H. Kubo and R. Nagata, “Considerations of propagation of light waves in photoelastic materials and crystals,” Optik 52, 37-47 (1978).

Proc. Soc. Exp. Stress Anal.

H. Lawrence and H. N. Lee, “Effects of rotation of principal stresses on photoelastic retardation,” Proc. Soc. Exp. Stress Anal. 21, 306-312 (1964).

Z. Angew. Math. Mech.

H. J. Menges, “Die experimentelle Ermittlung räumlicher Spannungszustände an durchsichtigen Modellen mit Hilfe des Tyndallefektes,” Z. Angew. Math. Mech. 20, 210-217 (1940).
[CrossRef]

Zh. Tekh. Fiz.

V. L. Ginzburg, “Investigation of stress by the optical method,” Zh. Tekh. Fiz. 14, 181-192 (1944) (in Russian).

Other

A. Kammerer, Recherches sur la Photoélasticimétrie (Herman et Cie, 1944).

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

S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Its Application to Physical Problems (Andhra University, 1951).

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

P. Hartman, Ordinary Differential Equations (Wiley, 1964).

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Equations (121)

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d z = C ( σ 1 σ 2 ) + 2 d φ d z sin   Δ   cot   κ ,
d κ d z = d φ d z cos   Δ ,
tan   κ = A 2 A 1 ,
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 = ( β 0 + i β 1 β 2 + i β 3 β 2 + i β 3 β 0 i β 1 ) ,
β 0 2 + β 1 2 + β 2 2 + β 3 2 = 1.
d U d z = V U
U ( z 0 ) = s 0 .
s 0 = ( 1 0 0 1 ) .
s 1 = ( i 0 0 i ) ,     s 2 = ( 0 1 1 0 ) ,     s 3 = ( 0 i i 0 ) .
s 0 s j = s j s 0 = s j ,     s j 2 = s 0 ,
s 1 s 2 = s 2 s 1 = s 3 ,     s 2 s 3 = s 3 s 2 = s 1 ,
s 3 s 1 = s 1 s 3 = s 2 ,     j = 1 , 2 , 3.
σ = 1 2 C ( σ 11 σ 22 ) ,     τ = C σ 12 .
V = σ s 1 τ s 3 ,
U = β 0 s 0 + β 1 s 1 + β 2 s 2 + β 3 s 3 .
β 0 s 0 + β 1 s 1 + β 2 s 2 + β 3 s 3 = ( σ s 1 τ s 3 ) ( β 0 s 0 + β 1 s 1 + β 2 s 2 + β 3 s 3 ) .
β 0 = σ β 1 + τ β 3 ,
β 1 = σ β 0 + τ β 2 ,
β 2 = σ β 3 τ β 1 ,
β 3 = σ β 2 τ β 0 .
β 0 ( z 0 ) = 1 ,     β 1 ( z 0 ) = β 2 ( z 0 ) = β 3 ( z 0 ) = 0.
E = P 0 Q 45 U Q 45 E 0 ,
P 0 = ( 1 0 0 0 ) ,
Q 45 = 1 + i 2 ( s 0 s 3 ) ,     Q 45 = 1 + i 2 ( s 0 + s 3 ) .
E 0 = ( 0 1 ) .
E = ( 1 + i ) 2 4 ( 1 0 0 0 ) ( s 0 s 3 ) ( β 0 s 0 + β 1 s 1 + β 2 s 2 + β 3 s 3 ) ( s 0 + s 3 ) ( 0 1 ) ,
E = ( 1 + i ) 2 2 ( β 1 + β 3 0 ) .
I = E ̃ E ,
I = β 1 2 + β 3 2 .
β 1 = β 1 ( x 1 , x 2 , z ) ,     β 3 = β 3 ( x 1 , x 2 , z ) ,
I = I ( x 1 , x 2 , z ) .
I ( x 1 , x 2 ) = β 1 2 + β 3 2 ,
I l = β 0 2 + β 2 2 .
I ( x 1 , x 2 ) = const .
I [ x 1 , x 2 ( x 1 ) ] = const .
I x 1 + I x 2 d x 2 d x 1 = 0 ,
d x 2 d x 1 = I x 1 I x 2 .
d x 2 d x 1 = x ̇ 2 x ̇ 1 ,
x ̇ 1 = I x 2 ,     x ̇ 2 = I x 1 .
I x 1 = 0 ,     I x 2 = 0.
I x 1 = k ( x 1 , x 2 ) h ( x 1 , x 2 ) ,
I x 2 = l ( x 1 , x 2 ) h ( x 1 , x 2 ) ,
k ( x 1 , x 2 ) = 0 ,     l ( x 1 , x 2 ) = 0 ,
h ( x 1 , x 2 ) = 0.
I x 1 = 2 β 1 β 1 x 1 + 2 β 3 β 3 x 1 ,
I x 2 = 2 β 1 β 1 x 2 + 2 β 3 β 3 x 2 .
2 β 1 β 1 x 1 + 2 β 3 β 3 x 1 = k ( x 1 , x 2 ) h ( x 1 , x 2 ) ,
2 β 1 β 1 x 2 + 2 β 3 β 3 x 2 = l ( x 1 , x 2 ) h ( x 1 , x 2 ) .
β 0 = 1 σ 2 + τ 2 ( σ β 1 + τ β 3 ) ,
β 2 = 1 σ 2 + τ 2 ( τ β 1 σ β 3 ) ,
( σ σ 2 + τ 2 β 1 ) + σ β 1 + ( τ σ 2 + τ 2 β 3 ) + τ β 3 = 0 ,
( τ σ 2 + τ 2 β 1 ) + τ β 1 ( σ σ 2 + τ 2 β 3 ) σ β 3 = 0.
β 1 ( x 1 , x 2 , z 0 ) = 0 ,     β 3 ( x 1 , x 2 , z 0 ) = 0 ,
β 1 ( x 1 , x 2 , z 0 ) = σ 0 , ( x 1 , x 2 ) ,     β 3 ( x 1 , x 2 , z 0 ) = τ 0 ( x 1 , x 2 ) ,
σ 0 ( x 1 , x 2 ) = σ ( x 1 , x 2 , z 0 ) ,     τ 0 ( x 1 , x 2 ) = τ ( x 1 , x 2 , z 0 ) .
β 1 σ σ + τ τ σ 2 + τ 2 β 1 + ( σ 2 + τ 2 ) β 1 + σ τ τ σ σ 2 + τ 2 β 3 = 0 ,
β 3 σ σ + τ τ σ 2 + τ 2 β 3 + ( σ 2 + τ 2 ) β 3 σ τ τ σ σ 2 + τ 2 β 1 = 0.
( 1 σ 2 + τ 2 β 1 ) + σ 2 + τ 2 β 1 + σ τ τ σ ( σ 2 + τ 2 ) 3 / 2 β 3 = 0 ,
( 1 σ 2 + τ 2 β 3 ) + σ 2 + τ 2 β 3 σ τ τ σ ( σ 2 + τ 2 ) 3 / 2 β 1 = 0.
u = z 0 z σ 2 + τ 2 d z .
2 β 1 u 2 + β 1 + R β 3 u = 0 ,
2 β 3 u 2 + β 3 R β 1 u = 0 ,
R = σ τ u τ σ u σ 2 + τ 2 .
β 1 u = 0 = 0 ,     β 3 u = 0 = 0 ,
| β 1 u | u = 0 = σ 0 σ 0 2 + τ 0 2 ,     | β 3 u | u = 0 = τ 0 σ 0 2 + τ 0 2 .
R = 1 1 + τ 2 / σ 2 u ( τ σ )
R = u arctan ( τ σ ) .
τ σ = 2 σ 12 σ 11 σ 22 .
2 σ 12 σ 11 σ 22 = tan   2 φ ,
τ σ = tan   2 φ .
R = 2 φ u ,
2 β 1 u 2 + β 1 + 2 φ u β 3 u = 0 ,
2 β 3 u 2 + β 3 2 φ u β 1 u = 0.
β 1 u = 0 = 0 ,     β 3 u = 0 = 0 ,
| β 1 u | u = 0 = cos   2 φ 0 ,     | β 3 u | u = 0 = sin   2 φ 0 ,
I ( u ) = [ β 1 ( u ) ] 2 + [ β 3 ( u ) ] 2 .
I x 1 = I u u x 1 ,
I x 2 = I u u x 2 ,
| I u | z = z = 0.
2 β 1 u 2 β 1 + 2 β 3 u 2 β 3 + β 1 2 + β 3 2 2 φ u Ψ = 0 ,
Ψ = β 1 u β 3 β 3 u β 1 .
I u = 2 ( β 1 u β 1 + β 3 u β 3 ) ,
2 I u 2 = 2 [ ( β 1 u ) 2 + ( β 3 u ) 2 ] + 2 ( 2 β 1 u 2 β 1 + 2 β 3 u 2 β 3 ) .
2 β 1 u 2 β 1 + 2 β 3 u 2 β 3 = 1 2 2 I u 2 [ ( β 1 u ) 2 + ( β 3 u ) 2 ] .
u [ ( β 1 u ) 2 + ( β 3 u ) 2 ] + u ( β 1 2 + β 3 2 ) = 0.
( β 1 u ) 2 + ( β 3 u ) 2 + β 1 2 + β 3 2 = K .
( β 1 u ) 2 + ( β 3 u ) 2 = 1 ( β 1 2 + β 3 2 ) .
2 I u 2 + 4 I = 2 + ϕ ,
ϕ = 4 φ u Ψ .
J = I u .
2 J u 2 + 4 J = ϕ u .
ϕ u = 4 2 φ u 2 Ψ + 4 φ u Ψ u .
Ψ u = 2 β 1 u 2 β 3 2 β 3 u 2 β 1 .
ψ u = 2 φ u ( β 1 u β 1 + β 3 u β 3 ) .
ψ u = φ u J .
Ψ = 0 u φ u J d u .
ϕ u = 4 2 φ u 2 0 u φ u J d u 4 ( φ u ) 2 J .
2 J u 2 + 4 [ 1 + ( φ u ) 2 ] J + 4 2 φ u 2 0 u φ u J d u = 0.
u = z 0 z σ 2 + τ 2 d z .
J ( u ) = 0.
J ( 0 ) = 0 ,     | J u | u = 0 = 2.
2 J u 2 + 4 J = 0.
J = sin   2 u .
sin   2 u = 0.
z 0 z σ 2 + τ 2 d z = n π 2 ,     n = 1 , 2 , .
I = sin 2 u .
0 I 1.
φ u = a ( x 1 , x 2 ) .
2 J u 2 + 4 ( 1 + a 2 ) J = 0.
J = 1 1 + a 2 sin   2 1 + a 2 u .
sin   2 1 + a 2 u = 0.
z 0 z σ 2 + τ 2 d z = n π 2 1 + a 2 ,     n = 1 , 2 , .
I = 1 1 + a 2 sin 2 1 + a 2 u .
0 I 1 1 + a 2 .
2 J u 2 + 4 [ 1 + ( φ u ) 2 ] J = 0.
0 m 4 [ 1 + ( φ u ) 2 ] M ,
π M u 2 u 1 π m .

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