Abstract

This research aims to find a new way to get the intensity equations for the phase-shifting model in digital photoelasticity. The procedure is based on the rotation of the analyzer itself. From the intensity equations, the isoclinic and isochromatic equations parameters are deduced by applying a new numerical technique. This approach can be done to calculate how many images allow the resolution of the polariscope. Each image indicates the stress forces in the object. In this study the plane polariscope was used. The amount of images will determinate the number of errors and uncertainties of the study, due to the observation that the veracity of the equations increases considerably with a large amout of images. Several analyses are performed with different amounts of photographic images. The results showed the possibility to measure stress forces with high precision using plane polariscopes.

© 2016 Optical Society of America

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References

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  1. C. A. Magalhães and P. A. A. Magalhães Júnior, “New numerical methods for the photoelastic technique with high accuracy,” J. Appl. Phys. 112(8), 083111 (2012).
    [Crossref]
  2. M. Ramji, V. Y. Gadre, and K. Ramesh, “Comparative study of evaluation of primary isoclinic data by various spatial domain methods in digital photoelasticity,” J. Strain Anal. Eng. 41(5), 333–348 (2006).
    [Crossref]
  3. P. Pinit and E. Umezaki, “Absolute fringe order determination in digital photoelasticity,” JSMME 2(4), 519–529 (2008).
  4. M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – Issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
    [Crossref]
  5. K. Ramesh, Digital Photoelasticity: Advanced Techniques and Applications (Springer-Verlag, 2000).
  6. M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
    [Crossref]
  7. K. Gallo and G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” J. Opt. Soc. Am. B 16(2), 267–269 (1999).
    [Crossref]
  8. J. A. Quiroga and A. González-Cano, “Method of error analysis for phase-measuring algorithms applied to photoelasticity,” Appl. Opt. 37(20), 4488–4495 (1998).
    [Crossref] [PubMed]
  9. C. Fang, Y. Xiang, K. Qi, C. Zhang, and C. Yu, “An 11-frame phase shifting algorithm in lateral shearing interferometry,” Opt. Express 21(23), 28325–28333 (2013).
    [Crossref] [PubMed]
  10. F. Liu, Y. Wu, and F. Wu, “Phase shifting interferometry from two normalized interferograms with random tilt phase-shift,” Opt. Express 23(15), 19932–19946 (2015).
    [Crossref] [PubMed]
  11. L. Fei, X. Lu, H. Wang, W. Zhang, J. Tian, and L. Zhong, “Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms,” Opt. Express 22(25), 30910–30923 (2014).
    [Crossref] [PubMed]
  12. O. Medina, J. C. Estrada, and M. Servin, “Robust adaptive phase-shifting demodulation for testing moving wavefronts,” Opt. Express 21(24), 29687–29694 (2013).
    [Crossref] [PubMed]
  13. Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express 20(15), 16471–16479 (2012).
    [Crossref]
  14. H. Guo and B. Lü, “Phase-shifting algorithm by use of Hough transform,” Opt. Express 20(23), 26037–26049 (2012).
    [Crossref] [PubMed]
  15. C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. Arévalo Aguilar, G. Rodriguez-Zurita, and V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express 21(14), 17228–17233 (2013).
    [Crossref] [PubMed]
  16. F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
    [Crossref] [PubMed]
  17. N. Plouzennec and A. Lagarde, “Two-wavelengh method for full-field automated photoelasticity,” Exp. Mech. 39(4), 274–277 (1999).
    [Crossref]
  18. M. Ramji and R. G. R. Prasath, “Sensivity of isoclinic data using various phase shifting techniques in digital photoelasticity towards generalized error sources,” Opt. Lasers Eng. 49(9-10), 1153–1167 (2011).
    [Crossref]

2015 (2)

2014 (1)

2013 (3)

2012 (3)

2011 (1)

M. Ramji and R. G. R. Prasath, “Sensivity of isoclinic data using various phase shifting techniques in digital photoelasticity towards generalized error sources,” Opt. Lasers Eng. 49(9-10), 1153–1167 (2011).
[Crossref]

2008 (3)

P. Pinit and E. Umezaki, “Absolute fringe order determination in digital photoelasticity,” JSMME 2(4), 519–529 (2008).

M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – Issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
[Crossref]

M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
[Crossref]

2006 (1)

M. Ramji, V. Y. Gadre, and K. Ramesh, “Comparative study of evaluation of primary isoclinic data by various spatial domain methods in digital photoelasticity,” J. Strain Anal. Eng. 41(5), 333–348 (2006).
[Crossref]

1999 (2)

N. Plouzennec and A. Lagarde, “Two-wavelengh method for full-field automated photoelasticity,” Exp. Mech. 39(4), 274–277 (1999).
[Crossref]

K. Gallo and G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” J. Opt. Soc. Am. B 16(2), 267–269 (1999).
[Crossref]

1998 (1)

Arévalo Aguilar, L. M.

Assanto, G.

Du, Y.

Estrada, J. C.

Fang, C.

Fei, L.

Feng, G.

Gadre, V. Y.

M. Ramji, V. Y. Gadre, and K. Ramesh, “Comparative study of evaluation of primary isoclinic data by various spatial domain methods in digital photoelasticity,” J. Strain Anal. Eng. 41(5), 333–348 (2006).
[Crossref]

Gallo, K.

González-Cano, A.

Guerrero-Sánchez, F.

Guo, H.

Ixba-Santos, V.

Juarez-Salazar, R.

Lagarde, A.

N. Plouzennec and A. Lagarde, “Two-wavelengh method for full-field automated photoelasticity,” Exp. Mech. 39(4), 274–277 (1999).
[Crossref]

Li, H.

Liu, F.

Lu, X.

Lü, B.

Magalhães, C. A.

C. A. Magalhães and P. A. A. Magalhães Júnior, “New numerical methods for the photoelastic technique with high accuracy,” J. Appl. Phys. 112(8), 083111 (2012).
[Crossref]

Magalhães Júnior, P. A. A.

C. A. Magalhães and P. A. A. Magalhães Júnior, “New numerical methods for the photoelastic technique with high accuracy,” J. Appl. Phys. 112(8), 083111 (2012).
[Crossref]

Medina, O.

Meneses-Fabian, C.

Pinit, P.

P. Pinit and E. Umezaki, “Absolute fringe order determination in digital photoelasticity,” JSMME 2(4), 519–529 (2008).

Plouzennec, N.

N. Plouzennec and A. Lagarde, “Two-wavelengh method for full-field automated photoelasticity,” Exp. Mech. 39(4), 274–277 (1999).
[Crossref]

Prasath, R. G. R.

M. Ramji and R. G. R. Prasath, “Sensivity of isoclinic data using various phase shifting techniques in digital photoelasticity towards generalized error sources,” Opt. Lasers Eng. 49(9-10), 1153–1167 (2011).
[Crossref]

Qi, K.

Quiroga, J. A.

Ramesh, K.

M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
[Crossref]

M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – Issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
[Crossref]

M. Ramji, V. Y. Gadre, and K. Ramesh, “Comparative study of evaluation of primary isoclinic data by various spatial domain methods in digital photoelasticity,” J. Strain Anal. Eng. 41(5), 333–348 (2006).
[Crossref]

Ramji, M.

M. Ramji and R. G. R. Prasath, “Sensivity of isoclinic data using various phase shifting techniques in digital photoelasticity towards generalized error sources,” Opt. Lasers Eng. 49(9-10), 1153–1167 (2011).
[Crossref]

M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – Issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
[Crossref]

M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
[Crossref]

M. Ramji, V. Y. Gadre, and K. Ramesh, “Comparative study of evaluation of primary isoclinic data by various spatial domain methods in digital photoelasticity,” J. Strain Anal. Eng. 41(5), 333–348 (2006).
[Crossref]

Robledo-Sanchez, C.

Rodriguez-Zurita, G.

Servin, M.

Tian, J.

Umezaki, E.

P. Pinit and E. Umezaki, “Absolute fringe order determination in digital photoelasticity,” JSMME 2(4), 519–529 (2008).

Vargas, J.

Wang, H.

Wu, F.

Wu, Y.

Xiang, Y.

Yu, C.

Zhang, C.

Zhang, W.

Zhong, L.

Zhou, S.

Appl. Opt. (1)

Exp. Mech. (1)

N. Plouzennec and A. Lagarde, “Two-wavelengh method for full-field automated photoelasticity,” Exp. Mech. 39(4), 274–277 (1999).
[Crossref]

J. Appl. Phys. (1)

C. A. Magalhães and P. A. A. Magalhães Júnior, “New numerical methods for the photoelastic technique with high accuracy,” J. Appl. Phys. 112(8), 083111 (2012).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Strain Anal. Eng. (1)

M. Ramji, V. Y. Gadre, and K. Ramesh, “Comparative study of evaluation of primary isoclinic data by various spatial domain methods in digital photoelasticity,” J. Strain Anal. Eng. 41(5), 333–348 (2006).
[Crossref]

JSMME (1)

P. Pinit and E. Umezaki, “Absolute fringe order determination in digital photoelasticity,” JSMME 2(4), 519–529 (2008).

Opt. Express (8)

C. Fang, Y. Xiang, K. Qi, C. Zhang, and C. Yu, “An 11-frame phase shifting algorithm in lateral shearing interferometry,” Opt. Express 21(23), 28325–28333 (2013).
[Crossref] [PubMed]

F. Liu, Y. Wu, and F. Wu, “Phase shifting interferometry from two normalized interferograms with random tilt phase-shift,” Opt. Express 23(15), 19932–19946 (2015).
[Crossref] [PubMed]

L. Fei, X. Lu, H. Wang, W. Zhang, J. Tian, and L. Zhong, “Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms,” Opt. Express 22(25), 30910–30923 (2014).
[Crossref] [PubMed]

O. Medina, J. C. Estrada, and M. Servin, “Robust adaptive phase-shifting demodulation for testing moving wavefronts,” Opt. Express 21(24), 29687–29694 (2013).
[Crossref] [PubMed]

Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express 20(15), 16471–16479 (2012).
[Crossref]

H. Guo and B. Lü, “Phase-shifting algorithm by use of Hough transform,” Opt. Express 20(23), 26037–26049 (2012).
[Crossref] [PubMed]

C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. Arévalo Aguilar, G. Rodriguez-Zurita, and V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express 21(14), 17228–17233 (2013).
[Crossref] [PubMed]

F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
[Crossref] [PubMed]

Opt. Lasers Eng. (3)

M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – Issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
[Crossref]

M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – issues, implementation and application,” Opt. Lasers Eng. 46(3), 257–271 (2008).
[Crossref]

M. Ramji and R. G. R. Prasath, “Sensivity of isoclinic data using various phase shifting techniques in digital photoelasticity towards generalized error sources,” Opt. Lasers Eng. 49(9-10), 1153–1167 (2011).
[Crossref]

Other (1)

K. Ramesh, Digital Photoelasticity: Advanced Techniques and Applications (Springer-Verlag, 2000).

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Figures (4)

Fig. 1
Fig. 1 Optical arrangement of a plane polariscope.
Fig. 2
Fig. 2 Flowchart showing each processing stage.
Fig. 3
Fig. 3 Average error of θ and δ.
Fig. 4
Fig. 4 Results obtained through experimental measurements using the new equations with N = 4 and step = 4 of: σx, σy, τxy.

Tables (2)

Tables Icon

Table 1 Values of the real coefficients (bj, cj, dj, ej, fj, gj) when N = 4 and step = 4

Tables Icon

Table 2 Values of the real coefficients (bj, cj, dj, ej, fj, gj) when N = 3 and step = 4

Equations (10)

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

I= I o [ ( cos δ 2 ) 2 ( cos( βα ) ) 2 + ( sin δ 2 ) 2 ( cos( β+α2θ ) ) 2 ].
δ= 2πt F ( σ 1 σ 2 ).
θ= 1 2 tan 1 ( 2 τ xy σ x σ y ).
θ= 1 2 tan 1 ( j=1 N b j I j j=1 N c j I j ).
δ= cos 1 ( cos2( θα )( sin2θ j=1 N d j I j +cos2θ j=1 N e j I j ) sin2( θα )( cos2θ j=1 N f j I j +sin2θ j=1 N g j I j ) ).
Maximize j=1 N b j + c j subject to { Quantities 1) tan(2 θ ν ) j=1 N c j . I j ν = j=1 N b j . I j ν ν>2N 2) 1 b j 1, 1 c j 1 j=1..N 3) b j , c j are real numbers j=1..N
Maximize j=1 N d j + e j + f j + g j subject to { Quantities 1) cos( δ ν )sin[ 2.( θ ν α ) ].[ cos(2. θ ν ). j=1 N f j . I j ν +sin(2. θ ν ). j=1 N g j . I j ν ]= cos[ 2.( θ ν α ) ].[ sin(2. θ ν ). j=1 N d j . I j ν +cos(2. θ ν ). j=1 N e j . I j ν ] ν>4N 2) 1 d j 1, 1 e j 1, 1 f j 1, 1 g j 1 j=1..N 3) d j , e j , f j , g j are real number j=1..N
{ I j v = I o v .[ cos 2 ( δ v 2 ). cos 2 ( β j α )+ sin 2 ( δ v 2 ). cos 2 ( β j +α2. θ v ) ],j=1..N I o v [0; 255] random and real θ υ [ π/4, π/4 ] random and real δ υ [ 0, π ] random and real β j =[ (UpperAngle LowerAngle) π(j1) 180(step1) +π LowerAngle 180 ],j=1..N step=divisors (UpperAngle LowerAngle)+1 step is increase the angle of the analyzer (Integer in degrees) N=number of images from 3 until the step value (Integer) j=1..N β j =analyzer angle α=polarizer angle θ=isoclinic angle δ=isochromatic parameter LowerAngle = Lower angle analyzer in degrees (Integer) UpperAngle =Upper angle analyzer in degrees (Integer) Input: LowerAngle , UpperAngle and α Output: For each value of N and step: b j ,c j ,d j ,e j ,f j and g j with j=1..N
E θ = 1 M i=1 M | θ i exact θ i | | θ i exact | .
E δ = 1 M i=1 M | δ i exact δ i | | δ i exact | .

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