The Grid Method Professor Fabrice Pierron University of
The Grid Method Professor Fabrice Pierron University of Southampton
Summary § Basic principle § Spatial phase-shifting § Cross-grid processing § Phase unwrapping: spatial vs temporal § Grid bonding § Examples of application – In-plane displacements (bonded grid) – Slopes (reflected grid) § Comparison between GM and DIC Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 2/35
Basic principle (1/4) § Grid – Periodic pattern of contrasted intensity lines Unidirectional grid Bidirectional grid Unidirectional grid Mean intensity Contrast pitch Phase is a 2 p periodical function (shape of grid) Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 3/35
Basic principle (2/4) § Spatial phase is displacement Initial state After deformation x intensity φi = 0 x x φf 0 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method x 4/35
Basic principle (3/4) § In fact, when displacements are larger recursive § Need for an iterative scheme, see Grédiac, M. , F. Sur, and B. Blaysat, The Grid Method for in-plane displacement and strain measurement: a review and analysis. Strain, 2016. 52(3): p. 205 -243. Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 5/35
Basic principle (4/4) § Spatial frequency is strain § Frequency is the derivative of phase § Two main routes of grid processing – Measure frequency (FFT, wavelets, etc…): strain – Measure phase (phase shifting, correlation): displacements Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 6/35
Spatial phase shifting (1/10) I( x ) Ipix 2 Ipix 1 M = 4 (number x Ipix 3 p/4 x=0 At point P(x=0), three unknowns of intensity samples) Ipix 4 x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 7/35
Spatial phase shifting (2/10) I( x ) Ipix 1 Ipix 2 M = 4 (number x Ipix 3 x=0 of intensity samples) Ipix 4 x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 8/35
Spatial phase shifting (3/10) I( x ) Ipix 1 Ipix 2 Ipix 3 Ipix 4 0 x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 9/35
Spatial phase shifting (4/10) § General formulation With: • a 0 = 0 ; a 1 = -1 ; a 2 = 0 ; a 3 = 1 • b 0 = 1 ; b 1 = 0 ; b 2 = -1 ; b 3 = 0 § Complex number formulation with: c 0 = i ; c 1 = -1 ; c 2 = -i ; c 3 = 1 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 10/35
Spatial phase shifting (5/10) § Discrete Fourier Transform M=4 DFT § Problem: algorithm sensitive to “miscalibration” Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 11/35
Spatial phase shifting (6/10) § Non-sine grid – Phase shifting algorithm considers the grid as a sine function. – In real life: more like a sharp pattern – Need to take this into account Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 12/35
Spatial phase shifting (7/10) § N-bucket algorithm or DFT algorithm – Choice of d = 2 p / N – Eliminates harmonics up to N-2 – In the grid method, N is the sampling (i. e. the number of pixels period) Considered pixel where f is calculated Pixels where the data needed to calculate f are taken Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 13/35
Spatial phase shifting (8/10) § How to visualize miscalibration? – Undersample image (also known as ‘pixel binning’) – If match is exact: flat image – Otherwise, moiré fringes Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 14/35
Spatial phase shifting (9/10) § Miscalibration – Phase shift is p/M where M is the number of phase samples – Here, M is also the number of pixels sampling each period – If image pixel size not exactly p/M, then phase shift is not exactly 2 p/M – Introduces an error on the phase, hence on the displacement Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 15/35
Spatial phase shifting (10/10) § Windowed N-bucket algorithm – Same algorithm but with M = 2 N-1 and a triangular windowing Minimizes calibration errors (uncertainty on the value of d) Considered pixel where f is calculated Pixels where the data needed to calculate f are taken Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 16/35
Cross-grid processing § Suppression of the horizontal and vertical frequencies by averaging Averaging over a period in the y direction Calculation of fy Calculation of fx Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 17/35
Phase unwrapping (1/3) § Due to the nature of the mesurand (a phase) the value is known modulo 2 p § Unwrap the phase = eliminate the steps of 2 p to obtain an « absolute » value of the phase -p 0 p Phase step Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 18/35
Phase unwrapping (2/3) -p 0 p -8 Wrapped phase f +p -p -5 -2 Unwrapped phase Step of 2 p x -2 p Spatial phase unwrapping Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 19/35
Phase unwrapping (3/3) § Phase steps can be observed on Df maps (and so on displacement maps) if, on a point, the displacement is > to the pitch of the grid § Displacement maps need to be unwrapped § A simple solution: make load increments such as the displacement is always < to the pitch § Incremental computation of displacements § This method is known as « temporal phase unwrapping » Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 20/35
Grid bonding § Print grids on a photosensitive film on a polymer backing § Bond grid onto specimen (print side), white glue (contrast) § Cure (about 36 hours) § Peel off backing § Pitch down to 20 mm (50800 DPI printer) J. -L. Piro, M. Grédiac, Producing and transferring low-spatialfrequency grids for measuring displacement fields with moiré and grid methods, Experimental Techniques, 28(4): 23 -26, 2004. Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 21/35
Grid printing § Print grids directly on specimen § Flat bed printer Canon Océ Arizona 1260 XT § Grids down to 0. 33 mm pitch § Print either white or black dots Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 22/35
Applications (1/11) § Detection of cracks in an open-hole composite SHPB tensile specimen Moulart R. , Pierron F. , Hallett S. R. , Wisnom M. R. , Full-field strain measurement and identification of mechanical properties at high strain rate, Experimental Mechanics, vol. 51, n° 4, pp. 509 -536, 2011. http: //dx. doi. org/10. 1007/s 11340 -010 -9433 -4 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 23/35
Applications (2/11) § Longitudinal displacement (mm) – Interframe time of 3. 3 ms Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 24/35
Applications (3/11) § Onset of microplasticity in steel (microgrids) Pitch of 5 mm Moulart R. , Rotinat R. , Pierron F. , Lerondel G. , On the realization of microscopic grids for local strain measurement by direct interferometric photolithography, Optics and Lasers in Engineering , vol. 45, n° 12, pp. 1131 -1147, 2007. http: //dx. doi. org/10. 1016/j. optlaseng. 2007. 06. 009 Moulart R. , Rotinat R. , Pierron F. , Full-field evaluation of the onset of microplasticity in a steel specimen, Mechanics of Materials, vol. 41, pp. 1207 -1222, 2009. http: //dx. doi. org/10. 1016/j. mechmat. 2009. 07. 002 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 25/35
Applications (4/11) § Slope measurements: deflectometry d Q da P M CCD d = 2 h da O h Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 26/35
Applications (5/11) § Slope measurements: deflectometry Devivier, C. , Pierron, F. , & Wisnom, M. (2012). Damage detection in composite materials using full-field slope measurements. Composites Part A, 43(10), 1650 -1666. Devivier, C. , Pierron, F. , & Wisnom, M. R. (2013). Impact damage detection in composite plates using deflectometry and the Virtual Fields Method. Composites Part A, 48, 201 -218. Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 27/35
Applications (6/11) § Composite plate in bending Exp. FE y x 0° 20 N x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 28/35
Applications (7/11) § Impacted composite beams Longitudinal equivalent strains Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 29/35
Applications (8/11) § Slope measurements: deflectometry in vibration High speed camera shaker tested plate Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method grid light 30/35
Applications (9/11) § Strains PMMA, 100 Hz y x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 31/35
Applications (10/11) § Lamb waves Shimadzu HPV-X @ 1 000 fps, 200 ns exposure Devivier C. , Pierron F. , Glynne-Jones P. , Hill M. , Time-resolved full-field imaging of ultrasonic Lamb waves using deflectometry, Experimental Mechanics, vol. 56, n° 3, pp. 345 -357, 2016. Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 32/35
Applications (11/11) § Glass mirror excited at 43 k. Hz deflection (nm) Slope x (mm/km) Slope y (mm/km) strain x (mm/m) strain y (mm/m) strain s (mm/m) Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 33/35
Comparison between GM and DIC § DIC – Subset of say 25 by 25 pixels: 1 independent displacement value for 625 pixels § Grid – 5 pixels/period: one independent measurement for 9 by 9 pixels, ie, 81 pixels (2 N-1, N-bucket algorithm) – One order of magnitude more independent data points with the grid method – Cost: bond a grid, limited to 2 D – No need for commercial software… Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 34/35
Comparison between GM and DIC § Practical example: T-shaped specimen DIC, 32 x 32 subset GM, 11 x 11 subset Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Grid method 35/35
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