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Completely automated photoelastic fringe analysis
Optics and Lasers in Engineering 21 (1994) 133-149 © 1994 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0143-8166/94/$7.00 ELSEVIER
Completely Automated Photoelastic Fringe Analysis J. C a r a z o - A l v a r e z , S. J. H a a k e & E. A. P a t t e r s o n Department of Mechanical and Process Engineering, University of Sheffield, Mappin St, Sheffield, UK, S1 4DU (Received 26 November 1993; revised version received 28 January 1994; accepted 2 February 1994)
ABSTRACT The integration of two automated systems of photoelastic analysis has been performed. This combines the results obtained with a full-field polariscope based on phase-stepping techniques with those from spectral contents analysis. The latter technique identifies the absolute value of the isochromatic parameter at a point. This information is used to calibrate maps of relative retardation produced by the phase-stepping method. Thus, completely automatic determination of calibrated isochromatic and isoclinic patterns in photoelastic specimens is achieved. The evaluation of the whole system, establishing limits of validity and influences of parameters, is performed. Results show that the analysis carried out with the combined system, incurred errors no larger than those of each system working independently.
INTRODUCTION Photoelasticity, as an experimental stress technique, has been used for many years for the analysis of complex components. It is virtually the only experimental technique that can provide data on the threedimensional stress state throughout a component. This is achieved by the slicing and subslicing of stress-frozen specimens. Recently, numerical techniques have become the most frequently used stress analysis tool. However, the need for experimental results to verify complex computer models has renewed interest in photoelasticity. With the advent of personal computers, automated photoelasticity is becoming a reality, with the advantage of increases in speed, accuracy and repeatability of analysis of photoelastic models. Recent a u t o m a t e d polariscopes have concentrated on two ap133
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J. Carazo-AIvarez, S. J. Haake, E. A. Patterson
proaches: point-by-point analysis, and full-field analysis. Point-by-point analysis has used various techniques including rotation of the polarizer and analyser. 13 The other major point-by-point technique is that of spectral analysis, 4~ which identifies uniquely and exactly the isochromatic parameter at a point on a sample by its spectral signature. The first proponents of the m e t h o d of spectral contents analysis were Redner, 4 Sanford and Iyengar 6 and Voloshin and R e d n e r ? A further system, based on this technique, has been proposed by Haake and Patterson? A u t o m a t e d full-field systems have been developed by Mtiller and Saackel, ~° Volshin and Burger It and by Brown and SullivanY The a u t o m a t e d polariscope developed by Mtiller and SaackeP ° used a TV camera as an image store and used fringe thinning to give the integer fringes of the specimen. Counting and identification of the fringes was carried out by the operator. Voloshin and Burger ~' developed the concept of 'half fringe photoelasticity' while Brown and Sullivan ~2 developed the computer-aided holophotoelastic method. Takashi ~3 used numerical techniques to represent experimental data with mathematical functions. Four images were collected and used to construct a final image of the isoclinic and isochromatic parameter. Patterson and Wang 14 used six phase-stepped images of the specimen to construct full-field isoclinic and isochromatic data maps for both simple and complex components. The major disadvantage of this technique is that it requires the operator to define the absolute isochromatic fringe order at a point in the field of view, since it provides relative retardation. In this paper it is proposed to combine the full-field capabilities and the high quality results of the full-field system developed by Patterson and Wang 14 with the accuracy of the point-by-point system using spectral contents analysis to identify the absolute value of the isochromatic parameter. S P E C T R A L C O N T E N T S A N A L Y S I S A P P L I E D TO PHOTOELASTICITY The m e t h o d of spectral contents analysis essentially uses the colour of the light from the specimen to determine the difference of principal stresses at the point being analysed. The mathematical procedure can be described in the following manner: the light intensity for a point on a specimen observed in a circular, transmission polariscope set for a light field with a white light source is given by, I =
f
~' I,, cos 2 -~~ d~
(1)
Completely automated photoelastic fringe analysis
135
where /,, is the light intensity emerging from the polariscope in zero stress conditions, 6 is the retardation related to the fringe order by 6 = NA = C(o-~ - o-2)t
(2)
and C is the stress-optic coefficient, o., and o2 are the principal stresses, t is the thickness of the specimen and A, and A2 are the lower and upper wavelengths of the white light used. In practice, the experimental spectrum is modified by the mismatch of the quarter-wave plates with the wavelength of the light used, ~ and by the dispersion of birefringence, ~5 i.e. the dependence of the stress-optic coefficient on the wavelength of light employed. Thus, eqn (1) becomes, I=
1,, cos: - T - -
1 - 0.5 sin: L~
1
d,~
(3)
where Ca is the stress-optic coefficient at wavelength ,~ and Co is the value of the stress-optic coefficient at the matching wavelength of the quarter wave plates A,,. Equation (3) gives the theoretical intensity spectrum emerging from the polariscope for a point on a specimen with a retardation & The retardation is found by comparing the spectrum achieved experimentally with that expected from theory, and described by eqn (3). The experimental spectrum is obtained using the system shown in Fig. 1, which includes a Monolite 6000 series spectral analyser. A series
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Fig. 1. Apparatus of the automated photoelastic analysis with phase-stepping and spectral analysis. The dashed line frames the apparatus of the spectral contents analyser. The computer is common to both systems.
136
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of two lenses and a pinhole causes a 0.14 mm diameter beam of circular polarised light to pass through the specimen. A light fibre conveys the point of light from the analyser to a rotating diffraction grating, causing the light to diffract into a spectral band. This spectrum of light is passed across the detection window of a photomultiplier tube, and this output is amplified, digitised and displayed by the computer as a plot of intensity versus wavelength. In order to allow for the wavelength-dependent transmission of the optical elements and the overall spectral factor of the apparatus, two spectra are measured, one of an unstressed sample made from identical material, and having identical thickness to the model, and the second of a stressed point on the model. The transmission spectrum is then calculated as their ratio and converted to percentage values. The range of wavelength is limited by the light source used, which in this case was 450 to 750 nm. The plot is in 5-rim steps, i.e. 61 points for this spectrum. The optical elements of the system that produces the light beam, i.e. the pinhole, lenses, polarizer, analyser, quarter-wave plates and light guides were chosen and mounted in such a way that the total length of the spectral contents head allowed its insertion into the full-field polariscope set up for the phase-stepping analysis. The total length of the insertion head was 83.5 mm. After the experimental spectrum has been recorded, it is possible by varying the value of the retardation 6 in eqn (3) to match the theoretical to the measured spectrum. A match was assumed to occur when the difference function below takes its minimum value, A,
rJifference function = ~'~ I,h~,,(6, A) - I~×p(A)
(4)
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where 1,h~,,,(6, A) represents the theoretical spectrum, which is retardation and wavelength dependent, and l~x~,(A) represents the measured spectrum. An example of this function versus retardation is shown in Fig. 2, the difference functions using 5-nm and 40-nm spectrum steps are shown. Also the theoretical function for the retardation where the minimum occurs is compared with the experimental one in Fig. 3. The minimisation of the difference function is carried out by a computer program, which uses standard algorithms for the absolute function minimum without the use of derivatives. The program uses the parabolic interpolation of the Brent's method and the Golden Section search (Press et al. ~) if the parabolic fit is found to be nonconverging. A bracketing procedure is carried out before performing the convergence and all the local minima are reached, keeping the lowest one as the
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final result. An analysis with a 40-nm spectrum step takes around 0.1 s when processed with a 486-33 MHz computer.
P H A S E S T E P P I N G A P P L I E D TO P H O T O E L A S T I C I T Y The phase stepping concept has been simply described as changing the absolute phase of the reference wave in equal steps and measuring the local light intensity after each step. ~7 This change in phase is achieved
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138
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in practice by rotation of the o u t p u t optical elements of the polariscope, i.e. analyser and o u t p u t quarter-wave plate. The general equation for the light intensity emerging at a point (x, y) on a specimen viewed in circularly polarised light is given by H i(x, y ) = i,,, + <(sin 2(/3 - q~) cos a - sin 2(0 - q)) cos (/3 - ~) sin a )
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where i,. is the intensity emerging when all the axes of the polariscope and the specimen are parallel and the t e r m im takes account of stray light. The angles /3, q~ and 0 are defined as the angles b e t w e e n the reference axis and the slow axis of the analyser, the o u t p u t quarterwave plate and specimen respectively. T h e intensities i(/3, q~) m e a s u r e d relative to these angles by Patterson and W a n g '4 are i,(0, Jr/4), i2(0,-1v/4), i3(0, 0), i4(rc/4, Jr/4), i5(7c/2, ~/2) and i6(3~/4, 3~/4). T h e expressions of the relative retardation ~, and isoclinic p a r a m e t e r 0, as functions of these intensities, take the form: '4 c~ : arctan
i,) cos 20 ): i(i~ - i4-i
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The analysis based on these expressions yields the isoclinic p a r a m e t e r and the relative retardation, at individual points in the field of view, without reference to neighbouring points. T h e relative retardation is a periodic function, which is related to the fringe order N as, c~ = 2;rN
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T o obtain the continuous m a p of the isochromatic parameter, an arbitrary point must be calibrated with the absolute value of the fringe order. Scanning the field of view point by point, the absolute fringe order is calculated for the rest of the points. The experimental configuration of the system d e v e l o p e d by Patterson and Wang is shown in Fig. 1. T h e full-field polariscope has a m o n o c h r o m a t i c C C D camera that records six images from the o u t p u t of the polariscope corresponding to the six orientations of the o u t p u t quarter-wave plate and analyscr. T h e images are amplified, digitised and transferred to a personal c o m p u t e r that processes the data. An unwrapping algorithm is e m p l o y e d to produce continuous isoclinic and relative retardation dala. The algorithm identilies zones of isoclinic data bclonging to different periods which have different signs and accounts for lhis in lhe produclion of the maps of continuous data.
Completely automated photoelastic fringe analys&
139
The digitiser is controlled by the computer and digitises a 256 by 256 pixel image from the camera. The computer also controls the optical element controller, which rotates the analyser and quarter-wave plate to the desired positions using stepper motors. A TV monitor is used to aid the positioning of the photoelastic samples.
LIMITS OF S P E C T R A L C O N T E N T S A N A L Y S I S Some tests were performed in order to determine the influence of the magnitude of the fringe order and of the stress gradient on accuracy of results obtained with the spectral contents analysis. To determine the maximum fringe order measurable by the system, the central point of a compressed disc loaded diametrically was studied. The stress gradient at this point is zero. The study was carried out on a disc (disc A) of 70 m m diameter and 15 m m thickness, made of epoxy resin (Araldite MY750 with HY901 hardener) and m o u n t e d in a loading frame. The load was increased incrementally and the retardation measured using spectral contents analysis. It is possible to reduce the n u m b e r of points used to describe the spectrum in order to increase the speed of analysis. Consequently two sets of data were obtained, one with eight points to describe the spectrum (40 nm between points) and one with 61 points to describe the spectrum (5 nm between points). The results are shown in Fig. 4. The maximum value which could be measured using 61 points was 21.82 fringe orders. This upper limit was determined by the onset of
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J. Carazo-Alvarez, S. J. Haake, E. A. Patterson
140
plasticity in the disc. The maximum value measured using the eightpoint spectrum was 7.83. Above this value the program to determine the retardation was frequently unable to converge (Fig. 4). The maximum difference between the retardation determined using the two sets of data was 0.04 fringes orders below N = 7.83. To determine the influence of the stress gradient and establish the limits of the spectral contents analysis in terms of a high density of fringes, a point near to the loading point of a second disc (disc B) in diametrical compression was studied. Again the load was increased incrementally. The disc had a diameter of 78ram and a thickness of 5.75 mm and was made from a different epoxy resin (CT200) in order to achieve greater fringe densities. The stress gradient was determined from the theoretical solution for the principal stresses at a point in a disc. ~s The point of interest was chosen to lie on the y-axis through the centre of the disc and the differential of the theoretical principal stress difference equation with respect to y was used to find the fringe gradient. The data obtained are shown in Fig. 5. The maximum fringe order measured with the 61-point spectrum was found to be 6.71 corresponding to a fringe gradient of 2.45 fringes/mm. For the study with the eight-point spectrum, it was found that for a fringe density of greater than 1.4 fringes/mm the fringe order did not always converge to the same value found using 61 points. It can be seen in Fig. 3 that the experimental spectrum does not exactly match the theoretical one/' There are two main causes for this occurrence. The first relates to the finite bandwidth that has been used 8 o 61
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Completely automated photoelastic fringe analysis
141
for the experimental results. The second is related to the nonuniformity of the birefringence across the point studied in areas of high stress gradient. These factors cause similar effects on the spectra, notably a reduction of the peaks and an increase in the troughs of the transmission spectrum. This difference between the spectra is detected by the difference function (eqn (4)) resulting in a higher o p t i m u m value in areas of high stress gradient for the same range of fringe order. This can be minimised by decreasing the diameter of the light beam.
E V A L U A T I O N OF B O T H A U T O M A T E D SYSTEMS WORKING INDEPENDENTLY The formal evaluation of each independent system has been performed comparing the results obtained with each system with those obtained by manual analysis using Tardy compensation. Phase stepping has been evaluated previously. ~9 The evaluation of the spectral contents analysis was carried out on a compressed disc loaded diametrically (disc C), scanning along the horizontal diameter. The disc was 76 m m diameter and 6 m m thickness, and was made of epoxy resin (MY750 with HY901). The model used is stress-frozen with a load of 70 N. The results are shown in Fig. 6. The maximum difference between results was found to be ± 1 % . 5.5
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J. Carazo-Alvarez, S. J. Haake, E. A. Patterson
142
C A L I B R A T I O N OF P H A S E STEPPING WITH S P E C T R A L CONTENTS ANALYSIS The physical integration of both systems is accomplished by a mechanical system that allows the head of the spectral contents analyser to be placed into the full-field polariscope and be removed once the data for the calibration point have been obtained. The analysis of the full-field image can then be performed. The use of the spectral contents system to calibrate the isochromatic patterns relies on the knowledge of the exact position of the light beam on the specimen, and its equivalent position on the full-field image. The probe for the spectral contents analyser was positioned in the polariscope using a locating pin. This ensured that the light beam was positioned in exactly the same place each time. The output section of the probe was removed so that the beam of light was incident on the CCD camera and appeared on the screen of the computer. The position of the beam light was noted to within 1 pixel. This position was then used in the subsequent analysis as the calibration point. The technique was evaluated by using two or three different points for the calibration of the full-field image. Any shift in the value of N at the calibration point is also added to all points in the full-field image. Three models were studied, a disc in diametrical compression used previously (disc C) and a turbine blade and slot (supplied by SNECMA). The last two models were made of epoxy resin (CT200) with 1 mm thickness.
Disc in diametrical compression The lens used with the camera was a 25-mm standard lens with an extension tube of 5 mm. For this configuration, the scale of the image taken by the camera was 0.212 mm/pixel thus giving an accuracy of 0.212 mm for the positioning of the calibration point. It was found that the maximum shift in the isochromatic fringe order map was 0.022 fringes between the three calibration points.
Turbine slot A 75-mm standard lens with an extension tube of 35 mm was used to take an image of the specimen. For this configuration, the scale of the image taken by the camera was 0.035 mm/pixel. The two points along
Completely automated photoelastic fringe analysis
143
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the line A B shown in Fig. 7(a), were used to calibrate the full-field image. The shift in isochromatic fringe o r d e r m a p using each point was 0.019 fringe orders. Full-field c o n t o u r maps of the isochromatic and isoclinic p a r a m e t e r s are shown in Figs 8(a) and (b). C o n t o u r maps have b e e n used since the colour images p r o d u c e d by the system do not r e p r o d u c e well in black and white.
144
J. Carazo-Alvarez, S. J. Haake, E. A. Patterson
~o----~,~ ~"----._ ~o~
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Full field contour maps traced from the computer screen of (a) isochromatic and (b) isoclinic parameters for the turbine slot.
Turbine blade The lens used in this case is a 75-mm standard lens with an extension tube of 1 5 m m . The scale of the image taken by the camera was 0.066 mm/pixel. Two points along the line CD shown in Fig. 9(a) were used for the calibration. The variation of isochromatic fringe order between the two full-field images obtained was 0.008 fringes. Full-field contour maps of the isochromatic and isoclinic parameters are shown in Figs 10(a) and (b).
DISCUSSION It has been found previously that the m e t h o d of spectral contents analysis can be used to give isochromatic values with errors of about +0.005 fringes. In an evaluation of the phase-stepping system it was found that the isochromatic error was about +0.007 fringes, and that the limiting fringe gradient of the system was 0.1 fringe/pixel. For disc C analysed in this investigation, the fringe gradient limit of 1.4 fringes/mm found for the spectral contents system corresponds to 0.7 fringes/pixel. In this case, the phase-stepping system is clearly the limiting factor in the analysis. For the analysis of the turbine slot and blade a longer focal length was used resulting in a higher image magnification (approximately six
Completely automated photoelastic fringe analysis
145
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was performed.
times that of the disc image). The fringe gradient limit of the spectral analyser corresponds to 0.05 and 0.1 fringes/pixel for the slot and blade, respectively. Thus, as the image magnification is increased the limits of the two systems converge. The fringe gradients that can be measured with this system compare favourably with that of 2.5 fringes/mm found by Allison. 2°
146
J. Carazo-Alvarez, S. J. Haake, E. A. Patterson
(a)
(b)
Fig. 10. Full-field contour maps traced from the computer screen of (a) isochromatic and (b) isoclinic parameters for the turbine blade.
When using only eight points to describe the spectrum, the maximum fringe order and fringe gradient which could be measured by the spectral contents system was 7.8 fringes and 1.4 fringes/mm, respectively. Since this is an acceptable performance for which only eight values of wavelength are required, the relatively complex apparatus used for the spectral contents analysis could be replaced by a simplified and cheaper arrangement composed of eight narrow band filters and a single pickup diode. The full-field analysis of a component can be carried out in about 20 min, using a 386 personal computer and providing 256 × 256 points of isochromatic and isoclinic data. A manual analysis of a line of data points may take at least fis long. The method therefore represents a dramatic decrease in analysis time and a increase of three orders of magnitude of the volume of data being handled. At present, automated polariscopes can be grouped into three categories. The systems in the first category require the operator to identify a fringe order at a point in the field Of view. Systems based on phase-stepping fall into this category. Some techniques that use phasestepping do not appear to address the problem of calibration, ~2~3'21 whilst Sarma e t al. 25 could obtain only fractional fringe orders. Those techniques based on fringe thinning 1°'22 24 require the operator to identify and assign fringe orders to isochromatic fringes. The second category contains those system which have limited fringe order capability. Voloshin and Redner 5 placed specimens with known isochromatic values in the field of view of a system using 'half-fringe' photoelasticity. The intensity in the calibration sample was used to
Completely automated photoelastic fringe analys&
147
calibrate the intensities found in the specimen to be analysed. The m e t h o d is limited to a m a x i m u m fringe order of 0.5. Finally, a third group can be formed from those systems that take images from two or more different wavelengths and make an identification of the zero-order fringe by detecting it as the only fringe that has suffered no displacement with the change in wavelength. If there is no zero-order fringe present then the technique fails. Umezaki et al. 2~" and Kihara 2~ used this technique. The system described here represents an advance on the systems currently available since full-field maps of both the isoclinic parameter and calibrated isochromatic parameter can be obtained for general complex components. CONCLUSION Combined use of phase-stepping and spectral contents methods provides a photoelastic fringe analysis that requires no operator input other than simple commands for the m a n a g e m e n t of the computer and the mounting of the specimen. It is possible to carry out a spectral contents analysis of a point on a specimen with a fringe order less than 7.8 fringes and a maximum gradient of 1.4 fringes/mm using only eight points to describe the spectrum. The final results are full-field data of the isoclinic angle and absolute isochromatic fringe order for the photoelastic model. This system gives a dramatic decrease in the time required for the photoelastic analysis of both simple and complex models. ACKNOWLEDGEMENTS The authors would like to thank the Science and Engineering Research Council (SERC) for their support and Le Soci6te National d'6tude et de Construction de Moteurs D'Aviation ( S N E C M A ) for supplying models for analysis. REFERENCES 1. Allison, I. M. & Nurse, P., Automatic acquisition of photoelastic data~ Proc. JBCSA Conf. on the Recording and Interpretation of Engineering Measurements, Inst. Mar. Engrs, London, 1972, 203-7. 2. Redner, A. S., A new automatic polariscope system. Experimental Mechanics, 14 (1974) 486-91.
3. Fcssler, H., Marston, R. E. & Ollerton, E., A micropolariscope for automatic stress analysis. J. Strain Analysis, 22 (1) (1987) 25-35.
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4. Redner, A. S., Photoelastic measurements by means of computer assisted spectral-contents analysis. Proc. 5th Int. Conf. on Exp. Mech., Montreal, 1984, 421-7. 5. Voloshin, A. S. & Redner, A. S., Automated measurement of birefringence: development and experimental evaluation of the techniques. Experimental Mechanics, 28 (1989) 252-7. 6. Sanford, R. J. & Iyengar, V., The measurement of the complete photoelastic fringe order using a spectral scanner. Proc. 1985 SEM Spring Conf. on Exp. Mech., 1985, 160-8. 7. Sanford, R. J., On the range and accuracy of spectrally scanned white light photoelasticity. Proc. 1986 SEM Spring Conf. on Exp. Mech., 1986, 901-8. 8. Iyengar, V., New method for birefringence measurement using a spectral scanner. MS Thesis, Univ. of Maryland, Aug. 1984. 9. Haake, S. J. & Patterson, E. A., Photoelastic analysis of frozen stressed specimens using spectral contents analysis. Experimental Mechanics, 32 (1992) 266-72. 10. Mtiller, R. K. & Saackel, L. R., Complete automatic analysis of photoelastic fringes. Experimental Mechanics, 19 (1979) 245-51. 11. Voloshin, A. S. & Burger, C. P., Half-fringe photoelasticity. A new approach to whole-field analysis, Experimental Mechanics, 23 (1983) 304-14. 12. Brown, G. M. & Sullivan, J. L., The computer-aided holophotoelastic method. Experimental Mechanics, 30 (1990) 135-44. 13. Takashi, M., Mawatara, S., Toyoda, Y. & Kunio, T., A new computeraided system for photoelastic stress analysis with structure driven type image processing. In Applied Stress Analysis, ed. T. H. Hyde & E. Ollerton. Elsevier Applied Science, London, 1990, pp. 516-25. 14. Patterson, E. A. & Wang, Z. F., Towards full-field automated photoelastic analysis of complex components. Strain, 27 (1991) 49-53. 15. Haake, S. J. & Patterson, E. A., The dispersion of birefringence in photoelastic materials. Strain, 29 (1) (1993) 3-7. 16. Press, W. H., Flannery, B. P., Tcnkolsky, S. A. & Vetterling, W. T., Numerical Recipes in Pascal--The Art of Scientific Computing. Cambridge University Press, 1990, 759 pp. 17. Hecker, F. W. & Morche, B., Computer measurement of relative retardations in plane photoelasticity. In Expt. Stress Anal., cd. H. Wieringa. Martinuus Nijhoff Publishers, Dordrecht, The Netherlands, 1986, pp. 532-42. 18. Frocht, M. M., Photoelasticity, Vol. 1. John Wiley and Sons, Inc., London, 1941. 19. Haake, S. J., Wang, Z. F. & Patterson, E. A., Evaluation of full field automated photoelastic analysis based on phase stepping, Experimental Techniques, 17 (6) (1993) 19-25. 20. Allison, I. M., Computation and interpretation of photoelastic test data. In Applied Solid Mechanic 3, ed. I. M. Allison & C. Ruiz. Elsevier Applied Sciences, London, 1989, pp. 258-68. 21. Asundi, A., Phase shifting in photoelasticity. Experimental Techniques, 17 (1) (1993) 19-23. 22. Seguchi, Y., Tomita, Y. & Watanbe, M., Computer aided fringe pattern
Completely automated photoelastic fringe analysis
23. 24. 25. 26. 27.
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analysis, a case of photoelastic fringe. Experimental Mechanics, 19 (1979) 362-70. Chen, T. Y. & Taylor, C. E., Computerized fringe analysis in photomechanics. Experimental Mechanics, 29 (3) (1989). Ramesh, K., Ganesan, V. R. & Mullick, S. K., Digital image processing of photoelastic fringes--A new approach. Experimental Techniques, 15 (5) (1991) 41-6. Sarma, A. V. S. S. S. R., Pillai, S. A., Subramanian, G. & Varadan, T. K., Computerized image processing for whole-field determination of isoclinics and isochrornatics. Experimental Mechanics, 32 (1) (1992) 24-9. Umezaki, E., Tamaki, T. & Takahashi, S., Automatic stress analysis of photoelastic experiment by use of image processing. Experimental Techniques, 13 (12) (1989) 22-7. Kihara, T., Automatic whole-field measuremenet of photoelasticity using linear polarized incident light. Proc. 9th Int. Conf. on Experimental Mechanics, Copenhagen, Denmark, 1990, Vol. 2, pp. 821-7.
Completely Automated Photoelastic Fringe Analysis J. C a r a z o - A l v a r e z , S. J. H a a k e & E. A. P a t t e r s o n Department of Mechanical and Process Engineering, University of Sheffield, Mappin St, Sheffield, UK, S1 4DU (Received 26 November 1993; revised version received 28 January 1994; accepted 2 February 1994)
ABSTRACT The integration of two automated systems of photoelastic analysis has been performed. This combines the results obtained with a full-field polariscope based on phase-stepping techniques with those from spectral contents analysis. The latter technique identifies the absolute value of the isochromatic parameter at a point. This information is used to calibrate maps of relative retardation produced by the phase-stepping method. Thus, completely automatic determination of calibrated isochromatic and isoclinic patterns in photoelastic specimens is achieved. The evaluation of the whole system, establishing limits of validity and influences of parameters, is performed. Results show that the analysis carried out with the combined system, incurred errors no larger than those of each system working independently.
INTRODUCTION Photoelasticity, as an experimental stress technique, has been used for many years for the analysis of complex components. It is virtually the only experimental technique that can provide data on the threedimensional stress state throughout a component. This is achieved by the slicing and subslicing of stress-frozen specimens. Recently, numerical techniques have become the most frequently used stress analysis tool. However, the need for experimental results to verify complex computer models has renewed interest in photoelasticity. With the advent of personal computers, automated photoelasticity is becoming a reality, with the advantage of increases in speed, accuracy and repeatability of analysis of photoelastic models. Recent a u t o m a t e d polariscopes have concentrated on two ap133
134
J. Carazo-AIvarez, S. J. Haake, E. A. Patterson
proaches: point-by-point analysis, and full-field analysis. Point-by-point analysis has used various techniques including rotation of the polarizer and analyser. 13 The other major point-by-point technique is that of spectral analysis, 4~ which identifies uniquely and exactly the isochromatic parameter at a point on a sample by its spectral signature. The first proponents of the m e t h o d of spectral contents analysis were Redner, 4 Sanford and Iyengar 6 and Voloshin and R e d n e r ? A further system, based on this technique, has been proposed by Haake and Patterson? A u t o m a t e d full-field systems have been developed by Mtiller and Saackel, ~° Volshin and Burger It and by Brown and SullivanY The a u t o m a t e d polariscope developed by Mtiller and SaackeP ° used a TV camera as an image store and used fringe thinning to give the integer fringes of the specimen. Counting and identification of the fringes was carried out by the operator. Voloshin and Burger ~' developed the concept of 'half fringe photoelasticity' while Brown and Sullivan ~2 developed the computer-aided holophotoelastic method. Takashi ~3 used numerical techniques to represent experimental data with mathematical functions. Four images were collected and used to construct a final image of the isoclinic and isochromatic parameter. Patterson and Wang 14 used six phase-stepped images of the specimen to construct full-field isoclinic and isochromatic data maps for both simple and complex components. The major disadvantage of this technique is that it requires the operator to define the absolute isochromatic fringe order at a point in the field of view, since it provides relative retardation. In this paper it is proposed to combine the full-field capabilities and the high quality results of the full-field system developed by Patterson and Wang 14 with the accuracy of the point-by-point system using spectral contents analysis to identify the absolute value of the isochromatic parameter. S P E C T R A L C O N T E N T S A N A L Y S I S A P P L I E D TO PHOTOELASTICITY The m e t h o d of spectral contents analysis essentially uses the colour of the light from the specimen to determine the difference of principal stresses at the point being analysed. The mathematical procedure can be described in the following manner: the light intensity for a point on a specimen observed in a circular, transmission polariscope set for a light field with a white light source is given by, I =
f
~' I,, cos 2 -~~ d~
(1)
Completely automated photoelastic fringe analysis
135
where /,, is the light intensity emerging from the polariscope in zero stress conditions, 6 is the retardation related to the fringe order by 6 = NA = C(o-~ - o-2)t
(2)
and C is the stress-optic coefficient, o., and o2 are the principal stresses, t is the thickness of the specimen and A, and A2 are the lower and upper wavelengths of the white light used. In practice, the experimental spectrum is modified by the mismatch of the quarter-wave plates with the wavelength of the light used, ~ and by the dispersion of birefringence, ~5 i.e. the dependence of the stress-optic coefficient on the wavelength of light employed. Thus, eqn (1) becomes, I=
1,, cos: - T - -
1 - 0.5 sin: L~
1
d,~
(3)
where Ca is the stress-optic coefficient at wavelength ,~ and Co is the value of the stress-optic coefficient at the matching wavelength of the quarter wave plates A,,. Equation (3) gives the theoretical intensity spectrum emerging from the polariscope for a point on a specimen with a retardation & The retardation is found by comparing the spectrum achieved experimentally with that expected from theory, and described by eqn (3). The experimental spectrum is obtained using the system shown in Fig. 1, which includes a Monolite 6000 series spectral analyser. A series
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136
J. Carazo-AIvarez, S. J. Haake, E. A. Patterson
of two lenses and a pinhole causes a 0.14 mm diameter beam of circular polarised light to pass through the specimen. A light fibre conveys the point of light from the analyser to a rotating diffraction grating, causing the light to diffract into a spectral band. This spectrum of light is passed across the detection window of a photomultiplier tube, and this output is amplified, digitised and displayed by the computer as a plot of intensity versus wavelength. In order to allow for the wavelength-dependent transmission of the optical elements and the overall spectral factor of the apparatus, two spectra are measured, one of an unstressed sample made from identical material, and having identical thickness to the model, and the second of a stressed point on the model. The transmission spectrum is then calculated as their ratio and converted to percentage values. The range of wavelength is limited by the light source used, which in this case was 450 to 750 nm. The plot is in 5-rim steps, i.e. 61 points for this spectrum. The optical elements of the system that produces the light beam, i.e. the pinhole, lenses, polarizer, analyser, quarter-wave plates and light guides were chosen and mounted in such a way that the total length of the spectral contents head allowed its insertion into the full-field polariscope set up for the phase-stepping analysis. The total length of the insertion head was 83.5 mm. After the experimental spectrum has been recorded, it is possible by varying the value of the retardation 6 in eqn (3) to match the theoretical to the measured spectrum. A match was assumed to occur when the difference function below takes its minimum value, A,
rJifference function = ~'~ I,h~,,(6, A) - I~×p(A)
(4)
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where 1,h~,,,(6, A) represents the theoretical spectrum, which is retardation and wavelength dependent, and l~x~,(A) represents the measured spectrum. An example of this function versus retardation is shown in Fig. 2, the difference functions using 5-nm and 40-nm spectrum steps are shown. Also the theoretical function for the retardation where the minimum occurs is compared with the experimental one in Fig. 3. The minimisation of the difference function is carried out by a computer program, which uses standard algorithms for the absolute function minimum without the use of derivatives. The program uses the parabolic interpolation of the Brent's method and the Golden Section search (Press et al. ~) if the parabolic fit is found to be nonconverging. A bracketing procedure is carried out before performing the convergence and all the local minima are reached, keeping the lowest one as the
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final result. An analysis with a 40-nm spectrum step takes around 0.1 s when processed with a 486-33 MHz computer.
P H A S E S T E P P I N G A P P L I E D TO P H O T O E L A S T I C I T Y The phase stepping concept has been simply described as changing the absolute phase of the reference wave in equal steps and measuring the local light intensity after each step. ~7 This change in phase is achieved
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138
J. Carazo-Alvarez. S..I. Haake, E. A. Patterson
in practice by rotation of the o u t p u t optical elements of the polariscope, i.e. analyser and o u t p u t quarter-wave plate. The general equation for the light intensity emerging at a point (x, y) on a specimen viewed in circularly polarised light is given by H i(x, y ) = i,,, + <(sin 2(/3 - q~) cos a - sin 2(0 - q)) cos (/3 - ~) sin a )
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where i,. is the intensity emerging when all the axes of the polariscope and the specimen are parallel and the t e r m im takes account of stray light. The angles /3, q~ and 0 are defined as the angles b e t w e e n the reference axis and the slow axis of the analyser, the o u t p u t quarterwave plate and specimen respectively. T h e intensities i(/3, q~) m e a s u r e d relative to these angles by Patterson and W a n g '4 are i,(0, Jr/4), i2(0,-1v/4), i3(0, 0), i4(rc/4, Jr/4), i5(7c/2, ~/2) and i6(3~/4, 3~/4). T h e expressions of the relative retardation ~, and isoclinic p a r a m e t e r 0, as functions of these intensities, take the form: '4 c~ : arctan
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The analysis based on these expressions yields the isoclinic p a r a m e t e r and the relative retardation, at individual points in the field of view, without reference to neighbouring points. T h e relative retardation is a periodic function, which is related to the fringe order N as, c~ = 2;rN
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T o obtain the continuous m a p of the isochromatic parameter, an arbitrary point must be calibrated with the absolute value of the fringe order. Scanning the field of view point by point, the absolute fringe order is calculated for the rest of the points. The experimental configuration of the system d e v e l o p e d by Patterson and Wang is shown in Fig. 1. T h e full-field polariscope has a m o n o c h r o m a t i c C C D camera that records six images from the o u t p u t of the polariscope corresponding to the six orientations of the o u t p u t quarter-wave plate and analyscr. T h e images are amplified, digitised and transferred to a personal c o m p u t e r that processes the data. An unwrapping algorithm is e m p l o y e d to produce continuous isoclinic and relative retardation dala. The algorithm identilies zones of isoclinic data bclonging to different periods which have different signs and accounts for lhis in lhe produclion of the maps of continuous data.
Completely automated photoelastic fringe analys&
139
The digitiser is controlled by the computer and digitises a 256 by 256 pixel image from the camera. The computer also controls the optical element controller, which rotates the analyser and quarter-wave plate to the desired positions using stepper motors. A TV monitor is used to aid the positioning of the photoelastic samples.
LIMITS OF S P E C T R A L C O N T E N T S A N A L Y S I S Some tests were performed in order to determine the influence of the magnitude of the fringe order and of the stress gradient on accuracy of results obtained with the spectral contents analysis. To determine the maximum fringe order measurable by the system, the central point of a compressed disc loaded diametrically was studied. The stress gradient at this point is zero. The study was carried out on a disc (disc A) of 70 m m diameter and 15 m m thickness, made of epoxy resin (Araldite MY750 with HY901 hardener) and m o u n t e d in a loading frame. The load was increased incrementally and the retardation measured using spectral contents analysis. It is possible to reduce the n u m b e r of points used to describe the spectrum in order to increase the speed of analysis. Consequently two sets of data were obtained, one with eight points to describe the spectrum (40 nm between points) and one with 61 points to describe the spectrum (5 nm between points). The results are shown in Fig. 4. The maximum value which could be measured using 61 points was 21.82 fringe orders. This upper limit was determined by the onset of
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J. Carazo-Alvarez, S. J. Haake, E. A. Patterson
140
plasticity in the disc. The maximum value measured using the eightpoint spectrum was 7.83. Above this value the program to determine the retardation was frequently unable to converge (Fig. 4). The maximum difference between the retardation determined using the two sets of data was 0.04 fringes orders below N = 7.83. To determine the influence of the stress gradient and establish the limits of the spectral contents analysis in terms of a high density of fringes, a point near to the loading point of a second disc (disc B) in diametrical compression was studied. Again the load was increased incrementally. The disc had a diameter of 78ram and a thickness of 5.75 mm and was made from a different epoxy resin (CT200) in order to achieve greater fringe densities. The stress gradient was determined from the theoretical solution for the principal stresses at a point in a disc. ~s The point of interest was chosen to lie on the y-axis through the centre of the disc and the differential of the theoretical principal stress difference equation with respect to y was used to find the fringe gradient. The data obtained are shown in Fig. 5. The maximum fringe order measured with the 61-point spectrum was found to be 6.71 corresponding to a fringe gradient of 2.45 fringes/mm. For the study with the eight-point spectrum, it was found that for a fringe density of greater than 1.4 fringes/mm the fringe order did not always converge to the same value found using 61 points. It can be seen in Fig. 3 that the experimental spectrum does not exactly match the theoretical one/' There are two main causes for this occurrence. The first relates to the finite bandwidth that has been used 8 o 61
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Completely automated photoelastic fringe analysis
141
for the experimental results. The second is related to the nonuniformity of the birefringence across the point studied in areas of high stress gradient. These factors cause similar effects on the spectra, notably a reduction of the peaks and an increase in the troughs of the transmission spectrum. This difference between the spectra is detected by the difference function (eqn (4)) resulting in a higher o p t i m u m value in areas of high stress gradient for the same range of fringe order. This can be minimised by decreasing the diameter of the light beam.
E V A L U A T I O N OF B O T H A U T O M A T E D SYSTEMS WORKING INDEPENDENTLY The formal evaluation of each independent system has been performed comparing the results obtained with each system with those obtained by manual analysis using Tardy compensation. Phase stepping has been evaluated previously. ~9 The evaluation of the spectral contents analysis was carried out on a compressed disc loaded diametrically (disc C), scanning along the horizontal diameter. The disc was 76 m m diameter and 6 m m thickness, and was made of epoxy resin (MY750 with HY901). The model used is stress-frozen with a load of 70 N. The results are shown in Fig. 6. The maximum difference between results was found to be ± 1 % . 5.5
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J. Carazo-Alvarez, S. J. Haake, E. A. Patterson
142
C A L I B R A T I O N OF P H A S E STEPPING WITH S P E C T R A L CONTENTS ANALYSIS The physical integration of both systems is accomplished by a mechanical system that allows the head of the spectral contents analyser to be placed into the full-field polariscope and be removed once the data for the calibration point have been obtained. The analysis of the full-field image can then be performed. The use of the spectral contents system to calibrate the isochromatic patterns relies on the knowledge of the exact position of the light beam on the specimen, and its equivalent position on the full-field image. The probe for the spectral contents analyser was positioned in the polariscope using a locating pin. This ensured that the light beam was positioned in exactly the same place each time. The output section of the probe was removed so that the beam of light was incident on the CCD camera and appeared on the screen of the computer. The position of the beam light was noted to within 1 pixel. This position was then used in the subsequent analysis as the calibration point. The technique was evaluated by using two or three different points for the calibration of the full-field image. Any shift in the value of N at the calibration point is also added to all points in the full-field image. Three models were studied, a disc in diametrical compression used previously (disc C) and a turbine blade and slot (supplied by SNECMA). The last two models were made of epoxy resin (CT200) with 1 mm thickness.
Disc in diametrical compression The lens used with the camera was a 25-mm standard lens with an extension tube of 5 mm. For this configuration, the scale of the image taken by the camera was 0.212 mm/pixel thus giving an accuracy of 0.212 mm for the positioning of the calibration point. It was found that the maximum shift in the isochromatic fringe order map was 0.022 fringes between the three calibration points.
Turbine slot A 75-mm standard lens with an extension tube of 35 mm was used to take an image of the specimen. For this configuration, the scale of the image taken by the camera was 0.035 mm/pixel. The two points along
Completely automated photoelastic fringe analysis
143
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the line A B shown in Fig. 7(a), were used to calibrate the full-field image. The shift in isochromatic fringe o r d e r m a p using each point was 0.019 fringe orders. Full-field c o n t o u r maps of the isochromatic and isoclinic p a r a m e t e r s are shown in Figs 8(a) and (b). C o n t o u r maps have b e e n used since the colour images p r o d u c e d by the system do not r e p r o d u c e well in black and white.
144
J. Carazo-Alvarez, S. J. Haake, E. A. Patterson
~o----~,~ ~"----._ ~o~
(a) Fig. 8.
(b)
Full field contour maps traced from the computer screen of (a) isochromatic and (b) isoclinic parameters for the turbine slot.
Turbine blade The lens used in this case is a 75-mm standard lens with an extension tube of 1 5 m m . The scale of the image taken by the camera was 0.066 mm/pixel. Two points along the line CD shown in Fig. 9(a) were used for the calibration. The variation of isochromatic fringe order between the two full-field images obtained was 0.008 fringes. Full-field contour maps of the isochromatic and isoclinic parameters are shown in Figs 10(a) and (b).
DISCUSSION It has been found previously that the m e t h o d of spectral contents analysis can be used to give isochromatic values with errors of about +0.005 fringes. In an evaluation of the phase-stepping system it was found that the isochromatic error was about +0.007 fringes, and that the limiting fringe gradient of the system was 0.1 fringe/pixel. For disc C analysed in this investigation, the fringe gradient limit of 1.4 fringes/mm found for the spectral contents system corresponds to 0.7 fringes/pixel. In this case, the phase-stepping system is clearly the limiting factor in the analysis. For the analysis of the turbine slot and blade a longer focal length was used resulting in a higher image magnification (approximately six
Completely automated photoelastic fringe analysis
145
(a)
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was performed.
times that of the disc image). The fringe gradient limit of the spectral analyser corresponds to 0.05 and 0.1 fringes/pixel for the slot and blade, respectively. Thus, as the image magnification is increased the limits of the two systems converge. The fringe gradients that can be measured with this system compare favourably with that of 2.5 fringes/mm found by Allison. 2°
146
J. Carazo-Alvarez, S. J. Haake, E. A. Patterson
(a)
(b)
Fig. 10. Full-field contour maps traced from the computer screen of (a) isochromatic and (b) isoclinic parameters for the turbine blade.
When using only eight points to describe the spectrum, the maximum fringe order and fringe gradient which could be measured by the spectral contents system was 7.8 fringes and 1.4 fringes/mm, respectively. Since this is an acceptable performance for which only eight values of wavelength are required, the relatively complex apparatus used for the spectral contents analysis could be replaced by a simplified and cheaper arrangement composed of eight narrow band filters and a single pickup diode. The full-field analysis of a component can be carried out in about 20 min, using a 386 personal computer and providing 256 × 256 points of isochromatic and isoclinic data. A manual analysis of a line of data points may take at least fis long. The method therefore represents a dramatic decrease in analysis time and a increase of three orders of magnitude of the volume of data being handled. At present, automated polariscopes can be grouped into three categories. The systems in the first category require the operator to identify a fringe order at a point in the field Of view. Systems based on phase-stepping fall into this category. Some techniques that use phasestepping do not appear to address the problem of calibration, ~2~3'21 whilst Sarma e t al. 25 could obtain only fractional fringe orders. Those techniques based on fringe thinning 1°'22 24 require the operator to identify and assign fringe orders to isochromatic fringes. The second category contains those system which have limited fringe order capability. Voloshin and Redner 5 placed specimens with known isochromatic values in the field of view of a system using 'half-fringe' photoelasticity. The intensity in the calibration sample was used to
Completely automated photoelastic fringe analys&
147
calibrate the intensities found in the specimen to be analysed. The m e t h o d is limited to a m a x i m u m fringe order of 0.5. Finally, a third group can be formed from those systems that take images from two or more different wavelengths and make an identification of the zero-order fringe by detecting it as the only fringe that has suffered no displacement with the change in wavelength. If there is no zero-order fringe present then the technique fails. Umezaki et al. 2~" and Kihara 2~ used this technique. The system described here represents an advance on the systems currently available since full-field maps of both the isoclinic parameter and calibrated isochromatic parameter can be obtained for general complex components. CONCLUSION Combined use of phase-stepping and spectral contents methods provides a photoelastic fringe analysis that requires no operator input other than simple commands for the m a n a g e m e n t of the computer and the mounting of the specimen. It is possible to carry out a spectral contents analysis of a point on a specimen with a fringe order less than 7.8 fringes and a maximum gradient of 1.4 fringes/mm using only eight points to describe the spectrum. The final results are full-field data of the isoclinic angle and absolute isochromatic fringe order for the photoelastic model. This system gives a dramatic decrease in the time required for the photoelastic analysis of both simple and complex models. ACKNOWLEDGEMENTS The authors would like to thank the Science and Engineering Research Council (SERC) for their support and Le Soci6te National d'6tude et de Construction de Moteurs D'Aviation ( S N E C M A ) for supplying models for analysis. REFERENCES 1. Allison, I. M. & Nurse, P., Automatic acquisition of photoelastic data~ Proc. JBCSA Conf. on the Recording and Interpretation of Engineering Measurements, Inst. Mar. Engrs, London, 1972, 203-7. 2. Redner, A. S., A new automatic polariscope system. Experimental Mechanics, 14 (1974) 486-91.
3. Fcssler, H., Marston, R. E. & Ollerton, E., A micropolariscope for automatic stress analysis. J. Strain Analysis, 22 (1) (1987) 25-35.
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.I. Carazo-Alvarez, S. J. Haake, E. A. Patterson
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