improved determination of retardation in digital photoelasticity by load stepping

*Corresponding author. Tel.:#91-512-597149; fax:#91-512-590007.E-mail address: [email protected] (K. Ramesh).

Optics and Lasers in Engineering 33 (2000) 387}400

Improved determination of retardation in digitalphotoelasticity by load stepping

K. Ramesh*, D.K. TamrakarDepartment of Mechanical Engineering, Indian Institute of Technology, Kanpur IIT, Kanpur 208016, India

Received 24 February 2000; received in revised form 15 August 2000; accepted 4 September 2000

Abstract

Use of phase shifting has revolutionized the acquisition of photoelastic data. Even since itsintroduction, various researchers have improved the methodology. In phase shifting theproblem of ambiguity exists in the sign of fractional retardation in some zones of the model.This has been overcome with the development of load stepping. However, the conventionalapproach to data processing in load stepping introduces several noise points in the phase map.In this paper a new methodology for data reduction in load stepping, which removes the noisepoints in the domain is presented. Results from four- and six-step algorithms are analyzed.A normalization scheme to improve the results has been proposed. A comparative study hasalso been carried out on the evaluation of phase map by phase shifting, load stepping and thenew methodology. ( 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

The main advantage of photoelasticity is that, it can give the distribution ofprincipal stress di!erence (isochromatics) and their orientation (isoclinics) for thewhole "eld. But it gives this result as an image of fringe patterns. The main task inphotoelasticity is the interpretation of these fringe patterns. In conventional methods,tedious compensation techniques are needed to obtain the fractional fringe orders atany point other than the fringe areas. The number of data points that could beconsidered is also less. These procedures are quite involved and requires skill in theidenti"cation of isochromatics. Hence, automation of data acquisition and analysis tominimize these problems and to provide fast and most accurate results has become

0143-8166/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 7 6 - 2

essential. With the advent of PC-based digital image processing systems, con-siderable success has been achieved in automating the data acquisition from the entire"eld.

In the early development of digital photoelasticity, the data acquisition is limited tofringe areas. The real potential of DIP hardware was realized only when the conceptof identifying the fringe "elds as phase maps came into existence. This introduced newconcepts in data acquisition. It is now possible to evaluate the total fringe order atevery pixel in the domain and not restricted to just fringe skeletons alone [1].

One of the early approaches in this was the method of phase shifting. In this,intensity data recorded for a few optical arrangements are judiciously used to solve forthe isochromatic and isoclinic fringe order. Following the lines of phase shifting [1],several variants of these such as polarization stepping [2] and load stepping appeared[3]. In polarization stepping the polarizer and analyzer are kept crossed for recordingintensity data. Load stepping essentially adopts one of the existing phase shiftingtechniques for intensity recording. Instead of recording intensity data for one load, thedata are recorded for small steps of loads on either side of the load of interest.

The methodologies such as phase shifting, polarization stepping or load steppinghave their own advantages and inherent limitations. Polarization stepping is bettersuited for evaluating isoclinic data rather than isochromatic data [1]. In this paper,the limitations of phase shifting and load stepping are initially highlighted. A newmethodology has been proposed which combines the merits of these two techniques.The approach is validated for the problem of a disk under diametral compression. Theapplicability of the new approach for evaluating the retardation in a complex stress"eld such as the problem of a ring under diametral compression is demonstrated.

2. Intensity of light transmitted for a generic arrangement of a circular polariscope

A generic optical arrangement of a circular polariscope is shown in Fig. 1 in whichthe second quarter-wave plate and the analyzer are kept at arbitrary positions. UsingJones calculus, the components of light vector along the analyzer axis and perpendicu-lar to the analyzer axis (for m"1353) are obtained as

GEbEb`p@2

H"1

2Ccosb sinb

!sinb cosbDC1!i cos 2g !i sin 2g

!i sin 2g 1#i cos 2gD

]Ccos

d2!i sin

d2

cos 2h !i sind2

sin 2h

!i sind2

sin 2h cosd2#i sin

d2

cos 2hDC1 i

i 1DG0

1Hke*ut (1)

Intensity of light transmitted is given by

Ii"I

"#

I!2#

I!2

[sin 2(bi!g

i) cos d!sin 2(h!g

i) cos 2(b

i!g

i) sin d] (2)

388 K. Ramesh, D.K. Tamrakar / Optics and Lasers in Engineering 33 (2000) 387}400

Fig. 1. Generic optical arrangement of a circular polariscope.

where I!

accounts for the amplitude of light vector and proportionality constant andI"

to account for background illumination/stray light. Iidenotes the intensity of light

transmitted for the ith optical arrangement.

3. Six-step phase shifting algorithm

One of the most widely used approaches of phase shifting to evaluate isochromaticand isoclinic parameter involves the use of six di!erent optical arrangements [1]. Thevarious positions and the respective intensity equations are summarized in Table 1.The arrangement given also accounts for the mismatch of quarter-wave plate error[1]. From the equations of Table 1, the values of h and d can be obtained as [4]

h#"

1

2tan~1A

I5!I

3I4!I

6B"

1

2tan~1A

I!sin d sin 2h

I!sin d cos 2hB, (3)

d#"tan~1A

(I5!I

3) sin 2h

##(I

4!I

6) cos 2h

#(I

1!I

2) B"tan~1A

I!sin d

#I!cos d

#B. (4)

Eq. (4) always guarantees a high modulation over the "eld and d evaluation will bequite accurate. In Eqs. (3) and (4) the subscript c indicates that the principal values ofthe inverse trigonometric function is referred. In subsequent discussions too the samenotation will be used.

The exact calculation of isoclinic angle and fractional retardation is not trivial fromEqs. (3) and (4). Eq. (3) shows that the isoclinic value is indeterminate at those points

K. Ramesh, D.K. Tamrakar / Optics and Lasers in Engineering 33 (2000) 387}400 389

Table 1Polariscope arrangement and the intensity equations for phase shifting/load stepping algorithm

No. m g b Intensity equation

a 3p/4 p/4 p/2I1"I

"#

I!2

(1#cos d)

b 3p/4 p/4 0I2"I

"#

I!2

(1!cos d)

c 3p/4 0 0I3"I

"#

I!2

(1!sin 2h sin d)

d 3p/4 p/4 p/4I4"I

"#

I!2

(1#cos 2h sin d)

e p/4 0 0I5"I

"#

I!2

(1#sin 2h sin d)

f p/4 3p/4 p/4I6"I

"#

I!2

(1!cos 2h sin d)

where d#"0, p, 2p, 3p,2. The implication is that the isoclinic contour will not be

continuous over the domain. When d#is not exactly equal to 0 or p or 2p, etc., but very

close to these values, h#does not become indeterminate but the value of h

#determined

will be unreliable and appears as noise in isoclinic fringe when plotted. This is knownas isochromatic}isoclinic interaction in the calculation of the isoclinic parameter.Apart from this, the principal value of h

#evaluated by Eq. (3) lies in the range !p/4

to p/4 and an ambiguity exists on whether h#

corresponds to p1

or p2

direction overthe domain [1]. Eq. (4) needs the correct values of isoclinic for evaluating theretardation. It is demonstrated in Ref. [1] that, the local oscillation on the values ofisoclinic does not adversely a!ect the retardation calculation. However, the ambiguityof h not representing uniformly either in p

1or p

2direction introduces an ambiguity

on the sign of fractional retardation in some zones of the model [1].A sequence of six images as per Table 1 is recorded for a disk under diametral

compression (diameter 60.10mm, thickness 6.28mm and load 773.0N). The materialstress fringe value is calculated by a digital technique [5] and is found to be11.9973N/mm/fr Fig. 2a shows the phase map obtained by Eq. (4). On examination,one can see that near the load-application points the phase map is not good. This isnot just due to high-stress gradient but is due to the interaction of isoclinic on theisochromatic evaluation. The principal values of h

#of Eq. (3) lies in the region

}p/4)h#)p/4 whereas, physically, the principal stress direction lies in the range

!p/2)h)p/2 and this introduces an ambiguity on d evaluation. Referring to Fig.2a, in the central portion of the disk, the value of h

#coincides with the p

1direction

and in zones close to the load-application point it corresponds to p2

direction. Onclose examination one can observe that the phase map has reversed in these zones. Ifthe direction of loading is changed with respect to the polarizer axis, these zones alsochange. For example, if the loading direction is horizontal then the zones in whichh#

corresponds to p1

and p2

are swapped. In the experiment such a situation can be


Fig. 2. Phase map for the problem of disk under diametral compression obtained by phase shifting: (a)using image sequence as in Table 1; (b) using image sequence as a, b, e, f, c, d.

obtained by just swapping the intensity maps recorded for I3

to I5

and I4

to I6

andvice versa. Fig. 2b shows the phase map obtained from swapping the intensity maps.

4. Photoelastic parameter evaluation by load stepping

Load stepping essentially uses the "rst four optical arrangements of the six-stepphase shifting technique (Table 1) for intensity recording. Instead of recording inten-sity data for only one load (p), intensity data is also recorded for small steps of loadson either side of the load of interest (i.e., p#dp and p!dp).

From the intensity equations of Table 1, one can de"ne the quantities I4, I,, Il and

I. as

I4"(I

1#I

2)/2"I

"#

I!2

,

I,"(I1!I

2)/2"

I!2

cos d,(5)

Il"I4!I

3"

I!2

sin d sin 2h,

I."I4!I

4"

I!2

sin d cos 2h.

The fractional retardation corresponding to load p for the whole "eld is obtained by[3]

d#"tan~1A

(I,p~$p

!I,p`$p

)/sin Ld#

(I,p~$p

#I,p`$p

)/cos Ld#B"tan~1A

I!sin d

#I!cos d

#B, (6)


Fig. 3. Phase map for the problem of disk under diametral compression obtained by loadstepping from: (a)experimental images, (b) theoretically-simulated images.

where Rd#

is given by

Ld#"cos~1A

I,p`$p

#I,p~$p

2I,p

B. (7)

The value of isoclinic angle is obtained by

h#"

1

2tan~1A

Ilp/sin d

#I.p/sin d

#B"tan~1A

I!/2 sin 2h

I!/2 cos 2hB. (8)

Intensity maps for loads p#dp (811.65N), p (773.0N) and p!dp (734.35N) for the"rst two optical arrangements shown in Table 1 are recorded for the problem of diskunder diametral compression referred in the previous section. Fig. 3a shows the phasemap obtained from these sets of images. The inspection of phase map indicates that itdoes not contain any zone, where the sign of d is ambiguous as in phase shifting (Fig.2). However, the phase map has signi"cant noise points. The existence of noise pointsis not just due to problems in recording intensity data but is signi"cantly due to thenature of Eqs. (6) and (7). Fig. 3b shows the phase map obtained from theoreticallysimulated images as per Table 1. Here again one observes the presence of several noisepoints.

The presence of noise could be attributed to the following factors. Eq. (7) shows thatthe value of Rd

#is indeterminate at those points where I,

pis zero (i.e.,

d"p/2, 3p/2,2). When d is not exactly equal to p/2 or 3p/2, etc., but very close tothese values, Rd

#does not become indeterminate but the value of Rd

#determined will

be unreliable. This eventually leads to an error in calculating d. The regions where thefringe gradient is low, the phase map is particularly very bad because of very lowintensity di!erence (I,

p}$p!I,

p`$p) in that region. A method to overcome this is to use

a higher load step. Thus, the extraction of data by load stepping is basically fringe-gradient dependent.


5. Noise-free determination of fractional retardation

The emphasis in this approach is to eliminate the presence of noise in retardationcalculation. In load stepping, only the dark- and bright-"eld images are processed forevaluating the fractional retardation and the images corresponding to other opticalarrangements are used for evaluating the isoclinic parameter. The new approachproposed uses the intensity maps corresponding to all the six arrangements forretardation calculation. For each load, fractional retardation is calculated as per thenormal phase shifting procedure. Using this, noise-free dark-"eld images are obtainedfor the three loads, namely p, p#dp and p!dp. From these dark-"eld images, thephase map corresponding to load p which is free of noise and ambiguity is thenobtained.

In six-step phase shifting, it is well established that d could be calculated moreaccurately by the following expression [1]:

d#"tan~1A

$J(I5!I

3)2#(I

4!I

6)2

(I1!I

2) B"tan~1A

$I!sin d

#I!cos d

#B. (9)

Eq. (9), which is same as Eq. (4), also provides a high modulation over the "eld. Byusing a tan 2( ) function and using only the #ve square root values one gets thefractional retardation, d

#in the range of 0(d

#(p. Noise-free dark-"eld isochro-

matics can be plotted using

g(x, y)"255

p(d

#), (10)

where g(x, y) is the grey level of the pixel at the location (x, y). On similar lines, thesimulated dark-"eld images are to be obtained for three loads p#dp, p and p!dp.From these three images, the phase map corresponding to the load p can be construc-ted.

Figs. 4a, b and c show the dark-"eld noise-free isochromatic plots corresponding toloads p#dp, p and p!dp, respectively, obtained for the disk referred in Section 4.Let g

p`$p, g

pand g

p~$pbe the gray levels of the pixel at the location (x, y) correspond-

ing to loads p#dp, p and p!dp, respectively. The variation of gp`$p

, gp

andgp~$p

for the diametral line of the circular disk under diametral compression areshown in Fig. 5a. If the sign of the gray-levels di!erence (g

p`$p!g

p~$p) is calculated

over the line of interest, then one can obtain a square-wave mask as shown in Fig. 5a.One gets the phase map corresponding to load p from

g#(x, y)"G

gp2

if (gp`$p

*gp~$p

),

255!gp2

if (gp`$p

(gp~$p

).(11)

The fractional fringe order dN

at any point (x, y) is obtained by

dN(x, y)"

g#(x, y)

255(12)


Fig. 4. Noise-free dark-"eld isochromatics corresponding to various loads: (a) p#dp (b) p (c) p!dp.

From the phase map thus obtained, total retardation can be obtained by usingstandard phase unwrapping algorithms [1]. It is to be noted that Eqs. (10)}(12) arewritten in grey levels which helps in simplifying the software as phase data can behandled as an image. Eq. (11) is not a!ected by such a representation as only gradientinformation is used for processing. However, the use of Eq. (10) for converting phasedata to an image and re-conversion by Eq. (12) introduces some error in phase dataand a simple calculation reveals that the maximum error thus introduced is about$0.0019 fringe orders. In view of the elegance in handling data, the use of grey-levelrepresentation is adopted in this work.

6. Results and discussions

Fig. 6a shows the phase map corresponding to load p obtained by the newapproach. On analyzing the phase map one can see that the phase map is free of noiseand is very good in the complete model domain. One can also get the fractionalretardation corresponding to loads p#dp, p and p!dp using only the "rst four


Fig. 5. Intensity variation along the diametral line for the problem of disk under diametral compression: (a)grey levels plot of dark-"eld for three incremental loads p#dp, p and p!dp, (b) saw tooth wave form offractional retardation corresponding to load p.

position of the six-step phase shifting algorithm. In such a case the retardation has tobe calculated by

d#"tan~1A

$J(I1#I

2!2I

3)2#(2I

4!I

1!I

2)2

(I1!I

2) B"tan~1A

I!sin d

I!cos dB (13)

Fig. 6b shows the phase map obtained from the intensity map corresponding to the"rst four positions of the six-step algorithm.


Fig. 6. Phase map for the problem of disk under diametral compression for load p obtained by the newapproach by using: (a) all the six optical positions, (b) "rst four optical positions.

Ideally, the variation of grey levels in the dark-"eld isochromatics should be 0}255but from Fig. 5a one can see that the minimum intensity is more than 0 and maximumintensity is less then 255 at some locations (circles marked on the graph). This isbecause of a "xed spatial discretization over the model domain and errors could bedi!erent for di!erent fringe gradients. Ideally, one has to do sub-pixel calculations toovercome this. A simpler approach may be just to provide a normalization. It isdesirable to have #exibility in selecting the level of normalization. This is achieved asfollows

I.!9

"g.!9

#factor](255!g.!9

),

I.*/

"g.*/

(1!factor). (14)

The output grey level is then obtained by

g065165

"I.*/

#AI.!9

!I.*/

g.!9

!g.*/B(g*/165!g

.*/), (15)

where g.!9

and g.*/

are the maximum and minimum grey levels within a mask andI.!9

, I.*/

are output maximum and minimum grey levels. If the factor is 0 it means nonormalization and if the factor equals to 1 it means normalization to 0}255 grey levels.

Fig. 7a shows the e!ect of normalization on the total fringe order for the R/2 line byusing six-step algorithm and Fig. 7b shows the same for the four-step algorithm. Theresult shows that normalization is required if four-step algorithm is used.

Fig. 8 shows the total fringe order variation along the diametral line and along the3R/4 line for the phase shifting and the new approach. It shows that the new approachgives the result as accurate as phase shifting method and is uniformly applicable forthe entire model domain. It is important to note that for y"3R/4 line, the six-stepphase shifting fails.

The applicability of the new methodology for the problem of a ring under diametralcompression (outside diameter 79.5mm, thickness 5.1mm and load 546.9N) is shown


Fig. 7. Total fringe order plot for normalization factors 0 and 1 along the R/2 line using: (a) all six imagescorresponding to optical position listed in Table 1, (b) "rst four images corresponding to optical positionlisted in Table 1.

in Fig. 9. Fig. 9a gives the phase map by six-step phase shifting algorithm, Fig. 9b givesthe phase map by conventional load stepping, Fig. 9c gives the phase map by modi"edprocess of load stepping using six-step algorithm and Fig. 9d by four-step algorithm.Fig. 9a shows that in several zones there is discontinuity in the phase map ascalculated by phase shifting whereas the new methodology has no such discontinuityor noise as in conventional load stepping.

7. Closure

A new methodology for data reduction by load stepping is presented in this paper.A comparative study has also been carried out on the evaluation of phase map by


Fig. 8. Comparison of total fringe order for the problem of disk under diametral compression by variousmethod for two di!erent lines: (a) y"0, (b) y"3R/4.

phase shifting, conventional load stepping and the new methodology. The study hasrevealed that the new methodology guarantees unambiguous and noise-free deter-mination of isochromatic fringe order over the model domain.

It is to be noted that load stepping is not applicable for analyzing stress-frozenmodels. For such problems one may resort to the use of processing phase shifting datafor three wavelengths. In such a case, one has only a limited #exibility in the selectionof wavelengths, which restricts the perturbation possible on the retardation values toe!ectively cover di!erent fringe gradient zones. This problem can be overcome byjudiciously combining the results of phase shifting and load stepping or improve theresults of phase shifting by interactive processing [6] with the help of load steppingresults.


Fig. 9. Phase map for the problem of ring under diametral compression obtained by various methods: (a)phase shifting, (b) load stepping, (c) new approach with six images, (d) new approach with four images.

Acknowledgements

This research was sponsored in part by the Aeronautics Research and Deve-lopment Board of the Government of India (Project No. 919) and the Departmentof Mechanical Engineering, IIT, Kanpur. The authors thank Prof. B. Dattaguru,Prof. K. Rajaiah and Prof. N. S. Venkataraman for their interest in thiswork.

References

[1] Ramesh K. Digital photoelasticity: advance techniques and applications. Berlin, Germany: Springer,2000.

[2] Buckberry C, Towers D. New approach to the full-"eld analysis of photoelastic stress patterns. OptLasers Engng 1996;24:415}28.

[3] Ekman MJ, Nurse AD. Completely automated determination of two-dimensional photoelastic para-meters using load stepping. Opt Engng 1998;37(6):1845}51.


[4] Quiroga JA, GonazaH lez-Cano A. Phase measuring algorithm for extraction of isochromatics ofphotoelastic fringe patterns. Appl Opt 1997;36(32):8397}402.

[5] Ramesh K, Ganesan VR, Mullick SK. Digital image processing of photoelastic fringes * a newapproach. Exp Technics 1991;15(5):41}6.

[6] Ramesh K, Mangal SK. Phase shifting calculations in 2-D photoelasticity: revisited. J Aeronaut SocIndia 2000;52(2):121}36.


improved determination of retardation in digital photoelasticity by load stepping

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