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Multiaperture speckle shearing arrangements for stress analysis
Volume 49, number 5
OPTICS COMMUNICATIONS
1 April 1984
MULTIAPERTURE SPECKLE SHEARING ARRANGEMENTS FOR STRESS ANALYSIS D.K. SHARMA, R.S. SIROHI and M.P. KOTHIYAL Engineering Design Centre, Indian Institute of Technology, Madras 600 036, India Received 6 December 1983 Revised manuscript received 24 January 1984 The investigations are made on two, four-aperture speckle shearing arrangements designed with custom built shearing elements. The first arrangement allows the measurement of inplane displacement along with the slope and curvature required for the evaluation of bending moments. The second arrangement is specifically designed for the measurement of second order cross derivatives that are needed for the evaluation of twisting moment for flexural members. It also yields slope information in two other orthogonal directions inclined at 45° to x or y directions. The main advantage of the two arrangements is that only two exposures are necessary and there is no shift of recording plate or repetition of loading conditions involved in the process.
1. Introduction The determination o f strain, stress and bending moments requires differentiation o f the measured distribution of surface displacement. This can be conveniently done with the help o f speckle shear interferometry. Here the wavefront from a diffuse object surface is first divided into two or more wavefronts using an optical arrangement and then recombined at the recording plane with a relative lateral displacement (shear) between them. This results in any point on the image plane receiving contributions from two or more points on the object. The first derivative o f displacement or slope can be obtained once the contributions from two adjacent object points are reaching the same point on the image. Several authors [ 1 - 6 ] have reportedly obtained the slope information using (a) defocussing technique [1,2] and ( b ) u s e of discrete optical elements for introduction o f necessary shear [3,4,6]. As far as the evaluation o f second derivatives is concerned the various techniques have been variation o f the moir6 method. In one case two transparencies containing the same slope information are superposed and sheared in a particular direction. The resultant moir6 pattern yields the curvature in that direction [9]. In the second approach the two slope patterns are recorded on the same photographic plate with plate being displaced in between the two recordings [5]. The meth0 030-4018/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
od is cumbersome in that it involves four exposures, replication of loading conditions and a displacement of the recording plate. Besides, the information extraction involves a Fourier filtering stage. Recent work o f Krishnamurthy et al. [7] has led to the conclusion that only three sheared fields are necessary for obtaining the curvature (d 2 w/dx 2) information. They obtained curvature with the help o f a three segment lens. We have recently reported [8] a three aperture speckle shearing interferometer making use o f two small angle wedges as shearing elements. This arrangement was shown to yield not only curvature information with a two exposure shear specklegram but also pure slope information. Here we are reporting two, four-aperture speckle shearing arrangements. The first one yields slope and curvature information and simultaneously senses any inplane displacement occuring during the experiment. The second is specifically designed for evaluation o f second order cross derivatives needed for evaluating twisting moment. It however, also yields slope information in two other orthogonal directions inclined at 45 ° to x or y directions.
2. The four-aperture speckle shearing interferometer This arrangement is designed for obtaining slope 313
Volume 49, number 5
OPTICS COMMUNICATIONS
(a)
t April 1984 (b)
Fig. 1. (a) Four aperture arrangement for slope (dw/dx) and curvature d2w/dx 2 measurement. (b) The arrangement of halos in the Fourier transform plane.
and curvature information simultaneously in one step. It has a circular opaque plate with 4 holes drilled at 90 ° intervals as shown in the fig. 1. Holes 1 and 3 are covered with a glass plate. Two small angled wedges are mounted in front of the holes 2 and 4 with orientation of the wedges along x-axis as shown in fig. 2. This device is mounted in front of a standard camera lens of a suitable focal length (fig. 3). Three sheared speckle fields are seen in the image plane. The double
(b) (o)
(c)
(d)
Fig. 2. (a) Half plate and half wedge elements. (b) The arrangement of the shearing elements over the four aperture plate. (c) Four sheared fields seen in the image plane. (d) The resultant arrangement of halos in the Fourier plane.
314
exposed shear specklegram when subjected to Fourier filtering yields 9 distinct halos fig. 1 (b). Since there is no shearing element in front of holes 1 and 3, the arrangement is equivalent to Duffys' arrangement, and filtering through 1 or 5 halos corresponding to these holes yields only in-plane displacement information. The sensitivity is determined by s o = Xp/Dwhere s o is the apparent grid spacing on the object and X is the wavelength of light used, p is the object distance from the lens and D is the distance between the two apertures. Diffraction halos 2 and 6 are resulting from the superposition of information from holes 1,4, 2 and 3. Since there is a relative shear between 1,4 as well as 2, 3 this results in superposition of two slopes patterns (relatively displaced) resulting in moir6 pattern yielding the curvature information. Likewise halos 4 and 8 also contain slope fringes together with relatively broad, low visibility curvature fringes. The halos 3 and 7 corresponding to holes 2 and 4 contain pure slope information with sensitivity double that of the slope fringes obtained from halos 2, 4, 6 and 8. Thus we have a very interesting system which could be used for study of objects undergoing flexural deformations to yield slope and curvature from a single double exposed shear specklegram. This, in addition provides a check on whether or not any inplane motion has taken place during the experiment.
3. Four-aperture arrangement for obtaining cross second order derivatives. In this arrangement we again start with an opaque circular plate with four drilled holes as shown in fig.
Volume 49, number 5
OPTICS COMMUNICATIONS
1 April 1984
Xt COMPENSATING~GLASS PLATE~
-
IMAGEPLANE ~
!i~
CAMERA LENS
SPECIM'~EN ILLUMINATION
Fig. 3. Schematic of the optical layout.
2(a). We mount in front o f these holes a circular glass plate with half of it having a small wedge angle while the rest half is a parallel plate as shown in fig. 2 (a). This results in introduction o f equal shear between A and D, and B and C, consequently, we obtain two sheared images in the x-direction. We mount another identical plate infront o f the above arrangement with the wedge angle turned at right angles to the first glass plate as shown in fig. 2(b). This shears two previously x-sheared images in the y-direction. Thus we get four sheared images of the specimen which can be thought o f as two x-sheared pairs further sheared in y-direction or vice versa. This device is mounted in front of the standard lens of a camera. Two exposures are recorded on a plate located in the image plane of the camera, one before and the other after the deformation. The resultant shear specklegram after development, when subjected to Fourier filtering results in 9 distinct diffraction spots as in the previous case. The cross derivatives d2w/dxdy and d2w/dydx appear in halos 1,5 and 3,7 respectively. Halos 2,6 and 4,8 yield pure slope information in two orthogonal directions inclined at 45 ° to x o r y directions with a sensitivity X/~ times that o f the slope fringes in other four halos.
4. Experimental
Glass wedges with 1° angles were used for the fabri-
cation of the devices. Four holes 6 mm in diameter at the corners of a 20 mm square were used on a hylam plate. The two aforementioned devices were tested by mounting infront of a standard "Linhof" camera with f = 150, f/5.6 lens. The specimen was a centrally loaded phosphor bronze diaphram with edges clamped. The specimen was illuminated normally with collimated beam from a 7 mW He-Ne laser. The exposures were made on 1OE-75 holographic plates. Two exposures were made with a loading of 15/.trn in between the exposures. The resulting shear specklegrams were filtered on a standard setup and the information from various halo spots was recorded. The results are presented in figs. 4(a), (b), (c) for the first arrangement. Fig. 4(a) shows slope fringes together with curvature fringes (d2w/dx2) o f low visibility, fig. 4(b) shows slope fringe with double sensitivity as compared to that shown in 4(a) while fig. 4(c) shows diaphragm without any fringes indicating that no inplane motion has taken place. The cross derivatives d2w/dxdy and d2w/dydx are shown in figs. 5(a) and 5(b) as the moir6 fringes o f low visibility resulting from the super-position of two slope patterns. The slope information obtained in addition is presented in fig. 5(c). Although one can use wedges with unequal angles in front of the aperture, the errors in measurement are minimum when the two shears are of the same magnitude as pointed out by Boone [9]. Therefore it is re315
Volume 49, n u m b e r 5
OPTICS COMMUNICATIONS
Fig. 4. (a) Slope and curvature fringes for a centrally loaded p h o s p h o r bronze diaphragm with edges clamped; central deflection ~- 15 t~m. (b) Pure slope fringes with double sensitivity from the same specklegram. (c) Diaphragm photographed through halo 1 showing no inplane m o t i o n during the experiment.
316
1 April 1984
Fig. 5. (a) Cross second order derivative d2w/dxdy for a centrally loaded diaphragm with central deflection ~- 15 tzm, with equal x and y shears. (b) Cross derivative (d2w/dydx)obtained from same specklegram. (c) Pure slope information obtained with x / ~ times the sensitivity o f slope fringes in the previous case inclined in a direction at 45 ° to x- ory-axis.
Volume 49, number 5
OPTICS COMMUNICATIONS
commended to use wedges of equal angles in the experiment. We have thus presented two arrangements which can be used in conjunction with a standard camera to yield slope and curvature information for a class of objects such as thin plates subjected to flexural loading. Thus bending moments and twisting moment can be directly obtained. However, their application is limited by the poor visibility of the moir6 fringes as also the requirement that the specimen is grossly planar. One of the arrangement also serves to check if any unwanted inplane m o t i o n results during the experiment.
1 April 1984
[2] F.P. Chiang and R.M. Juang, Appl. Optics 15 (1976) 2199. [3] J.A. Leendertz and J.N. Butters, J. Phys. E. Scient. Instr. 6 (1973) 1107. [4] Y.Y. Hung and C.Y. Liang, Appl. Optics 18 (1979) 1046. [5] A. Assa, J. Politch and A.P. Betser, Expt. Mech. 19 (1979) 126. [6] S. Nakadate, T. Yatagai and H. Saito, Appl. Optics 19 (1980) 4241. [7] R. Krishnamurthy, R.S. Sirohi and M.P. Kothiyal, presented at Eleventh All India Symposium on Optics and Optoelectronics (Opt. Soc. of India) paper 2.1 January (1983). [8] D.K. Sharma, R.S. Sirohi and M.P. Kothiyal, Simultaneous measurement of slope and curvature with three aperture speckle shear interferometer, submitted. [9] P.M. Boone, Expt. Mech. 15 (1975) 295.
References [ 1] Y.Y. Hung, I.M. Daniel and R.E. Rowland, Appl. Optics 19 (1980) 618.
317
OPTICS COMMUNICATIONS
1 April 1984
MULTIAPERTURE SPECKLE SHEARING ARRANGEMENTS FOR STRESS ANALYSIS D.K. SHARMA, R.S. SIROHI and M.P. KOTHIYAL Engineering Design Centre, Indian Institute of Technology, Madras 600 036, India Received 6 December 1983 Revised manuscript received 24 January 1984 The investigations are made on two, four-aperture speckle shearing arrangements designed with custom built shearing elements. The first arrangement allows the measurement of inplane displacement along with the slope and curvature required for the evaluation of bending moments. The second arrangement is specifically designed for the measurement of second order cross derivatives that are needed for the evaluation of twisting moment for flexural members. It also yields slope information in two other orthogonal directions inclined at 45° to x or y directions. The main advantage of the two arrangements is that only two exposures are necessary and there is no shift of recording plate or repetition of loading conditions involved in the process.
1. Introduction The determination o f strain, stress and bending moments requires differentiation o f the measured distribution of surface displacement. This can be conveniently done with the help o f speckle shear interferometry. Here the wavefront from a diffuse object surface is first divided into two or more wavefronts using an optical arrangement and then recombined at the recording plane with a relative lateral displacement (shear) between them. This results in any point on the image plane receiving contributions from two or more points on the object. The first derivative o f displacement or slope can be obtained once the contributions from two adjacent object points are reaching the same point on the image. Several authors [ 1 - 6 ] have reportedly obtained the slope information using (a) defocussing technique [1,2] and ( b ) u s e of discrete optical elements for introduction o f necessary shear [3,4,6]. As far as the evaluation o f second derivatives is concerned the various techniques have been variation o f the moir6 method. In one case two transparencies containing the same slope information are superposed and sheared in a particular direction. The resultant moir6 pattern yields the curvature in that direction [9]. In the second approach the two slope patterns are recorded on the same photographic plate with plate being displaced in between the two recordings [5]. The meth0 030-4018/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
od is cumbersome in that it involves four exposures, replication of loading conditions and a displacement of the recording plate. Besides, the information extraction involves a Fourier filtering stage. Recent work o f Krishnamurthy et al. [7] has led to the conclusion that only three sheared fields are necessary for obtaining the curvature (d 2 w/dx 2) information. They obtained curvature with the help o f a three segment lens. We have recently reported [8] a three aperture speckle shearing interferometer making use o f two small angle wedges as shearing elements. This arrangement was shown to yield not only curvature information with a two exposure shear specklegram but also pure slope information. Here we are reporting two, four-aperture speckle shearing arrangements. The first one yields slope and curvature information and simultaneously senses any inplane displacement occuring during the experiment. The second is specifically designed for evaluation o f second order cross derivatives needed for evaluating twisting moment. It however, also yields slope information in two other orthogonal directions inclined at 45 ° to x or y directions.
2. The four-aperture speckle shearing interferometer This arrangement is designed for obtaining slope 313
Volume 49, number 5
OPTICS COMMUNICATIONS
(a)
t April 1984 (b)
Fig. 1. (a) Four aperture arrangement for slope (dw/dx) and curvature d2w/dx 2 measurement. (b) The arrangement of halos in the Fourier transform plane.
and curvature information simultaneously in one step. It has a circular opaque plate with 4 holes drilled at 90 ° intervals as shown in the fig. 1. Holes 1 and 3 are covered with a glass plate. Two small angled wedges are mounted in front of the holes 2 and 4 with orientation of the wedges along x-axis as shown in fig. 2. This device is mounted in front of a standard camera lens of a suitable focal length (fig. 3). Three sheared speckle fields are seen in the image plane. The double
(b) (o)
(c)
(d)
Fig. 2. (a) Half plate and half wedge elements. (b) The arrangement of the shearing elements over the four aperture plate. (c) Four sheared fields seen in the image plane. (d) The resultant arrangement of halos in the Fourier plane.
314
exposed shear specklegram when subjected to Fourier filtering yields 9 distinct halos fig. 1 (b). Since there is no shearing element in front of holes 1 and 3, the arrangement is equivalent to Duffys' arrangement, and filtering through 1 or 5 halos corresponding to these holes yields only in-plane displacement information. The sensitivity is determined by s o = Xp/Dwhere s o is the apparent grid spacing on the object and X is the wavelength of light used, p is the object distance from the lens and D is the distance between the two apertures. Diffraction halos 2 and 6 are resulting from the superposition of information from holes 1,4, 2 and 3. Since there is a relative shear between 1,4 as well as 2, 3 this results in superposition of two slopes patterns (relatively displaced) resulting in moir6 pattern yielding the curvature information. Likewise halos 4 and 8 also contain slope fringes together with relatively broad, low visibility curvature fringes. The halos 3 and 7 corresponding to holes 2 and 4 contain pure slope information with sensitivity double that of the slope fringes obtained from halos 2, 4, 6 and 8. Thus we have a very interesting system which could be used for study of objects undergoing flexural deformations to yield slope and curvature from a single double exposed shear specklegram. This, in addition provides a check on whether or not any inplane motion has taken place during the experiment.
3. Four-aperture arrangement for obtaining cross second order derivatives. In this arrangement we again start with an opaque circular plate with four drilled holes as shown in fig.
Volume 49, number 5
OPTICS COMMUNICATIONS
1 April 1984
Xt COMPENSATING~GLASS PLATE~
-
IMAGEPLANE ~
!i~
CAMERA LENS
SPECIM'~EN ILLUMINATION
Fig. 3. Schematic of the optical layout.
2(a). We mount in front o f these holes a circular glass plate with half of it having a small wedge angle while the rest half is a parallel plate as shown in fig. 2 (a). This results in introduction o f equal shear between A and D, and B and C, consequently, we obtain two sheared images in the x-direction. We mount another identical plate infront o f the above arrangement with the wedge angle turned at right angles to the first glass plate as shown in fig. 2(b). This shears two previously x-sheared images in the y-direction. Thus we get four sheared images of the specimen which can be thought o f as two x-sheared pairs further sheared in y-direction or vice versa. This device is mounted in front of the standard lens of a camera. Two exposures are recorded on a plate located in the image plane of the camera, one before and the other after the deformation. The resultant shear specklegram after development, when subjected to Fourier filtering results in 9 distinct diffraction spots as in the previous case. The cross derivatives d2w/dxdy and d2w/dydx appear in halos 1,5 and 3,7 respectively. Halos 2,6 and 4,8 yield pure slope information in two orthogonal directions inclined at 45 ° to x o r y directions with a sensitivity X/~ times that o f the slope fringes in other four halos.
4. Experimental
Glass wedges with 1° angles were used for the fabri-
cation of the devices. Four holes 6 mm in diameter at the corners of a 20 mm square were used on a hylam plate. The two aforementioned devices were tested by mounting infront of a standard "Linhof" camera with f = 150, f/5.6 lens. The specimen was a centrally loaded phosphor bronze diaphram with edges clamped. The specimen was illuminated normally with collimated beam from a 7 mW He-Ne laser. The exposures were made on 1OE-75 holographic plates. Two exposures were made with a loading of 15/.trn in between the exposures. The resulting shear specklegrams were filtered on a standard setup and the information from various halo spots was recorded. The results are presented in figs. 4(a), (b), (c) for the first arrangement. Fig. 4(a) shows slope fringes together with curvature fringes (d2w/dx2) o f low visibility, fig. 4(b) shows slope fringe with double sensitivity as compared to that shown in 4(a) while fig. 4(c) shows diaphragm without any fringes indicating that no inplane motion has taken place. The cross derivatives d2w/dxdy and d2w/dydx are shown in figs. 5(a) and 5(b) as the moir6 fringes o f low visibility resulting from the super-position of two slope patterns. The slope information obtained in addition is presented in fig. 5(c). Although one can use wedges with unequal angles in front of the aperture, the errors in measurement are minimum when the two shears are of the same magnitude as pointed out by Boone [9]. Therefore it is re315
Volume 49, n u m b e r 5
OPTICS COMMUNICATIONS
Fig. 4. (a) Slope and curvature fringes for a centrally loaded p h o s p h o r bronze diaphragm with edges clamped; central deflection ~- 15 t~m. (b) Pure slope fringes with double sensitivity from the same specklegram. (c) Diaphragm photographed through halo 1 showing no inplane m o t i o n during the experiment.
316
1 April 1984
Fig. 5. (a) Cross second order derivative d2w/dxdy for a centrally loaded diaphragm with central deflection ~- 15 tzm, with equal x and y shears. (b) Cross derivative (d2w/dydx)obtained from same specklegram. (c) Pure slope information obtained with x / ~ times the sensitivity o f slope fringes in the previous case inclined in a direction at 45 ° to x- ory-axis.
Volume 49, number 5
OPTICS COMMUNICATIONS
commended to use wedges of equal angles in the experiment. We have thus presented two arrangements which can be used in conjunction with a standard camera to yield slope and curvature information for a class of objects such as thin plates subjected to flexural loading. Thus bending moments and twisting moment can be directly obtained. However, their application is limited by the poor visibility of the moir6 fringes as also the requirement that the specimen is grossly planar. One of the arrangement also serves to check if any unwanted inplane m o t i o n results during the experiment.
1 April 1984
[2] F.P. Chiang and R.M. Juang, Appl. Optics 15 (1976) 2199. [3] J.A. Leendertz and J.N. Butters, J. Phys. E. Scient. Instr. 6 (1973) 1107. [4] Y.Y. Hung and C.Y. Liang, Appl. Optics 18 (1979) 1046. [5] A. Assa, J. Politch and A.P. Betser, Expt. Mech. 19 (1979) 126. [6] S. Nakadate, T. Yatagai and H. Saito, Appl. Optics 19 (1980) 4241. [7] R. Krishnamurthy, R.S. Sirohi and M.P. Kothiyal, presented at Eleventh All India Symposium on Optics and Optoelectronics (Opt. Soc. of India) paper 2.1 January (1983). [8] D.K. Sharma, R.S. Sirohi and M.P. Kothiyal, Simultaneous measurement of slope and curvature with three aperture speckle shear interferometer, submitted. [9] P.M. Boone, Expt. Mech. 15 (1975) 295.
References [ 1] Y.Y. Hung, I.M. Daniel and R.E. Rowland, Appl. Optics 19 (1980) 618.
317